From c2f07d311bfdad0d89bf961e4da133ee97bb78de Mon Sep 17 00:00:00 2001 From: hcg890 <53488436+hcg890@users.noreply.github.com> Date: Fri, 9 Oct 2026 21:19:19 -0700 Subject: [PATCH] Publish verified multicut delayed-inverse checkpoint (conditional kappa 7.110455476244e-4) --- .../BANK-PROOF.md | 129 + .../HERITAGE.md | 121 + .../LICENSE | 201 + .../MANIFEST.json | 141 + .../NEGATIVE-RESULTS.md | 108 + .../NOTICE.md | 23 + .../PR259-PROVENANCE.md | 27 + .../PROOF.md | 121 + .../README.md | 65 + .../UPSTREAM-PR234-NOTICE.md | 13 + .../UPSTREAM-PR234-PROOF.md | 37 + .../bank_check.py | 265 + .../bank_template.py | 180 + .../base_two_moment.py | 63 + .../code/complex_basis.py | 35 + .../code/complex_labels.py | 176 + .../code/complex_scalars.py | 216 + .../code/complex_splice.py | 277 + .../code/geometry527.py | 103 + .../code/global_lowering.py | 315 + .../code/physical527.py | 102 + .../expected/certificate.json | 221 + .../finite_check.py | 141 + .../hybrid-selection.json | 7384 + 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The bundled +[bank_check.py](bank_check.py) and [bank_template.py](bank_template.py) check the +actual retimed source context and global ledger. The complete construction's +evidence and inherited assumptions are stated in [PROOF.md](PROOF.md). + +## Literal capacity and role assignment + +The focused source has 16,587 independent helper roles, of which 2,827 have a +nonzero entrance. Its residual rank counts are + +``` +4:2200, 6:22, 7:7, 9:4, 10:8, 11:55, 12:30, 13:14, 14:2, +15:20, 16:1, 18:1, 19:31, 20:19, 21:3, 22:41, 23:369, 24:13760. +``` + +Their sum is 16,587 and their residual volume is 351,267 per replica. For each +residual width r other than 4 and 24, use 10 times its role count banks with four +blocks of width r and 30-r blocks of width 4. Each such bank has exactly +4r+4(30-r)=120 coordinates. This uses 65,830 of the 88,000 available width-4 +occurrences across 40 replicas. The remaining 22,170 width-4 occurrences fill +739 banks of 30 blocks. The 550,400 width-24 occurrences fill 110,080 banks of +five blocks. There is no rounding, unused capacity, or fractional bank. + +This gives 117,089 banks per stage and 585,445 stage-private bank families across +five stages. There are 40*4*1760=281,600 data families, for literal stock 867,045. +The one external work family is separate from the live-stock count. + +Roles retain their actual original IDs. Within each residual family they are +sorted, and occurrence q means role floor(q/40), replica q modulo 40. The segment +mapping in bank_template.py assigns every occurrence to one contiguous bank +block. bank_check.py independently enumerates sequential slots and compares all +3,317,400 stage/replica/role assignments against that callable mapping. The +bundled production census also checks the exact role set against physical ownership, +borrowed roles, and removed roles. No source producer or ownership is changed. + +## Generic charts and full addresses + +All 770 actual entrance frame IDs are rebuilt. For an actual source basis S and +its Euclidean annihilator row a, the residual vector is 15a-sum(a)*1, divided by +its integer row gcd. It is orthogonal to S for the actual form 9I-J. The residual +vectors followed by S form a 24-column chart B. The checker computes A,d with +BA=AB=dI, checks projector action on every chart column, and replays every exact +rational row factor to the identity. This establishes independence and exact +conjugacy without a norm-four or rank-one special case. The checked maxima are +576 chart factors, factor numerator 165, and denominator 1080. + +For a stage i and assigned block b, the normalizer is +N=(b+1) Pi embed_i(B^-1). Pi sends the actual residual coordinate window to its +assigned contiguous block. A column outside the stage window is fixed by the +embedded chart; its image under N is (b+1)e_(Pi(j)). Distinct blocks have distinct +positive scalars, so these witnesses distinguish the normalizers even when +chart matrices repeat. All full (family, route) namespaces are injective for all +200 stage/replica pairs, including both active data banks, all helpers, and the +external work register. The 34,965 intentional same-family coincidences retain +different full routes. Equality of family numbers alone is not an address test. + +The production callable lowering preserves every scalar, frame, projection, +stage and reversal field and changes only actual stream operands. ADD/COPY +operands are checked for full-address nonaliasing. Original helper class +d*tau_i is routed to bank class d*tau_i*N^-1. The inverse-normalizer multiplication +order and all denominators are retained. Every actual logical operand and all +five paid stage streams are bound by the fresh checker before acceptance. + +## Complete banks and completion discharge + +Every bank pattern partitions all 120 coordinates. Its disjoint projected block +swaps therefore give the full swap on all 240 arbitrary-dirty bank/work columns. +The inverse word restores all columns. bank_check.py tests omission and repetition +of every individual block in every one of the 18 patterns. The phase-major +schedule completes all cover classes for each replica before proceeding; +boundary phases follow all 40 replicas of their stage. Banks are stage-private. + +There are 2,827 independent entrance completions with width counts + +``` +5:369, 10:41, 15:3, 20:19, 25:31, 30:1, 40:1, 45:20, +50:2, 55:14, 60:30, 65:55, 70:8, 75:4, 85:7, 90:22, 100:2200. +``` + +The checker matches the exact tagged entrance list and every completion record +before subtracting these counts. Widths 5, 10, 15, 20 and 50 also have ordinary +paid children. It subtracts only the tagged count and deletes a histogram bucket +only if the remainder is zero. A whole-bucket deletion would fail the retained +ordinary-child assertions and exact ledger comparison. The address mapper rejects +completion opcode 6; discharge is explicit, never a silent mapper shortcut. + +## Retiming boundary and conservative tariff + +The focused source before the 25 compatible retimings has 494,670 retained +five-stage calls per replica and rank mass 2,596,735. Retimings preserve all +entrance gauges, charts, packing and rank mass. The final checked count is +494,665 per replica, reconstructed by bank_check.py from the actual live ledger. +Forty replicas have rank mass 103,869,400 and deficit +120*867045-103869400=176000. + +The normalizer bound remains the conservative 838: 576+119+120=815 is below it. +No faster route or selector tariff is claimed. The charged selector allowance is + +``` +2*5*40*((867045-1)+16587*120*838) = 667542305600 < 2^40. +``` + +## Actual retimed registry binding + +The final raw word is +acf510b1377a6bd96eea50577de1aa049382499335fa7e5cac9b0300ecb377c7. +The bundled checker reconstructs its initial frames and actual entrance bases. +The actual 770-chart hash is +98d3fbe41db017678b416bdffe4e79361ad08ceda99a84eed340a7dc9d1467d2. +These chart IDs contain 459 distinct entrance bases. An exact rebuild checks +both inverses and every rational factor for the actual registry. The maxima +are 576 factors, numerator 165 and denominator 1080. + +Annihilator row sign changes do not change the represented subspace. +Assignment and namespace separation use the actual reconstructed charts; +the outside-column separation proof is invariant under such sign choices. +No earlier chart presentation is an input to these checks. + +The live callable bank checks bind all five actual operand streams, all +2,827 tagged completions, the charts, the assignment and the full namespaces. +The actual global ledger confirms 494,665 retained calls and rank mass +2,596,735 per replica after subtracting precisely those completions. Across +forty replicas its literal histogram has 19,786,600 calls and rank mass +103,869,400. The complete verifier also checks the current prime inventory, +exact moments, outer inequalities and finite bill. These finite checks retain +the conditional theorem interfaces described in PROOF.md. diff --git a/research/multicut-delayed-inverse-checkpoint/HERITAGE.md b/research/multicut-delayed-inverse-checkpoint/HERITAGE.md new file mode 100644 index 0000000000..cba2634f9f --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/HERITAGE.md @@ -0,0 +1,121 @@ +# Heritage, attribution and exact source boundaries + +This construction builds on the contributors listed below. Credits distinguish the actual selected sources and witnesses, inherited mathematical mechanisms, and historical or parallel work. Independently reported exponent improvements are never added together. Attribution does not imply review or endorsement of this submission. + +PR links identify contributions; immutable commits and the packaged source manifests identify the versions actually consumed. A live PR head can advance beyond an imported snapshot. Evidence paths below are relative to this public package. + +## 1. Immediate construction and selected witnesses + +| Contribution | Author | Precisely what this construction retains | +|---|---|---| +| [PR259](https://github.com/CrocSwap/integer-mult-bounds/pull/259), a53ec0ca4d59634f16630dd16adf035c9f12595c | **Dugongue** | Multi-cut response-kernel condensation, shared donor lifts and the frozen 518-cohort witness. This construction retains **513 cohorts**, removing the five conflicting rank-11 quartets at pivots 17207, 17222, 17246, 17255 and 17273 before introducing seven delayed-inverse target intervals. | +| [PR249](https://github.com/CrocSwap/integer-mult-bounds/pull/249), 96495746c786d6d0339dbb38c7f553d4af3f88ed | **eumemic** | The exact source527 baseline, F₂ compensation transport and **21 transported entrances**. These 21 entrances are retained. | +| [PR244](https://github.com/CrocSwap/integer-mult-bounds/pull/244), a568d94f941929232ab393c7d33dc5a30e017892 | **eumemic** | The preceding common-frame retiming and its physical-word reconstruction, inherited through PR249. The baseline retains its original 364 retimings. | +| [PR258](https://github.com/CrocSwap/integer-mult-bounds/pull/258), cf82ca09322d1327ab887cfd1cce97591b8d3cfa | **amanforindia** | The frozen 25-gate rational-frame retiming witness, originally composed with PR254. The selected scalar keys and old/new basis data are retained and rebound to this word. | +| [PR260](https://github.com/CrocSwap/integer-mult-bounds/pull/260), 87aa555a455cca84a3e3296ced2440623d39fc4d | **Rohan Arun / rohanarun** | The independently published concave-descent retiming and Python-verifier composition on PR251. It identifies the **same 25 scalar gates and replacement subspaces** as PR258. Both contributions are credited; their retiming is applied **once**. | +| [PR254](https://github.com/CrocSwap/integer-mult-bounds/pull/254), 9ce32efb421132420242cfdb91ddd9eabe2c3795 | **Dugongue** | The earlier 132-pair response-kernel rewrite, a mathematical and implementation predecessor of PR259 and the baseline used by PR258. Its old 132-pair improvement is **not added separately** to the selected PR259 cohorts. | +| [PR251](https://github.com/CrocSwap/integer-mult-bounds/pull/251), 2b030c06a811473ae8ed49dce07bc7ea50b72e9e | **eumemic** | Shared-frame-intersection entrance transport and the earlier checkpoint on which the seven-interval experiment was developed. Its **16 entrances beyond PR249 are excluded** from the present word because they conflict with PR259. | + +The new contribution here is the source-bound compatibility construction: replace five conflicting PR259 quartets with seven paid closed-target delayed-inverse intervals, rebind the once-counted 25 retimings, and reconstruct the resulting physical word, bank assignment and full price. It does not claim the inherited kernel, retiming, source-reuse, frame, bank or five-stage mechanisms as new. + +Package evidence: PR259-PROVENANCE.md, hybrid-selection.json, kernel-candidates.json.gz, regauge-selection.json, retiming25-selection.json, hybrid_transform.py, retiming25.py and PROOF.md. + +The retained historical notices describe earlier intermediate constructions, including one with all 37 PR251 entrances and one with the PR254 paired rewrite. Those historical descriptions do not specify the active selection. The current selection is **21 PR249 entrances, 513 PR259 cohorts, seven intervals and 25 once-counted retimings**. + +## 2. Direct supplier, architecture and source ancestry + +| Contribution | Author | Inherited role and source boundary | +|---|---|---| +| [PR210](https://github.com/CrocSwap/integer-mult-bounds/pull/210) | **eumemic** | Source borrowing, shared and nested gauges, target-response transforms, the source527 helper and F₂ movement fusion. The source471 loader is pinned specifically to df95878d11190518e45ef9717c9ee05011f88ace; later source527 material is inherited through the selected PR249 package. A newer live PR210 head is not a substitute for these pins. | +| [PR230](https://github.com/CrocSwap/integer-mult-bounds/pull/230) | **sennemmi** | Nineteen operation-frame improvements retained by source527. source527/pr230/PROVENANCE.json records imported head fee4c05b91d4cdebc0b9d8cbb266a98069928a9a, the original witness hashes and the distinct active composed frame witness. | +| [PR216](https://github.com/CrocSwap/integer-mult-bounds/pull/216) | **Dugongue** | Composed cascade frames, global-cut refinements and finite-bootstrap work acknowledged in the source527 lineage. | +| [PR211](https://github.com/CrocSwap/integer-mult-bounds/pull/211) | **Rohan Arun / rohanarun** | Cascade-optimized operation frames, inherited through the source471/source527 construction; the retained notice pins c63e50a5dde96fe1459d6b29e55e47f42f104347. | +| [PR207](https://github.com/CrocSwap/integer-mult-bounds/pull/207) | **Dugongue** | Coordinated frame optimization, completed entrance banks, offline source packaging and independent bank admission; retained at cd14825023b75af4f5919a30e5e6d548b6ade5bc. | +| [PR200](https://github.com/CrocSwap/integer-mult-bounds/pull/200) | **Chafik Boukhalfa / chafreaky** | The physical bit helper, local face-diagonal/fixed-coordinate circuit, regenerated physical layer and terminal interfaces. | +| [PR202](https://github.com/CrocSwap/integer-mult-bounds/pull/202) | **Chafik Boukhalfa / chafreaky** | The exact carrying snapshot of the consumed PR200 bit source: 8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d. Its historical composition also credits PR194; the present complex supplier is specified separately below. | +| [PR234](https://github.com/CrocSwap/integer-mult-bounds/pull/234), af3fe331ca60c936229e060681ffccbc1c208678 | **Henry Grant / hcg890** | Source-bound five-stage F₂ architecture, paid interfaces, finite accounting and the integration with Sussman’s published complex baseline. It also carries the primary OpenAI bit-scalar/common-frame source included under proof/primary/. | +| [PR209](https://github.com/CrocSwap/integer-mult-bounds/pull/209) and [the pinned five-stage development](https://github.com/jacobalansussman/wht-power-saving-lean/tree/f010392c923279e3dd59ef3aa5fedad23affc5fd) | **Jacob Sussman / jacobalansussman** | The five-stage bridged layout, general-frame Lean development, explicit gcert/1 complex certificate and associated sources. **PR209 is the public documentation cross-reference; the actual imported source is the separate repository at f010392c923279e3dd59ef3aa5fedad23affc5fd.** This submission does not claim a new Lean build or transfer Sussman’s Fourier theorem into an unconditional integer-multiplication theorem. | +| [PR193](https://github.com/CrocSwap/integer-mult-bounds/pull/193), 187e1010ac8b259af8e9b5166f68b64bc27b4b47 | **Avi Eisenberg / ikeboy** | Donor pairing that avoids parity erasure in the retained source-assisted complex helper. Sussman’s explicit certificate derives from the community’s PR168/PR193 data; the paired-cube circuit itself is not newly attributed to Sussman or this submission. | +| [PR191](https://github.com/CrocSwap/integer-mult-bounds/pull/191) | **Avi Eisenberg / ikeboy** | The move of PR184’s source-assisted complex construction to PR168 v4 modules, preceding PR193. | +| [PR184](https://github.com/CrocSwap/integer-mult-bounds/pull/184) | **icekylinx** | Source-assisted common-frame lifts, source-parity local circuits, compatible zero-fresh reuse, controlled affine cleanup and the retained finite bridge/assembly lineage. | +| [PR168](https://github.com/CrocSwap/integer-mult-bounds/pull/168), v4 4a3c769e5c5430e7114c4d3e099ff34664677f17 | **eumemic** | Annealed query modules, local paired-cube circuits, frozen carrier arcs, physical frames, compensated handoffs, terminal-sink integration and assembly. PR184’s own earlier fixed experiment used fd25adb7fbaa12ee761d02c733c54d1d2a7687ee; PR191/193 and the selected bit inputs carry the v4 continuation. | +| [PR144](https://github.com/CrocSwap/integer-mult-bounds/pull/144) | **icekylinx** | Paired-cube identity, signed overlap channels, original-source involution, coordinate-star centers, shared completed-core geometry and supplier/assembly framework. | +| [PR149](https://github.com/CrocSwap/integer-mult-bounds/pull/149) | **Mister-Meeseeks** | Reviewed repository integration of PR144, explicitly identified as the carrying integration in PR184’s heritage. | + +Package evidence: loader/SOURCE_PINS.json, loader/NOTICE.md, the notices under loader/source_inputs/, source527/pr230/PROVENANCE.json, source527/pr230/NOTICE, UPSTREAM-PR234-NOTICE.md, inputs/complex/source-pins.json, inputs/complex/NOTICE, inputs/complex/sources/tools__gx__ORIGIN.md, proof/primary/PROVENANCE.md and outer.py. + +## 3. Earlier retained implementation and mathematical lineage + +The following predecessors are explicitly credited by the retained source notices and the public heritage statements of PR144, PR168, PR184 and the direct suppliers. An inherited method or proof is not a claim that every historical selected graph, certificate or exponent remains active. + +### Paired-cube modules, physical execution and stopping + +- **eumemic**: [PR155](https://github.com/CrocSwap/integer-mult-bounds/pull/155), paired cubes with physical frames and the paired-cube bit word; [PR157](https://github.com/CrocSwap/integer-mult-bounds/pull/157), searched pair modules; [PR161](https://github.com/CrocSwap/integer-mult-bounds/pull/161), merged complex outputs and annealed bit pair modules; [PR131](https://github.com/CrocSwap/integer-mult-bounds/pull/131), physical frame descent and scratch reuse; [PR143](https://github.com/CrocSwap/integer-mult-bounds/pull/143), late compensation reads and joint frame refinement. PR168 expressly incorporates the PR161 construction and its predecessors. +- **DaysSky**: [PR162](https://github.com/CrocSwap/integer-mult-bounds/pull/162), closure-compiled carrier matching and scheduling. +- **James Chang / jamesyc**: [PR124](https://github.com/CrocSwap/integer-mult-bounds/pull/124), compatible birth-read slot recycling; [PR166](https://github.com/CrocSwap/integer-mult-bounds/pull/166), the terminal-target accumulation/substitution lemma. The retained PR200 bit source uses this terminal substitution. PR184’s new complex ledger does not import its predecessor’s old terminal deduction. +- **Chafik Boukhalfa / chafreaky**: [PR163](https://github.com/CrocSwap/integer-mult-bounds/pull/163), same-singleton output fusion and paid balanced transfer, credited by PR168. +- **geckods**: [PR153](https://github.com/CrocSwap/integer-mult-bounds/pull/153) and [PR158](https://github.com/CrocSwap/integer-mult-bounds/pull/158), phase-stop and atom-exponent-at-the-toll-edge refinements. +- **Avi Eisenberg / ikeboy**: [PR194](https://github.com/CrocSwap/integer-mult-bounds/pull/194), the historical PR193-complex/PR189-bit composition carried by PR202. +- **Chafik Boukhalfa / chafreaky**: [PR189](https://github.com/CrocSwap/integer-mult-bounds/pull/189), the preceding face-diagonal bit supplier. PR200 replaces its selected bit circuit, so its old price is not retained as an additional gain. + +### Frame, recursion, producer and ordinary-leaf foundations + +- **icekylinx**: [PR10](https://github.com/CrocSwap/integer-mult-bounds/pull/10), projector batching and controlled bases; [PR18](https://github.com/CrocSwap/integer-mult-bounds/pull/18), partial-swap frames; [PR24](https://github.com/CrocSwap/integer-mult-bounds/pull/24), endpoint gauges; [PR32](https://github.com/CrocSwap/integer-mult-bounds/pull/32), structured projector blocks and semantic/bulk assembly; [PR36](https://github.com/CrocSwap/integer-mult-bounds/pull/36), copied retained centers and two-stage assembly; [PR104](https://github.com/CrocSwap/integer-mult-bounds/pull/104), opposite-bank factorization and stopped product-ring interchange; [PR115](https://github.com/CrocSwap/integer-mult-bounds/pull/115), partial source gauges; [PR130](https://github.com/CrocSwap/integer-mult-bounds/pull/130), arbitrary-subspace Clifford frames, three-stage Cayley covers, weighted compiler and charged fallback. The explicit retained chain is PR10 → PR18 → PR24 → PR32 → PR36 → PR104 → PR115 → PR130 → PR144. +- **eumemic**: [PR3](https://github.com/CrocSwap/integer-mult-bounds/pull/3), compressed complex producers; [PR5](https://github.com/CrocSwap/integer-mult-bounds/pull/5), Gaussian resampling; [PR6](https://github.com/CrocSwap/integer-mult-bounds/pull/6), aligned circuits and cheaper centers; [PR13](https://github.com/CrocSwap/integer-mult-bounds/pull/13), auxiliary source-frame context; [PR117](https://github.com/CrocSwap/integer-mult-bounds/pull/117), the positive local DAG, module lineage and exact support/label replay. PR168’s later annealed modules replace particular older module selections. +- **Zhihao Chen / jacklightChen**: [PR7](https://github.com/CrocSwap/integer-mult-bounds/pull/7), finite ternary producers; [PR16](https://github.com/CrocSwap/integer-mult-bounds/pull/16), nested controlled-basis context; [PR21](https://github.com/CrocSwap/integer-mult-bounds/pull/21), translated complex endpoints; [PR23](https://github.com/CrocSwap/integer-mult-bounds/pull/23), semantic/bulk compatibility, scalar charge and product row stock; [PR29](https://github.com/CrocSwap/integer-mult-bounds/pull/29), unequal two-stage compatibility and paid interfaces; [PR97](https://github.com/CrocSwap/integer-mult-bounds/pull/97), deferred bit execution, reflected accounting and signed endpoint correction. +- **RaD project / hipotures**: [PR20](https://github.com/CrocSwap/integer-mult-bounds/pull/20), completed-child semantic precision, arbitrary-coordinate routing, phase-cell Gaussian inversion, bulk tape transfer and the balanced positional transform. +- **Rohan Arun / rohanarun**: [PR25](https://github.com/CrocSwap/integer-mult-bounds/pull/25) and [PR31](https://github.com/CrocSwap/integer-mult-bounds/pull/31), corner methods, incidence/rank partitions and exact corner certificates. +- **Dominik Scholz / DominikScholz**: [PR33](https://github.com/CrocSwap/integer-mult-bounds/pull/33), parameterized dimensions/corners; [PR35](https://github.com/CrocSwap/integer-mult-bounds/pull/35), two fixed local factors and CRT arguments; [PR38](https://github.com/CrocSwap/integer-mult-bounds/pull/38), copied-center/fixed-basis composition. +- **James Chang / jamesyc**: [PR34](https://github.com/CrocSwap/integer-mult-bounds/pull/34), reversed geometry and balanced assembly composition. +- **Rohan Arun / rohanarun**: [PR37](https://github.com/CrocSwap/integer-mult-bounds/pull/37), [PR39](https://github.com/CrocSwap/integer-mult-bounds/pull/39) and [PR40](https://github.com/CrocSwap/integer-mult-bounds/pull/40), copied reversed corners and fixed-basis compositions. The packaged independent rational moment engine retains its PR39-era provenance. + +### Completed banks and finite-leaf mechanisms + +- **Evan McKinney / evmckinney9**: [PR197](https://github.com/CrocSwap/integer-mult-bounds/pull/197), completed-bank packing and restored scratch-residual sharing, explicitly inherited by the later source/bank lineage. +- **Rohan Arun / rohanarun**: [PR185](https://github.com/CrocSwap/integer-mult-bounds/pull/185), finite ordinary-leaf bootstrap; [PR187](https://github.com/CrocSwap/integer-mult-bounds/pull/187), globally replayed source-assisted bit execution and finite-leaf composition, credited by PR197. +- **Dugongue**: [PR186](https://github.com/CrocSwap/integer-mult-bounds/pull/186), related earlier entrance-bank/frame construction credited by PR197. Its selected supplier is not separately imported into the current word. +- **Rohan Arun / rohanarun**: [PR237](https://github.com/CrocSwap/integer-mult-bounds/pull/237), the earlier published width-120 banking extension of PR234. The present lineage uses its own explicitly checked stage-private bank allocation and claims no exclusive priority for width-120 banking. +- **an664**: [PR128](https://github.com/CrocSwap/integer-mult-bounds/pull/128), the substantial completed-core workspace-sharing principle. The retained PR144 notice explicitly states that **no PR128 source code, partition or serialized witness was imported**. + +## 4. Earlier compiler and repository lineage preserved in the notices + +The source-specific notices also retain the following earlier contributions. These are historical implementation, proof and integration credits; their old scalar graphs, standalone certificates and claimed exponent increments should not be mistaken for additional active mechanisms in this submission. + +| Author | Credited PRs and roles | +|---|---| +| **RaD project / hipotures** | [PR41](https://github.com/CrocSwap/integer-mult-bounds/pull/41), alternating pair order and independent physical-role compiler; [PR51](https://github.com/CrocSwap/integer-mult-bounds/pull/51), enlarged frames, source partitions and paid whole-chain cloning. The retained notice distinguishes PR51’s research witness from machinery consumed by later compositions. | +| **Chafik Boukhalfa / chafreaky** | [PR43](https://github.com/CrocSwap/integer-mult-bounds/pull/43), changed-graph/fixed-basis checks; [PR46](https://github.com/CrocSwap/integer-mult-bounds/pull/46), exact data recovery and matching integration; [PR48](https://github.com/CrocSwap/integer-mult-bounds/pull/48), reordered exclusions; [PR54](https://github.com/CrocSwap/integer-mult-bounds/pull/54), original-envelope clones; [PR58](https://github.com/CrocSwap/integer-mult-bounds/pull/58), joint dual-skip/frame composition; [PR60](https://github.com/CrocSwap/integer-mult-bounds/pull/60), ranked reclamation. | +| **Rohan Arun / rohanarun** | [PR44](https://github.com/CrocSwap/integer-mult-bounds/pull/44), weighted carrier matching; [PR47](https://github.com/CrocSwap/integer-mult-bounds/pull/47) and [PR49](https://github.com/CrocSwap/integer-mult-bounds/pull/49), climbed producer/exclusion orders; [PR56](https://github.com/CrocSwap/integer-mult-bounds/pull/56), positive-frame and paid-clone composition. | +| **Avi Eisenberg / ikeboy** | [PR53](https://github.com/CrocSwap/integer-mult-bounds/pull/53), skip-prefix strips; [PR62](https://github.com/CrocSwap/integer-mult-bounds/pull/62), interval strips and core-aware pair primitives. | +| **Rohan Gupta / gupt1156** | [PR55](https://github.com/CrocSwap/integer-mult-bounds/pull/55), dual-suffix strip graph. | +| **eumemic** | [PR57](https://github.com/CrocSwap/integer-mult-bounds/pull/57), joint reversible frame compiler, physical replay and paid signal reclamation. | +| **Rohan Garg / rohangar1** | [PR59](https://github.com/CrocSwap/integer-mult-bounds/pull/59), singleton/pair recursion and paid-clone composition. | +| **Alejandro Zarzuelo Urdiales / alejandrozu** | [PR45](https://github.com/CrocSwap/integer-mult-bounds/pull/45), Gaussian parity, canonical arithmetic and mixed-center checks; [PR61](https://github.com/CrocSwap/integer-mult-bounds/pull/61), parameter refinement and scoped arithmetic/search tools. | +| **Ryan S / princezuda** | [PR26](https://github.com/CrocSwap/integer-mult-bounds/pull/26), historical Lean certificate arithmetic, algebraic frame/movement/guard lemmas and independent paired-circuit checks. These are not a new formal certification of the present construction. | + +These credits are traceable in loader/source_inputs/base_bit/NOTICE, loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE and source527/pr230/NOTICE. + +## 5. Related ideas and parallel work, without an import claim + +- **Bortlesboat**, [PR2](https://github.com/CrocSwap/integer-mult-bounds/pull/2): aligned pairing; **dleen**, [PR4](https://github.com/CrocSwap/integer-mult-bounds/pull/4): retained totals and stage sharing. The inherited notices distinguish these related ideas from imported PR3/PR5 sources. +- **Rohan Gupta / gupt1156**, [PR50](https://github.com/CrocSwap/integer-mult-bounds/pull/50), and **Rohan Arun / rohanarun**, [PR52](https://github.com/CrocSwap/integer-mult-bounds/pull/52): parallel order-search records. The retained notice explicitly says they did not replace the earlier climbed-48 implementation. +- **eumemic**, [PR164](https://github.com/CrocSwap/integer-mult-bounds/pull/164): parallel connected bit-frame descent, acknowledged as parallel work by PR168. +- Sussman’s retained notice also credits **Swapnil Jain’s NStar3 development** and its transfer to the Fourier problem by **shea256** as related ideas; it explicitly distinguishes these from imported source files. + +## 6. Non-PR sources and original framework + +- **OpenAI**: the original [Integer multiplication below n log n manuscript](https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf) and its scalar/frame framework. The preserved base notice records the original openai/math source pin adc7f1241b42e322a6451854ab7e4b4c146bf78a. The source carried under proof/primary/ is identified separately by the PR234 carrying snapshot; that carrying commit is not presented as an original OpenAI commit. +- **Douglas Colkitt**: the original integer-multiplication research repository and maintainer integration/arithmetic. The retained notice expressly states that the community release is not solely his work. +- **David Harvey and Joris van der Hoeven**: the underlying analytic multiplication lineage credited by the retained sources. +- **Aurel Prosz / Paureel**: [two-stage topology and paid copied-stream endpoint correction](https://github.com/Paureel/integer-mult-bounds/tree/c82d09eb4781b68435e3fa3eb6beea72fa900fab), inherited through PR29 and the ordinary-leaf lineage. +- **Swapnil Jain**: [two-stage development](https://github.com/Swapnil-jain/integer-mult-kappa/tree/ae405eb474d1486b2d8aef90139869f927f4836a) and the [round-seven deferred-word witness](https://github.com/Swapnil-jain/integer-mult-kappa/tree/741e7aa078392553815df7926ee17ac5e25a8c38) underlying PR97, including deferred readouts, V leaves and lifted frames. The source notices distinguish conceptual two-stage credit from actual later frozen-witness lineage. +- **RaD / hipotures**: the [pinned semantic/bulk sources](https://github.com/hipotures/rad/tree/45d9b60355872041f9275f77c057514def0d45bc/research/integer-multiplication-bounds), with their separate CC0 terms. +- **Baker, Harman and Pintz**: the prime-interval theorem credited by the adopted packing argument. +- **OpenAI’s Fourier development**: Sussman’s notice records the separate openai/math pin fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb for the Fourier/RAM development on which his work builds. This is distinct from the integer-multiplication source pin above. + +## 7. Licenses, assistance disclosures and scope + +Preserve the original Apache-2.0 licenses, notices, source headers and separately applicable CC0 notices. Preserve each contributor’s original AI-assistance disclosure, including the Claude disclosures associated with Sussman’s development, eumemic’s module work, Avi Eisenberg’s work and Rohan Arun’s PR260, and the original OpenAI/Codex disclosures elsewhere. Assistance with the present integration does not transfer authorship of inherited contributions. + +The present construction and its review remain conditional on the stated all-size compiler, common-chart/GL-cover, restored-row, routing, prime, precision/recovery and analytic interfaces. Heritage records attribution and source boundaries; it does not independently establish those hypotheses or imply contributor endorsement. + diff --git a/research/multicut-delayed-inverse-checkpoint/LICENSE b/research/multicut-delayed-inverse-checkpoint/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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"verify.py": "489bee6d23fbed6732e60d62f8fa98e7d2dfb2d2b7f75fd2bd3bc4a920155f10" + }, + "schema": "source-bound-pr259-hybrid-package/1" +} diff --git a/research/multicut-delayed-inverse-checkpoint/NEGATIVE-RESULTS.md b/research/multicut-delayed-inverse-checkpoint/NEGATIVE-RESULTS.md new file mode 100644 index 0000000000..987154661d --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/NEGATIVE-RESULTS.md @@ -0,0 +1,108 @@ +# Research findings and scoped negative results + +These results distinguish exact incompatibility of fixed witnesses from a paid-cost exclusion at a specified rate. They do not establish global optimality. The standalone verifier checks the selected construction; the earlier screens below concern declared alternative families. + +The alternative-family screens and exploratory controls summarized here are +reports of separate bounded research checks. Their test programs and execution +reports are not bundled in this minimal construction package, and verify.py +does not replay them. The displayed case counts and measured operation +inventories report those separate checks. In contrast, the selected hybrid's +compatibility, retiming and final paid ledger are reconstructed by the bundled +verifier from the current witnesses. These research reports do not add new +transformations to the certified construction. + +## 1. Fixed-star inverse at the high compensation frame + +**Baseline:** PR249 at `96495746c786d6d0339dbb38c7f553d4af3f88ed`, raw word SHA256 `2e0bc2a5c54cf8caf26fa30a1a2949702d842f620b28023ca2ba59140ecbed1a`. + +**Rate:** `a = 44431075156777 / 62500000000000000`, with paid weights `r(120/r)^a` and the complete `32·120²/10¹⁶` rank-one fallback allowance per positive child. + +**Family:** the 128 declared two-helper closures with equal transported signed response columns; one fixed invertible target star-basis preconditioner; each helper compensated exactly once at a positive entrance frame; nested compensation frames L ⊆ H; literal target inverse executed at H. Producers, read successors and external observations are unchanged. Every allowed integer rank, helper order and target pivot is screened, including optimistic rank choices without a demonstrated nondegenerate realizing subspace. + +Of the 128 helper pairs, 121 have zero common allowed entrance dimension. For the other seven, all 2,940 rank/order/pivot cases have strictly positive paid residual under optimistic fractional bank volume. The minimum outward-rational lower bound exceeds 0.00963624. Even after including the baseline's available moment slack, the minimum full-recurrence residual numerator exceeds 0.04818120. Those exact signs exclude improving the certified rate within this family. Extra nonnegative packing, routing or chart charges cannot repair the rejection. + +The exclusion requires paying the target paths: omitting their tolls makes all 2,940 cases look favorable. It does not exclude a later inverse at a common K containing H. The selected rank-20 K changes the family by letting satellite targets avoid the intermediate H visit. This is why the delayed-inverse construction can improve the price. + +## 2. Direct union of the seven intervals with all 518 PR259 cohorts + +**Pinned witness:** PR259 at `a53ec0ca4d59634f16630dd16adf035c9f12595c`, candidate gzip SHA256 `5f6863ad27eb9f2fcb14db10320c00607451c01e12da303def13e48eda7a5d2d`. + +The low helpers of the seven intervals overlap exactly five rank-11 quartets, with pivots 17207, 17222, 17246, 17255 and 17273. Two low helpers are quartet pivots with nonzero entrance frames, and five are written donors. The unchanged intervals require those helpers to remain zero-initial and untouched through their cut. Keeping both fixed witnesses violates that entry contract, so the independent union is inadmissible. + +The selected construction removes all five quartets, checks the exact signed helper responses and their zero columns on every retained cohort operand at all four cuts, then inserts the seven intervals. Removing five quartets loses 55 entrance-rank units; the seven intervals add 133, for net 78. Normalization is applied after composition: the final new entrance rank is 1422, divisible by three. Normalizing the partially removed witness first would accidentally discard two additional rank-one cohorts. + +All sixteen entrances that PR251 adds beyond PR249 also overlap the pinned PR259 selection, across ten cohort blocks. They are excluded here. This is a conflict between unchanged witnesses, not an impossibility proof for redesigned entrances or a reoptimized cohort selection. + +## 3. PR258 and PR260 do not give two retiming increments + +PR258 at `cf82ca09322d1327ab887cfd1cce97591b8d3cfa` and PR260 at `87aa555a455cca84a3e3296ced2440623d39fc4d` select the same 25 scalar events and exact rational replacement subspaces. Their record offsets and basis representations differ because their baselines differ. + +The adapter matches each scalar event uniquely and checks its exact old basis before changing the frame. Applying this plan twice fails those old-frame checks. The resulting local histogram delta is applied once: + +`{2: −2, 4: +24, 5: −48, 6: +25, 11: −1, 13: +2, 17: −1}`. + +It removes one local positive-rank child with zero change to rank mass. Across forty replicas and five stages, that is 200 literal children. + +## 4. Boundary of the replacement-subset comparison + +Before the 25 final retimings, the exact paid screen examines all 127 nonempty subsets of the seven fixed interval witnesses, removing every PR259 quartet touched by each subset. The all-seven/five-removal choice has the best certified price in that declared ideal-stock model. At its certified coarse rate every other subset has moment greater than one. + +Only 43 subsets satisfy the same forty-replica volume-integrality condition. The remaining 84 are optimistic fractional-volume screens and require a different replication or separately charged packing adjustment. Rounding stock alone is not a valid repair. + +This comparison uses the fixed witnesses, cuts, inherited fallback and pricing model. It does not establish optimality after reoptimizing intervals, cohorts, banks or retimings, and its pre-retiming conclusion is not silently transferred to the final rate. + +## 5. Persistent transformed blocks: real reuse, unsuitable immediate insertion + +**Evidence status:** exact finite control and source-specific interface screen; no full-supplier replacement or total-runtime benchmark. + +The control receives two growing radix-four digit streams, retains each completed two-digit block as a length-four transform over Z/97, and treats the open boundary separately. It checks every required coefficient observation, exact final product, carries, ordinary output order, input preservation and owned-storage cleanup. All 1,585 cases pass: 1,364 exhaustive binary-string pairs of lengths 1–5, 204 deterministic radix-four cases, and 17 maximal-digit carry cases. Contamination, omitted product, missing carry, premature observation and missing cleanup are rejected. + +Both comparators retain the two newest transforms within each batch. At n=8, persistent retention reduces forward transforms from 20 to 8; the charged seven-tape data-access subtotal rises from 2,520 to 4,192 bit steps. The exact difference is 1,672 data-access steps minus 96 residue additions and 12 constant multiplications, plus the control-cost difference. Arithmetic primitives and control costs are not declared free or equal. Thus the access subtotal alone proves neither an overall win nor an overall loss. The constant-size blocked schedule also has quadratic block-product count; it is not the published multilevel relaxed-multiplication algorithm. + +The inspected PR249 five-stage schedule has successive source values x, x+y, y+z, y+z+w and y+w on four independent bank inputs. Repeated register names do not mean equal operands. Its stage routes occupy different full address windows, and independent dirty helper contents prevent silently adding cached representations without compensation and restoration. The one-shot outer multiplication already transforms each operand once. + +**Future use:** a caller that repeatedly consumes unchanged completed blocks, such as a growing-precision or power-series calculation, with a paid multilevel schedule, exact observation boundaries, physical access and cleanup. No such unchanged-operand repetition was found in the inspected supplier contract. Linear relations among stages might still support a different fully paid representation; this screen does not prove algebraic reuse impossible. + +## 6. Larger helper cohorts: a fixed-line family is excluded + +**Evidence status:** exact bounded enumeration and directed-rational price exclusion on an earlier checkpoint, plus a tiny basis/inverse control. No global impossibility theorem or synthesized new supplier. + +The baseline is the earlier PR254+PR251+seven-interval construction, raw SHA256 `cfd71e5ff0ba2316137ed7fb40cef1a28be733036417a206f79b4c23f5a1fe30`. Its pre-kernel word is `9f03fa69a06434d28f8c27fc06da7bc7db05f5a316d7f88b664c73dc6b45a887`; the fixed terminal cut is record 727473. The screen fixes the producer, observations, first-use and final frames, and the independent-completed-bank interface. Added helpers must be initially zero and untouched, and their shared entrance must lie on one of the 132 actual PR254 meeting lines. + +Among 3,337 eligible helpers, all 32,670 three-role extensions retain only the original pair's one-dimensional response kernel. Of 4,998,237 four-role extensions, 60 gain a second kernel; every such kernel splits into two disjoint equal-response pairs. The 24 distinct additional pairs have first-rank profiles (5,17) or (15,12), and both exact optimistic prices lose even with free perfect packing and maximal fractional stock credit. + +There are also 288 unrelated three-role relations on those lines. The cheapest standard one-pivot realization has first ranks (2,2,15), with paid histogram change `4[1]−2[2]+[14]−[15]`. The small integer basis and inverse, all 1,760 F2 response rows and three dirty columns are checked. At the pinned coarse rate `142206175346067/200000000000000000`, even its relaxed full moment exceeds one by more than `2275/10^12`; the other enumerated standard triple realizations cost more. Scalar read savings cannot erase an already losing recursive moment. + +This is exhaustive within the stated enumerated family, rather than a claim about all helper cohorts. **Future use:** change cuts, first-use geometry, meeting subspaces, observation contracts or completion coupling, with their new costs paid. PR259's multicut method is a different construction and is not ruled out by this result. + +## 7. Lift and contraction: useful identities, no free bilinear work + +**Evidence status:** algebraic statements under an explicit model and an exhaustive finite coefficient test; physical integration and an improved complete cost bound are absent. + +For radix digits a and b, the lifted matrix has entries T_ij=a_i b_j and coefficient observation C(T)_k=Σ_(i+j=k)T_ij. Dense storage can be avoided by immediate accumulation or factored bilinear algorithms. Calling T rank one does not remove the products, transform work or coefficient widths. + +For four radix-four digits, all 65,536 operand pairs give identical coefficients under dense contraction, fused accumulation and recursive Karatsuba, and the coefficients evaluate to the exact integer product. The dense method uses 16 scalar products and nine accumulating additions; the factored one uses nine products and 24 additions/subtractions, with wider intermediate values. These are operation/width comparisons, not a tape-runtime ranking. + +For a fixed projection P of lifted state, preserving every coefficient map requires CP=C, since elementary outer products span the lifted matrix space. Period-four exponent folding violates this condition on six lifted basis columns and corrupts 62,208 of the 65,536 tested products. Carrying the aliased coefficients cannot reconstruct the lost exponents. Guarded packing recovers the coefficients, but its tested operands grow from eight to twenty bits, so it is not a reduction to a smaller integer multiplication. + +At the separate linear supplier boundary, the full arbitrary-input endpoint S is invertible. If a fixed full-state projection must preserve that endpoint and there is no other retained recovery information, SP=S implies P=I. This excludes permanent rank loss under that unchanged contract. Temporary invertible basis changes, dependent copies and revised observation/retirement boundaries remain possible. A nonlinear lift also cannot inherit the linear formal-column proof: xy over F2 vanishes on both input basis vectors but not on their sum. + +**Future use:** an explicitly changed coefficient/observation boundary with a paid bilinear factorization and exact recovery. The bill must include transforms, products, widths, physical routes, carries, cleanup and any retained recovery information. No admissible lower-cost replacement of the present supplier has been established. + +## 8. A 79-factor address router: a local improvement without exponent gain + +**Evidence status:** independently checked exact semantic helper; a proposed finite-tariff specialization is conditional and unadmitted. + +The canonical 48-coordinate rational address router has an explicit 79-factor realization: 56 additions and 23 swaps, compared with the pinned Gaussian program's 195 additions, 44 swaps and 27 scalings. It uses no clean extra coordinate. All rational columns, inverse factors and the actual right-multiplication orientation are checked. Denominators introduce no prime exclusions beyond two and three. + +The source-bound PR249 adapter preserves the actual one-block permutations and stage windows. Checks cover all 1,760 port adapters, 100,540 owner/address sites, exact rational port cases, local-ring low/high carry semantics, inverse return and unchanged spectators. Missing adapters, carries and reversed factor order are rejected. This establishes a reusable semantic implementation and a smaller operation inventory, without assigning equal machine costs to unlike primitives. + +The current finite checker charges the generic high-address envelope `16(m²+1)²` per route. The 266-factor reconstruction is a diagnostic, not that coefficient's multiplier. Replacing it by 79 therefore leaves the emitted finite coefficient and recursive moment unchanged. A proposed dimension-two sparse-affine tariff still needs a compiler proof that includes nonadjacent relocation, descriptors, low-class selection and tape movement. + +Even granting that additional tariff on the pinned PR249 profile, its conditional coefficient screen retains the same 56-bit envelope. The term `36/δ²` dominates all three displayed cutoff levels, so those cutoffs remain equal. The full inherited primitive constants are still required. No smaller all-size threshold, runtime percentage or κ improvement follows. + +**Future use:** a justified tariff-parametric sparse compiler or a practical implementation study under an explicitly chosen cost model. A local address-factor improvement should not be credited as a recursive-child saving or attributed the entire hypothetical generic-to-sparse reduction. + +## Evidence boundaries + +The construction verifier checks the selected hybrid. These additional findings have their own bounded source snapshots and controls; they are not extra transformations in its paid ledger. Historical prices are tied to the stated baseline rather than silently repriced at the final κ. Exact finite tests, algebraic exclusions, unresolved compiler obligations and future opportunities are intentionally distinguished. No global search exhaustion or total-runtime conclusion is inferred from them. diff --git a/research/multicut-delayed-inverse-checkpoint/NOTICE.md b/research/multicut-delayed-inverse-checkpoint/NOTICE.md new file mode 100644 index 0000000000..4f30bb4fd8 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/NOTICE.md @@ -0,0 +1,23 @@ +# Attribution and licence + +Apache-2.0 applies. Original headers, licences, notices and assistance disclosures are preserved. Credits do not imply review or endorsement. The complete identified heritage is recorded in `HERITAGE.md`; original five-stage credits remain in `UPSTREAM-PR234-NOTICE.md` and the individual source notices. + +## Present construction + +This package retains 513 of Dugongue's 518 PR259 multicut cohorts, replaces five conflicting rank-11 quartets with seven closed target intervals, and applies the 25 gate-frame retimings represented by amanforindia's PR258 and Rohan Arun's PR260 once. It retains eumemic's PR249 compensation transport and 21 entrances. PR251's sixteen additional entrances and the earlier separate PR254 package are not additional contributions to its paid ledger. + +Henry Grant / hcg890's PR234 supplies the source-bound five-stage architecture and finite interfaces. Rohan Arun's PR237 published the width-120 bank extension; no exclusive priority is claimed for that extension. The callable bank assignment and address adapter here are checked against the actual forty-replica word. + +The seven delayed-inverse intervals, this Python composition and its checks were assisted with ChatGPT. No exclusive priority is claimed for target recoding, borrowed workspace, completed banks, cohort sharing or inherited kernel mechanisms. + +## Inherited sources and preceding contributions + +The source527 construction builds on eumemic's PR210 source reuse and target transforms, sennemmi's PR230 operation-frame changes, Dugongue's PR216 and PR207, Rohan Arun's PR211, Chafik Boukhalfa's PR200, Evan McKinney's completed bank work, and their retained dependencies. The complex input preserves Jacob Sussman's published five-stage certificate and Avi Eisenberg / ikeboy's PR193 helper, with the original Claude assistance disclosures. + +The 31 zero-response source aliases, 48 retained dirty-anchor paths, physical event tagging and source527 integration were prepared by eumemic with substantial OpenAI Codex assistance. The parity-filtered common-frame retiming in PR244 and its exact reconstruction were also prepared by eumemic with substantial OpenAI Codex assistance. Its 353 unborrowed pair mixes use their existing common delivery frames. + +The predecessor PR251 extension combined the 21 PR249 transported entrances with sixteen further shared-allowed-frame entrances, adding 224 entrance-rank units. Its compensation transport, cloned producer/execution contexts and completed-bank charts were prepared by eumemic with substantial OpenAI Codex assistance. Only the inherited 21 PR249 entrances are used in the present construction. + +Earlier interval development retained all 37 PR251 entrances in a ten-replica checkpoint; an intervening composition also used Dugongue's PR254 paired-helper rewrite at `9ce32efb421132420242cfdb91ddd9eabe2c3795`. Those experiments, packing witnesses, adversarial checks and Python integration were prepared with substantial OpenAI assistance. They are historical antecedents. The construction and counts in `README.md` and `PROOF.md` govern this package. + +All earlier contributor credits and original assistance disclosures remain in the supplied source notices. diff --git a/research/multicut-delayed-inverse-checkpoint/PR259-PROVENANCE.md b/research/multicut-delayed-inverse-checkpoint/PR259-PROVENANCE.md new file mode 100644 index 0000000000..f4c50d9886 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/PR259-PROVENANCE.md @@ -0,0 +1,27 @@ +# PR259 lineage and selected inputs + +PR249: 96495746c786d6d0339dbb38c7f553d4af3f88ed. +PR259: a53ec0ca4d59634f16630dd16adf035c9f12595c, repository +Dugongue/integer-mult-bounds, research/multicut-kernel-condensation/. +Its candidate gzip is included byte-for-byte as kernel-candidates.json.gz: +SHA256 5f6863ad27eb9f2fcb14db10320c00607451c01e12da303def13e48eda7a5d2d; +Git blob ca1360f111e32a699ae687d979dec1d473083377. + +hybrid-selection.json declares the five removals and seven exact interval +witnesses. retiming25-selection.json declares the uniquely scalar-keyed 25 +retimings and their exact old/new subspaces. No optimizer or external source +path is required during verification. Optional annihilator annotations in +the upstream candidate search data are not trusted: they are reconstructed +from the basis, matching the actual upstream compiler's behavior. + +The sixteen extra PR251 entrances are not included: all conflict with the +new PR259 selection. Historical copied interval/kernel witnesses are omitted. +The current interval data are in hybrid-selection.json, and the byte-exact +upstream PR259 witness is kernel-candidates.json.gz. portable_bit.py calls +the current hybrid_transform.py and retiming25.py for this construction. + +PR258: cf82ca09322d1327ab887cfd1cce97591b8d3cfa. +PR260: 87aa555a455cca84a3e3296ced2440623d39fc4d. +These independently published retiming witnesses identify the same 25 +scalar gates and replacement subspaces; the changes are applied once. +See [HERITAGE.md](HERITAGE.md) for contributor attribution and exact boundaries. diff --git a/research/multicut-delayed-inverse-checkpoint/PROOF.md b/research/multicut-delayed-inverse-checkpoint/PROOF.md new file mode 100644 index 0000000000..bee151eb55 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/PROOF.md @@ -0,0 +1,121 @@ +# Source-bound hybrid construction + +## Exact lineage and reconstruction + +The defining source527 producer, its signed integer decoder, owned source +streams and arbitrary dirty roles are unchanged. Fresh preparation, physical +emission, parity fusion, the original 364 retimings and the 21 PR249 +transported entrances reproduce local raw word +2e0bc2a5c54cf8caf26fa30a1a2949702d842f620b28023ca2ba59140ecbed1a. +hybrid_transform.py reads the pinned 518-candidate PR259 witness, removes +pivots 17207, 17222, 17246, 17255 and 17273, and constructs the selected +513 kernel cohorts and seven target intervals. retiming25.py then applies +the separate 25 scalar-keyed changes. The resulting exact raw word is +acf510b1377a6bd96eea50577de1aa049382499335fa7e5cac9b0300ecb377c7. + +## Kernel and observation boundaries + +For each retained cohort the pivot and donors have zero initial/full final +frames and are untouched through their declared cut except for zero-frame +dirty-to-target reads. Its exact nondegenerate entrance basis is contained +in every actual first-use frame. Pivot supports are unique and disjoint +from donor supports, and the emitted donor frame sequences are nested. +At the cut, donor += pivot gates are applied at that entrance; corresponding +literal inverses occur at the common full endpoint. Every surviving gate +is retained in order. The complete physical word and its literal reverse +are checked on all 20,107 arbitrary source, old-target and dirty columns. + +The old seven target intervals cannot simply be added: their low helpers +overlap five PR259 quartets. Those five quartets are removed in full. Every +one of their twenty helpers belongs to that cohort only. All fourteen +interval helpers then remain independent, zero-initial and untouched up to +the terminal semantic cut. Their exact signed prefix columns are each +minus one on their four targets, with no surviving external helper/source +observer. At all four retained-cohort cuts, these columns vanish on every +retained cohort operand. This checks the interaction rather than assuming +that disjoint helper identifiers are sufficient. + +Each interval preconditions three targets by subtracting a pivot target at +frame zero. It compensates that pivot from the low/high helper at nested +frames L/H of ranks 5/14, then applies the target inverse at a legal common +rank-20 frame K. Exact six-column integer matrix multiplication gives the +original two-helper contribution to all four targets and preserves arbitrary +old target contents; the literal inverse restores all six columns. Every +subspace inclusion, Gram determinant, target successor and intervening +observation is checked. The later inverse frame changes the paid movement +partition. Restricting the inverse to H is a different, previously rejected +family and cannot exclude this construction. + +The replacement word before final retiming has hash +6ab38175783163e729a77032e7e637299b2341aa9ab282134c22796b66699112. +The total new entrance rank is 1344-5*11+7*19=1422. Modulo-three normalization +is checked only after composition; no additional rank-one candidates are +silently dropped. Within each cut the adapter orders cohorts by +(dimension,pivot), and checks that actual order. No claim of byte identity +with unspecified C++ equal-rank sort order is needed. + +## Paid retiming and restoration + +Each of the 25 selected scalar events is uniquely keyed and checked against +its old exact basis. Its new frame equals an existing successor frame, so +no extra chart is introduced. Both operand predecessor/new/successor +containments are checked; COPY source values and frames remain immutable +through their windows. Re-emitted MOVE partitions change the local histogram +by 2:-2, 4:+24, 5:-48, 6:+25, 11:-1, 13:+2, 17:-1. Scalar/COPY projection and +all endpoints are byte-identical. Local paid calls fall by one; rank mass +does not change. PR258 and PR260 identify the same 25 gates, applied once. + +The final local word has 96,117 paid children, rank mass 432,755, 760,383 +scalar events and 763,023 unit-expanded additions. Its chosen signed-lift +majorants are 73,095 forward and 3,396,290 inverse. The F2 identity and +the signed-lift majorant are distinct obligations; no old integer endpoint +equality is claimed. + +## Global assembly, banks and exact accounting + +The callable five-stage lowering freshly checks every literal frame path, +copied-center deadline, reflected orientation, data bridge and terminal +exchange. It emits exactly 2,827 tagged independent completions. The actual +bank plan consumes those tags by count. Completion widths 5, 10, 15, 20 +and 50 also contain ordinary children, which are never discarded wholesale. +The retained per-replica ledger is 494,665 calls / rank mass 2,596,735. + +See BANK-PROOF.md for the exact 18-pattern forty-replica packing, all 770 +actual chart IDs, 3,317,400 ownership assignments and 200 injective complete +namespaces. Their 459 distinct entrance bases and every other physically +consumed or defining source basis receive exact determinant/prime checks. +Repeated frame IDs are charged separately even when their basis bytes agree. +Every new chart uses the generic actual-form construction; a norm-four +rank-one shortcut would be invalid for 239 of the rank-one entrances. + +Two independent directed-rational moment engines include the complete rare +fallback and separate adjacent-grid rejection. Three ordinary bootstrap +levels and all 47 outer inequalities yield the stated rational exponent. +The forty-replica finite bill includes all literal children, unit additions, +bridges, route families, high/low affine factors, wrappers, matrix preparation, +copies, selectors, chart normalizers, fallback and row restoration. Its +coefficient is 62,644,197,182,478,801, below 2^80; payload prefix upper is +288305228198350917154735200000, a 98-bit integer below 2^104. The exact +cutoff rule still includes every inherited constant, not just this displayed +finite coefficient. + +## Evidence scope + +The earlier cloud check freshly executed changed source/physical/global/geometry/ +F2-prefix/prime/bank/moment/finite stages and reused unchanged signed-source +and complex evidence under exact input/code and decoder-AST bindings. +The later clean macOS reproduction freshly executed all eight mandatory +stages from primary inputs; see the redacted summary in [README.md](README.md). +The standalone [verify.py](verify.py) always reconstructs all eight stages. +This public extraction changes documentation and excludes unused historical +files, while retaining the exact code, current witnesses and generated +[certificate](expected/certificate.json) of that verified construction. + +The 127-subset screen establishes the chosen replacement's best price in +that declared family before adding the 25 retimings. It is not a claim of +global optimality or untested family optimality at another rate. Inherited +weighted compiler, GL-cover/common-ancestor chart, restored-row/routing, +prime supply, recovery, complex-symbolic/analytic and full-cutoff interfaces +remain explicit assumptions of this conditional construction. + +See [NEGATIVE-RESULTS.md](NEGATIVE-RESULTS.md) for the precise alternative-family boundaries. diff --git a/research/multicut-delayed-inverse-checkpoint/README.md b/research/multicut-delayed-inverse-checkpoint/README.md new file mode 100644 index 0000000000..7fb7dda594 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/README.md @@ -0,0 +1,65 @@ +# Multicut kernels, delayed target inverses and paid retiming + +Conditional κ = **0.000711045547624400**, exactly +1777613869061 / 2500000000000000. + +The construction starts from exact PR249 source, retains 513 of PR259's 518 +multicut kernel cohorts, replaces five conflicting rank-11 quartets with seven +closed-target delayed-inverse intervals, and applies the 25 PR258/PR260 +retimings exactly once. It does not add the sixteen extra PR251 entrances. +The full price is recomputed from the resulting physical word. + +The literal forty-replica construction uses stock 867,045 and 19,786,600 +paid children, rank mass 103,869,400 and deficit 176,000. All 770 actual +bank chart IDs, 3,317,400 assignments and 200 full namespaces are checked. +The complete finite coefficient is 62,644,197,182,478,801; every scalar, +movement, copied center, selector, normalizer, fallback and ordinary cutoff +charge is included. + +Run in a new isolated directory with Python 3.11+, SymPy 1.14.0 and +assertions enabled: + + PYTHONDONTWRITEBYTECODE=1 python3 -B verify.py --output /new/output/directory + +The output must be new and outside this package. Never use -O. +The standalone verifier reconstructs all eight mandatory stages from inputs; +it does not consume saved execution receipts and makes no network calls. + +On 10 October 2026, a clean macOS reproduction of the source checkpoint +completed all eight mandatory stages freshly in 580.9 seconds, with 25 +scalar controls, six prime controls and eight omitted-stage controls. +The tested source manifest is +9f6d8637b9a119cfe4566a02ab2d704f23c502e5e4357b40c28ef85a3fd4de89. +This public extraction preserves all 117 retained numerical pins, removes +seven unused historical numerical files and changes only documentation. +It reuses that numerical execution evidence; it is not a claim that the +documentation-only extracted tree underwent a second full numerical run. +The exact generated certificate is included as [expected/certificate.json](expected/certificate.json), +SHA256 7eb58ec449c932b7092481746390c776365f302a742836cddef5643c0351802b. + +This is a deliberately redacted reproduction summary. It omits local +paths, process identifiers and checkout details. The original reproduction +receipt has SHA256 +327b8b0d5b7b27bc3f25e17222168262671364a3a698e4915a249bed863a8b06. +Python 3.14.2 and SymPy 1.14.0 were used, with assertions enabled and no +source drift. The raw receipt is not a package dependency. + +The payload contract is F2 with separately charged chosen signed-lift prefix +majorants. It is not equality to the prior signed integer endpoint. See +[PROOF.md](PROOF.md), [BANK-PROOF.md](BANK-PROOF.md) and +[PR259-PROVENANCE.md](PR259-PROVENANCE.md). No exhaustive global +optimality or practical runtime claim is made. + +The all-size compiler, complete GL-cover/common-ancestor charts, restored +rows, routing, prime supply, precision/recovery, complex-symbolic/analytic +interfaces and full inherited-constant cutoff remain theorem dependencies. + +## Checkpoint status + +This is an older verified construction checkpoint, not a current-best claim. +As checked on 10 October 2026, [PR266](https://github.com/CrocSwap/integer-mult-bounds/pull/266) +at b2b2a33dc95eb2d4ad9ddfade6bb9416ab9ec38e reports the stronger +same-model conditional bound 44463833535321/62500000000000000 = +0.000711421336565136. That report is a separate construction and is not +substituted into this package's source freeze, certificate or price. + diff --git a/research/multicut-delayed-inverse-checkpoint/UPSTREAM-PR234-NOTICE.md b/research/multicut-delayed-inverse-checkpoint/UPSTREAM-PR234-NOTICE.md new file mode 100644 index 0000000000..5f8fbb25c1 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/UPSTREAM-PR234-NOTICE.md @@ -0,0 +1,13 @@ +# Attribution and licence + +This contribution uses Apache-2.0. Original source headers, licences and notices are preserved. These credits do not imply review or endorsement by the named contributors or OpenAI. + +- Jacob Sussman: five-stage complex geometry and published gcert/1 export in wht-power-saving-lean at f010392c923279e3dd59ef3aa5fedad23affc5fd. Original Claude assistance disclosure retained in inputs/complex/NOTICE. +- Avi Eisenberg / ikeboy: retained PR193 source-assisted complex helper and donor selection at 187e1010ac8b259af8e9b5166f68b64bc27b4b47. +- eumemic: current PR210 frames, gauges, source borrowing and retimed bit helper at df95878d11190518e45ef9717c9ee05011f88ace. +- Chafik Boukhalfa / chafreaky: PR200 physical bit construction carried by PR202 at 8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d. +- Rohan Arun, Dugongue, evmckinney9, icekylinx, James Chang, Douglas Colkitt, OpenAI and earlier contributors retain the attribution and original assistance disclosures in the supplied source notices. The exact outer assembly also credits Zhihao Chen and the RaD / hipotures balanced reduction in its original header. + +The paired-cube helper design predates Sussman’s certificate export. The new contribution here is the explicit source-bound five-stage composition, adapters, paid accounting and reproducible finite verification. No exclusive priority is claimed for the inherited designs or theorem interfaces. + +Assisted with ChatGPT. diff --git a/research/multicut-delayed-inverse-checkpoint/UPSTREAM-PR234-PROOF.md b/research/multicut-delayed-inverse-checkpoint/UPSTREAM-PR234-PROOF.md new file mode 100644 index 0000000000..0524507982 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/UPSTREAM-PR234-PROOF.md @@ -0,0 +1,37 @@ +# Historical PR234 five-stage finite composition and retained hypotheses + +This is the historical PR234 proof. Its source471 parameters and numerical +examples describe the predecessor construction. The current source527 hybrid +construction, exact counts and exponent are in [PROOF.md](PROOF.md). + +The historical conditional saving is κ = 703701743496697 / 10^18. The fresh verifier establishes the finite construction and exact arithmetic under the inherited source and all-size interfaces below. + +## Bit construction + +The pinned source471 helper is checked on all 20,163 independent F2 source, target and dirty columns. Its physical realization is the exact input/output frame telescope, with all source loans, donor/recipient identifications, compensated reads, terminal substitutions and cleanup retained. Copied centers are complete temporary streams; their reverse implementation copies the original anew, transforms and scatters that copy, then erases it. It does not reverse an erasure. + +The five phase-major invocations use the explicitly routed source owners X1, Y1, X1, Y2 and X2. Each helper's five separated residuals occupy disjoint windows. One split rank-5a exterior correction completes each genuine independent entrance of rank a. All 2,279 corrections remain paid, and source-owned gauges stay within the visible-data paths. Actual rational matrix checks establish bridge equality, nested idle changes and terminal exchanges. Cover expansion has a barrier after each phase, across every class. + +The fresh program has m=120, W=23,683, 495,304 children, rank mass 2,837,560 and deficit 4,400. All 120 rank-22 copied-center calls are included. Exact cleared-Gram determinants and factor witnesses are recomputed for every consumed basis. Fivefold direct sums introduce no new excluded primes. The retained weighted q-local compiler, unit-pivot fallback, prime-supply and complete-stream tape proofs remain hypotheses. + +The proper child ratio 5/6 requires halving degree four. The one-level weighted recurrence is retained; its histogram is not multiplied by four. The finite accounting rederives route factors, scalar work, internal row stock, strict moment/linear gaps and explicit finite cutoff expressions. Unknown constants of inherited primitive implementations remain quantified constants, not invented measured costs. See proof/FINITE_BIT_ACCOUNTING.md and the included original bit/row/copied-center source texts. + +## Complex construction + +The published gcert/1 baseline is pinned to Sussman's f010392 revision and the PR193 paired-cube helper. Its live-label walk, scalar programs, copied-center lifetimes, final climbs before cleanup, exact symbolic splice and finite guard are regenerated. Exact complex frame identities retain their phase representatives. Physical work stock 24,127 is separately charged; the live recursion denominator remains 14,692. + +The pinned scalar identities hAi, hBi, hx, hy and hid, the Lab.inv dirty-lifting theorem and five-window geometry/NetCert theorems are inherited source proofs. The command does not rebuild their Lean proof chain or claim a fresh all-column complex scalar proof. General Clifford synthesis, charged complete-stream movement, common-grid precision and restored-row proofs remain stated analytic dependencies. Their consumed declarations and sources are hash-pinned under inputs/complex/. + +## Arithmetic and outer reduction + +Both rational moments are freshly enclosed. Three finite ordinary-leaf levels start from the retained completed ordinary saving 384599/10^10; the unattained coarse limiting saving is not used as an ordinary leaf. Exact outer assembly checks all 47 strict inequalities and seven margins and rejects the next 10^-18 κ grid point. Precision/recovery, prime supply, completed ordinary leaves and the fixed-tape all-size reduction remain hypotheses. + +[proof/INTEGRATED_PROOF.md](proof/INTEGRATED_PROOF.md) preserves the detailed +historical composition argument. The current executable dependency closure +is pinned by [MANIFEST.json](MANIFEST.json); [verify.py](verify.py) rebuilds +its mandatory stages from these inputs. Development receipts and historical +source trees are not runtime inputs. + +No claim is made of unconditional all-size formalization, global optimality or measured multiplier performance. Original lineage and assistance disclosures are preserved in NOTICE.md and the supplied source notices. + +Assisted with ChatGPT. diff --git a/research/multicut-delayed-inverse-checkpoint/bank_check.py b/research/multicut-delayed-inverse-checkpoint/bank_check.py new file mode 100644 index 0000000000..291383b6f9 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/bank_check.py @@ -0,0 +1,265 @@ +"""Fresh context-bound stage-private120-coordinate completed bank admission. + +PR259 replacement plus25 compatible retimings, using forty +literal replicas. No saved chart, receipt, physical census or old source tree is consumed. +The fraction-free inverse routine derives from the retained PR197/PR200 bank +checker. Inherited attribution/notices apply. Apache-2.0. +Prepared with substantial OpenAI Codex assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from math import gcd +from pathlib import Path +import hashlib,json,struct,time +from bank_template import BankPlan + +assert __debug__, 'Do not run bank admission under -O.' +HERE=Path(__file__).resolve().parent + +def inverse_integer(a): + """Fraction-free Gauss-Jordan, returning C,d with aC=Ca=dI.""" + n=len(a);m=[list(row)+[int(i==j)for j in range(n)]for i,row in enumerate(a)] + previous=1;factors=[] + for k in range(n): + pivot=next((i for i in range(k,n)if m[i][k]),None) + assert pivot is not None,'singular actual bank chart' + m[k],m[pivot]=m[pivot],m[k] + if pivot!=k:factors.append(('swap',k,pivot,1,1)) + p=m[k][k] + if p!=previous: + ratio=Q(previous,p);factors.append(('scale',k,k,ratio.numerator,ratio.denominator)) + for i in range(n): + if i==k:continue + u=m[i][k] + if u: + ratio=Q(-u,previous);factors.append(('add',i,k,ratio.numerator,ratio.denominator)) + for j in range(2*n): + if j==k:continue + value=p*m[i][j]-u*m[k][j];assert value%previous==0;m[i][j]=value//previous + m[i][k]=0 + previous=p + d=m[0][0];assert d and all(m[i][j]==d*int(i==j)for i in range(n)for j in range(n)) + c=[row[n:]for row in m];common=abs(d) + for row in c: + for x in row:common=gcd(common,x) + d//=common;c=[[x//common for x in row]for row in c] + if d<0:d=-d;c=[[-x for x in row]for row in c] + return c,d,factors + +def census(context): + W,C,roles=context['W'],context['C'],context['regs'] + assert len(roles)==len(set(roles))==16587 and roles==sorted(roles) + assert set(roles)==set(W.phys.values())-set(context['borrow'])-context['removed'] + assert not(set(roles)&set(context['borrow'])) + gauge_frames={role:W.gauge[role]['frame']for role in roles if role in W.gauge} + assert Counter(C.dimf[f]for f in gauge_frames.values())=={1:369,2:41,3:3,4:19,5:31,6:1,8:1,9:20,10:2,11:14,12:30,13:55,14:8,15:4,17:7,18:22,20:2200} + assert all(C.dimf[gauge_frames[r]]==14 for r in (14212,14221,14230,14239,14248,14266,14284)) + assert all(C.dimf[gauge_frames[r]]==5 for r in (14214,14223,14232,14241,14250,14268,14286)) + families={} + for role in roles: + rank=24-C.dimf[gauge_frames[role]]if role in gauge_frames else 24 + families.setdefault(rank,[]).append(role) + assert {r:len(xs)for r,xs in families.items()}=={4:2200,6:22,7:7,9:4,10:8,11:55,12:30,13:14,14:2,15:20,16:1,18:1,19:31,20:19,21:3,22:41,23:369,24:13760} + return families,gauge_frames + +class BankLowerer: + """Final full-address lowering of the already checked logical IR. + + Scalar/projection/geometry fields are retained byte-for-byte; only stream + operands change to the proven bank namespaces. Completion discharge is + checked separately by run(), and no completion opcode is emitted. + """ + def __init__(self,context,lower): + families,gauges=census(context) + self.plan=BankPlan(families,gauges,context['regs']);self.lower=lower + def iter_stage(self,stage,replica): + for row in self.lower.iter_stage(stage):yield self.plan.map_stage_row(stage,replica,row) + def iter_idle(self,which,replica): + for row in self.lower.iter_idle(which):yield self.plan.map_boundary_row(replica,row) + def iter_bridges(self,which,replica): + for row in self.lower.iter_bridges(which):yield self.plan.map_boundary_row(replica,row) + def iter_exchanges(self,replica): + for row in self.lower.iter_exchanges():yield self.plan.map_boundary_row(replica,row) + def phases(self): + """Each iterator is expanded over all cover classes before next phase.""" + for stage in range(5): + for replica in range(self.plan.replicas): + yield('helper',stage,replica,lambda s=stage,r=replica:self.iter_stage(s,r)) + if stage in(1,2,4): + which={1:0,2:1,4:2}[stage] + for replica in range(self.plan.replicas): + yield('idle',which,replica,lambda i=which,r=replica:self.iter_idle(i,r)) + yield('bridge',which,replica,lambda i=which,r=replica:self.iter_bridges(i,r)) + for replica in range(self.plan.replicas):yield('terminal_exchange',0,replica,lambda r=replica:self.iter_exchanges(r)) + +def bind(context,lower):return BankLowerer(context,lower) + +def run(context,global_result,lower,progress=lambda text:None): + begun=time.monotonic();W,C=context['W'],context['C'] + api_pin=hashlib.sha256((HERE/'bank_template.py').read_bytes()).hexdigest() + families,gauge_frames=census(context);bound=bind(context,lower);plan=bound.plan + assert lower.W is W and lower.C is C + # Derive all actual chart bases and exact inverse/factor programs afresh. + charts=[];maxops=maxnum=maxden=0 + for frame,count in sorted(Counter(gauge_frames.values()).items()): + residual=[] + for a in C.A[frame]: + row=[15*x-sum(a)for x in a];d=gcd(*row);assert d;residual.append([x//d for x in row]) + source=C.B[frame];rank=len(residual) + assert rank+len(source)==24 and len(source)==C.dimf[frame] + assert all(9*sum(x*y for x,y in zip(a,b))-sum(a)*sum(b)==0 for a in residual for b in source) + B=list(map(list,zip(*(residual+list(source)))));A,d,ops=inverse_integer(B) + for L,R in((A,B),(B,A)): + assert all(sum(x*y for x,y in zip(row,col))==d*int(i==j)for i,row in enumerate(L)for j,col in enumerate(zip(*R))) + P=[[sum(B[i][k]*A[k][j]for k in range(rank))for j in range(24)]for i in range(24)] + assert sum(P[i][i]for i in range(24))==d*rank + assert all(sum(P[i][k]*B[k][j]for k in range(24))==(d*B[i][j]if j=n;H[r]-=n + if not H[r]:del H[r] + assert all(H[r]>0 for r in(5,10,15,20,50)) + assert sum(before_completion_removal.values())-sum(H.values())==2827 + assert max(H)==50 and sum(H.values())==494665 and sum(r*n for r,n in H.items())==2596735 + literal=Counter({r:plan.replicas*n for r,n in H.items()}) + assert 120*plan.live_families-sum(r*n for r,n in literal.items())==plan.replicas*4400 + # Independently enumerate each physical role/replica/stage slot. + incidence=hashlib.sha256();assignments=0;normalizers_checked=set();endpoint_controls=[] + for stage in range(5): + used=Counter();bank=0;assigned=set() + outside=next(j for j in range(120)if not 24*stage<=j<24*(stage+1)) + for pattern,(widths,count)in enumerate(plan.patterns): + offset=0;witnesses=[] + for block,rank in enumerate(widths): + original=list(range(24*stage,24*stage+rank));target=list(range(offset,offset+rank)) + src=original+[j for j in range(120)if j not in original];dst=target+[j for j in range(120)if j not in target];pi=dict(zip(src,dst)) + witnesses.append((pi[outside],block+1));offset+=rank + assert len(set(witnesses))==len(widths) + if stage==0: + offsets=[sum(widths[:b])for b in range(len(widths))] + def endpoint(blocks): + state=list(range(240)) + for b in blocks: + for j in range(offsets[b],offsets[b]+widths[b]):state[j],state[120+j]=state[120+j],state[j] + return state + blocks=list(range(len(widths)));full=list(range(120,240))+list(range(120)) + assert endpoint(blocks)==full and endpoint(blocks+blocks[::-1])==list(range(240)) + assert endpoint(blocks[:-1])!=full and endpoint(blocks+[0])!=full + endpoint_controls.append(dict(pattern=pattern,forward_columns=240,inverse_columns=240,omitted_rejected=True,repeated_rejected=True)) + for b in range(bank,bank+count): + offset=0 + for block,rank in enumerate(widths): + q=used[rank];role=families[rank][q//plan.replicas];replica=q%plan.replicas;used[rank]+=1 + assert(role,replica)not in assigned;assigned.add((role,replica)) + a=plan.assignment(stage,replica,role);frame=gauge_frames.get(role) + assert a==dict(family=plan.data_families+stage*plan.banks_per_stage+b,stage_bank=b,pattern=pattern,block=block,offset=offset,rank=rank,frame=frame,scalar=block+1) + key=stage,pattern,block,frame + if key not in normalizers_checked: + N=plan.normalizer(stage,replica,role) + assert N['unit_column_witness']==witnesses[block]and N['inverse_chart_frame']==frame and N['chart_window']==stage + assert N['permutation'][24*stage:24*stage+rank]==tuple(range(offset,offset+rank)) + normalizers_checked.add(key) + incidence.update(f'{stage},{pattern},{role},{replica},{b},{block},{offset},{rank}\n'.encode()) + assignments+=1;offset+=rank + assert offset==120 + bank+=count + assert bank==plan.banks_per_stage and len(assigned)==plan.replicas*len(context['regs'])and used=={r:plan.replicas*len(xs)for r,xs in families.items()} + assert assignments==3317400 + progress('Matched all3317400 literal bank assignments') + namespace=hashlib.sha256();collisions=0 + for stage in range(5): + for replica in range(plan.replicas): + addresses=[plan.local_address(stage,replica,i)for i in range(plan.local_work+1)] + assert len(set(addresses))==len(addresses)and addresses[-1]==(plan.work_family,('external_work',0)) + assert all(f=4*self.v:return 2*self.v+family-4*self.v + bank,port=divmod(family,self.v) + if bank==self.active[stage][0]:return port + assert bank==self.active[stage][1], 'stage references an inactive data bank' + return self.v+port + + def map_stage_row(self, stage, replica, row, cover='d'): + """Lower an unchanged logical8-field MOVE/ADD/COPY/ERASE record. + + All mathematical fields remain unchanged. The two operand slots now + contain exact (family,route) keys when they denote streams; frame IDs + in MOVE's old/new slots are not mistaken for stream operands. + """ + op,a,b,c,f,z,s,rev=row + assert op in(0,1,2,3) and s==stage and rev==int(stage in(1,3)) + def address(family): + return self.local_address(stage,replica,self.local_from_logical_family(stage,family),cover) + physical_a=address(a) + physical_b=address(b) if op in(1,2) else b + if op in(1,2):assert physical_a!=physical_b + return (op,physical_a,physical_b,c,f,z,s,rev) + + def map_boundary_row(self, replica, row, cover='d'): + """Lower an IDLE/BRIDGE/EXCHANGE with its existing exact projector. + + Boundary projectors already include the checked data-route conjugacy; + their physical class is d. Independent completion rows must be + discharged explicitly by the bank endpoint verifier, never skipped + implicitly by this address mapper. + """ + op,a,b,c,f,z,s,rev=row + assert op in(4,5,7), 'only proved bank completion may discharge opcode6' + def address(family): + assert 0<=family<4*self.v + bank,port=divmod(family,self.v) + return self.data_family(replica,bank,port),('class',cover) + physical_a=address(a) + physical_b=address(b) if op in(5,7) else b + if op in(5,7):assert physical_a!=physical_b + return (op,physical_a,physical_b,c,f,z,s,rev) + + def phase_schedule(self): + """Phase-major schedule; every cover class is completed per replica. + + Each emitted stage phase means: for the indicated replica, iterate + every cover class and execute the entire local scalar telescope. + Boundary phases occur after all40 replicas of that stage finish. + """ + for stage in range(self.stages): + for replica in range(self.replicas): + yield ('stage',stage,replica,'all_cover_classes') + if stage in (1,2,4): + for replica in range(self.replicas): + yield ('idle',stage,replica,'all_cover_classes') + yield ('bridge',stage,replica,'all_cover_classes') + for replica in range(self.replicas): + yield ('terminal_exchange',replica,'all_cover_classes') diff --git a/research/multicut-delayed-inverse-checkpoint/base_two_moment.py b/research/multicut-delayed-inverse-checkpoint/base_two_moment.py new file mode 100644 index 0000000000..14a90fa0e7 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/base_two_moment.py @@ -0,0 +1,63 @@ +#!/usr/bin/env python3 +"""Maintainer arithmetic cross-check of the pinned PR39 certificate. + +Uses no producer/checker imports. Counts are inputs, not proved by this script. +80-term rational logarithms and 12-term exponentials independently enclose +the two moments. The accompanying written review addresses their realization. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +GRID = 10**30 + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def log_unit(x): + require(1 <= x <= 2, "logarithm range") + z = (x-1)/(x+1) + lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0)) + return lo, lo+2*z**161/(161*(1-z*z)) + + +def log_bounds(x): + k = 0 + while x > 2: + x /= 2 + k += 1 + lo, hi = log_unit(x) + l2, u2 = log_unit(Q(2)) + lo, hi = lo+k*l2, hi+k*u2 + # Outward rational rounding keeps later certificates compact. + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + + +def exp_bounds(x): + require(0 <= x < 1, "exponential range") + term = total = Q(1) + for j in range(1, 13): + term *= x/j + total += term + tail = term*x/13/(1-x/14) + return total, total+tail + + +def moment(m, W, entries, saving): + lo = hi = Q(0) + for t, count in entries: + t, count = int(t), int(count) + require(0 < t < m and count > 0, "improper child") + l, u = log_bounds(Q(m, t)) + lower = exp_bounds(saving*l)[0] + upper = exp_bounds(saving*u)[1] + lo += Q(t*count, m*W)*lower + hi += Q(t*count, m*W)*upper + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + diff --git a/research/multicut-delayed-inverse-checkpoint/code/complex_basis.py b/research/multicut-delayed-inverse-checkpoint/code/complex_basis.py new file mode 100644 index 0000000000..8c400798a5 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/code/complex_basis.py @@ -0,0 +1,35 @@ +"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers.""" +from collections import Counter +from copy import deepcopy +from fractions import Fraction +Q=Fraction +from hashlib import sha256 +from pathlib import Path +from math import prod +import gzip,json,sys + +def need(b, why): + if not b: raise ValueError(why) + +def gl_count_test(): + # Exhaustively test a literal binary Gaussian-elimination implementation + # through m=4. Row swaps cost three selected-bit additions. + out=[] + for m in range(1,5): + good=mx=0 + for word in range(1<<(m*m)): + rows=[(word>>(m*i))&((1<>k&1),None) + if p is None: break + if p!=k: rows[p],rows[k]=rows[k],rows[p];ops+=3 + for i in range(m): + if i!=k and rows[i]>>k&1: rows[i]^=rows[k];ops+=1 + else: + need(rows==[1<0, f"{where}: coefficient shape") + q=Fraction(a,b) + odd=q.denominator + while odd%2==0: + odd//=2 + need(odd in (1,3), f"{where}: literal coefficient outside dyadic/divisor-three grid") + return q + +def replay_label_counts(c): + """Full finite LABEL walk only. Never multiplies the 12,052-square scalar map.""" + need(c.get("format")=="gcert/1", "format") + need(all(c.get(k)==v for k,v in BASE.items()), "baseline parameter/count mismatch") + h,v,R=c["h"],c["v"],c["R"]; n=2*v+R + need(c.get("ext")==[], "baseline has no exterior-gauged helper inputs") + frames=c["frames"]; dim=list(map(len,frames)) + need(len(frames)==18248, "baseline frame-table count") + need(frames[0]==[] and frames[1]==[1<piv[i+1] for i in range(len(piv)-1)), "frame pivot order") + need(all(not(b>>p&1) for i,b in enumerate(f) for j,p in enumerate(piv) if i!=j), + "frame is not reduced echelon") + ports=c["ports"] + need(len(ports)==v==len(set(ports)), "port count/uniqueness") + need(all(type(p) is int and 0=4 and g[0] in ("out","in"), where+": gate shape") + tag,f,pivot,terms=g[:4] + need(len(g)==(5 if tag=="out" else 4), where+": gate arity") + need(type(f) is int and 0<=f{tr}"]+=1 + totals[phase]["scalar_additions"]+=1 + if sr==tr=="y": + ybracket[(source,target)]+=q;ypiv.add(source);ytarget.add(target) + for r in regs: + climb(r,f,phase) + if phase=="A": + cut=cur.copy() + need(all(cur[v+t]==0 for t in range(v)), "scatter: Y role moved too early") + for k,r,f in c["ret"]: + need(type(r) is int and 2*v<=r>k&1 else c["scat"]["outside"]),"scatter")) + for p in ports for k in range(22)) + return { + "frame_count":len(frames),"tested_distinct_nested_pairs":len(pairs), + "live_registers_per_invocation":n,"temporary_centers":22, + "ranked_live_transitions":transitions, + "phase_stats":totals,"role_histograms":measured, + "phase_histograms":{p:hist(x) for p,x in by_phase.items()}, + "invocation_histogram":hist(combined),"invocation_calls":sum(combined.values()), + "invocation_rank_mass":mass,"retained_centers":c["ret"], + "phase_state_sha256":{"start":digest(start),"after_A":digest(cut), + "after_B":digest(before_final),"final":digest(final)}, + "literal_gate_coefficients":dict(sorted(literal.items())), + "scalar_role_directions":dict(sorted(kind.items())), + "scatter_coefficients":dict(sorted(scatter.items())), + "scatter_additions":sum(scatter.values()),"Cc_copy_additions":22, + "two_hosts_metadata_count":len(c["two_hosts"]), + "two_hosts_note":"Metadata only; not an independent allocator or host-use proof.", + "Y_bracket_pairs":len(ybracket), + "literal_inverse":"Each nonself additive shear inverts by negating its coefficient; inverse phase reverses order; inverse transpose keeps chronological order and swaps source/target with negation.", + "limits":["No scalar all-column replay","No old-response or cleanup matrix formed", + "No generated Lean data comparison or kernel build","No physical stream compiler"] + } + +def scalar_word(ops): + matrix=[[int(i==j) for j in range(4)] for i in range(4)] + for target,source,q in ops: + matrix[target]=[a+q*b for a,b in zip(matrix[target],matrix[source])] + return matrix diff --git a/research/multicut-delayed-inverse-checkpoint/code/complex_scalars.py b/research/multicut-delayed-inverse-checkpoint/code/complex_scalars.py new file mode 100644 index 0000000000..771057b567 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/code/complex_scalars.py @@ -0,0 +1,216 @@ +"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers.""" +from collections import Counter +from copy import deepcopy +from fractions import Fraction +Q=Fraction +from hashlib import sha256 +from pathlib import Path +from math import prod +import gzip,json,sys + +V,R,H=1320,9412,22 +def need(x,msg): + if not x:raise ValueError(msg) + +def kind(r):return "X" if r>k&1 else c["scat"]["outside"])) + yield add(live(V+t),tmp(r),q) + yield from part(B) + +def old_word(orientation,K): + if orientation=="forward": + yield from (copy(("T",q),live(2*V+q)) for q in range(R)) + for e in K: + op,t,s,a,b=e + yield add(t,s,Q(-a if t[0]=="L" else a,b)) + else: + yield from (clear(("T",q)) for q in range(R)) + for op,t,s,a,b in reversed(K): + yield add(s,t,Q(a,b)) + yield from (add(live(2*V+q),("T",q),Q(1)) for q in range(R)) + yield from (clear(("T",q)) for q in range(R)) + +def cleanup(orientation,AB): + for t,s,q in reversed(AB): + if kind(t)=="Y":continue + if orientation=="forward":yield add(live(t),live(s),-q) + else:yield add(live(s),live(t),q) + +def main_scalar(orientation,A,B,c): + def phase(gs): + for t,s,q in gs: + if orientation=="forward":yield add(live(t),live(s),q) + else:yield add(live(s),live(t),-q) + yield from phase(A) + # Every center is a complete temporary stream and one rank20 child, + # initialized/read/discarded before the next center uses the work stock. + for k,r,_ in c["ret"]: + yield clear(center(k)) + if orientation=="forward":yield add(center(k),live(r),Q(1)) + for t,p in enumerate(c["ports"]): + q=Q(*(c["scat"]["inside"] if p>>k&1 else c["scat"]["outside"])) + if orientation=="backward":yield add(center(k),live(V+t),q) + yield ("recursive_center",k,20) + if orientation=="forward": + for t,p in enumerate(c["ports"]): + q=Q(*(c["scat"]["inside"] if p>>k&1 else c["scat"]["outside"])) + yield add(live(V+t),center(k),q) + else:yield add(live(r),center(k),Q(-1)) + yield clear(center(k)) + yield from phase(B) + +def inv_schedule(orientation,A,B,c): + K=list(old_positive(A,B,c)) + yield from ((p,e) for p,es in [ + ("old_response",old_word(orientation,K)), + ("main",main_scalar(orientation,A,B,c)), + ("cleanup",cleanup(orientation,A+B))] for e in es) + +def coeff_steps(q): + """Non-destructive coefficient expansion using one complete work stream. + Copy source, double/halve/divide by3 as needed, add/subtract, clear. + Counts Gaussian operations; multiply by2 for real/imag component primitives.""" + a,b=abs(q.numerator),q.denominator + need(a in (1,2) and b in (1,2,3,6),"unsupported new coefficient") + return 3+int(a==2)+int(b%2==0)+int(b%3==0) + +def bill(schedule): + phases={p:Counter() for p in ("old_response","main","cleanup")} + coeffs={p:Counter() for p in phases} + tape=sha256() + for p,e in schedule: + tape.update((json.dumps((p,e),separators=(",",":"))+"\n").encode()) + phases[p]["all_macros"]+=1 + if e[0]=="recursive_center": + phases[p]["recursive_centers"]+=1 + phases[p]["recursive_rank_mass"]+=e[2] + continue + phases[p]["scalar_macros"]+=1 + phases[p][e[0]]+=1 + if e[0]=="add": + q=Q(e[3],e[4]);coeffs[p][str(q)]+=1 + phases[p]["Gaussian_primitive_steps"]+=coeff_steps(q) + phases[p]["scalar_log2_magnitude_charge"]+=1+int(abs(q)==2) + phases[p]["halving_exposure"]+=int(q.denominator%2==0) + phases[p]["divisor3_exposure"]+=int(q.denominator%3==0) + else:phases[p]["Gaussian_primitive_steps"]+=1 + for p in phases:phases[p]["real_component_primitive_steps"]=2*phases[p]["Gaussian_primitive_steps"] + total=sum((Counter({k:v for k,v in x.items() if k not in ["recursive_rank_mass"]}) for x in phases.values()),Counter()) + return {"phases":{p:dict(x) for p,x in phases.items()},"coefficient_histograms":{p:dict(x) for p,x in coeffs.items()}, + "totals":dict(total),"schedule_sha256":tape.hexdigest(), + "other_main_recursive_calls":"70147 existing live-label transitions, unchanged; not scalar macros"} + +def mm(A,B):return [[sum((x*y for x,y in zip(row,col)),Q(0)) for col in zip(*B)] for row in A] + +def lifecycle(schedule): + """Verify that new scalar-only work stock never survives into a child.""" + initialized=set();dirty=set();active_centers=set();seen_main=False;children=0 + for phase,e in schedule: + if phase=="main" and not seen_main: + need(not dirty,"old-response tableau survives into recursive main phase") + seen_main=True + need(e[0] in ("add","copy","clear","recursive_center"),"unknown or unpaid recursive operation") + if e[0]=="recursive_center": + k=e[1] + need(type(k) is int and 0<=k<22 and e[2]==20,"new temporary entered uncharged child") + need(phase=="main" and not dirty,"old-response temporary used during child") + need(active_centers<=set([k]),"other center was not discarded before next child") + children+=1 + continue + target=e[1] + source=e[2] if e[0] in ("add","copy") else None + if source and source[0]=="T":need(source[1] in initialized,"uninitialized temporary read") + if target[0]=="T": + need(phase=="old_response","scalar-only tableau touched after recursive phase begins") + initialized.add(target[1]) + if e[0]=="clear":dirty.discard(target[1]) + else:dirty.add(target[1]) + if target[0]=="C": + if e[0]=="clear":active_centers.discard(target[1]) + else:active_centers.add(target[1]) + if e[0]=="add": + # Coefficient expansion is deliberately local: copy into W, + # optionally double/halve/divide3, accumulate, and clear W. + expanded=["copy_work"]+["scale"]*(coeff_steps(Q(e[3],e[4]))-3)+["accumulate","clear_work"] + need(expanded[-1]=="clear_work","coefficient work survived scalar macro") + need(not dirty and not active_centers,"temporary stock not erased at invocation end") + need(children==22,"center child count changed") + return {"tableau_cleared_before_entire_main_phase":True,"tableau_absent_from_all_children":True, + "coefficient_work_cleared_per_add":True,"center_children":children, + "new_recursive_children":0,"moment_live_width":14692, + "new_work_stock_only":9413,"complete_stream_fraction":"1/(14692*|Orth(110,2)|)"} + +def ident(n):return [[Q(int(i==j)) for j in range(n)] for i in range(n)] + +def matrix(n,gs): + a=ident(n) + for t,s,q in gs:a[t]=[x+q*y for x,y in zip(a[t],a[s])] + return a + +def toy_checks(): + # 2 X, 3 S, 2 Y. Explicit nontrivial X mixing is undone; helper map + # and K are otherwise arbitrary. Exact Fraction checks cover every column + # of THIS TINY TEMPLATE, not of the published 12052-register source. + n=7;X={0,1};S={2,3,4};Y={5,6} + gs=[(2,0,Q(1)),(3,2,Q(2)),(5,3,Q(-1,6)),(0,1,Q(1,2)), + (4,0,Q(2)),(2,4,Q(1,2)),(6,2,Q(1,3)),(0,1,Q(-1,2)), + (3,1,Q(-1)),(5,4,Q(1)),(6,0,Q(1))] + M=matrix(n,gs) + need(all(M[i]==ident(n)[i] for i in X),"toy source rows not restored") + Ugs=[e for e in gs if e[0] not in Y] + U=matrix(n,Ugs) + Uinv=matrix(n,[(t,s,-q) for t,s,q in reversed(Ugs)]) + need(mm(Uinv,U)==ident(n),"restricted inverse identity") + Mi=matrix(n,[(t,s,-q) for t,s,q in reversed(gs)]) + g8=[Mi[i] if i in S else ident(n)[i] for i in range(n)] + h8=[M[i] if i in S else ident(n)[i] for i in range(n)] + need(g8==Uinv and h8==U,"cleanup projection identities") + Ut=matrix(n,[(s,t,q) for t,s,q in reversed(Ugs)]) + need(Ut==[list(x) for x in zip(*U)],"backward cleanup transpose") + Kgs=[e for e in gs if e[1] not in X] + Kmat=matrix(n,Kgs) + need(all(Kmat[i][j]==M[i][j] for i in Y for j in S),"zero-source old response") + Kt=matrix(n,[(s,t,q) for t,s,q in reversed(Kgs)]) + need(all(Kt[i][j]==M[j][i] for i in S for j in Y),"old response adjoint") + bad=matrix(n,[(t,s,q) for t,s,q in reversed(Ugs)]) + need(bad!=g8,"wrong inverse sign must fail") + bad_order=matrix(n,[(s,t,q) for t,s,q in Ugs]) + need(bad_order!=Ut,"unreversed transpose must fail") + return {"tiny_exact_template_columns_checked":7,"checks_passed":7, + "wrong_inverse_sign_rejected":True,"unreversed_transpose_rejected":True, + "published_full_scalar_replay":False} diff --git a/research/multicut-delayed-inverse-checkpoint/code/complex_splice.py b/research/multicut-delayed-inverse-checkpoint/code/complex_splice.py new file mode 100644 index 0000000000..11373d1d57 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/code/complex_splice.py @@ -0,0 +1,277 @@ +"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers.""" +from collections import Counter +from copy import deepcopy +from fractions import Fraction +Q=Fraction +from hashlib import sha256 +from pathlib import Path +from math import prod +import gzip,json,sys + +V,R,H,N,W=1320,9412,22,12052,14692 +PHASES=["old_response","ad0","A","centers","B","final","adF","cleanup","done"] +def need(x,s): + if not x:raise ValueError(s) + +def blobhash(p):return sha256(p.read_bytes()).hexdigest() + +def digest(x):return sha256(json.dumps(x,separators=(',',':'),sort_keys=True).encode()).hexdigest() + +def role(r):return 'X' if r0 else -1),clear(work)] + +def emit(c,ori): + """Construct the complete local parametric invocation, before cover routing. + One role stream is its full complete address rectangle. Frame labels denote + exact canonical representatives, never just rank or binary matrices. + """ + A=list(flat(c['A']));B=list(flat(c['B']));cur=c['start'].copy() + K=[] + for part in (A,None,B): + if part is None: + for t,p in enumerate(c['ports']): + for k,r,f in c['ret']: + K.append(add(L(V+t),T(r),Q(*(c['scat']['inside'] if p>>k&1 else c['scat']['outside'])))) + else: + for t,s,q in part: + if role(s)=='X':continue + K.append(add(T(t) if role(t)=='S' else L(t),T(s),q)) + yield ('phase','old_response') + if ori=='forward': + for q in range(R):yield scal('old_response',copy(('T',q),L(2*V+q)),'old0') + for _,t,s,a,b in K:yield scal('old_response',add(t,s,Q(-a if t[0]=='L' else a,b)),'old0') + else: + for q in range(R):yield scal('old_response',clear(('T',q)),'old0') + for _,t,s,a,b in reversed(K):yield scal('old_response',add(s,t,Q(a,b)),'old0') + for q in range(R):yield scal('old_response',add(L(2*V+q),('T',q),Q(1)),'old0') + for q in range(R):yield scal('old_response',clear(('T',q)),'old0') + yield ('phase','ad0') + for r,f in enumerate(c['start']):yield ('adapter','old_to_new',r,f) + for part in ['A','centers','B','final']: + yield ('phase',part) + if part in ['A','B']: + for g in c[part]: + regs=[g[2]]+[x[0] for x in g[3]]+(g[4] if g[0]=='out' else []) + for r in regs: + if cur[r]!=g[1]: + old=cur[r];cur[r]=g[1] + yield ('climb',part,r,old,g[1],len(c['frames'][g[1]])-len(c['frames'][old])) + for t,s,q in flat([g]): + e=add(L(t),L(s),q) if ori=='forward' else add(L(s),L(t),-q) + yield scal('main',e,g[1]) + elif part=='centers': + for k,r,f in c['ret']: + yield scal('main',clear(C(k)),f if ori=='forward' else 0) + if ori=='forward':yield scal('main',add(C(k),L(r),Q(1)),f) + else: + for t,p in enumerate(c['ports']): + yield scal('main',add(C(k),L(V+t),Q(*(c['scat']['inside'] if p>>k&1 else c['scat']['outside']))),0) + yield ('center',k,f if ori=='forward' else 0,0 if ori=='forward' else f,20) + if ori=='forward': + for t,p in enumerate(c['ports']): + yield scal('main',add(L(V+t),C(k),Q(*(c['scat']['inside'] if p>>k&1 else c['scat']['outside']))),0) + else:yield scal('main',add(L(r),C(k),Q(-1)),f) + yield scal('main',clear(C(k)),0 if ori=='forward' else f) + else: + for r,f in enumerate(c['final']): + if cur[r]!=f: + old=cur[r];cur[r]=f + yield ('climb','final',r,old,f,len(c['frames'][f])-len(c['frames'][old])) + yield ('phase','adF') + for r,f in enumerate(c['final']):yield ('adapter','new_to_old',r,f) + yield ('phase','cleanup') + for t,s,q in reversed(A+B): + if role(t)=='Y':continue + e=add(L(t),L(s),-q) if ori=='forward' else add(L(s),L(t),q) + yield scal('cleanup',e,'old1') + yield ('phase','done') + +def contains(U,V): + for u in U: + for v in V: + if u>>(v.bit_length()-1)&1:u^=v + if u:return False + return True + +def validate(events,c,ori,manifest,bill): + phase=None;seen=[];cur=c['start'].copy();cent={};dirty=set();initialized=set() + climbed=Counter();hist=Counter();projection=sha256();whole=sha256();counts=Counter();ad0=[];adF=[] + pairs=set();states={};all_frame_incidence=0;primitive=sha256();primitive_count=0 + for ev in events: + whole.update((json.dumps(ev,separators=(',',':'))+'\n').encode()) + tag=ev[0];counts[tag]+=1 + if tag=='phase': + if phase=='A':states['after_A']=digest(cur) + if phase=='B':states['after_B']=digest(cur) + if phase=='final':states['final']=digest(cur) + phase=ev[1];seen.append(phase) + need(seen==PHASES[:len(seen)],'strict old/A/copy/B/final/adapter/cleanup chronology') + if phase=='ad0':need(not dirty,'old-response tableau survived before label recursion') + if phase=='A':need(ad0==list(enumerate(c['start'])),'missing or reordered exact entry adapter') + if phase=='adF':need(cur==c['final'],'final climbs omitted before cleanup') + if phase=='cleanup':need(adF==list(enumerate(c['final'])),'missing or reordered exact final adapter') + continue + if tag=='adapter': + _,which,r,f=ev + if which=='old_to_new':need(phase=='ad0','entry adapter position');ad0.append((r,f)) + else:need(which=='new_to_old' and phase=='adF','final adapter position');adF.append((r,f)) + elif tag=='climb': + _,p,r,old,new,d=ev + need(phase==p and p in ['A','B','final'],'live child outside source label walk') + need(not dirty and not cent,'new work stock or center active during live child') + need(cur[r]==old and d==len(c['frames'][new])-len(c['frames'][old]) and d>0,'wrong exact child endpoints/rank') + pair=(old,new) + if pair not in pairs:need(contains(c['frames'][old],c['frames'][new]),'nonnested child');pairs.add(pair) + cur[r]=new;climbed[p]+=1;hist[d]+=1 + elif tag=='center': + _,k,old,new,d=ev + need(phase=='centers' and not dirty and set(cent)=={k},'center child ownership') + need(cent[k]==old and (k,c['ret'][k][1],c['ret'][k][2])==tuple(c['ret'][k]),'center input frame') + f=c['ret'][k][2] + need((old,new,d)==((f,0,20) if ori=='forward' else (0,f,20)),'center child direction/rank') + cent[k]=new;hist[d]+=1 + projection.update((json.dumps(('main',('recursive_center',k,d)),separators=(',',':'))+'\n').encode()) + elif tag=='scalar': + _,p,e,f=ev + need(p==('main' if phase in ['A','centers','B'] else phase),'scalar phase binding') + need(e[0] in ['add','copy','clear'],'unpaid scalar operation') + target=e[1];source=e[2] if e[0] in ['add','copy'] else None + for x in [target]+([source] if source else []): + bank,r=x + if bank=='T':need(phase=='old_response' and 0<=r=0 + if f:stage_paid[f]+=1 + if a==WORK: + assert copy_open and c==lower.ZERO and f==22 + elif op==ADD: + assert a in live_image and (b in live_image or b==WORK) and a!=b + assert not(a==WORK) + assert (b==WORK)==(z==center_category) + if b==WORK:assert copy_open and f==lower.ZERO;copy_reads+=1 + adds+=1;coefficient[abs(c)]+=1 + elif op==COPY: + assert not copy_open and a==WORK and b in live_image + assert b>=4*V and lower.C.dimf[f]==22 + copy_open=True;copy_reads=0;copies+=1 + elif op==ERASE: + assert copy_open and a==WORK and f==lower.ZERO and copy_reads==220 + copy_open=False;erases+=1 + else:raise AssertionError('unexpected stage opcode') + data=PACK.pack(*row);digest.update(data);program.update(data);opcode_counts[op]+=1 + assert not copy_open and adds==physical['weighted_scalar_events'] and copies==erases==24 + assert stage_paid==Counter({int(k):n for k,n in physical['paid_histogram'].items()}) + assert coefficient=={int(k):v for k,v in physical['coefficient_histogram'].items()} + paid.update(stage_paid) + stages.append(dict(stage=stage,sha256=digest.hexdigest(),weighted_additions=adds, + fresh_copies=copies,erasures=erases,paid_histogram=dict(sorted(stage_paid.items())), + source_owner=BANKS[ACTIVE[stage][0]],target_owner=BANKS[ACTIVE[stage][1]], + complemented_reverse=stage in(1,3),work_family=WORK)) + if stage in(1,2,4): + which={1:0,2:1,4:2}[stage] + for row in lower.iter_idle(which): + paid[row[4]]+=1;program.update(PACK.pack(*row));opcode_counts[IDLE]+=1 + for row in lower.iter_bridges(which): + program.update(PACK.pack(*row));opcode_counts[BRIDGE]+=1 + for row in lower.iter_completions(): + paid[row[4]]+=1;program.update(PACK.pack(*row));opcode_counts[COMPLETE]+=1 + for row in lower.iter_exchanges(): + program.update(PACK.pack(*row));opcode_counts[EXCHANGE]+=1 + assert opcode_counts[EXCHANGE]==2*V + expected=Counter({int(k):5*n for k,n in raw['one_stage_helper_histogram_including_copies'].items()}) + expected.update({r:2*V for r in(46,23,50,4)}) + expected.update({5*int(a):n for a,n in raw['auxiliary_entrance_rank_histogram'].items()}) + assert paid==expected and sum(paid.values())==raw['five_stage_profile']['calls'] + assert sum(r*n for r,n in paid.items())==raw['five_stage_profile']['rank_mass'] and LIVE*M-sum(r*n for r,n in paid.items())==4400 + assert opcode_counts[BRIDGE]==6*V and opcode_counts[COMPLETE]==2827 + # Independent four-bank all-column F2 composition, no source helper replay. + def bank_endpoint(skip_bridge=None): + banks=[1<','exec'),m.__dict__);scalar=m.replay();scalar_sha=scalar_digest.hexdigest();print('PASS independent F2 event observer',scalar_count,scalar_sha,dict(scalar_coeff),flush=True) + state={i:W.w['source_frame'][i]for i in range(v)};state.update({v+t:ZERO for t in range(v)});state.update({2*v+j:W.gauge[r]['frame']if r in W.gauge else ZERO for j,r in enumerate(regs)});initial_state=state.copy();caps={t:W.register(W.module.kernel([C.cov[t]],W.h)[0])for t in range(v)} + records=array('i');hist=Counter();categories=Counter();coefficients=Counter();centers=[];center=None;used_frames=set(state.values());subcache={};count=0;tag_hash=hashlib.sha256();projection=hashlib.sha256() + # The line binding is checked against the exact generated source digest below. + tags={43:("group['frame']",'equal_inverse'),50:("group['frame']",'rank_inverse'),56:("group['frame']",'echelon_inverse'),62:("W.gauge[s]['frame']",'paired_alias_corruption'),67:("W.gauge[s]['frame']",'fresh_source_read'),81:("W.gauge[s]['frame']",'echelon_read'),96:("read_frames[s,k]",'equal_read'),98:("read_frames[s,k]",'equal_corruption'),108:("W.gauge[s]['frame']if s in W.gauge else ZERO",'dirty_read'),113:("W.opframe[i]if sign==1 else FULL",'gate'),116:("W.w['source_frame'][n]",'inject'),120:("kpair[r['source']]['mix_frame']",'gauge_mix'),127:("kpair[b]['mix_frame']",'partner_mix'),133:('ZERO','center'),137:('ZERO','echelon_setup'),142:('ZERO','rank_setup'),147:('ZERO','equal_setup'),151:('ZERO','terminal_pre'),159:('f','anchor_inject'),166:('W.opframe[i]','terminal_write'),172:("e['root_frame']",'terminal_post'),181:("W.w['root_frame'][j]",'side_root'),184:("kpair[b]['mix_frame']",'partner_mix'),186:("e['deliver_frame']",'partner_delivery'),189:('FULL','partner_cleanup'),192:('FULL','partner_cleanup'),201:('FULL','anchor_restore'),204:('FULL','partner_cleanup'),205:('FULL','uninject')} + category_names=sorted(set(k for f,k in tags.values())|{'forward_gate','cleanup_gate'});kind_ids={k:i for i,k in enumerate(category_names)} + def move(i,f): + used_frames.add(f);old=state[i] + if old==f:return + if(old,f)not in subcache:subcache[old,f]=C.sub(old,f) + assert subcache[old,f],('physical descent',i,old,f,C.dimf[old],C.dimf[f]) + d=C.dimf[f]-C.dimf[old];assert d>=0 + records.extend((0,i,old,f,d,0)) + if d:hist[d]+=1 + state[i]=f + def physical_add(x,c,y,f,kind): + nonlocal count + assert x!=y;move(x,f);physical_y=y + if kind=='center':assert center is not None and center['source']==y and f==ZERO;physical_y=n + else:move(y,f) + records.extend((1,x,physical_y,c,f,kind_ids[kind]));digest_add(projection,[x,y,c]);digest_add(tag_hash,[x,physical_y,c,f,kind]);count+=1;categories[kind]+=1;coefficients[abs(c)]+=1 + return x + def center_begin(source,frame): + nonlocal center + assert center is None;move(source,frame);rank=C.dimf[frame];assert rank==22;hist[rank]+=1 + center=dict(source=source,frame=frame,rank=rank,first_event=count,temporary=n,copy_output_frame=ZERO);centers.append(center);records.extend((2,source,n,frame,ZERO,rank)) + def center_end(): + nonlocal center + assert center is not None;center['after_event']=count;center['scatter_reads']=count-center['first_event'];assert center['scatter_reads']==220 + records.extend((3,center['source'],n,center['frame'],ZERO,center['rank']));center=None + class Tag(ast.NodeTransformer): + def visit_Call(self,node): + self.generic_visit(node) + if isinstance(node.func,ast.Name)and node.func.id=='add'and node.lineno<=205: + frame,kind=tags[node.lineno];node.func.id='physical_add';kindnode=ast.parse("'forward_gate'if sign==1 else'cleanup_gate'",mode='eval').body if kind=='gate'else ast.Constant(kind);node.args += [ast.parse(frame,mode='eval').body,kindnode] + return node + q=Tag().visit(ast.parse(source));fn=q.body[0];end=next(i for i,z in enumerate(fn.body)if isinstance(z,ast.If)and ast.unparse(z.test)=='binary');fn.body=fn.body[:end]+[ast.parse('return dict(center_reads=len(center_reads))').body[0]] + fn.body=[z for z in fn.body if not(isinstance(z,ast.FunctionDef)and z.name=='add')] + for node in ast.walk(fn): + if isinstance(node,ast.Assign)and any(isinstance(t,ast.Name)and t.id=='unit'for t in node.targets):node.value=ast.parse('lambda i:i',mode='eval').body + if isinstance(node,ast.FunctionDef)and node.name=='target_move':node.body.insert(0,ast.parse('move(v+t,f)').body[0]) + if isinstance(node,ast.FunctionDef)and node.name=='source_move':node.body.insert(0,ast.parse('move(physical_id(s),f)').body[0]) + # Modify the center loop without changing any scalar expression or event order. + centerloop=next(z for z in fn.body if isinstance(z,ast.For)and z.lineno==129) + centerloop.target=ast.parse('(j,(r,s))',mode='eval').body;centerloop.target.ctx=ast.Store();centerloop.target.elts[0].ctx=ast.Store();centerloop.target.elts[1].ctx=ast.Store() + for z in centerloop.target.elts[1].elts:z.ctx=ast.Store() + centerloop.iter=ast.Call(func=ast.Name(id='enumerate',ctx=ast.Load()),args=[centerloop.iter],keywords=[]) + block=centerloop.body[0];assert isinstance(block,ast.If);block.body.insert(0,ast.parse("center_begin(value(s),W.w['root_frame'][j])").body[0]);block.body.append(ast.parse('center_end()').body[0]) + ast.fix_missing_locations(q);body=ast.unparse(q); + if D is not None:(D/'generated-physical-word.py').write_text(body+'\n') + m.__dict__.update(dict(physical_id=physical_id,move=move,physical_add=physical_add,center_begin=center_begin,center_end=center_end,ZERO=ZERO,FULL=FULL)) + exec(compile(q,'','exec'),m.__dict__);local=m.replay('token');assert center is None + for i in range(v):move(i,FULL) + for t in range(v):move(v+t,caps[t]) + for j,r in enumerate(regs):move(2*v+j,FULL) + assert count==scalar_count and projection.hexdigest()==scalar_sha and coefficients==scalar_coeff + assert len(centers)==24 and local['center_reads']==5280 + result=dict(status='PASS_PHYSICAL527_EXACT_SCALAR_PROJECTION_AND_MONOTONE_FRAME_TAGS',scalar_source_sha256=sha(source.encode()),scalar_projection_sha256=scalar_sha,tagged_scalar_sha256=tag_hash.hexdigest(),weighted_scalar_events=count,coefficient_histogram=dict(coefficients),categories=dict(categories),category_names=category_names,physical_registers=n,independent_dirty_registers=len(regs),source_heads=len(m.borrow),positive_rank_moves=sum(hist.values())-24,copied_centers=len(centers),copied_center_blocks=centers,center_scatter_reads=local['center_reads'],paid_histogram=dict(sorted(hist.items())),paid_rank_mass=sum(k*v for k,v in hist.items()),initial_independent_entrances=dict(Counter(C.dimf[initial_state[2*v+j]]for j in range(len(regs))if C.dimf[initial_state[2*v+j]])),used_frames=[dict(frame_id=f,dimension=C.dimf[f],basis_sha256=sha(json.dumps(C.B[f],separators=(',',':')).encode()))for f in sorted(used_frames)],input_pins=dict(source_pins or {}),physical_emitter_sha256=sha(Path(__file__).read_bytes())) + + if D is not None: + (D/'physical.json').write_text(json.dumps(result,indent=2)+'\n') + (D/'records.bin.gz').write_bytes(gzip.compress(records.tobytes(),mtime=0)) + print('PASS PHYSICAL',count,'records',len(records)//6,'rankmass',result['paid_rank_mass'],'tagged',result['tagged_scalar_sha256'],flush=True) + + return dict(context=m.__dict__,W=W,C=C,records=records,physical=result,initial_state=initial_state,ZERO=ZERO,FULL=FULL,result=result,scalar_result=scalar) diff --git a/research/multicut-delayed-inverse-checkpoint/expected/certificate.json b/research/multicut-delayed-inverse-checkpoint/expected/certificate.json new file mode 100644 index 0000000000..b07c53647c --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/expected/certificate.json @@ -0,0 +1,221 @@ +{ + "assembly": { + "absorption_gap": "226517973821095735830462702853396040780237925082816259563387218561818767290563/375868220963471386443643705764673339874334101028814695902158520638627929850022292500000000000000", + "margins": { + "Gaussian_arithmetic": "171046032096340035546583134813480037267955322491671736969927330361011695758816028059991083/240555661416621687323931971689390937519573824658441405377381453208721875104014267200000000000", + "balanced_prefix": "106903770154179577457482305869335949733640411525878360863408255451171940008917/150347288385388554577457482305869335949733640411525878360863408255451171940008917", + "bulk_exposure": "2779498020107978009934755908622604204891895908082621174264779884079/3906250000000000000000000000000000000000000000000000000000000000000000", + "compact_phase_layer": "106903770003832289072093751291878467427771075576144720451882377090308531753465828059991083/150347288385388554577457482305869335949733640411525878360863408255451171940008917000000000000", + "coordinate_movement": "2779498020107978009934755908622604204891895908082621174264779884079/3906250000000000000000000000000000000000000000000000000000000000000000", + "dimension": "150240384615234375000000000000000000000000000000000000000000000000000000000000000/150347288385388554577457482305869335949733640411525878360863408255451171940008917", + "scalar_work": "855230161083089331274469892377230115563253956257293246495740165248512111815884828059991083/1202778307083108436619659858446954687597869123292207026886907266043609375520071336000000000000" + }, + "minimum_margin": "106903770003832289072093751291878467427771075576144720451882377090308531753465828059991083/150347288385388554577457482305869335949733640411525878360863408255451171940008917000000000000", + "parameters": { + "C0": { + "base": 2, + "exponent": 210000 + }, + "C1": 1, + "a_bit": "2779498020107978009934755908622604204891895908082621174264779884079/3906250000000000000000000000000000000000000000000000000000000000000000", + "a_complex": 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Apache-2.0. +Retains PR234exact-tape/prime/ordinary-leaf hypotheses, and pays bankroutes. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import importlib.util,json +HERE=Path(__file__).resolve().parent +THEOREM_PINS={ + 'proof/three-stage-cover-bit.tex':'47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511', + 'proof/three-stage-cover-rows.tex':'6ad97a5cd7d6aecf978a81226e1c2763d37840caca130c61f3bf5c4ac111c5c8', + 'proof/copied-centers-lemma.tex':'d0ce6d3d504996aadb098227885224e48df46101daec8893523a0a8b7364d8fe'} +def ceilq(x):return -((-x.numerator)//x.denominator) + +def cutoff_log2(delta,C): + assert 0=1 + return max(1,ceilq(36/delta**2),ceilq(2*(4+(C-1).bit_length())/delta)) + +def hist(x):return Counter({int(k):v for k,v in x.items()if v}) + +def route_template(): + """Recompute the actual 48-dimensional rational route, not its receipt.""" + h=24 + I=lambda n:[[Q(i==j)for j in range(n)]for i in range(n)] + q=[int(i<3)for i in range(h)] + P=[[Q(q[i])*(Q(q[j])-Q(1,3))/2 for j in range(h)]for i in range(h)] + K=[[Q(i==j)-P[i][j]for j in range(h)]for i in range(h)] + rho=[P[i]+K[i]for i in range(h)]+[K[i]+P[i]for i in range(h)] + a=[r[:]for r in rho];factors=[] + for j in range(2*h): + p=next(i for i in range(j,2*h)if a[i][j]) + if p!=j:a[j],a[p]=a[p],a[j];factors.append(('swap',j,p,None)) + if a[j][j]!=1: + c=1/a[j][j];a[j]=[c*x for x in a[j]];factors.append(('scale',j,None,c)) + for i in range(2*h): + if i!=j and a[i][j]: + c=-a[i][j];a[i]=[x+c*y for x,y in zip(a[i],a[j])];factors.append(('add',i,j,c)) + assert a==I(2*h) + def replay(omit=None): + a=I(2*h) + for z in reversed(range(len(factors))): + if z==omit:continue + kind,i,j,c=factors[z] + if kind=='swap':a[i],a[j]=a[j],a[i] + elif kind=='scale':a[i]=[x/c for x in a[i]] + else:a[i]=[x-c*y for x,y in zip(a[i],a[j])] + return a + assert replay()==rho + assert replay(next(i for i,f in enumerate(factors)if f[0]=='add'))!=rho + counts=dict(Counter(f[0]for f in factors));denoms=sorted({c.denominator for kind,i,j,c in factors if c is not None}) + assert counts==dict(scale=27,add=195,swap=44) and denoms==[1,2,3,6] + serialization=[[kind,i,j,None if c is None else str(c)]for kind,i,j,c in factors] + return dict(factors=len(factors),factor_counts=counts,denominators=denoms, + factor_program_sha256=sha256(json.dumps(serialization,separators=(',',':')).encode()).hexdigest(), + exact_inverse_reconstruction=True,missing_addition_rejected=True) + +def run(raw,physical,scalar,prime_result,math_result,banks,banked_result,global_result): + for name,h in THEOREM_PINS.items():assert sha256((HERE/name).read_bytes()).hexdigest()==h,('changed theorem',name) + bit=(HERE/'proof/three-stage-cover-bit.tex').read_text();rows=(HERE/'proof/three-stage-cover-rows.tex').read_text();copied=(HERE/'proof/copied-centers-lemma.tex').read_text() + for s in ('constants are independent of $w,n$ and ancestor values','absorbed by the complete $q^w$ target fiber','spectator counters','32m^2','only units are inverted'):assert s in bit + for s in ('W(w)^{D(n)}','padding factor is less than two','below $q^2$'):assert s in rows + assert 'Copying, pointwise reads and erasure are linear'in copied and'fixed work stream'in copied + assert raw['physical_R']==16587 and raw['v']==1760 and raw['h']==24 and raw['source_aliases']==527 + assert scalar['forward']['event_sha256']==physical['scalar_projection_sha256']==global_result['scalar_projection_sha256'] + assert physical['tagged_scalar_sha256']==global_result['local_tagged_sha256'] + f,b=scalar['forward'],scalar['inverse'];n=2*raw['v']+raw['physical_R'];assert n==20107 + for row in(f,b): + assert row['all_source_and_dirty_restored']and row['arbitrary_target_contents_preserved']and row['all_formal_columns']==n + assert row['weighted_additions']==physical['weighted_scalar_events']==760383 + assert hist(row['coefficient_counts'])==hist(physical['coefficient_histogram']) + assert row['literal_unit_additions']==sum(c*v for c,v in hist(row['coefficient_counts']).items())==763023 + assert f['max_intermediate_row_l1']==73095 and b['max_intermediate_row_l1']==3396290 + assert raw['parity_transform']['both_reflected_ledgers'] and raw['parity_transform']['remaining_payload_additions']==786621 + assert raw['parity_transform']['removed_even_add_count']==1783196 + assert raw['retiming_transform']['both_reflected_ledgers']and raw['retiming_transform']['unchanged_all_input_output_frames'] + assert raw['retiming_transform']['selected_gate_count']==364 and raw['retiming_transform']['unchanged_copy_lifetimes'] + assert raw['regauge_transform']['selected_gauge_count']==21 and raw['regauge_transform']['total_added_entrance_rank']==325 + assert raw['regauge_transform']['both_reflected_ledgers']and raw['regauge_transform']['transport']['producer_context_unchanged'] + assert raw['regauge_transform']['remaining_payload_additions']==786623 + h=raw['hybrid_transform'] + assert h['retained_cohorts']==513 and h['new_entrance_rank']==1422 and len(h['intervals'])==7 + assert h['removed_pivots']==[17207,17222,17246,17255,17273] and h['no_intermediate_rank1_drop'] + assert h['all14_signed_prefix_columns_exact'] and h['retained_kernel_operands_zero_on_chosen_columns'] + assert h['both_reflected_ledgers'] and h['all_external_observations_closed'] and h['producer_context_unchanged'] + assert h['selected_retimings']==h['retiming']['selected_gate_count']==25 + assert h['retiming']['scalar_COPY_projection_byte_identical'] and h['retiming']['unchanged_endpoints'] + assert h['retiming']['all_COPY_sources_immutable'] and h['retiming']['new_registered_frames']==0 + assert h['raw_sha256']=='acf510b1377a6bd96eea50577de1aa049382499335fa7e5cac9b0300ecb377c7' + assert physical['copied_centers']==24 and len(physical['copied_center_blocks'])==24 + assert all(z['rank']==22 and z['scatter_reads']==220 for z in physical['copied_center_blocks']) + assert hist(physical['paid_histogram'])==hist(raw['one_stage_helper_histogram_including_copies']) + assert prime_result['all_remaining_factors_below_2_power_80']and prime_result['physical_inventory_bound'] + assert prime_result['independent_entrances']==2827 and prime_result['bundled_unique_bases']==459 + assert hist(prime_result['entrance_rank_counts'])==hist(raw['auxiliary_entrance_rank_histogram']) + assert banks['conservative_extra_selector_calls']==2*5*40*((867045-1)+16587*120*banks['normalizer_factor_bound']) + assert banks['normalizer_factor_bound']==838 and banks['max_chart_factors']+119+120<=838 + assert banks['physical_replicas']==40 and banks['literal_stock']==867045 and banks['assignments']==3317400 + assert banked_result['all_assignments_match']and banked_result['all_logical_helpers_bound']and banked_result['independent_completions_removed']==2827 + m=120;v=raw['v'];R=raw['physical_R'];T=banks['physical_replicas'];stock=banks['literal_stock'];d=m*m;N=2*m + H=hist(raw['five_stage_profile']['histogram']);assert H==hist(global_result['paid_histogram']) + for a,cnt in hist(raw['auxiliary_entrance_rank_histogram']).items(): + assert H[5*a]>=cnt;H[5*a]-=cnt + if not H[5*a]:del H[5*a] + literal=Counter({r:T*cnt for r,cnt in H.items()});E=sum(literal.values());mass=sum(r*cnt for r,cnt in literal.items()) + assert m*stock-mass==176000 and max(literal)==50 and Q(max(literal),m)0 and delta1>0 + chain=[Q(384599,10**10)];gaps=[] + for j in range(3): + old=chain[-1];a=(1-c)*c+c*old;ds=dict(atom=c-a,borrowing=1-a-c,remainder=1-a-c*(1-old),stock=1-c);assert old0 + delta=min(ds.values());L=cutoff_log2(delta,coefficient);assert Q(L)*delta**2>=36 and Q(L)*delta>=2*(4+(coefficient-1).bit_length()) + gaps.append(dict(stage=j+1,gaps={k:str(x)for k,x in ds.items()},minimum_gap=str(delta),cutoff_log2_at_charged_coefficient=str(L),full_cutoff_rule='Apply cutoff_log2(delta, C_full), where C_full bounds the displayed coefficient times every inherited primitive, wrapper and preceding ordinary-level constant.'));chain.append(a) + assert list(map(Q,mathematics['ordinary_bootstrap_chain']))==chain + controls=[] + for name,test in [('zero cutoff gap',lambda:cutoff_log2(Q(0),coefficient)),('negative constant',lambda:cutoff_log2(Q(1,2),0))]: + try:test() + except AssertionError:controls.append(name) + else:raise AssertionError('bad finitecutoff accepted') + return dict(status='PASS_LITERAL_PARITY_FUSED_BANKED_FIVE_STAGE527_FINITE_BILL',theorem_pins=THEOREM_PINS,source_binding=dict(event=physical['scalar_projection_sha256'],tagged=physical['tagged_scalar_sha256'],global_program=global_result['program_sha256']),paid_inventory=dict(m=m,literal_stock=stock,physical_replicas=T,positive_rank_children=E,rank_mass=mass,weighted_additions=weighted,unit_expanded_additions=unit,bridge_additions=T*6*v,route_families=J,terminal_exchange_stream_movements=T*4*v,high_affine_factors=J*high,low_transposition_coefficient=J*(stock+24),generic_wrappers=E*good,matrix_preparation=E*128*N**3,copy_erase_episodes=120*T,fallback_per_child=32*m*m,extra_bank_selector_calls=K,normalizer_factor_bound=banks['normalizer_factor_bound'],simultaneous_extra_work_streams=1,payload_prefix_upper=payload,payload_prefix_bits=payloadbits),q_power_bound=dict(coefficient=coefficient,low_matrix_power=14400,strict_upper='q^14401'),rational_route=route,row_reserve=row_reserve,moment=dict(coarse=str(c),tau=str(1-c),delta_tau_lower=str(delta_tau),delta_linear=str(delta1),fallback_added_in_full=True),recurrence=dict(bound='A*(n+n^(1-c)*w^(1-a_j))',A='C*(1+1/delta_linear+1/delta_tau)',halving_degree=1,one_level_histogram=True),bootstrap=dict(chain=list(map(str,chain)),ordinary=str(chain[-1]),gaps=gaps),rejected_controls=controls,scope='Actual40replicas/literalbankstock andallfixedbanknormalizerschargedbefore finiteordinaryleaf absorption. 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-0,0 +1,206 @@ +"""Fresh PR259 replacement plus seven target intervals and25 retimings. +No saved execution receipt is consumed. Prepared with OpenAI assistance. +""" +from pathlib import Path +from array import array +from collections import Counter,defaultdict +from fractions import Fraction as Q +from copy import copy,deepcopy +from bisect import bisect_right +from math import gcd,lcm +import sys,json,gzip,hashlib,time,importlib.util +sys.dont_write_bytecode=True +BASE_RAW='2e0bc2a5c54cf8caf26fa30a1a2949702d842f620b28023ca2ba59140ecbed1a' +INTERVAL_RAW='6ab38175783163e729a77032e7e637299b2341aa9ab282134c22796b66699112' +FINAL_RAW='acf510b1377a6bd96eea50577de1aa049382499335fa7e5cac9b0300ecb377c7' +def sha(b):return hashlib.sha256(b).hexdigest() +def integral(row): + row=list(map(Q,row));d=lcm(*(x.denominator for x in row));v=[int(x*d)for x in row];g=gcd(*v);return [x//g for x in v]if g else v +def run(p,raw,package,output): + begun=time.monotonic();package=Path(package);W,C=p['W'],p['C'];old=p['records'];zero,full=p['ZERO'],p['FULL'];v=1760;n=20107 + assert sha(old.tobytes())==BASE_RAW + spec=json.loads((package/'hybrid-selection.json').read_text()) + allc=json.loads(gzip.decompress((package/'kernel-candidates.json.gz').read_bytes())) + assert len(allc)==518 and sum(z['dim']for z in allc)==1344 + drop=set(spec['removed_pivots']);assert drop=={17246,17222,17273,17255,17207} + rows=[z for z in allc if z['pivot']not in drop];assert len(rows)==513 + intervals=spec['intervals'];assert len(intervals)==7 + chosen={q for z in intervals for q in z['streams']};targets={q for z in intervals for q in z['targets']} + pivots={z['pivot']:z for z in rows};donors={q for z in rows for q in z['roles']if q!=z['pivot']} + assert len(pivots)==513 and not(set(pivots)&donors)and not chosen.intersection(set(pivots)|donors) + assert sum(z['dim']for z in rows)+133==1422 and 1422%3==0 + assert len(chosen)==14 and len(targets)==28 + meta=p['physical'];assert len(meta['category_names'])==28 + assert meta['category_names'][4]=='dirty_read'and meta['category_names'][25]=='terminal_pre' + cuts=[k//6 for k in range(0,len(old),6)if tuple(old[k:k+6])==(1,3491,3488,-1,zero,25)] + assert cuts==[727593];cut=cuts[0] + final=dict(p['initial_state']);uses=defaultdict(list);badwrite={};badread={};cols={q:{q:1}for q in chosen};temporary=None + at=defaultdict(list) + for z in rows:at[z['cut']].append(z) + for ci in at:at[ci].sort(key=lambda z:(z['dim'],z['pivot'])) + for k in range(0,len(old),6): + i=k//6;op,a,b,c,f,z=old[k:k+6] + if op==0:final[a]=c + elif op==1: + uses[a].append((i,f));uses[b].append((i,f));badwrite.setdefault(a,i) + if not(z==4 and f==zero and v<=a<2*v):badread.setdefault(b,i) + if i<=cut: + assert a not in chosen + if v<=b<2*v:assert v<=a<2*v + if b in chosen:assert z==4 and f==zero and c==-1 + source=temporary if b==n else b + for q,col in cols.items(): + value=col.get(a,0)+c*col.get(source,0) + if value:col[a]=value + else:col.pop(a,None) + elif op==2: + uses[a].append((i,c));badwrite.setdefault(a,i);badwrite.setdefault(b,i) + if i<=cut:assert temporary is None and a not in chosen and not v<=a<2*v;temporary=a + elif op==3 and i<=cut:assert temporary==a;temporary=None + if i in at: + assert temporary is None + for z in at[i]:assert all(not any(col.get(q,0)for q in z['roles'])for col in cols.values()) + assert temporary is None + def install(f,B): + assert f not in C.B + B=tuple(tuple(integral(b))for b in B);A,_=W.module.kernel(B,24) + C.B[f],C.A[f],C.dimf[f]=B,A,len(B) + assert C.nondeg(f)and len(A)==24-len(B) + assert all(not sum(x*y for x,y in zip(b,a))for b in B for a in A) + return f + entries=[] + for z in rows: + f=install(z['new_frame_id'],z['basis']);assert C.dimf[f]==z['dim'] + for q in z['roles']: + assert p['initial_state'][q]==zero and final[q]==full + assert badwrite.get(q,10**9)>z['cut']and badread.get(q,10**9)>z['cut'] + j=bisect_right(uses[q],(z['cut'],10**9));assert j=4)for j in range(6)]for i in range(6)] + for op,a,b,c,f,cat in reversed(local):M[slot[a]]=[x-c*y for x,y in zip(M[slot[a]],M[slot[b]])] + assert M==[[int(i==j)for j in range(6)]for i in range(6)] + blocks.append(dict(z,L=L,H=H,K=K,word=local)) + for q,r,f in((high,z['roles'][0],H),(low,z['roles'][1],L)): + assert p['initial_state'][q]==zero and r not in W.gauge + entries.append(dict(stream=q,role=r,frame=f,rank=C.dimf[f],basis=C.B[f],targets=[t-v for t in ts])) + initial=dict(p['initial_state']) + for z in entries:initial[z['stream']]=z['frame'] + state=dict(initial);out=array('i');hist=Counter();temporary=None;removed=Counter();added=Counter() + def move(q,f): + before=state[q] + if before==f:return + if temporary is not None:assert q!=temporary[0] + assert C.sub(before,f),(q,before,f) + d=C.dimf[f]-C.dimf[before];assert d>=0 + out.extend((0,q,before,f,d,0));state[q]=f + if d:hist[d]+=1 + def emit(e): + nonlocal temporary + op,a,b,c,f,z=e + if op==1: + if temporary is not None:assert a!=temporary[0] + move(a,f);move(b,f);out.extend(e) + elif op==2: + assert temporary is None;move(a,c);state[b]=f;temporary=(a,b,c,f);out.extend(e);hist[z]+=1 + else: + assert op==3 and temporary==(a,b,c,f)and state[a]==c and state[b]==f + out.extend(e);del state[b];temporary=None + for k in range(0,len(old),6): + i=k//6;e=tuple(old[k:k+6]);op,a,b,c,f,z=e + if op: + if op==1 and b in pivots and z==4 and f==zero and i<=pivots[b]['cut']:removed['kernel_reads']+=1 + elif op==1 and b in chosen and i<=cut:assert z==4 and f==zero and c==-1;removed['interval_reads']+=1 + else:emit(e) + for z in at.get(i,[]): + assert temporary is None + for q in z['roles']: + if q!=z['pivot']:emit((1,q,z['pivot'],1,z['new_frame_id'],28));added['Q']+=1 + if i==cut: + assert temporary is None + for block in blocks: + for e in block['word']:emit(e);added['interval']+=1 + for q in sorted(final):move(q,final[q]) + for pivot,z in sorted(pivots.items()): + for q in z['roles']: + if q!=pivot:emit((1,q,pivot,-1,full,29));added['Qinverse']+=1 + assert state==final and temporary is None and sha(out.tobytes())==INTERVAL_RAW + assert removed=={'kernel_reads':27752,'interval_reads':56}and added=={'Q':756,'Qinverse':756,'interval':56} + rt_spec=importlib.util.spec_from_file_location('hybrid_retiming25',package/'retiming25.py');rt=importlib.util.module_from_spec(rt_spec);rt_spec.loader.exec_module(rt) + F={f:dict(basis=C.B[f],annihilator=C.A[f])for f in set(initial.values())|{out[k+j]for k in range(0,len(out),6)if out[k]==0 for j in(2,3)}} + out,retiming=rt.transform(out,F,initial,final,json.loads((package/'retiming25-selection.json').read_text()),INTERVAL_RAW) + assert sha(out.tobytes())==FINAL_RAW + # Rebuild all required-use paths, not just the changed width histogram. + needs={q:[]for q in initial};copies=Counter();centers=[];center=None;scalar=hashlib.sha256();tagged=hashlib.sha256();coeff=Counter();counts=Counter();count=0 + cats=list(meta['category_names'])+['multicut_basis','multicut_inverse','closed_interval_precondition','closed_interval_compensation','closed_interval_inverse'] + def event(h,x):h.update(json.dumps(x,separators=(',',':')).encode()+b'\n') + for k in range(0,len(out),6): + op,a,b,c,f,z=out[k:k+6] + if op==1: + needs[a].append(f) + if b!=n:needs[b].append(f) + else:assert center is not None + source=center['source']if b==n else b;event(scalar,[a,source,c]);event(tagged,[a,b,c,f,cats[z]]);coeff[abs(c)]+=1;counts[cats[z]]+=1;count+=1 + elif op==2: + assert center is None;needs[a].append(c);copies[z]+=1;center=dict(source=a,frame=c,rank=z,temporary=b,first_event=count,copy_output_frame=f) + elif op==3: + assert center is not None and center['source']==a;center.update(after_event=count,scatter_reads=count-center['first_event']);assert center['scatter_reads']==220;centers.append(center);center=None + assert center is None and count==760383 and coeff=={1:759063,3:1320} + per={};pairs=set();hist=Counter(copies);sourceH=Counter();targetH=Counter();internalH=Counter(copies) + for q,before in initial.items(): + h=Counter() + for f in needs[q]+[final[q]]: + assert C.sub(before,f);d=C.dimf[f]-C.dimf[before];assert 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Apache-2.0. + +This bridge applies to a completed, physically proved round-13 profile. +It cannot make a numerically attractive but unconstructed profile valid. +The local source/target flags, controlled injections, old-response +subtraction, inverse cleanup, and retained alias boundaries are separate +construction obligations. assemble_profiles.py records their role as +proof dependencies and independently checks the exact arithmetic. + +1. The inherited uniform interfaces + +Retain the three completed-core cover of PR144/PR168 and all of its +analytic and fixed-tape assumptions. The complex cover is O(m,2), m<=72, +with order V<2^(m*m)<=2^5184. No new independent cover coordinates are +introduced by common-frame scalar lifts or source-controlled local +rewrites. Every stage still uses one completed invocation at each vertex. + +For a supplied profile with W0<10^7 roles per cover vertex, the full +physical stock W=V W0 is less than2^5208. Its total rank s is less +than mW<2^5215. The profile's actual proper-child maximum is retained; +the selected complex maximum must be less than m/2. The script checks +these inequalities rather than assuming that fewer auxiliary rows imply +a better contraction or a shorter recursion. + +The bit word uses the same weighted q-adic frame interface and the same +internally borrowed selectors as its pinned parent. New source controls +must use existing nondegenerate source-meeting planes or separately +certified new frames. All zero-rank adapters remain charged scalar work. +The group-size factor is not omitted and is not inserted into an external +polynomial row reserve. The inherited ordinary leaf saving is exactly +384599/10^10. The bad-row allowance is1/10^16; the charged fallback per +edge is32m^2. Its inequality2m^3/2^80<1/10^16 is checked explicitly. + +2. A deliberately loose finite scalar bound + +The pinned words and round-13 modifications are finite straight-line +programs with fewer than2^40 stored nodes, roles, coefficients, and +boundary entries. Scalar sum/difference lifts use actual invertible +matrices and are executed pointwise at common frame representatives. +For the selected dyadic rewrite, new elementary coefficients are +dyadic; the inherited center scatters retain their fixed odd divisor3. +No unrecorded new odd denominator may be included under this bridge. + +Fixed integer multipliers can be expanded into scalar doublings and +additions; dyadic divisions can be expanded into halvings. The explicit +low words and full-frame inverses are much smaller than2^4096 primitive +scalar/movement groups per invocation. Even constructing their fixed +old-response matrices by exact elimination is dominated by this bound: +matrix order is below2^40, original finite-word coefficient bit lengths +are below2^80, and the elementary elimination and coefficient expansion +bounds are below2^1024. No input-size-dependent dense elimination is +being performed. These are constants of the chosen finite construction. + +Consequently all3V invocation words, endpoint adapters, and finite role +bookkeeping fit within K<2^9300 logical groups. The retained universal +router is overcharged by a full W-role permutation for every group: + + 64 (m+1)^3 (K+1) (W+1)^2 < 2^20000. + +Choose the exact dominating scalar/router budget G=2^30000. This also +dominates the primitive arithmetic required to realize the fixed scalar +coefficients and source-controlled inverse cleanup. The router still +uses a fixed number of tapes, with sequential reuse of row stock. + +3. Explicit semantic constants + +Choose the exact integer E=2^100000 and put B=s+E. Then + + 2 G W^2 + 8s + 4W + 4 + 32m < 2^40500 < E, + E <= B < 2^100001. + +For the actual maximum child r= s+E >= 1. + +Choose C0=2^210000 and C1=1. These dominate both the retained semantic +requirement32m B^2 and2B+18. The reported strict_literal_gap=1 and +induction_gap_lower=1 are proved lower bounds on the positive actual +integer gaps; they are not claims that those gaps equal one. + +Use the inherited common dyadic grid with fixed3^(-K') exposure and +retain its precision/rounding contract. No child receives an independent +rounding allowance and no source-controlled injection is treated as an +unpaid numerical operation. + +4. External rows and stopping + +The complex child maximum below m/2 gives halving degree1, while +log2 W<5208<10000. Keep the deliberately conservative inherited bit +coarse reserve9909 and ordinary-leaf reserve252. Thus the external +row coefficient is bounded by + + A=10000+9909+252=20161. + +Choose degree D=1000000 and suffix slope4D. The exact row gap is + + D - (51/25)A > 0. + +The q-adic selector fields and source-control labels remain internally +borrowed and restored. Increasing these finite overcharges changes +constants and the all-size bridge, not the selected exponent saving. + +5. Exact supplier moments and balanced assembly + +For each actual histogram, assemble_profiles.py proves + + sum_r n_r r exp(a log(m/r))/(mW0) + bad_allowance < 1 + +using rational upper bounds, with upward rounding on the2^120 grid. +The bit atom stop is the first10^(-12) grid point above +a/(1+a-old), which implies both required strict toll inequalities. +Its effective saving is(1-beta)a+beta*old. This stopping rule is the +already reviewed PR158 edge rule, not a new local-word contribution. + +The final assembly uses the substantial PR168 balanced-prefix adapter, +with eta=10^(-8) and phase stop10^(-9), retaining all47 strict +constraints and all7 margins. The selected final kappa is rounded +down on the10^(-10) grid below the minimum complete margin. The +script checks every constraint exactly and reports whether the fixed +target0.0009218338 is reached. + +This is a conditional bound of the retained theorem form + + T(n)=O(n (log n)^(1-kappa)). + +The improvement ratio refers to kappa, the logarithmic exponent saving. +It is not a measurement of practical multiplication throughput. + +Credits + +The base supplier, scalar/frame verification, and actual profiles are +pinned to PR168 fd25adb7fbaa12ee761d02c733c54d1d2a7687ee, eumemic +(Anthropic Claude assistance), building on icekylinx's PR144. The +balanced prefix is RaD/hipotures; the semantic constraints are retained +from the reviewed PR23 line; completed-core sharing is credited to +an664 PR128; the atom-edge stop is credited to geckods PR158. Existing +source notices retain their more detailed upstream attributions. +The new source-controlled boundary construction, this conservative +bridge, and its integration are GPT-6 Astra's round-13 work. diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/compact-control-layout.tex b/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/compact-control-layout.tex new file mode 100644 index 0000000000..16f4d23347 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/compact-control-layout.tex @@ -0,0 +1,118 @@ +\paragraph{Rows and complete compact fields.} +Let $m,W,s$ denote the complex network constants and keep the layer's global +$p,d,K$ at every node. Put +\[ + G=4\lceil\log_2p\rceil+6,\quad H=dG,\quad + q_F=\left\lceil\frac{2H}{K}\right\rceil,\quad + q_B=\left\lceil\frac{H}{K}\right\rceil, +\] +\[ + q_0=\lceil\log_2W\rceil\lceil\log_m(2d)\rceil, + \quad k_0=\lceil\log_m d\rceil,\quad Q_0=q_0+q_F+q_B. +\] +If $D\le Q_0$, apply all $D$ selected kernels individually. Otherwise, +process individually the first $q_0+q_F$ and last $q_B$ selected axes. +This uses existing coordinates only. The remaining active axes are a +consecutive middle block of length $D-Q_0$. + +Regard the first $q_0$ chunks as the row index. Immediately after it, the +next $q_F$ chunks have at least $2H$ bits. Reinterpret these consecutive +bits as two $H$-bit front fields and a spectator remainder; a remainder +of length zero is omitted. The last $q_B$ chunks similarly supply one +$H$-bit back field and a spectator remainder. These reinterpretations move +no records and preserve the total address space. For a call with $f$ axes +per slot, carve $nG=(f-1)G\le H$ consecutive bits from each of these +fields; collapse their unused bits into spectators. The resulting field +list has a fixed number of entries. Both front fields precede all active +slots and the back field follows them, so the compact-control proposition +applies to every ordered pair of slots. For $f=1$ no compact fields are used. + +The original row range has length $R_{\rm row}=2^{q_0K}\ge W^{k_0}$. +For each original prefix, pad this row range to the next multiple of +$W^{k_0}$ by adding zero rows. A row includes the entire complete front +fields, active middle block, back field, all spectator coordinates and +polynomial record suffix. Padding adds whole such rows, so increases total +volume by at most two and does not change any within-row range. + +At a node write the row index as $u=Wg+w$, $0\le w2r_j^{t_j}$ for largest children $r_b=529,r_c=756$. +At width $e$ reserve the product +$Q=W_c^{D_c(e)}W_b^{D_b(e)}$, with +$D_j(e)=t_j\lceil\log_2e\rceil$. Its coefficient is +$9\cdot28+20\cdot30=852$, and +\[ + 2000>\frac{51}{25}\,852. +\] +The sufficient stock is therefore $p^{2000}$, supplied by the retained +physically preceding transformed prefix and a single complete-row +padding. Nested bit stock is restored before another complex child. +Sequential copies introduce no independent row field. The supplementary +suffix condition $b_{\rm in}^{1-\epsilon}>8000(\log_2b_{\rm in}+8)$ +is eventual for fixed $\epsilon<1$. + +\subsection{Exact conditional assembly} +Use the analytic proof map in +\path{notes/structured-bulk-assembly.tex}: arbitrary-source routing +with active masking and paid repair scans; one rounded phase-cell +matrix with preserved period cuts and downward-weight error bounds; +persistent target halos, sequential overlap tapes and factor pages; +Baker--Harman--Pintz prime packing; and the retained exact-recovery +interface. Their full costs are unchanged. In particular the bulk +bound retains the terms +\[ + O\!\left(Tp\left[p^\tau\operatorname{polylog}p+ + d\operatorname{polylog}p+dp^\delta+w^2p^\delta+ + dw^2p^\delta/L\right]\right)+n^{o(1)}, +\] +with $L=2^{\lceil8\log_2p\rceil}$, +$w^2=O(p^{1-r})$, and normalization exponent requiring +$\epsilon+r<1$. These arguments, not a change of arithmetic model, +justify the seven margins below. + +\begin{theorem}[Conditional fixed-tape multiplication bound] +Under the retained analytic and fixed-tape interfaces of PR23/PR29 +and the preceding proof map, the selected copied-center networks give +\[ + T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad + \boxed{\kappa=\frac{384569}{10000000000}=3.84569\times10^{-5}}. +\] +\end{theorem} +\begin{proof} +Take the strict finite savings $a=a_b=384599/10^{10}$ and +$b=a_c=717/10^7$, with $\tau=1-a$, $\sigma=1-b$, and put +\[ +\begin{gathered} + \eta=10^{-8},\quad\beta=\tfrac14,\quad q=a(1-2\eta),\quad + c=q(1+\eta),\quad\epsilon=\frac{1-\eta}{1+c+q},\\ + \lambda'=1-q,\quad\lambda=(\tau+\lambda')/2,\quad + g=\epsilon q,\quad r=(g+1-\epsilon)/2,\quad\delta=\eta/8. +\end{gathered} +\] +The stopped complex leaf saving $(1-\beta)b=53.775\times10^{-6}$ +is strictly above $a$, so paid internal adapters use exponent $\tau$ +and both layer exponents lie below $\lambda'$. The exact certificate +checks all 47 strict constraints of the retained semantic/bulk +composition, using the new literal scalar charge and product-row gap. +These include compact reservations, geometry, the linear guard, +$\epsilon+r<1$, phase-cell/band separation, prime packing, short-record +fallback and artificial-boundary allowances. + +The seven final savings are +\[ + 1-\epsilon(1+c),\quad a,\quad g,\quad a,\quad + \min(1-\epsilon-\delta,r-\delta),\quad + 1-\epsilon-\delta,\quad\epsilon. +\] +Their minimum is exactly $g$ and the prefix margin equals $g+\eta$. +Exact rational comparison gives +\[ + g-\frac{384569}{10^{10}}>4.0746056580\times10^{-11}>0. +\] +These positive gaps absorb fixed logarithms. The all-size conclusion +retains constructive finite setup, descriptor and catalogue domination, +prime existence and analytic recovery as eventual thresholds. The +strict bit and complex moments, copied-stream recurrence contracts and +these assembly inequalities prove the claimed conditional bound. +\end{proof} diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/copied-centers-lemma.tex b/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/copied-centers-lemma.tex new file mode 100644 index 0000000000..8933aaf508 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/copied-centers-lemma.tex @@ -0,0 +1,103 @@ +\subsection{Scalar, frame and temporary-stream contracts} +The retained source-copy, reversible mixer and scatter maps satisfy +$JLV=I$. Over $\mathbb F_2$ the chronological word +\[ + L,\ J,\ L^{-1},\ V,\ L,\ J,\ L^{-1},\ V +\] +adds the input to the target and restores every arbitrary auxiliary. +Over Gaussian dyadics use its corresponding old-value subtraction and +new-value addition. A designated retained output has a distinct terminal +output-use carrier: a matching may feed that carrier, but cannot merge +two designated terminals or append a later middle-producer consumer. +Its remaining uses before cleanup are its scatter reads, all of which +leave its scalar value unchanged. + +Within one tensor invocation suppress the fixed future line and put +\[ + D_0=B\otimes F_h,\quad D_1=A\otimes F_h,\quad + A=B\mathbin{\perp}P,\quad D_U=D_0\mathbin{\perp}(P\otimes U), +\] +where $P$ and the suppressed future factor are one-dimensional and +$U$ is a nondegenerate retained-center space of dimension $r$. +Thus a local change of rank $d$ has ambient rank $d$. +Write $\mathcal F_Q$ for the address frame: it is the rational partial +swap $D_Q$ in the bit construction, and the binary phase $C_Q$ in the +complex construction. Changing a copy from frame $Q_1$ to $Q_2$ means +applying $\mathcal F_{Q_2}\mathcal F_{Q_1}^{-1}$ to its physical stream. +Orthogonal backgrounds cancel in this quotient. + +Copies use Paureel's retained copy/transform/read/discard interface +\cite{paureel,pr29}. A copy has precisely the original complete role +stream, including all control and spectator ranges, and volume $V/W$. +It is processed while inactive original streams are parked on the fixed +depth-first stack tapes. Copying, pointwise reads and erasure are linear +in that stream's volume. Sequential processing needs one additional +fixed work stream, not an independently varying row coordinate. + +\begin{lemma}[Copied retained-center schedule] +In either invocation orientation a designated retained center's +physical rank profile can be changed from $z^r+z^h$ to +$z^r+z^{h-r}$, with the same scalar action, original role endpoints, +and arbitrary-scratch restoration. The rank-$r$ transform on the copy +is included in the recursive call list. +\end{lemma} +\begin{proof} +In the forward orientation the original center has scalar value $w$ +at frame $D_U$ after the middle mixer. Its old sequence was +$D_U\to D_0\to D_1$: scatter at $D_0$, then cleanup at $D_1$, +with ranks $r,h$. +Keep the original at $D_U$, copy its physical stream, and transform +only the copy to $D_0$, paying rank $r$. Supply every scatter read +from that transformed copy while its data targets remain at their +unchanged common frame $D_0$, then erase the temporary. The reads +are exactly those of $w$ in the old schedule. Since they do not change +$w$, move the original directly from $D_U$ to $D_1$ before its first +cleanup incidence; this costs $h-r$. Every subsequent original gate +therefore receives the same scalar value at its required frame. + +In the reverse complementary orientation the old sequence was +$D_0\to D_1\to D_{U^\perp}$, with ranks $h,r$ and scatter at $D_1$. +After the preceding all-$D_0$ operations, advance the original directly +to $D_{U^\perp}=D_0\mathbin{\perp}(P\otimes U^\perp)$, paying +$h-r$. Copy it and transform the copy to $D_1$, paying $r$. +Use that copy for all early scatter reads, then erase it. The original +already has the exact frame and unchanged scalar value required by the +following inverse mixer. Terminal output-use separation ensures no +intervening producer consumer has been bypassed. + +For partial swaps either direction of the rank-$r$ change is the same +involution. For complex phases the forward copy uses $C_U^{-1}$, +the direct original growth uses $C_{U^\perp}$, and the reverse copy +uses $C_FC_{U^\perp}^{-1}=C_U$. These are the same charged ranks, +implemented by the retained forward/inverse residual interface. +All tensor backgrounds cancel as stated. Every original scalar +incidence, final source-copy cancellation and auxiliary source/sink +operator is unchanged. Thus the scalar identity and all endpoints +remain exact on arbitrary initial auxiliary values. +\end{proof} + +For the selected centers $r=h-1$, and there are $h$ distinct retained +output uses per invocation. Their old histogram has rank mass +$hR_h+2\ell_h$, where $\ell_h=h(h-1)$. The new histogram is exactly +\[ + H'_{h,1}=H_{h,1}+h,\qquad H'_{h,h}=H_{h,h}-h, + \qquad H'_{h,r}=H_{h,r}\quad(r\ne1,h), +\] +and hence $\sum_r rH'_{h,r}=hR_h+\ell_h$. +Every old rank-$(h-1)$ residual remains as a paid transform of a copy; +its projector and ordinary bit pivot profile are unchanged. Only the +full-rank original transition becomes the rank-one complement. +The physical role count $R_h$ is unchanged. + +For two factors, put $N=v_av_b$, $B_a=v_bR_a$, $B_b=v_aR_b$, +$W=2N+B_a+B_b$ and $L=v_b\ell_a+v_a\ell_b$. +All exterior, data-front and endpoint-correction calls remain in place. +The resulting complete rank mass is +\[ + s=Wm-N+L,\qquad Wm-s=N-L. +\] +This follows by subtracting exactly $L$ from the retained two-stage +physical mass $Wm-N+2L$. The $N$ rank-one endpoint-copy calls are +separate and remain fully charged. Because center copies retain the +same complete role-volume fraction $1/W$, the existing complete-row, +integer-width remainder and fixed-tape recurrence proofs apply. diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/endpoint-gauge-complex.tex b/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/endpoint-gauge-complex.tex new file mode 100644 index 0000000000..907a0ceda8 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/retained/endpoint-gauge-complex.tex @@ -0,0 +1,225 @@ +\subsection{A normal form for every nondegenerate binary residual} +Let $E$ have binary basis matrix $V$ and nonsingular Gram matrix +$G=V^tV$. In dual Fourier coordinates put +$q(z)=\operatorname{wt}(VG^{-1}z)\pmod4$. +Its polarization is $2z^tG^{-1}w$. The same argument applies to mixed +signs on orthogonal summands, since reversing a sign preserves this +polarization. Write $S_q=\sum_z i^{q(z)}$. Completing the square gives +\[ +\begin{aligned} +K(x,y)&=2^{-r}\sum_z i^{q(z)}(-1)^{z^t(x+y)}\\ + &=2^{-r}S_q\,i^{-q(Gx)}(-1)^{x^tGy}i^{-q(Gy)}. +\end{aligned} +\] +The scaled Walsh kernel in the middle is a binary permutation of +$H_0^{\otimes r}$, where +\[ + H_0=\tfrac12\begin{pmatrix}1&1\\1&-1\end{pmatrix} + =bSCS,\qquad b=(1-i)/2,\quad S=\operatorname{diag}(1,i). +\] +The remaining scalar $S_qb^r$ is a fourth root of unity. To verify +this, decompose the nondegenerate quadratic module into odd +one-dimensional and alternating two-dimensional blocks. Their Gauss +sums are $1\pm i$ and $\pm2$, respectively; multiplying by $b$ and +$b^2$ gives a fourth root in each case. This is a proof device, not +an orthonormal-basis assumption on $E$. + +Consequently one rank-$r$ child suffices, with binary address changes, +diagonal fourth-root phases, and one scalar fourth-root phase. +On $f$ parallel columns it is one $C^{\otimes rf}$ child, the scalar +raised to power $f$, and columnwise quadratic phases modulo four. +The retained packed address operations evaluate these fixed formulas. +These wrappers have modulus one and introduce no denominator loss. + +\subsection{The selected Gaussian-dyadic producer} +Use paired triple exclusion with base threshold two, complete unsplit +disjoint and pair-star sums, and sharing by actual ordinary support. +For an ordinary support with at least two common points use the span +of its triple indicators; these form an orthonormal common-pair star. +Otherwise use the coordinate space of its union. Support inclusion +preserves these labels: losing the common pair enters a containing +coordinate space. Source lines use ordinary triple containment. + +Retain every full pair-star $P_{uv}=\sum_{w\ne u,v}X_{uvw}$ and form +$B_u=\sum_{v\ne u}P_{uv}=2\sum_{T\ni u}X_T$. +For even $h$ label these central additions by the nondegenerate +hyperplane $E_u=\{x:\sum_{j\ne u}x_j=0\}$; every contributing star +lies in it. Retain the global total $T$ with full-space label. +The central correction at target $S$ is +\[ + \frac14\sum_{u\in S}B_u-\frac12T + =\sum_U\frac{|S\cap U|-1}{2}X_U. +\] +Adding half the disjoint sum and subtracting half the intersection-two +sum gives exactly $X_S$. All coefficients are Gaussian dyadic. +The transparent retained-output wrapper and oriented carrier compiler +apply to these nested nondegenerate labels. The $h$ hyperplane centers +lose $h(h-1)$ and the global center loses $h$, so loss is $h^2$. +The selected counts, including all point centers, are +\[ +\begin{array}{c|rrrrrr} +h&v_h&c_h&q_h&|\mathcal M_h|&R_h&\text{loss}\\\hline +32&4960&122512&19873&19241&123144&1024\\ +30&4060&99153&16271&15873&99551&900\\ +40&9880&252023&39561&37642&253942&1600 +\end{array} +\] +Their internal histograms satisfy $\sum_r rH_{h,r}=hR_h+2h^2$. +They are distinct from the bit histograms. + +\subsection{Endpoint normalization and the complex characteristic} +For nondegenerate $U$, Fourier phases give +\[ + C_U^2=X_{P_U\mathbf1},\qquad + C_{U^\perp}C_U^{-1}=C_FX_{P_U\mathbf1}. +\] +Put $v=P_U\mathbf1$. On coordinates in $v$, this endpoint has +$C^{-1}$ instead of $C$. The identity $C^{-1}=-iZCZ$ shows that +conjugating by $Z_v$ and multiplying by $i^{\operatorname{wt}(v)}$ +normalizes it to $C_F$; for $f$ columns the exponent is +$f\operatorname{wt}(v)$. These are diagonal wrappers without a child. + +Choose stage-two source $C_{D_0}$ and sink $C_{D_0^\perp}$, where +$D_1=D_0\mathbin{\perp} A$ and $\dim A=b$. The same refined copy +schedule removes the entrance on every auxiliary role. With +$R=D_1^\perp$, the exit is $C_RC_{D_0}^{-1}$ on +$E=R\mathbin{\perp}D_0=A^\perp$. Its rank is $m-b$ and its +nondegenerate polar form is unchanged by the mixed signs. The normal +form above executes it as one child. Arbitrary auxiliary restoration +justifies the endpoint normalization on the entire $B_2$ bank. + +Use the complex counts to define $B_i=NR_{h_i}/v_{h_i}$, and now +\[ +\begin{aligned} +W&=2N+B_2+B_3=10390178240000,\\ +L&=\sum_i(N/v_{h_i})h_i^2=117399987200,\\ +s&=Wm-2N+2L=398982681296998400. +\end{aligned} +\] +The whole-residual macro list is +\[ +\begin{array}{c|rrrrr} +t&38328&38360&38370&37401&899\\\hline +n_t&B_1&B_3-B_1&B_2&2N&2N. +\end{array} +\] +Add $(N/v_h)H_{h,r}$ rank-$r$ children for each positive internal rank. +Leave the remaining physical-data rank, 39393978624000, as singleton +calls. The resulting list has weighted rank exactly $s$. +For $a_c=3/125000$ and $\sigma=1-a_c$, the rational logarithm bounds +above and $e^u\le(1-u)^{-1}$, $0\le u<1$, give +\[ + \Psi_c(\sigma)\le\sum_t\frac{n_tt}{Wm(1-a_c\ell_t)} + <1-\frac{792981575160764}{10^{23}}. +\] + +\subsection{The dependency-path guard} +Before gauging, nested producer labels telescope along every arithmetic +path, including wire switches at common-frame scalar gates. Retained +center drops occur on one return frontier per stage, crossed at most +once by a path, and each costs at most that axis dimension. Carrier +reuse adds only increasing continuations. This gives +$q_0=m+2(a+b+k)$. Gauging removes an early edge and adds $ab-b$ at +a late stage-two edge. A path meets at most one such edge, since it +terminates that stage-two role, which is not shared across stages. +Thus the conservative total rank budget is +\[ + q=m+2(a+b+k)+ab=39564. +\] +Every child except the four large macros is at most 899; two large +macros cannot fit within $q$. Set $\rho=19/10$, $\gamma=9997/10000$. +The integer comparisons $m^{19}>500000000^{10}$ and +$899^9<460^{10}$ give $m^\rho>500000000$ and +$899^{\rho-1}<460$. +If a path has one large child $t$, its moment is at most +\[ + (t/m)^\rho+(q-t)899^{\rho-1}/m^\rho. +\] +This expression increases over the large-child range, since its +numerator derivative is $\rho t^{\rho-1}-899^{\rho-1}>0$. +Use $t=m-b$ and the concavity bound +$(1-x)^{19/10}\le1-19x/10+9x^2/10$ to obtain +\[ + \sum_{\rm path}(r/m)^\rho + <1-\frac{19}{12800}+\frac9{10\cdot1280^2} + +\frac{1194\cdot460}{500000000} + =\frac{51180270301}{51200000000}<\gamma. +\] +Without a large macro the bound is $227493/6250000<\gamma$. + +For coefficient-depth accounting take $E_0=128(W+m+1)^3$. +There are at most $s$ positive-rank residual calls and $W$ endpoint +normalizations. Combining the quadratic phases with the $S$ factors, +each uses at most two diagonal phase passes and one scalar phase. +The inherited gate and remainder count is therefore bounded conservatively by +\[ + 36W^3+16s+16W+8m+4 card Γ * N.unit φ) + +Scalar chain of one class (`x1 y1 x2 y2` on `X1 Y1 X2 Y2`), as in `bridged-word.md`: + + stage 0 F on pair 1 Y1 = y1 + x1 + stage 1 B on pair 1 X1 = -y1 + gates A X1 += X2, Y2 -= Y1 + stage 2 F on pair 1 Y1 = x1 + x2 + gates B Y2 += Y1, X1 -= X2 (X1 = -y1, Y2 = y2 - y1 + x2) + stage 3 B on pair 2 X2 = y1 - y2 + stage 4 F on pair 2 Y2 = x2 + gates C Y1 -= Y2, X2 += X1 (X1 = -y1, X2 = -y2, Y1 = x1, Y2 = x2) + exchange X_i := Y_i, Y_i := -X_i +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace BR +open Binary Matrix Finset RAM SS CB RF +noncomputable section + +section Gates +variable {Γ T Sl C : Type} + +/-- source of the gates of the layers A and B: `X1 ← X2`, `Y2 ← Y1`. -/ +def pAB : BRole Γ T Sl C → BRole Γ T Sl C := + mkB (fun g t => kX2 (g,t)) (fun g t => kY1 (g,t)) (fun g t => kX2 (g,t)) + (fun g t => kY1 (g,t)) (fun g q => kS (g,q)) (fun g c => kC (g,c)) + +/-- layer A: `X1 += X2`, `Y2 -= Y1`. -/ +def kA : BRole Γ T Sl C → ℚ := + mkB (fun _ _ => 1) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => -1) (fun _ _ => 0) (fun _ _ => 0) + +/-- layer B: `X1 -= X2`, `Y2 += Y1`. -/ +def kB : BRole Γ T Sl C → ℚ := + mkB (fun _ _ => -1) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => 1) (fun _ _ => 0) (fun _ _ => 0) + +/-- source of the gates of the layer C: `Y1 ← Y2`, `X2 ← X1`. -/ +def pC : BRole Γ T Sl C → BRole Γ T Sl C := + mkB (fun g t => kX1 (g,t)) (fun g t => kY2 (g,t)) (fun g t => kX1 (g,t)) + (fun g t => kY2 (g,t)) (fun g q => kS (g,q)) (fun g c => kC (g,c)) + +/-- layer C: `Y1 -= Y2`, `X2 += X1`. -/ +def kCc : BRole Γ T Sl C → ℚ := + mkB (fun _ _ => 0) (fun _ _ => -1) (fun _ _ => 1) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => 0) + +/-- scalar map of a gate layer. -/ +def gsc (p : BRole Γ T Sl C → BRole Γ T Sl C) (k : BRole Γ T Sl C → ℚ) + (x : BRole Γ T Sl C → ℂ) : BRole Γ T Sl C → ℂ := fun i => x i + (k i : ℂ) * x (p i) + +/-- the signed exchange `X_i := Y_i`, `Y_i := -X_i` on both pairs. -/ +def bpull : BRole Γ T Sl C → BRole Γ T Sl C := + mkB (fun g t => kY1 (g,t)) (fun g t => kX1 (g,t)) (fun g t => kY2 (g,t)) + (fun g t => kX2 (g,t)) (fun g q => kS (g,q)) (fun g c => kC (g,c)) + +def bsgn : BRole Γ T Sl C → ℚ := + mkB (fun _ _ => 1) (fun _ _ => -1) (fun _ _ => 1) (fun _ _ => -1) (fun _ _ => 1) (fun _ _ => 1) + +variable [Fintype Γ] [DecidableEq Γ] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] + +def brecM : Matrix (BRole Γ T Sl C) (BRole Γ T Sl C) ℚ := + fun i j => if j = bpull i then bsgn i else 0 + +lemma actPoint_brecM (x : BRole Γ T Sl C → ℂ) (i : BRole Γ T Sl C) : + actPoint brecM x i = (bsgn i : ℂ) * x (bpull i) := by + unfold actPoint + rw [sum_eq_single (bpull i)] + · simp [brecM] + · intro j _ hj; simp [brecM, hj] + · simp + +/-- the scalar map of the whole network before the exchange, on one class. -/ +def chain (x : BRole Γ T Sl C → ℂ) : BRole Γ T Sl C → ℂ := + gsc pC kCc (lift2 GF (lift2 GB (gsc pAB kB (lift1 GF (gsc pAB kA (lift1 GB (lift1 GF x))))))) + +/-- **The scalar identity of the bridged chain**: `X_i = -y_i`, `Y_i = x_i` on both pairs, +helper slots unchanged. -/ +lemma chain_eq (x : BRole Γ T Sl C → ℂ) (l : BLive Γ T Sl) : + chain x (Sum.inl l) = mkB (fun g t => -x (kY1 (g,t))) (fun g t => x (kX1 (g,t))) + (fun g t => -x (kY2 (g,t))) (fun g t => x (kX2 (g,t))) + (fun g q => x (kS (g,q))) (fun g c => x (kC (g,c))) (Sum.inl l) := by + rcases l with (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩) + all_goals + simp only [chain, gsc, lift1, lift2, GF, GB, mkB, mkS, pAB, pC, kA, kB, kCc, pos1, pos2, + kX1, kY1, kX2, kY2, kS, kC, jX, jY, jS, jC, Function.comp_apply] + first + | (push_cast; ring) + | (push_cast; done) + | ring + +end Gates + +section Net +variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] + [Fintype Γ] [DecidableEq Γ] +variable (N : Bridge2 H T Sl C α Γ) + +/-- gate layer A, after stage 1 and the first idle climb of pair 2: twins have equal frames. -/ +lemma Bridge2.gateA : XRoute N.S3 N.S3 (gsc pAB kA) (fun _ => 0) := by + refine gateP pAB kA N.S3 ?_ + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk + · rfl + · exact (hk rfl).elim + · exact (hk rfl).elim + · rfl + · exact (hk rfl).elim + · exact (hk rfl).elim + +/-- gate layer B, after stage 2 and the second idle climb of pair 2. -/ +lemma Bridge2.gateB : XRoute N.S5 N.S5 (gsc pAB kB) (fun _ => 0) := by + refine gateP pAB kB N.S5 ?_ + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk + · exact N.x23 g t + · exact (hk rfl).elim + · exact (hk rfl).elim + · exact (N.y23 g t).symm + · exact (hk rfl).elim + · exact (hk rfl).elim + +/-- gate layer C, after the final climbs. -/ +lemma Bridge2.gateC : XRoute N.S8 N.S8 (gsc pC kCc) (fun _ => 0) := by + refine gateP pC kCc N.S8 ?_ + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk + · exact (hk rfl).elim + · rfl + · rfl + · exact (hk rfl).elim + · exact (hk rfl).elim + · exact (hk rfl).elim + +/-- the free shifts at the end. -/ +def Bridge2.corr : BRole Γ T Sl C → Space α := + mkB (fun _ _ => 0) (fun g t => N.u g t) (fun _ _ => 0) (fun g t => N.u g t) + (fun _ _ => 0) (fun _ _ => 0) + +def Bridge2.S9 : BRole Γ T Sl C → CMat α := fun r => shift (N.corr r) * N.S8 r + +/-- the whole word before the exchange: five stages, three idle-climb layers, three gate +layers. -/ +lemma Bridge2.route : + XRoute N.S0 N.S8 chain (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by + have R := ((((((((((N.stage0.trans N.stage1).trans N.climbA).trans N.gateA).trans + N.stage2).trans N.climbB).trans N.gateB).trans N.stage3).trans N.stage4).trans + N.fin).trans N.gateC) + refine (R.castg ?_).cast (fun φ => ?_) + · rfl + · simp only [Bridge2.unit] + ring + +/-- **The scratch certificate of the bridged five-stage network with shared helper sets**, +in whole blocks, with its exact price for every price list. -/ +theorem bridge2_certificate : + BCert α (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) + (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by + have R2 : XRoute N.S8 N.S9 id (fun _ => 0) := XRoute.shifts N.hm N.S8 N.corr + have R3 : XRoute N.S9 (fun r => N.S9 (bpull r)) (actPoint brecM) (fun _ => 0) := + XRoute.gate brecM N.S9 _ (fun i j h => by + have hj : j = bpull i := by + by_contra hne + exact h (by simp [brecM, hne]) + rw [hj]) + have Rt := ((N.route.trans R2).trans R3).cast (c' := fun φ => (Fintype.card Γ : ℝ) * N.unit φ) + (fun φ => by first | ring | simp) + have hSinv : ∀ r, ∃ M, N.S0 r * M = 1 := by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact N.inv0 g t + · refine ⟨1, ?_⟩ + show (N.M0 g).Φ 0 * 1 = 1 + rw [N.z0, Matrix.mul_one] + · exact N.inv0 g t + · refine ⟨1, ?_⟩ + show (N.M0 g).Φ 0 * 1 = 1 + rw [N.z0, Matrix.mul_one] + · exact ⟨1, Matrix.mul_one _⟩ + · exact ⟨1, Matrix.mul_one _⟩ + choose S' hS' using hSinv + have hT : ∀ l : BLive Γ T Sl, N.S9 (bpull (Sum.inl l)) = kernel α * N.S0 (Sum.inl l) := by + rintro (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩) + · show shift (N.u g t) * N.YF g t = kernel α * (N.M0 g).Φ (tt (N.I.tv t)) + exact (N.term g t).symm + · show shift (0 : Space α) * kernel α = kernel α * (N.M0 g).Φ 0 + rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one] + · show shift (N.u g t) * N.YF g t = kernel α * (N.M0 g).Φ (tt (N.I.tv t)) + exact (N.term g t).symm + · show shift (0 : Space α) * kernel α = kernel α * (N.M0 g).Φ 0 + rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one] + · show shift (0 : Space α) * kernel α = kernel α * 1 + rw [shift_zero, Matrix.one_mul, Matrix.mul_one] + have hg : ∀ x : BRole Γ T Sl C → ℂ, + (∀ s, (∀ l : BLive Γ T Sl, (Sum.inl l : BRole Γ T Sl C) ≠ s) → x s = 0) → + ∀ l : BLive Γ T Sl, + (actPoint brecM ∘ (id ∘ chain)) x (Sum.inl l) = x (Sum.inl l) := by + intro x _ l + change actPoint brecM (chain x) (Sum.inl l) = _ + rw [actPoint_brecM] + rcases l with (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩) + · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inr (g,t))))) = _ + rw [chain_eq] + show ((1:ℚ):ℂ) * x (kX1 (g,t)) = x (kX1 (g,t)) + first | (push_cast; ring) | simp + · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inl (g,t))))) = _ + rw [chain_eq] + show ((-1:ℚ):ℂ) * (-x (kY1 (g,t))) = x (kY1 (g,t)) + first | (push_cast; ring) | simp + · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inr (g,t))))) = _ + rw [chain_eq] + show ((1:ℚ):ℂ) * x (kX2 (g,t)) = x (kX2 (g,t)) + first | (push_cast; ring) | simp + · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inl (g,t))))) = _ + rw [chain_eq] + show ((-1:ℚ):ℂ) * (-x (kY2 (g,t))) = x (kY2 (g,t)) + first | (push_cast; ring) | simp + · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inr (g,q))) = _ + rw [chain_eq] + show ((1:ℚ):ℂ) * x (kS (g,q)) = x (kS (g,q)) + first | (push_cast; ring) | simp + exact XRoute.liveKernel (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) N.S0 S' + (fun r => N.S9 (bpull r)) hS' _ _ Rt hg hT + +end Net +end +end BR +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.BR.chain_eq +#print axioms OAI.PowerSaving.BR.Bridge2.route +#print axioms OAI.PowerSaving.BR.bridge2_certificate diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Bridge__Net.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Bridge__Net.lean new file mode 100644 index 0000000000..204f6d9cf9 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Bridge__Net.lean @@ -0,0 +1,345 @@ +import Work.Bridge.Inv + +/-! +# (key: bridge-net) The BRIDGED five-stage network B_2 with shared helper sets, abstract form + +Generalises `RF.Fold3S` / `RF.fold3s_certificate` (`Work.Reframe.SCert`) to the bridged word +of PLAN2.md B1: every class `(g,t)` has TWO bank pairs `(X1,Y1)`, `(X2,Y2)` with the same +line; five stages of invocations + + stage 0 forward on pair 1 invocation of the class (g,t): g + stage 1 backward on pair 1 r1 t g + stage 2 forward on pair 1 r2 t g + stage 3 backward on pair 2 r3 t g + stage 4 forward on pair 2 r4 t g + +and three layers of FREE gates between twins at equal frames: + + after stage 1 (pair 2 has climbed to the frames of pair 1): X1 += X2, Y2 -= Y1 + after stage 2 (pair 2 has climbed one more window): Y2 += Y1, X1 -= X2 + after the final climbs (all four roles at their end frames): Y1 -= Y2, X2 += X1 + +Result on both pairs: `X_i = -y_i`, `Y_i = x_i`; then the usual signed exchange. + +The idle pair does not move during a stage. Its climbs are ONE block per role: +`aX2`, `aY2` (pair 2 before the first gate layer, prices `caX`, `caY`; in B_2 ONE block of +rank `2h-2` each), +the window of stage 2 (pair 2 before the second gate layer, rank `card H - 1`, from the +package: `Inv.climbX`, `Inv.climbY`), `fX1`, `fY1` (pair 1 at the end, prices `cX1`, `cY1`; +in B_2 ONE block of rank `2h+2` each), `fX2`, `fY2` (pair 2 at the end, prices `cX2`, `cY2`; +in B_2 one block of rank `4` each). The prices are arbitrary functions of the price list, so +a geometry that makes an idle climb in several blocks can be plugged in as well. + +`Bridge2 H T Sl C α Γ` is everything the wiring needs to know about the geometry; +`bridge2_certificate`: a `BCert` on the live roles `BLive Γ T Sl` +(`card Γ * (4 card T + card Sl)` of them, `BR.card_BLive`) whose price, for EVERY price +list, is `card Γ * N.unit φ`, + + N.unit φ = 5 * N.I.cost φ + card T * ((caX φ + caY φ) + 2 * bcost φ (card H - 1) + + (cX1 φ + cY1 φ) + (cX2 φ + cY2 φ)). +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace BR +open Binary Matrix Finset RAM SS CB RF +noncomputable section + +/-- See the file header. Fields in the style of `RF.Fold3S`; stages are numbered 0 to 4. -/ +structure Bridge2 (H T Sl C α Γ : Type) [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] where + /-- the invocation (for a block circuit `X`: `X.inv`). -/ + I : Inv H T Sl C + /-- frame maps of the invocations of the five stages. -/ + M0 : Γ → XMap H α + M1 : Γ → XMap H α + M2 : Γ → XMap H α + M3 : Γ → XMap H α + M4 : Γ → XMap H α + /-- the class `(g,t)` is served by the invocation `g` in stage 0 and `r_l t g` in stage `l`. -/ + r1 : T → Γ ≃ Γ + r2 : T → Γ ≃ Γ + r3 : T → Γ ≃ Γ + r4 : T → Γ ≃ Γ + /-- the invocation `d` of stage `l` uses the helper set (and the copies) `s_l d`. -/ + s1 : Γ ≃ Γ + s2 : Γ ≃ Γ + s3 : Γ ≃ Γ + s4 : Γ ≃ Γ + /-- inverse of the frame of the base, and frame of the window, of every invocation. -/ + N1 : Γ → CMat α + W1 : Γ → CMat α + N2 : Γ → CMat α + W2 : Γ → CMat α + N3 : Γ → CMat α + W3 : Γ → CMat α + N4 : Γ → CMat α + W4 : Γ → CMat α + u : Γ → T → Space α + YF : Γ → T → CMat α + hm : 2 ≤ Fintype.card α + /-- prices of the idle climbs, for every price list. B_2 with merged climbs: + `caX = caY = fun φ => bcost φ (2h-2)`, `cX1 = cY1 = fun φ => bcost φ (2h+2)`, + `cX2 = cY2 = fun φ => bcost φ 4`. -/ + caX : (ℕ → ℝ) → ℝ + caY : (ℕ → ℝ) → ℝ + cX1 : (ℕ → ℝ) → ℝ + cY1 : (ℕ → ℝ) → ℝ + cX2 : (ℕ → ℝ) → ℝ + cY2 : (ℕ → ℝ) → ℝ + inv0 : ∀ g t, ∃ N : CMat α, (M0 g).Φ (tt (I.tv t)) * N = 1 + z0 : ∀ g, (M0 g).Φ 0 = 1 + hN1 : ∀ d, (M1 d).Φ 0 * N1 d = 1 + hW1 : ∀ d, (M1 d).Φ 1 = W1 d * (M1 d).Φ 0 + hN2 : ∀ d, (M2 d).Φ 0 * N2 d = 1 + hW2 : ∀ d, (M2 d).Φ 1 = W2 d * (M2 d).Φ 0 + hN3 : ∀ d, (M3 d).Φ 0 * N3 d = 1 + hW3 : ∀ d, (M3 d).Φ 1 = W3 d * (M3 d).Φ 0 + hN4 : ∀ d, (M4 d).Φ 0 * N4 d = 1 + hW4 : ∀ d, (M4 d).Φ 1 = W4 d * (M4 d).Φ 0 + /-- the bank frames a stage leaves are the bank frames the next stage starts from. -/ + x01 : ∀ g t, (M0 g).Φ 1 = (M1 (r1 t g)).Φ (tt (I.tv t)) + y01 : ∀ g t, (M0 g).Φ (1 + tt (I.tv t)) = (M1 (r1 t g)).Φ 0 + x12 : ∀ g t, (M1 (r1 t g)).Φ 1 = (M2 (r2 t g)).Φ (tt (I.tv t)) + y12 : ∀ g t, (M1 (r1 t g)).Φ (1 + tt (I.tv t)) = (M2 (r2 t g)).Φ 0 + x23 : ∀ g t, (M2 (r2 t g)).Φ 1 = (M3 (r3 t g)).Φ (tt (I.tv t)) + y23 : ∀ g t, (M2 (r2 t g)).Φ (1 + tt (I.tv t)) = (M3 (r3 t g)).Φ 0 + x34 : ∀ g t, (M3 (r3 t g)).Φ 1 = (M4 (r4 t g)).Φ (tt (I.tv t)) + y34 : ∀ g t, (M3 (r3 t g)).Φ (1 + tt (I.tv t)) = (M4 (r4 t g)).Φ 0 + /-- the five windows of a helper set make up the kernel. -/ + sfin : ∀ g, W4 (s4.symm g) * (W3 (s3.symm g) * (W2 (s2.symm g) * + (W1 (s1.symm g) * (M0 g).Φ 1))) = kernel α + /-- pair 2, idle in stages 0 and 1: ONE block from its start frame to the frame of its + twin after stage 1. -/ + aX2 : ∀ g t, XReach ((M0 g).Φ (tt (I.tv t))) ((M1 (r1 t g)).Φ 1) caX + aY2 : ∀ g t, XReach ((M0 g).Φ 0) ((M1 (r1 t g)).Φ (1 + tt (I.tv t))) caY + /-- pair 1, idle in stages 3 and 4: ONE block from its frame after stage 2 to the end. -/ + fX1 : ∀ g t, XReach ((M2 (r2 t g)).Φ 1) (kernel α) cX1 + fY1 : ∀ g t, XReach ((M2 (r2 t g)).Φ (1 + tt (I.tv t))) (YF g t) cY1 + /-- pair 2 after stage 4: ONE final block. -/ + fX2 : ∀ g t, XReach ((M4 (r4 t g)).Φ 1) (kernel α) cX2 + fY2 : ∀ g t, XReach ((M4 (r4 t g)).Φ (1 + tt (I.tv t))) (YF g t) cY2 + term : ∀ g t, kernel α * (M0 g).Φ (tt (I.tv t)) = shift (u g t) * YF g t + +section Net +variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] + [Fintype Γ] [DecidableEq Γ] +variable (N : Bridge2 H T Sl C α Γ) + +/-- price of the blocks of one unit (one element of `Γ`): five invocations and the idle +climbs of the four bank roles of every class. -/ +def Bridge2.unit (φ : ℕ → ℝ) : ℝ := + 5 * N.I.cost φ + (Fintype.card T : ℝ) * ((N.caX φ + N.caY φ) + + 2 * bcost φ (Fintype.card H - 1) + + (N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ)) + +/-- scratch copies always enter and leave with the matrix `1` (also: the helper slots at the +start, with `C := Sl`). -/ +def cOne (Γ C α : Type) [Fintype α] [DecidableEq α] : Γ → C → CMat α := fun _ _ => 1 + +/-- the bank index of stage 0: the class `(g,t)` belongs to the invocation `g`. -/ +abbrev bid (Γ T : Type) : T → Γ ≃ Γ := fun _ => Equiv.refl Γ + +/-! bank frames of the class `(g,t)`: `xi l`, `yi l` at the START of stage `l`; `xo l`, +`yo l` at the END of stage `l` (only where no next stage follows on that pair). -/ +def Bridge2.xi0 (g : Γ) (t : T) : CMat α := (N.M0 g).Φ (tt (N.I.tv t)) +def Bridge2.yi0 (g : Γ) (t : T) : CMat α := (N.M0 g).Φ 0 +def Bridge2.xi1 (g : Γ) (t : T) : CMat α := (N.M1 (N.r1 t g)).Φ (tt (N.I.tv t)) +def Bridge2.yi1 (g : Γ) (t : T) : CMat α := (N.M1 (N.r1 t g)).Φ 0 +def Bridge2.xi2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ (tt (N.I.tv t)) +def Bridge2.yi2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ 0 +def Bridge2.xo2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ 1 +def Bridge2.yo2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ (1 + tt (N.I.tv t)) +def Bridge2.xi3 (g : Γ) (t : T) : CMat α := (N.M3 (N.r3 t g)).Φ (tt (N.I.tv t)) +def Bridge2.yi3 (g : Γ) (t : T) : CMat α := (N.M3 (N.r3 t g)).Φ 0 +def Bridge2.xi4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ (tt (N.I.tv t)) +def Bridge2.yi4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ 0 +def Bridge2.xo4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ 1 +def Bridge2.yo4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ (1 + tt (N.I.tv t)) + +/-- helper slots after the stages 0, 1, 2, 3, 4 (at the start: `cOne Γ Sl α`). -/ +def Bridge2.Z1 : Γ → Sl → CMat α := fun g _ => (N.M0 g).Φ 1 +def Bridge2.Z2 : Γ → Sl → CMat α := fun g q => N.W1 (N.s1.symm g) * N.Z1 g q +def Bridge2.Z3 : Γ → Sl → CMat α := fun g q => N.W2 (N.s2.symm g) * N.Z2 g q +def Bridge2.Z4 : Γ → Sl → CMat α := fun g q => N.W3 (N.s3.symm g) * N.Z3 g q +def Bridge2.Z5 : Γ → Sl → CMat α := fun g q => N.W4 (N.s4.symm g) * N.Z4 g q + +/-- frames: at the start; after stage 0; after stage 1; after the first idle climb of pair 2 +(first gate layer here); after stage 2; after the second idle climb of pair 2 (second gate +layer here); after stage 3; after stage 4; after the final climbs (third gate layer here). -/ +def Bridge2.S0 : BRole Γ T Sl C → CMat α := mkB N.xi0 N.yi0 N.xi0 N.yi0 (cOne Γ Sl α) (cOne Γ C α) +def Bridge2.S1 : BRole Γ T Sl C → CMat α := mkB N.xi1 N.yi1 N.xi0 N.yi0 N.Z1 (cOne Γ C α) +def Bridge2.S2 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi0 N.yi0 N.Z2 (cOne Γ C α) +def Bridge2.S3 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi2 N.yi2 N.Z2 (cOne Γ C α) +def Bridge2.S4 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi2 N.yi2 N.Z3 (cOne Γ C α) +def Bridge2.S5 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi3 N.yi3 N.Z3 (cOne Γ C α) +def Bridge2.S6 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi4 N.yi4 N.Z4 (cOne Γ C α) +def Bridge2.S7 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xo4 N.yo4 N.Z5 (cOne Γ C α) +def Bridge2.S8 : BRole Γ T Sl C → CMat α := + mkB (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) (cOne Γ C α) + +/-- stage 0: forward on pair 1, every helper set starts at the matrix `1`. -/ +lemma Bridge2.stage0 : + XRoute N.S0 N.S1 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + have h := istage_fwd N.I (bid Γ T) (Equiv.refl Γ) N.M0 (fun _ => 1) (fun g => (N.M0 g).Φ 1) + (fun d => by rw [N.z0, Matrix.mul_one]) (fun d => by rw [N.z0, Matrix.mul_one]) + (cOne Γ Sl α) (cOne Γ C α) (cOne Γ C α) + have e : bOut N.I.tv (bid Γ T) N.M0 + (fun g (q : Sl) => (N.M0 ((Equiv.refl Γ).symm g)).Φ 1 * cOne Γ Sl α g q) (cOne Γ C α) + = mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α) := by + funext r + rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩) + · exact N.x01 g t + · exact N.y01 g t + · show (N.M0 g).Φ 1 * 1 = (N.M0 g).Φ 1 + rw [Matrix.mul_one] + · rfl + have h' : XRoute (bIn N.I.tv (bid Γ T) N.M0 (cOne Γ Sl α) (cOne Γ C α)) + (mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α)) GF (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + rw [← e] + exact h + exact blift1 N.xi0 N.yi0 h' + +/-- stage 1: backward on pair 1; the helper set `g` serves the invocation `s1⁻¹ g`. -/ +lemma Bridge2.stage1 : + XRoute N.S1 N.S2 (lift1 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + have h := istage_bwd N.I N.r1 N.s1 N.M1 N.N1 N.W1 N.hN1 N.hW1 N.Z1 (cOne Γ C α) (cOne Γ C α) + have e : bOut N.I.tv N.r1 N.M1 (fun g q => N.W1 (N.s1.symm g) * N.Z1 g q) (cOne Γ C α) + = mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α) := by + funext r + rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩) + · exact N.x12 g t + · exact N.y12 g t + · rfl + · rfl + have h' : XRoute (bIn N.I.tv N.r1 N.M1 N.Z1 (cOne Γ C α)) + (mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + rw [← e] + exact h + exact blift1 N.xi0 N.yi0 h' + +/-- stage 2: forward on pair 1. -/ +lemma Bridge2.stage2 : + XRoute N.S3 N.S4 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := + blift1 N.xi2 N.yi2 + (istage_fwd N.I N.r2 N.s2 N.M2 N.N2 N.W2 N.hN2 N.hW2 N.Z2 (cOne Γ C α) (cOne Γ C α)) + +/-- stage 3: backward on pair 2. -/ +lemma Bridge2.stage3 : + XRoute N.S5 N.S6 (lift2 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + have h := istage_bwd N.I N.r3 N.s3 N.M3 N.N3 N.W3 N.hN3 N.hW3 N.Z3 (cOne Γ C α) (cOne Γ C α) + have e : bOut N.I.tv N.r3 N.M3 (fun g q => N.W3 (N.s3.symm g) * N.Z3 g q) (cOne Γ C α) + = mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α) := by + funext r + rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩) + · exact N.x34 g t + · exact N.y34 g t + · rfl + · rfl + have h' : XRoute (bIn N.I.tv N.r3 N.M3 N.Z3 (cOne Γ C α)) + (mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + rw [← e] + exact h + exact blift2 N.xo2 N.yo2 h' + +/-- stage 4: forward on pair 2. -/ +lemma Bridge2.stage4 : + XRoute N.S6 N.S7 (lift2 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := + blift2 N.xo2 N.yo2 + (istage_fwd N.I N.r4 N.s4 N.M4 N.N4 N.W4 N.hN4 N.hW4 N.Z4 (cOne Γ C α) (cOne Γ C α)) + +/-- first idle climb of pair 2 (stages 0 and 1 in ONE block per role), to the frames of +pair 1 after stage 1. -/ +lemma Bridge2.climbA : + XRoute N.S2 N.S3 id (fun φ => (Fintype.card Γ : ℝ) * + ((Fintype.card T : ℝ) * (N.caX φ + N.caY φ))) := by + have H1 := XRoute.reach_all N.S2 N.S3 + (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ => N.caX) + (fun _ _ => N.caY) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact XReach.refl _ + · exact XReach.refl _ + · show XReach ((N.M0 g).Φ (tt (N.I.tv t))) ((N.M2 (N.r2 t g)).Φ (tt (N.I.tv t))) N.caX + rw [← N.x12 g t] + exact N.aX2 g t + · show XReach ((N.M0 g).Φ 0) ((N.M2 (N.r2 t g)).Φ 0) N.caY + rw [← N.y12 g t] + exact N.aY2 g t + · exact XReach.refl _ + · exact XReach.refl _) + refine H1.cast (fun φ => ?_) + simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)] + rw [sum_mkB] + simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add] + ring + +/-- second idle climb of pair 2: the window of stage 2, ONE block of rank `card H - 1` per +role, to the frames of pair 1 after stage 2. -/ +lemma Bridge2.climbB : + XRoute N.S4 N.S5 id (fun φ => (Fintype.card Γ : ℝ) * + ((Fintype.card T : ℝ) * (2 * bcost φ (Fintype.card H - 1)))) := by + have H1 := XRoute.reach_all N.S4 N.S5 + (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ φ => bcost φ (Fintype.card H - 1)) + (fun _ _ φ => bcost φ (Fintype.card H - 1)) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact XReach.refl _ + · exact XReach.refl _ + · show XReach ((N.M2 (N.r2 t g)).Φ (tt (N.I.tv t))) ((N.M3 (N.r3 t g)).Φ (tt (N.I.tv t))) + (fun φ => bcost φ (Fintype.card H - 1)) + rw [← N.x23 g t] + exact N.I.climbX (N.M2 (N.r2 t g)) t + · show XReach ((N.M2 (N.r2 t g)).Φ 0) ((N.M3 (N.r3 t g)).Φ 0) + (fun φ => bcost φ (Fintype.card H - 1)) + rw [← N.y23 g t] + exact N.I.climbY (N.M2 (N.r2 t g)) t + · exact XReach.refl _ + · exact XReach.refl _) + refine H1.cast (fun φ => ?_) + simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)] + rw [sum_mkB] + simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add] + ring + +/-- the final climbs: pair 1 from its frames after stage 2 (ONE block per role for the +stages 3, 4 and the rest), pair 2 from its frames after stage 4; the helper slots are +already at the kernel. -/ +lemma Bridge2.fin : + XRoute N.S7 N.S8 id (fun φ => (Fintype.card Γ : ℝ) * + ((Fintype.card T : ℝ) * ((N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ)))) := by + have H1 := XRoute.reach_all N.S7 N.S8 + (mkB (fun _ _ => N.cX1) (fun _ _ => N.cY1) (fun _ _ => N.cX2) (fun _ _ => N.cY2) + (fun _ _ _ => 0) (fun _ _ _ => 0)) (by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact N.fX1 g t + · exact N.fY1 g t + · exact N.fX2 g t + · exact N.fY2 g t + · show XReach (N.W4 (N.s4.symm g) * (N.W3 (N.s3.symm g) * (N.W2 (N.s2.symm g) * + (N.W1 (N.s1.symm g) * (N.M0 g).Φ 1)))) (kernel α) (fun _ => 0) + rw [N.sfin g] + exact XReach.refl _ + · exact XReach.refl _) + refine H1.cast (fun φ => ?_) + simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)] + rw [sum_mkB] + simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add] + ring + +end Net +end +end BR +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.BR.Bridge2.stage0 +#print axioms OAI.PowerSaving.BR.Bridge2.stage1 +#print axioms OAI.PowerSaving.BR.Bridge2.stage2 +#print axioms OAI.PowerSaving.BR.Bridge2.stage3 +#print axioms OAI.PowerSaving.BR.Bridge2.stage4 +#print axioms OAI.PowerSaving.BR.Bridge2.climbA +#print axioms OAI.PowerSaving.BR.Bridge2.climbB +#print axioms OAI.PowerSaving.BR.Bridge2.fin diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Carrier__Live.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Carrier__Live.lean new file mode 100644 index 0000000000..b8306527d6 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Carrier__Live.lean @@ -0,0 +1,232 @@ +import Work.Carrier.Pack +import Work.Carrier.Ring + +/-! +# (key: carrier-thm) The live roles of an invocation, and the four dirt gates as matrices + +`L3 T Sl = (T ⊕ T) ⊕ Sl`: the bank roles `x` (`lX`), `y` (`lY`) and the helper slots (`lS`), all +on the same footing (no scratch copies). `PX`, `PY`: the coordinate projectors. + +For the matrix `M` of all gates of the main phase and its inverse `Mi`: + + g1 M = 1 - PY * M * PS `y -= K s` g1i M = 1 + PY * M * PS (its inverse) + g8 Mi = PX + PY + PS * Mi slot rows of `M⁻¹` h8 M = PX + PY + PS * M (its inverse) + +(`PS = 1 - PX - PY`). This file: unit rows / columns of these gates (hence between which roles +their entries can be nonzero), the scatter with the copies eliminated (`scatM`), and the scalar +maps of `1 + PY * M * PX` and of the transpose of `1 - PY * M * PX`. + +No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace CR +open Binary Matrix Finset RAM SS CB BR +noncomputable section + +section Diag +variable {ρ : Type*} [Fintype ρ] [DecidableEq ρ] + +lemma col_zero_of {d : ρ → ℚ} {G : Matrix ρ ρ ℚ} + (h : G * Matrix.diagonal d = Matrix.diagonal d) {i j : ρ} (hj : d j = 1) (hij : i ≠ j) : + G i j = 0 := by + have e := congrFun (congrFun h i) j + rw [Matrix.mul_diagonal, Matrix.diagonal_apply_ne _ hij, hj, mul_one] at e + exact e + +lemma row_zero_of {d : ρ → ℚ} {G : Matrix ρ ρ ℚ} + (h : Matrix.diagonal d * G = Matrix.diagonal d) {i j : ρ} (hi : d i = 1) (hij : i ≠ j) : + G i j = 0 := by + have e := congrFun (congrFun h i) j + rw [Matrix.diagonal_mul, Matrix.diagonal_apply_ne _ hij, hi, one_mul] at e + exact e + +lemma diag_mul_of_rows {d : ρ → ℚ} {G : Matrix ρ ρ ℚ} + (h : ∀ i j, d i ≠ 0 → G i j = if i = j then 1 else 0) : + Matrix.diagonal d * G = Matrix.diagonal d := by + ext i j + rw [Matrix.diagonal_mul] + by_cases hd : d i = 0 + · rw [hd, zero_mul] + by_cases hij : i = j + · subst hij; rw [Matrix.diagonal_apply_eq, hd] + · rw [Matrix.diagonal_apply_ne _ hij] + · rw [h i j hd] + by_cases hij : i = j + · subst hij; rw [if_pos rfl, mul_one, Matrix.diagonal_apply_eq] + · rw [if_neg hij, mul_zero, Matrix.diagonal_apply_ne _ hij] + +lemma mul_diag_of_cols {d : ρ → ℚ} {G : Matrix ρ ρ ℚ} + (h : ∀ i j, d j ≠ 0 → G i j = if i = j then 1 else 0) : + G * Matrix.diagonal d = Matrix.diagonal d := by + ext i j + rw [Matrix.mul_diagonal] + by_cases hd : d j = 0 + · rw [hd, mul_zero] + by_cases hij : i = j + · subst hij; rw [Matrix.diagonal_apply_eq, hd] + · rw [Matrix.diagonal_apply_ne _ hij] + · rw [h i j hd] + by_cases hij : i = j + · subst hij; rw [if_pos rfl, one_mul, Matrix.diagonal_apply_eq] + · rw [if_neg hij, zero_mul, Matrix.diagonal_apply_ne _ hij] + +/-- scalar map of a diagonal gate matrix. -/ +lemma act_diag (d : ρ → ℚ) (u : ρ → ℂ) : + actPoint (Matrix.diagonal d) u = fun i => ((d i : ℚ) : ℂ) * u i := by + funext i + show ∑ j, ((Matrix.diagonal d i j : ℚ) : ℂ) * u j = ((d i : ℚ) : ℂ) * u i + rw [Finset.sum_eq_single i] + · rw [Matrix.diagonal_apply_eq] + · intro j _ hj + rw [Matrix.diagonal_apply_ne _ (Ne.symm hj), Rat.cast_zero, zero_mul] + · intro h + exact absurd (Finset.mem_univ i) h + +lemma actPoint_mul (A B : Matrix ρ ρ ℚ) (u : ρ → ℂ) : + actPoint (A * B) u = actPoint A (actPoint B u) := + ap_mul A B u + +end Diag + +section Live +variable {T Sl : Type} [Fintype T] [DecidableEq T] [Fintype Sl] [DecidableEq Sl] + +/-- live roles of one invocation: `x`, `y`, helper slots. -/ +abbrev L3 (T Sl : Type) := (T ⊕ T) ⊕ Sl +abbrev lX (t : T) : L3 T Sl := Sum.inl (Sum.inl t) +abbrev lY (t : T) : L3 T Sl := Sum.inl (Sum.inr t) +abbrev lS (q : Sl) : L3 T Sl := Sum.inr q + +/-- indicator of the x roles. -/ +def iX : L3 T Sl → ℚ := Sum.elim (Sum.elim (fun _ => 1) (fun _ => 0)) (fun _ => 0) +/-- indicator of the y roles. -/ +def iY : L3 T Sl → ℚ := Sum.elim (Sum.elim (fun _ => 0) (fun _ => 1)) (fun _ => 0) + +def PX : Matrix (L3 T Sl) (L3 T Sl) ℚ := Matrix.diagonal iX +def PY : Matrix (L3 T Sl) (L3 T Sl) ℚ := Matrix.diagonal iY + +lemma iX_cases (i : L3 T Sl) : iX i = 0 ∨ iX i = 1 := by + rcases i with ((t|t)|q) + · exact Or.inr rfl + · exact Or.inl rfl + · exact Or.inl rfl + +lemma iY_cases (i : L3 T Sl) : iY i = 0 ∨ iY i = 1 := by + rcases i with ((t|t)|q) + · exact Or.inl rfl + · exact Or.inr rfl + · exact Or.inl rfl + +lemma PX_sq : (PX : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PX = PX := by + unfold PX + have e : (fun i : L3 T Sl => iX i * iX i) = iX := by + funext i + rcases i with ((t|t)|q) <;> simp [iX] + rw [Matrix.diagonal_mul_diagonal, e] + +lemma PY_sq : (PY : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PY = PY := by + unfold PY + have e : (fun i : L3 T Sl => iY i * iY i) = iY := by + funext i + rcases i with ((t|t)|q) <;> simp [iY] + rw [Matrix.diagonal_mul_diagonal, e] + +lemma PX_PY : (PX : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PY = 0 := by + unfold PX PY + rw [Matrix.diagonal_mul_diagonal] + have e : (fun i : L3 T Sl => iX i * iY i) = fun _ => 0 := by + funext i + rcases i with ((t|t)|q) <;> simp [iX, iY] + rw [e] + exact Matrix.diagonal_zero + +lemma PY_PX : (PY : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PX = 0 := by + unfold PX PY + rw [Matrix.diagonal_mul_diagonal] + have e : (fun i : L3 T Sl => iY i * iX i) = fun _ => 0 := by + funext i + rcases i with ((t|t)|q) <;> simp [iX, iY] + rw [e] + exact Matrix.diagonal_zero + +lemma PX_T : (PX : Matrix (L3 T Sl) (L3 T Sl) ℚ)ᵀ = PX := Matrix.diagonal_transpose _ +lemma PY_T : (PY : Matrix (L3 T Sl) (L3 T Sl) ℚ)ᵀ = PY := Matrix.diagonal_transpose _ + +/-- "no gate into an x role": the x rows are unit rows. -/ +lemma PX_mul_of {M : Matrix (L3 T Sl) (L3 T Sl) ℚ} + (hx : ∀ t j, M (lX t) j = if lX t = j then 1 else 0) : PX * M = PX := by + unfold PX + apply diag_mul_of_rows + intro i j hi + rcases i with ((t|t)|q) + · exact hx t j + · exact absurd rfl hi + · exact absurd rfl hi + +/-- "no gate reads a y role": the y columns are unit columns. -/ +lemma mul_PY_of {M : Matrix (L3 T Sl) (L3 T Sl) ℚ} + (hy : ∀ i t, M i (lY t) = if i = lY t then 1 else 0) : M * PY = PY := by + unfold PY + apply mul_diag_of_cols + intro i j hj + rcases j with ((t|t)|q) + · exact absurd rfl hj + · exact hy i t + · exact absurd rfl hj + +/-! ### the four dirt gates -/ + +def g1 (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ := + 1 - PY * M * (1 - PX - PY) +def g1i (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ := + 1 + PY * M * (1 - PX - PY) +def g8 (Mi : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ := + PX + PY + (1 - PX - PY) * Mi +def h8 (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ := + PX + PY + (1 - PX - PY) * M + +/-- a gate with unit x rows and unit x columns joins two different roles only if neither is an +x role. -/ +lemma offdiag_x {G : Matrix (L3 T Sl) (L3 T Sl) ℚ} (h1 : G * PX = PX) (h2 : PX * G = PX) + {i j : L3 T Sl} (hij : G i j ≠ 0) (e : i ≠ j) : iX i = 0 ∧ iX j = 0 := by + constructor + · by_contra h + exact hij (row_zero_of h2 ((iX_cases i).resolve_left h) e) + · by_contra h + exact hij (col_zero_of h1 ((iX_cases j).resolve_left h) e) + +/-- a gate with unit x rows, unit y rows and unit y columns: a nonzero entry off the diagonal +is in a slot row and not in a y column. -/ +lemma offdiag_s {G : Matrix (L3 T Sl) (L3 T Sl) ℚ} (hx : PX * G = PX) (hy : PY * G = PY) + (hyc : G * PY = PY) {i j : L3 T Sl} (hij : G i j ≠ 0) (e : i ≠ j) : + iX i = 0 ∧ iY i = 0 ∧ iY j = 0 := by + refine ⟨?_, ?_, ?_⟩ + · by_contra h + exact hij (row_zero_of hx ((iX_cases i).resolve_left h) e) + · by_contra h + exact hij (row_zero_of hy ((iY_cases i).resolve_left h) e) + · by_contra h + exact hij (col_zero_of hyc ((iY_cases j).resolve_left h) e) + +lemma g1_x (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : g1 M * PX = PX ∧ PX * g1 M = PX := by + unfold g1 + constructor + · rw [sub_mul, one_mul, side_n_right (PX_sq (T := T) (Sl := Sl)) PY_PX, sub_zero] + · rw [mul_sub, mul_one, side_n_left (PX_PY (T := T) (Sl := Sl)), sub_zero] + +lemma g1i_x (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : g1i M * PX = PX ∧ PX * g1i M = PX := by + unfold g1i + constructor + · rw [add_mul, one_mul, side_n_right (PX_sq (T := T) (Sl := Sl)) PY_PX, add_zero] + · rw [mul_add, mul_one, side_n_left (PX_PY (T := T) (Sl := Sl)), add_zero] + +end Live +end +end CR +end PowerSaving +end OAI diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Carrier__Phases.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Carrier__Phases.lean new file mode 100644 index 0000000000..305da15b5d --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Carrier__Phases.lean @@ -0,0 +1,192 @@ +import Work.Carrier.Glue + +/-! +# (key: carrier-thm) A carrier circuit given by its phases: hypotheses and scalar identities + +`Phased H T Sl C` is the hypothesis structure of the carrier invocation theorem in the general +scope (any ordered mix of gates `(x, slot)`, `(slot, slot)`, `(x, y)`, `(slot, y)`): + +* phase A (before the scatter), phase B (after it) and the final climbs F are exact block routes + on the LIVE roles `L3 T Sl` (`x`, `y`, slots: all on the same footing), for every block frame + map, with the gate products `MA`, `MB` (and the transposed inverses for the backward word); +* the scatter goes through the scratch copies: `c += Cc s` at the labels `cen`, the copies come + down, `y += Jr c` at 0; +* `hx`: no gate INTO an x role, `hy`: no gate READS a y role, `hid`: the y roles receive exactly + the x roles — three facts about the ONE matrix `Mt = MB * scatM (Jr * Cc) * MA`. + +This file: `Mt`, its inverse `Mti`, and the scalar identities of the forward and of the backward +word (`fwd_scalar`, `bwd_scalar`): with the two dirt gates `g1`, `g8` (resp. the transposes of +`g1i`, `h8`) the word does `y += x` (resp. `x -= y`) for ARBITRARY slot contents. + +No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace CR +open Binary Matrix Finset RAM SS CB BR +noncomputable section + +/-- labels of the live roles at the START of an invocation. -/ +def lab3 {H T Sl : Type} (tv : T → Space H) : L3 T Sl → Lbl H := + Sum.elim (Sum.elim (fun t => lineL (tv t)) (fun _ => zeroL)) (fun _ => zeroL) + +/-- **The hypotheses of the carrier invocation theorem, by phases.** See the file header. -/ +structure Phased (H T Sl C : Type) [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] where + tv : T → Space H + kx : ∀ t, Climb (lineL (tv t)) fullL + ky : ∀ t, Climb zeroL (perpL (tv t)) + cen : C → Lbl H + kc : ∀ k, Climb zeroL (cen k) + Cc : Matrix C Sl ℚ + Jr : Matrix T C ℚ + labA : L3 T Sl → Lbl H + labB : L3 T Sl → Lbl H + labF : L3 T Sl → Lbl H + hAy : ∀ t, (labA (lY t)).P = 0 + hAc : ∀ k q, Cc k q ≠ 0 → (cen k).P = (labA (lS q)).P + hFx : ∀ t, (labF (lX t)).P = 1 + hFy : ∀ t, (labF (lY t)).P = 1 + tt (tv t) + hFs : ∀ q, (labF (lS q)).P = 1 + MA : Matrix (L3 T Sl) (L3 T Sl) ℚ + MAi : Matrix (L3 T Sl) (L3 T Sl) ℚ + MB : Matrix (L3 T Sl) (L3 T Sl) ℚ + MBi : Matrix (L3 T Sl) (L3 T Sl) ℚ + hAi : MAi * MA = 1 + hBi : MBi * MB = 1 + hx : ∀ t j, (MB * scatM (Jr * Cc) * MA) (lX t) j = if lX t = j then 1 else 0 + hy : ∀ i t, (MB * scatM (Jr * Cc) * MA) i (lY t) = if i = lY t then 1 else 0 + hid : ∀ S t, (MB * scatM (Jr * Cc) * MA) (lY S) (lX t) = if S = t then 1 else 0 + rA : List ℕ + rB : List ℕ + rF : List ℕ + pA : ∀ {α : Type} [Fintype α] [DecidableEq α] (Xm : XMap H α), + XRoute (fun r => Xm.Φ (lab3 tv r).P) (fun r => Xm.Φ (labA r).P) (actPoint MA) + (fun φ => rcost φ rA) ∧ + XRoute (fun r => Xm.Φ (lab3 tv r).P) (fun r => Xm.Φ (labA r).P) (actPoint MAiᵀ) + (fun φ => rcost φ rA) + pB : ∀ {α : Type} [Fintype α] [DecidableEq α] (Xm : XMap H α), + XRoute (fun r => Xm.Φ (labA r).P) (fun r => Xm.Φ (labB r).P) (actPoint MB) + (fun φ => rcost φ rB) ∧ + XRoute (fun r => Xm.Φ (labA r).P) (fun r => Xm.Φ (labB r).P) (actPoint MBiᵀ) + (fun φ => rcost φ rB) + pF : ∀ {α : Type} [Fintype α] [DecidableEq α] (Xm : XMap H α), + XRoute (fun r => Xm.Φ (labB r).P) (fun r => Xm.Φ (labF r).P) id (fun φ => rcost φ rF) + +section Alg +variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] + +namespace Phased +variable (K : Phased H T Sl C) + +/-- the product of ALL gates of the main phase (scatter with the copies eliminated). -/ +def Mt : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MB * scatM (K.Jr * K.Cc) * K.MA +/-- its inverse. -/ +def Mti : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MAi * scatMi (K.Jr * K.Cc) * K.MBi + +/-- price of one invocation: the blocks of the three phases and one block per copy. -/ +def cost (φ : ℕ → ℝ) : ℝ := + rcost φ K.rA + (∑ k, bcost φ (K.cen k).d) + rcost φ K.rB + rcost φ K.rF + +lemma im : K.Mti * K.Mt = 1 := by + unfold Mti Mt + have hB : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ, K.MBi * (K.MB * X) = X := fun X => by + rw [← Matrix.mul_assoc, K.hBi, Matrix.one_mul] + have hS : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ, + scatMi (K.Jr * K.Cc) * (scatM (K.Jr * K.Cc) * X) = X := fun X => by + rw [← Matrix.mul_assoc, scat_inv, Matrix.one_mul] + rw [Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc, hB, hS, K.hAi] + +lemma mi : K.Mt * K.Mti = 1 := mul_eq_one_comm.mp K.im + +lemma hxm : PX * K.Mt = PX := PX_mul_of (M := K.Mt) K.hx +lemma hmy : K.Mt * PY = PY := mul_PY_of (M := K.Mt) K.hy +lemma hiy : K.Mti * PY = PY := dirt_iy K.hmy K.im + +/-- **the dirt rule, forward**, for the matrices of the circuit. -/ +lemma hF : g8 K.Mti * K.Mt * g1 K.Mt = 1 + PY * K.Mt * PX := by + unfold g8 g1 + exact dirt_fwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi + +/-- **the dirt rule, backward.** -/ +lemma hB : g1i K.Mt * K.Mti * h8 K.Mt = 1 - PY * K.Mt * PX := by + unfold g1i h8 + exact dirt_bwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi + +/-- live part of the forward word before the scatter: `y -= K s`, then phase A. -/ +def f1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MA (actPoint (g1 K.Mt) u) +/-- after the scatter: phase B, then the slot rows of the inverse. -/ +def f2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (g8 K.Mti) (actPoint K.MB u) +/-- the same for the backward word (transposed inverses). -/ +def b1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u) +def b2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ u) + +lemma fwd_live (u : L3 T Sl → ℂ) : + K.f2 (actPoint (scatM (K.Jr * K.Cc)) (K.f1 u)) = actPoint (1 + PY * K.Mt * PX) u := by + rw [← K.hF] + show actPoint (g8 K.Mti) (actPoint K.MB (actPoint (scatM (K.Jr * K.Cc)) + (actPoint K.MA (actPoint (g1 K.Mt) u)))) + = actPoint (g8 K.Mti * (K.MB * scatM (K.Jr * K.Cc) * K.MA) * g1 K.Mt) u + rw [actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul] + +lemma bwd_live (u : L3 T Sl → ℂ) : + K.b2 (actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 u)) + = actPoint ((1 - PY * K.Mt * PX)ᵀ) u := by + rw [← K.hB] + show actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ (actPoint (scatMi (K.Jr * K.Cc))ᵀ + (actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u)))) + = actPoint ((g1i K.Mt * (K.MAi * scatMi (K.Jr * K.Cc) * K.MBi) * h8 K.Mt)ᵀ) u + rw [Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul, + actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul] + +lemma liveAct_glue (φ : (L3 T Sl → ℂ) → (L3 T Sl → ℂ)) (u : L3 T Sl → ℂ) (c : C → ℂ) : + liveAct φ (glue u c) = glue (φ u) c := + congrArg (fun v => glue (φ v) c) (glue_up u c) + +/-- **Scalar identity of the forward word**: `y += x`, slots restored, for ARBITRARY slot +contents (the copies are erased before and after). -/ +theorem fwd_scalar (z : Box T Sl C → ℂ) : + eraseC (liveAct K.f2 (gateS bY bC K.Jr (gateS bC bS K.Cc (liveAct K.f1 (eraseC z))))) + = shearF z := by + have h1 : liveAct K.f1 (eraseC z) = glue (K.f1 (z ∘ upE)) (fun _ => 0) := by + rw [erase_glue z, liveAct_glue] + have hs : gateS bY bC K.Jr (gateS bC bS K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0))) ∘ upE + = actPoint (scatM (K.Jr * K.Cc)) (K.f1 (z ∘ upE)) := by + have h := scat_glue K.Jr K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0)) rfl + rw [glue_up] at h + exact h + rw [h1] + unfold liveAct + rw [erase_of_glue, hs, K.fwd_live, act_shear K.Mt K.hid] + exact glue_shearF z + +/-- **Scalar identity of the backward word**: `x -= y`, slots restored. -/ +theorem bwd_scalar (z : Box T Sl C → ℂ) : + eraseC (liveAct K.b2 (gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (liveAct K.b1 (eraseC z))))) + = shearB z := by + have h1 : liveAct K.b1 (eraseC z) = glue (K.b1 (z ∘ upE)) (fun _ => 0) := by + rw [erase_glue z, liveAct_glue] + have hs : gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (glue (K.b1 (z ∘ upE)) (fun _ => 0))) ∘ upE + = actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 (z ∘ upE)) := by + have h := scat_glueT K.Jr K.Cc (glue (K.b1 (z ∘ upE)) (fun _ => 0)) rfl + rw [glue_up] at h + exact h + rw [h1] + unfold liveAct + rw [erase_of_glue, hs, K.bwd_live, act_shearT K.Mt K.hid] + exact glue_shearB z + +end Phased +end Alg +end +end CR +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.CR.Phased.fwd_scalar +#print axioms OAI.PowerSaving.CR.Phased.bwd_scalar diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Fourier__Main.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Fourier__Main.lean new file mode 100644 index 0000000000..9aa06b5bf2 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__Fourier__Main.lean @@ -0,0 +1,176 @@ +import Work.Fourier.Asymptotics +import OAI.Computability.FourierTransform.RegisterBounds +import Work.Fourier.Goal + +/- +Modified copy of lean/OAI/Computability/FourierTransform/Main.lean of github.com/openai/math +(commit fd4aeeb, Apache-2.0), generated by `tools/fourier/mkchain.py`. OpenAI's file itself is +unmodified in this repository and still imported. Changed for this repository, all mechanically: + * every declaration of this file (and of the other 10 copied files) has the suffix `Z` + on its name, so that it does not collide with OpenAI's original; + * OpenAI's exponent `alpha = 1 - 2/10^11` and envelope `hills` are replaced by + `alphaZ = 1 - 7474547/10^10` and `hillsZ` (Work/Fourier/Seam.lean), and the single use of + OpenAI's `hills_program` by `hillsZ_program`; + * `decimalExponent`, `decimalTime`, `TimeBounds`, `DFTGoal`, `ConvGoal` are replaced by their + `Z` versions from Work/Fourier/Goal.lean (exponent `1 - 7474546/10^10`); + * imports point to the copies. + * literal `let d : ℝ := 2/(10:ℝ)^11` -> `let d : ℝ := 7474547/(10:ℝ)^10` (alpha_thetaZ: the local constant d must equal 1 - alphaZ) + * `theorem transform_mainZ : DFTGoalZ :=`: docstring added + * `theorem convolution_mainZ : ConvGoalZ :=`: docstring added +Comments and docstrings below are OpenAI's and describe OpenAI's network, not ours. +-/ + +namespace OAI + +/-! +Exact uniform Fourier transforms and convolution. + +The construction formalizes the exact arithmetic algorithms of +*An explicit power saving for the exact discrete Fourier transform* (OpenAI). +The finite phase network originates in the complex construction of +*Integer multiplication below n log n*. + +`transform_mainZ` and `convolution_mainZ` give fixed finite RAM programs, including +preparation and index work, polynomial storage bounds, and a canonical root whose +order is computed from the input length. Each includes the paper exponent, decimal +power saving, and little-o bound. The `_order` variants also specify the root order. + +The implementation uses a smaller concrete tensor network (32 binary coordinates +per factor, 4060 triples, and a `2^50` role envelope) with a stronger saving that +implies the stated paper constants. The general rational-algebra extraction, +symbolic certificate search, circuit printer, and bounded-printer uniformization +from the synthesis appendix are outside this formalization. +-/ +namespace PowerSaving +open RAM Bench Selection Filter Asymptotics Real +noncomputable section +namespace Skies + +lemma multiplier_rangeZ : 1 < paperMultiplier ∧ paperMultiplier < paperM := by + norm_num [paperMultiplier,paperM,paperDelta,paperW] + +lemma theta_ltZ : theta < 1 := by + have h := multiplier_rangeZ + have hg : 0 by rw [speedZ,Real.rpow_one]; rfl + rw [h] at H; exact H + +lemma speed_paperZ : speedZ theta =O[atTop] paperTime := by + rw [isBigO_iff] + refine ⟨1,?_⟩ + filter_upwards [TtopZ.eventually_ge_atTop 1,QtopZ.eventually_ge_atTop 1] with n hn hh + have he : 0 (lunchZ a b N n:ℝ)) =O[atTop] decimalTimeZ := hd.isBigO + exact ⟨hp.isBigO.trans speed_paperZ,ht,ht.trans_isLittleO decimal_littleZ⟩ + +end Skies + +/-! Main results. Provide formula for the order besides the existential specification. -/ +theorem transform_main_orderZ : ∃ order solve W, + DFTProgram order solve W ∧ TimeBoundsZ W ∧ + ∀ n, 0 ⟨le_rfl,by dsimp; omega⟩) + a b,fun n hn => ⟨ho n hn,(ho n hn).symm ▸ (receipt_bound n hn).2⟩⟩ + +theorem convolution_main_orderZ : ∃ order solve W, + ConvProgram order solve W ∧ TimeBoundsZ W ∧ + ∀ n, 0 + ⟨ho n hn,(ho n hn).symm ▸ (receipt_bound _ (stretch_posZ hn)).2⟩⟩ + +/-- **The discrete Fourier transform of every length in `O(n (log n)^(1 - 7474546/10^10))`.** +OpenAI's `transform_main` (one fixed program, OpenAI's own `DFTProgram`, `paperTime` and `nlogn`) +with the decimal exponent `1 - 1/10^13` replaced by `decimalExponentZ = 1 - 7474546/10^10` +(Work/Fourier/Goal.lean). Sibling of the Walsh-Hadamard theorem `wht_main_block_B2Gp193x`: both +follow from one kernel program, `RAM.hillsZ_program` (Work/Fourier/Seam.lean). -/ +theorem transform_mainZ : DFTGoalZ := + let ⟨o,s,W,h,g,_⟩ := transform_main_orderZ + ⟨o,s,W,h,g⟩ + +/-- **Convolution of every length with the same bounds** (OpenAI's `convolution_main`, same change). -/ +theorem convolution_mainZ : ConvGoalZ := + let ⟨o,s,W,h,g,_⟩ := convolution_main_orderZ + ⟨o,s,W,h,g⟩ + +lemma matrix_definitionZ (n : ℕ) (x : Vec n) : dft n x = Matrix.mulVec (dftMatrix n) x := + rfl + +end +end PowerSaving + +end OAI diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__Geom.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__Geom.lean new file mode 100644 index 0000000000..eb4490af8b --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__Geom.lean @@ -0,0 +1,210 @@ +import Work.GCert.Chain.NetCert +import Work.GCert.Chain.WinB +import Work.BridgeGeom.B2 + +/-! +# (key: gx-chain) THE NETWORK LEMMA (spec statement 6): the bridged word B_2 delivers a `GCert` + +GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/BridgeGeom/Net.lean (section Net): +the geometry of `BG.geomBridge2`, word for word, with the windows `gwinB` (`.Xm = BG.fmapB`). + +`bridge2_gcert I hH`: for every generalised invocation package, a `GCert` on +`card Γ * (4 card T + card Sl)` live roles, label dimension `5h`, unit price +`5 * I.cost φ + card T * BG.bankPrice h φ` (unchanged). No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB RF BR GF BG +noncomputable section + +section Net +variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] +variable (I : GInv H T Sl C) (base : T → OBase H) (l : H) + (hbase : ∀ t, I.tv t = (base t).v l) (hH : 1 ≤ Fintype.card H) +include hbase + +lemma ggeom_x01 (g : Orth (L5 H)) (t : T) : + PhiB g ∅ 1 = PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) (tt (I.tv t)) := by + rw [hbase t] + exact bankT_x0 (base t) l (0 : Fin 4) g + +lemma ggeom_y01 (g : Orth (L5 H)) (t : T) : + PhiB g ∅ (1 + tt (I.tv t)) = PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) 0 := by + rw [hbase t] + exact bankT_y0 (base t) l (0 : Fin 4) g + +/-- consecutive stages `c ∈ bs`, `c' ∉ bs`: `X` leaves where it enters. -/ +lemma ggeom_xx (bs : Finset (Fin 4)) (c c' : Fin 4) (hc : c ∈ bs) (hc' : c' ∉ bs) + (g : Orth (L5 H)) (t : T) : + PhiB (RrB (base t) l c g) (qS bs l) 1 + = PhiB (RrB (base t) l c' g) (qS (insert c' bs) l) (tt (I.tv t)) := by + rw [hbase t] + exact bankT_xx (base t) l bs c c' hc hc' g + +/-- consecutive stages: `Y` leaves where it enters. -/ +lemma ggeom_yy (bs : Finset (Fin 4)) (c c' : Fin 4) (hc : c ∈ bs) (hc' : c' ∉ bs) + (g : Orth (L5 H)) (t : T) : + PhiB (RrB (base t) l c g) (qS bs l) (1 + tt (I.tv t)) + = PhiB (RrB (base t) l c' g) (qS (insert c' bs) l) 0 := by + rw [hbase t] + exact bankT_yy (base t) l bs c c' hc hc' g + +/-- terminal identity of the bank pairs of the class `(g,t)`. -/ +lemma ggeom_term5 (g : Orth (L5 H)) (t : T) : + kernel (L5 H) * PhiB g ∅ (tt (I.tv t)) + = shift ((BBB (base t) g).v (Sum.inl l)) + * frame (BBB (base t) g) (univ \ {Sum.inl l}) := by + rw [hbase t] + exact class_term (base t) l g + +include hH + +/-- pair 2 before the first gate layer, `X`: ONE block of rank `2h-2`. -/ +lemma ggeom_aX2 (g : Orth (L5 H)) (t : T) : + XReach (PhiB g ∅ (tt (I.tv t))) (PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) 1) + (fun φ => bcost φ (2 * Fintype.card H - 2)) := by + rw [hbase t] + exact idle_X0 (base t) l hB4 hH (0 : Fin 4) g _ (by omega) + +/-- pair 2 before the first gate layer, `Y`: ONE block of rank `2h-2`. -/ +lemma ggeom_aY2 (g : Orth (L5 H)) (t : T) : + XReach (PhiB g ∅ 0) (PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) (1 + tt (I.tv t))) + (fun φ => bcost φ (2 * Fintype.card H - 2)) := by + rw [hbase t] + exact idle_Y0 (base t) l hB4 hH (0 : Fin 4) g _ (by omega) + +/-- pair 1 at the end (idle in stages 3, 4, then the last four moves), `X`: ONE block `2h+2`. -/ +lemma ggeom_fX1 (g : Orth (L5 H)) (t : T) : + XReach (PhiB (RrB (base t) l (1 : Fin 4) g) (qS bs2 l) 1) (kernel (L5 H)) + (fun φ => bcost φ (2 * Fintype.card H + 2)) := + idle_Xend (base t) l hB4 hH bs2 (1 : Fin 4) (Finset.mem_insert_self _ _) g _ + (by rw [card_bs2, Fintype.card_fin]; omega) + +/-- pair 1 at the end, `Y`: ONE block of rank `2h+2`. -/ +lemma ggeom_fY1 (g : Orth (L5 H)) (t : T) : + XReach (PhiB (RrB (base t) l (1 : Fin 4) g) (qS bs2 l) (1 + tt (I.tv t))) + (frame (BBB (base t) g) (univ \ {Sum.inl l})) + (fun φ => bcost φ (2 * Fintype.card H + 2)) := by + rw [hbase t] + exact idle_Yend (base t) l hB4 hH bs2 (1 : Fin 4) (Finset.mem_insert_self _ _) g _ + (by rw [card_bs2, Fintype.card_fin]; omega) + +/-- pair 2 after stage 4, `X`: ONE block of rank `4`. -/ +lemma ggeom_fX2 (g : Orth (L5 H)) (t : T) : + XReach (PhiB (RrB (base t) l (3 : Fin 4) g) (qS bs4 l) 1) (kernel (L5 H)) + (fun φ => bcost φ 4) := + idle_Xend (base t) l hB4 hH bs4 (3 : Fin 4) (Finset.mem_insert_self _ _) g _ + (by rw [card_bs4, Fintype.card_fin]; omega) + +/-- pair 2 after stage 4, `Y`: ONE block of rank `4`. -/ +lemma ggeom_fY2 (g : Orth (L5 H)) (t : T) : + XReach (PhiB (RrB (base t) l (3 : Fin 4) g) (qS bs4 l) (1 + tt (I.tv t))) + (frame (BBB (base t) g) (univ \ {Sum.inl l})) (fun φ => bcost φ 4) := by + rw [hbase t] + exact idle_Yend (base t) l hB4 hH bs4 (3 : Fin 4) (Finset.mem_insert_self _ _) g _ + (by rw [card_bs4, Fintype.card_fin]; omega) + +/-- **The bridged word B_2 as a term of `BR.Bridge2`**: five stages on the label space +`H ⊕ (Fin 4 × H)`, for any invocation package whose lines are the vectors of index `l` of +orthonormal bases. -/ +def ggeomBridge2 : GBridge2 H T Sl C (L5 H) (Orth (L5 H)) where + I := I + G0 := fun g => gwinB hB4 hH g ∅ + G1 := fun d => gwinB hB4 hH d (qS bs1 l) + G2 := fun d => gwinB hB4 hH d (qS bs2 l) + G3 := fun d => gwinB hB4 hH d (qS bs3 l) + G4 := fun d => gwinB hB4 hH d (qS bs4 l) + r1 := fun t => RrB (base t) l (0 : Fin 4) + r2 := fun t => RrB (base t) l (1 : Fin 4) + r3 := fun t => RrB (base t) l (2 : Fin 4) + r4 := fun t => RrB (base t) l (3 : Fin 4) + s1 := (Orth.colPerm (tauB (0 : Fin 4))).symm + s2 := (Orth.colPerm (tauB (1 : Fin 4))).symm + s3 := (Orth.colPerm (tauB (2 : Fin 4))).symm + s4 := (Orth.colPerm (tauB (3 : Fin 4))).symm + N1 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs1 l)) * d.1ᵀ) + W1 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ) + N2 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs2 l)) * d.1ᵀ) + W2 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ) + N3 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs3 l)) * d.1ᵀ) + W3 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ) + N4 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs4 l)) * d.1ᵀ) + W4 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ) + u := fun g t => (BBB (base t) g).v (Sum.inl l) + YF := fun g t => frame (BBB (base t) g) (univ \ {Sum.inl l}) + hm := two_le_card_LB hB4 hH + caX := fun φ => bcost φ (2 * Fintype.card H - 2) + caY := fun φ => bcost φ (2 * Fintype.card H - 2) + cX1 := fun φ => bcost φ (2 * Fintype.card H + 2) + cY1 := fun φ => bcost φ (2 * Fintype.card H + 2) + cX2 := fun φ => bcost φ 4 + cY2 := fun φ => bcost φ 4 + inv0 := fun g t => ⟨unframeM (g.1 * Matrix.fromBlocks (tt (I.tv t)) 0 0 (diagI ∅) * g.1ᵀ), + frameM_unframeM _⟩ + z0 := fun g => PhiB_zero_empty g + hN1 := fun d => frameM_unframeM _ + hW1 := fun d => PhiB_one d (qS bs1 l) + hN2 := fun d => frameM_unframeM _ + hW2 := fun d => PhiB_one d (qS bs2 l) + hN3 := fun d => frameM_unframeM _ + hW3 := fun d => PhiB_one d (qS bs3 l) + hN4 := fun d => frameM_unframeM _ + hW4 := fun d => PhiB_one d (qS bs4 l) + x01 := fun g t => ggeom_x01 I base l hbase g t + y01 := fun g t => ggeom_y01 I base l hbase g t + x12 := fun g t => ggeom_xx I base l hbase bs1 0 1 (Finset.mem_singleton_self _) mem1 g t + y12 := fun g t => ggeom_yy I base l hbase bs1 0 1 (Finset.mem_singleton_self _) mem1 g t + x23 := fun g t => ggeom_xx I base l hbase bs2 1 2 (Finset.mem_insert_self _ _) mem2 g t + y23 := fun g t => ggeom_yy I base l hbase bs2 1 2 (Finset.mem_insert_self _ _) mem2 g t + x34 := fun g t => ggeom_xx I base l hbase bs3 2 3 (Finset.mem_insert_self _ _) mem3 g t + y34 := fun g t => ggeom_yy I base l hbase bs3 2 3 (Finset.mem_insert_self _ _) mem3 g t + sfin := fun g => helper_fin5 g + aX2 := fun g t => ggeom_aX2 I base l hbase hH g t + aY2 := fun g t => ggeom_aY2 I base l hbase hH g t + fX1 := fun g t => ggeom_fX1 I base l hbase hH g t + fY1 := fun g t => ggeom_fY1 I base l hbase hH g t + fX2 := fun g t => ggeom_fX2 I base l hbase hH g t + fY2 := fun g t => ggeom_fY2 I base l hbase hH g t + term := fun g t => ggeom_term5 I base l hbase g t + +end Net + +section End +variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] + +/-- an orthonormal basis through a unit vector with a zero coordinate, at any index. -/ +theorem base_ex (z : Space H) (hz : dot z z = 1) (hg : ∃ p, z p = 0) (l : H) : + ∃ A : OBase H, z = A.v l := by + obtain ⟨p, hp⟩ := hg + obtain ⟨A, k, hA⟩ := exists_base_one z hz p hp + refine ⟨reindex A (Equiv.swap k l), ?_⟩ + rw [← hA] + show A.v k = A.v (Equiv.swap k l l) + rw [Equiv.swap_apply_right] + +/-- **Generalised certificate of the bridged word B_2** for a generalised invocation package. -/ +theorem bridge2_gcert (I : GInv H T Sl C) (hH : 1 ≤ Fintype.card H) : + GCert (L5 H) (Sum.inl : BLive (Orth (L5 H)) T Sl → BRole (Orth (L5 H)) T Sl C) + (fun φ => (Fintype.card (Orth (L5 H)) : ℝ) + * (5 * I.cost φ + (Fintype.card T : ℝ) * bankPrice (Fintype.card H) φ)) := by + obtain ⟨l⟩ : Nonempty H := Fintype.card_pos_iff.mp (by omega) + have hb : ∀ t, ∃ A : OBase H, I.tv t = A.v l := + fun t => base_ex (I.tv t) (I.hunit t) (I.hgap t) l + choose base hbase using hb + exact gbridge2_certificate (ggeomBridge2 I base l hbase hH) + +end End +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.ggeomBridge2 +#print axioms OAI.PowerSaving.GX.bridge2_gcert diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__InvThm.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__InvThm.lean new file mode 100644 index 0000000000..d69fcb4992 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__InvThm.lean @@ -0,0 +1,164 @@ +import Work.GCert.Chain.Inv + +/-! +# (key: gx-chain) THE CARRIER INVOCATION THEOREM on the generalised engine: `Lab.inv` + +`CR.Phased.inv` (`Work/Carrier/PhasedInv.lean`) with subspace labels and `GRoute`. Forward word: + + erase | y -= K s (old frame 0) | free adapters to the labels of the certificate | phase A | + c += Cc s | copies down | y += Jr c | phase B | final climbs | free adapters to the old + frames | slots := slot rows of the inverse (old frame 1) | erase + +The backward word is the transposed inverse, gate by gate, on the same labels. The scalar +identities are `Scal.fwd_scalar` / `Scal.bwd_scalar` (those of `CR.Phased`). + +No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB BR CR GF RF +noncomputable section + +section Thm +variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] +variable {α : Type} [Fintype α] [DecidableEq α] +variable {S : Scal T Sl C} (L : Lab H S) (W : GWin H α) + +/-- **Forward word on clean frames.** -/ +theorem Lab.fwd_erased (c0 c1 : C → CMat α) : + GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0) + (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) shearF L.cost := by + have a1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) + (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) + (actPoint (g1 S.Mt)) (fun _ => 0) := + GRoute.gate (g1 S.Mt) _ _ (fun i j hij => by + by_cases e : i = j + · subst e; rfl + · obtain ⟨h1, h2⟩ := offdiag_x (g1_x S.Mt).1 (g1_x S.Mt).2 hij e + exact start_frames W.Xm L.tv i j h1 h2) + have b3 : GRoute (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) + (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) + (actPoint (g8 S.Mti)) (fun _ => 0) := + GRoute.gate (g8 S.Mti) _ _ (fun i j hij => by + by_cases e : i = j + · subst e; rfl + · have f := g8_facts S.hiy + obtain ⟨h1, h2, h3⟩ := offdiag_s f.1 f.2.1 f.2.2 hij e + exact end_old W.Xm L.tv i j h1 h2 h3) + have seg1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun r => W.X.Φ (L.labA r)) + S.f1 (fun φ => rcost φ L.rA) := + ((a1.trans (L.ad0 W)).trans (L.pA W.X W.big).1).cast (fun φ => by simp) + have seg2 : GRoute (fun r => W.X.Φ (L.labA r)) (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) + S.f2 (fun φ => rcost φ L.rB + rcost φ L.rF) := + ((((L.pB W.X W.big).1.trans (L.pF W.X W.big)).trans (L.adF W)).trans b3).cast + (fun φ => by simp) + have e0 : GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0) + (glue (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun k => W.X.Φ (L.cen k))) + eraseC (fun _ => 0) := + GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl)) + have L1 := glift_live (fun k => W.X.Φ (L.cen k)) seg1 + have m1 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k))) + (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k))) + (fun z => addBlk bC (ap S.Cc (z ∘ bS)) z) (fun _ => 0) := + GRoute.ofX (xgate_shear bC bS bS_inj S.Cc (fun k q h => congrArg W.X.Φ (L.hAc k q h))) + have m2 := L.cdown W (fun r => W.X.Φ (L.labA r)) + have m3 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0)) + (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0)) + (fun z => addBlk bY (ap S.Jr (z ∘ bC)) z) (fun _ => 0) := + GRoute.ofX (xgate_shear bY bC bC_inj S.Jr (fun t k _ => congrArg W.X.Φ (L.hAy t))) + have L2 := glift_live (fun _ : C => W.X.Φ L.z0) seg2 + have e1 : GRoute (glue (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) (fun _ : C => W.X.Φ L.z0)) + (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) eraseC (fun _ => 0) := + GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl)) + have tot := (((((e0.trans L1).trans m1).trans m2).trans m3).trans L2).trans e1 + refine (tot.castg (g' := shearF) (funext fun z => S.fwd_scalar z)).cast (fun φ => ?_) + simp only [Lab.cost, zero_add, add_zero] + ring + +/-- **Backward word on clean frames**: the transposed inverse gates, same labels; the copies +start at the zero frame and climb. -/ +theorem Lab.bwd_erased (c0 c1 : C → CMat α) : + GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0) + (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) shearB L.cost := by + have a1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) + (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) + (actPoint (g1i S.Mt)ᵀ) (fun _ => 0) := + GRoute.gate (g1i S.Mt)ᵀ _ _ (fun i j hij => by + by_cases e : i = j + · subst e; rfl + · have hij' : g1i S.Mt j i ≠ 0 := hij + obtain ⟨h1, h2⟩ := offdiag_x (g1i_x S.Mt).1 (g1i_x S.Mt).2 hij' (Ne.symm e) + exact start_frames W.Xm L.tv i j h2 h1) + have b3 : GRoute (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) + (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) + (actPoint (h8 S.Mt)ᵀ) (fun _ => 0) := + GRoute.gate (h8 S.Mt)ᵀ _ _ (fun i j hij => by + by_cases e : i = j + · subst e; rfl + · have hij' : h8 S.Mt j i ≠ 0 := hij + have f := h8_facts S.hmy + obtain ⟨h1, h2, h3⟩ := offdiag_s f.1 f.2.1 f.2.2 hij' (Ne.symm e) + exact (end_old W.Xm L.tv j i h1 h2 h3).symm) + have seg1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun r => W.X.Φ (L.labA r)) + S.b1 (fun φ => rcost φ L.rA) := + ((a1.trans (L.ad0 W)).trans (L.pA W.X W.big).2).cast (fun φ => by simp) + have seg2 : GRoute (fun r => W.X.Φ (L.labA r)) (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) + S.b2 (fun φ => rcost φ L.rB + rcost φ L.rF) := + ((((L.pB W.X W.big).2.trans (L.pF W.X W.big)).trans (L.adF W)).trans b3).cast + (fun φ => by simp) + have e0 : GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0) + (glue (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun _ : C => W.X.Φ L.z0)) + eraseC (fun _ => 0) := + GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl)) + have L1 := glift_live (fun _ : C => W.X.Φ L.z0) seg1 + have m1 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0)) + (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0)) + (fun z => addBlk bC (ap S.Jrᵀ (z ∘ bY)) z) (fun _ => 0) := + GRoute.ofX (xgate_shear bC bY bY_inj S.Jrᵀ (fun k t _ => (congrArg W.X.Φ (L.hAy t)).symm)) + have m2 := L.cup W (fun r => W.X.Φ (L.labA r)) + have m3 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k))) + (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k))) + (fun z => addBlk bS (ap (-S.Ccᵀ) (z ∘ bC)) z) (fun _ => 0) := + GRoute.ofX (xgate_shear bS bC bC_inj (-S.Ccᵀ) + (fun q k h => (congrArg W.X.Φ (L.hAc k q (ne_of_negT S.Cc h))).symm)) + have L2 := glift_live (fun k => W.X.Φ (L.cen k)) seg2 + have e1 : GRoute (glue (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) (fun k => W.X.Φ (L.cen k))) + (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) eraseC (fun _ => 0) := + GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl)) + have tot := (((((e0.trans L1).trans m1).trans m2).trans m3).trans L2).trans e1 + refine (tot.castg (g' := shearB) (funext fun z => S.bwd_scalar z)).cast (fun φ => ?_) + simp only [Lab.cost, zero_add, add_zero] + ring + +/-- **THE CARRIER INVOCATION THEOREM on the generalised engine.** The two halves of a +certificate are an invocation package for the bridged network, with the price `L.cost`. -/ +def Lab.inv : GInv H T Sl C where + tv := L.tv + hunit := L.hunit + hgap := L.hgap + cost := L.cost + fwd := fun W N0 Wn hN hW Z c0 c1 => + gdirty_of_erased W.Xm L.tv N0 Wn hN hW Z c0 c1 (L.fwd_erased W c0 c1) + (shearF_comm (fun _ => 1) (fun q => N0 * Z q) (fun _ => 1)) + bwd := fun W N0 Wn hN hW Z c0 c1 => + gdirty_of_erased W.Xm L.tv N0 Wn hN hW Z c0 c1 (L.bwd_erased W c0 c1) + (shearB_comm (fun _ => 1) (fun q => N0 * Z q) (fun _ => 1)) + +@[simp] lemma Lab.inv_cost (φ : ℕ → ℝ) : L.inv.cost φ = L.cost φ := rfl +@[simp] lemma Lab.inv_tv : L.inv.tv = L.tv := rfl + +end Thm +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.Lab.fwd_erased +#print axioms OAI.PowerSaving.GX.Lab.bwd_erased +#print axioms OAI.PowerSaving.GX.Lab.inv diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetCert.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetCert.lean new file mode 100644 index 0000000000..6336ba0766 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetCert.lean @@ -0,0 +1,157 @@ +import Work.GCert.Chain.NetStage +import Work.GFrame.Top.Assemble + +/-! +# (key: gx-chain) The bridged word B_2 on the generalised engine: the certificate `GCert` + +GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/Bridge/Cert.lean (section Net). +`gbridge2_certificate`: a `GCert` on the live roles `BLive Γ T Sl` with price `card Γ * N.unit φ`. +The gate layers, the idle climbs and the exchange are the old routes (`GRoute.ofX`). No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB RF BR GF +noncomputable section + +section Net +variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] + [Fintype Γ] [DecidableEq Γ] +variable (N : GBridge2 H T Sl C α Γ) + +/-- gate layer A, after stage 1 and the first idle climb of pair 2: twins have equal frames. -/ +lemma GBridge2.gateA : XRoute N.S3 N.S3 (gsc pAB kA) (fun _ => 0) := by + refine gateP pAB kA N.S3 ?_ + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk + · rfl + · exact (hk rfl).elim + · exact (hk rfl).elim + · rfl + · exact (hk rfl).elim + · exact (hk rfl).elim + +/-- gate layer B, after stage 2 and the second idle climb of pair 2. -/ +lemma GBridge2.gateB : XRoute N.S5 N.S5 (gsc pAB kB) (fun _ => 0) := by + refine gateP pAB kB N.S5 ?_ + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk + · exact N.x23 g t + · exact (hk rfl).elim + · exact (hk rfl).elim + · exact (N.y23 g t).symm + · exact (hk rfl).elim + · exact (hk rfl).elim + +/-- gate layer C, after the final climbs. -/ +lemma GBridge2.gateC : XRoute N.S8 N.S8 (gsc pC kCc) (fun _ => 0) := by + refine gateP pC kCc N.S8 ?_ + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk + · exact (hk rfl).elim + · rfl + · rfl + · exact (hk rfl).elim + · exact (hk rfl).elim + · exact (hk rfl).elim + +/-- the free shifts at the end. -/ +def GBridge2.corr : BRole Γ T Sl C → Space α := + mkB (fun _ _ => 0) (fun g t => N.u g t) (fun _ _ => 0) (fun g t => N.u g t) + (fun _ _ => 0) (fun _ _ => 0) + +def GBridge2.S9 : BRole Γ T Sl C → CMat α := fun r => shift (N.corr r) * N.S8 r + +/-- the whole word before the exchange: five stages, three idle-climb layers, three gate +layers. -/ +lemma GBridge2.route : + GRoute N.S0 N.S8 chain (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by + have R := ((((((((((N.stage0.trans N.stage1).trans (GRoute.ofX N.climbA)).trans + (GRoute.ofX N.gateA)).trans N.stage2).trans (GRoute.ofX N.climbB)).trans + (GRoute.ofX N.gateB)).trans N.stage3).trans N.stage4).trans + (GRoute.ofX N.fin)).trans (GRoute.ofX N.gateC)) + refine (R.castg ?_).cast (fun φ => ?_) + · rfl + · simp only [GBridge2.unit] + ring + +/-- **The scratch certificate of the bridged five-stage network with shared helper sets**, +in whole blocks, with its exact price for every price list. -/ +theorem gbridge2_certificate : + GCert α (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) + (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by + have R2 : GRoute N.S8 N.S9 id (fun _ => 0) := GRoute.shifts N.hm N.S8 N.corr + have R3 : GRoute N.S9 (fun r => N.S9 (bpull r)) (actPoint brecM) (fun _ => 0) := + GRoute.gate brecM N.S9 _ (fun i j h => by + have hj : j = bpull i := by + by_contra hne + exact h (by simp [brecM, hne]) + rw [hj]) + have Rt := ((N.route.trans R2).trans R3).cast (c' := fun φ => (Fintype.card Γ : ℝ) * N.unit φ) + (fun φ => by first | ring | simp) + have hSinv : ∀ r, ∃ M, N.S0 r * M = 1 := by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact N.inv0 g t + · refine ⟨1, ?_⟩ + show (N.G0 g).Xm.Φ 0 * 1 = 1 + rw [N.z0, Matrix.mul_one] + · exact N.inv0 g t + · refine ⟨1, ?_⟩ + show (N.G0 g).Xm.Φ 0 * 1 = 1 + rw [N.z0, Matrix.mul_one] + · exact ⟨1, Matrix.mul_one _⟩ + · exact ⟨1, Matrix.mul_one _⟩ + have hT : ∀ l : BLive Γ T Sl, N.S9 (bpull (Sum.inl l)) = kernel α * N.S0 (Sum.inl l) := by + rintro (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩) + · show shift (N.u g t) * N.YF g t = kernel α * (N.G0 g).Xm.Φ (tt (N.I.tv t)) + exact (N.term g t).symm + · show shift (0 : Space α) * kernel α = kernel α * (N.G0 g).Xm.Φ 0 + rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one] + · show shift (N.u g t) * N.YF g t = kernel α * (N.G0 g).Xm.Φ (tt (N.I.tv t)) + exact (N.term g t).symm + · show shift (0 : Space α) * kernel α = kernel α * (N.G0 g).Xm.Φ 0 + rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one] + · show shift (0 : Space α) * kernel α = kernel α * 1 + rw [shift_zero, Matrix.one_mul, Matrix.mul_one] + have hg : ∀ x : BRole Γ T Sl C → ℂ, + (∀ s, (∀ l : BLive Γ T Sl, (Sum.inl l : BRole Γ T Sl C) ≠ s) → x s = 0) → + ∀ l : BLive Γ T Sl, + (actPoint brecM ∘ (id ∘ chain)) x (Sum.inl l) = x (Sum.inl l) := by + intro x _ l + change actPoint brecM (chain x) (Sum.inl l) = _ + rw [actPoint_brecM] + rcases l with (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩) + · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inr (g,t))))) = _ + rw [chain_eq] + show ((1:ℚ):ℂ) * x (kX1 (g,t)) = x (kX1 (g,t)) + first | (push_cast; ring) | simp + · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inl (g,t))))) = _ + rw [chain_eq] + show ((-1:ℚ):ℂ) * (-x (kY1 (g,t))) = x (kY1 (g,t)) + first | (push_cast; ring) | simp + · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inr (g,t))))) = _ + rw [chain_eq] + show ((1:ℚ):ℂ) * x (kX2 (g,t)) = x (kX2 (g,t)) + first | (push_cast; ring) | simp + · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inl (g,t))))) = _ + rw [chain_eq] + show ((-1:ℚ):ℂ) * (-x (kY2 (g,t))) = x (kY2 (g,t)) + first | (push_cast; ring) | simp + · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inr (g,q))) = _ + rw [chain_eq] + show ((1:ℚ):ℂ) * x (kS (g,q)) = x (kS (g,q)) + first | (push_cast; ring) | simp + exact gcert_of_groute (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) N.S0 + (fun r => N.S9 (bpull r)) _ _ Rt hSinv hg hT + +end Net + +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.GBridge2.route +#print axioms OAI.PowerSaving.GX.gbridge2_certificate diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetDef.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetDef.lean new file mode 100644 index 0000000000..c13042eb43 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetDef.lean @@ -0,0 +1,157 @@ +import Work.GCert.Chain.Stage + +/-! +# (key: gx-chain) The bridged five-stage word B_2 on the generalised engine: data and frames + +GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/Bridge/Net.lean (`BR.Bridge2`): the +invocation is a `GInv`, the frame maps of the stages are windows `GWin` (`G0 .. G4`; the old +frame map of a window is `.Xm`). Everything else word for word. No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB RF BR GF +noncomputable section + +/-- See the file header. Fields in the style of `RF.Fold3S`; stages are numbered 0 to 4. -/ +structure GBridge2 (H T Sl C α Γ : Type) [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] where + /-- the invocation (for a block circuit `X`: `X.inv`). -/ + I : GInv H T Sl C + /-- frame maps of the invocations of the five stages. -/ + G0 : Γ → GWin H α + G1 : Γ → GWin H α + G2 : Γ → GWin H α + G3 : Γ → GWin H α + G4 : Γ → GWin H α + /-- the class `(g,t)` is served by the invocation `g` in stage 0 and `r_l t g` in stage `l`. -/ + r1 : T → Γ ≃ Γ + r2 : T → Γ ≃ Γ + r3 : T → Γ ≃ Γ + r4 : T → Γ ≃ Γ + /-- the invocation `d` of stage `l` uses the helper set (and the copies) `s_l d`. -/ + s1 : Γ ≃ Γ + s2 : Γ ≃ Γ + s3 : Γ ≃ Γ + s4 : Γ ≃ Γ + /-- inverse of the frame of the base, and frame of the window, of every invocation. -/ + N1 : Γ → CMat α + W1 : Γ → CMat α + N2 : Γ → CMat α + W2 : Γ → CMat α + N3 : Γ → CMat α + W3 : Γ → CMat α + N4 : Γ → CMat α + W4 : Γ → CMat α + u : Γ → T → Space α + YF : Γ → T → CMat α + hm : 2 ≤ Fintype.card α + /-- prices of the idle climbs, for every price list. B_2 with merged climbs: + `caX = caY = fun φ => bcost φ (2h-2)`, `cX1 = cY1 = fun φ => bcost φ (2h+2)`, + `cX2 = cY2 = fun φ => bcost φ 4`. -/ + caX : (ℕ → ℝ) → ℝ + caY : (ℕ → ℝ) → ℝ + cX1 : (ℕ → ℝ) → ℝ + cY1 : (ℕ → ℝ) → ℝ + cX2 : (ℕ → ℝ) → ℝ + cY2 : (ℕ → ℝ) → ℝ + inv0 : ∀ g t, ∃ N : CMat α, (G0 g).Xm.Φ (tt (I.tv t)) * N = 1 + z0 : ∀ g, (G0 g).Xm.Φ 0 = 1 + hN1 : ∀ d, (G1 d).Xm.Φ 0 * N1 d = 1 + hW1 : ∀ d, (G1 d).Xm.Φ 1 = W1 d * (G1 d).Xm.Φ 0 + hN2 : ∀ d, (G2 d).Xm.Φ 0 * N2 d = 1 + hW2 : ∀ d, (G2 d).Xm.Φ 1 = W2 d * (G2 d).Xm.Φ 0 + hN3 : ∀ d, (G3 d).Xm.Φ 0 * N3 d = 1 + hW3 : ∀ d, (G3 d).Xm.Φ 1 = W3 d * (G3 d).Xm.Φ 0 + hN4 : ∀ d, (G4 d).Xm.Φ 0 * N4 d = 1 + hW4 : ∀ d, (G4 d).Xm.Φ 1 = W4 d * (G4 d).Xm.Φ 0 + /-- the bank frames a stage leaves are the bank frames the next stage starts from. -/ + x01 : ∀ g t, (G0 g).Xm.Φ 1 = (G1 (r1 t g)).Xm.Φ (tt (I.tv t)) + y01 : ∀ g t, (G0 g).Xm.Φ (1 + tt (I.tv t)) = (G1 (r1 t g)).Xm.Φ 0 + x12 : ∀ g t, (G1 (r1 t g)).Xm.Φ 1 = (G2 (r2 t g)).Xm.Φ (tt (I.tv t)) + y12 : ∀ g t, (G1 (r1 t g)).Xm.Φ (1 + tt (I.tv t)) = (G2 (r2 t g)).Xm.Φ 0 + x23 : ∀ g t, (G2 (r2 t g)).Xm.Φ 1 = (G3 (r3 t g)).Xm.Φ (tt (I.tv t)) + y23 : ∀ g t, (G2 (r2 t g)).Xm.Φ (1 + tt (I.tv t)) = (G3 (r3 t g)).Xm.Φ 0 + x34 : ∀ g t, (G3 (r3 t g)).Xm.Φ 1 = (G4 (r4 t g)).Xm.Φ (tt (I.tv t)) + y34 : ∀ g t, (G3 (r3 t g)).Xm.Φ (1 + tt (I.tv t)) = (G4 (r4 t g)).Xm.Φ 0 + /-- the five windows of a helper set make up the kernel. -/ + sfin : ∀ g, W4 (s4.symm g) * (W3 (s3.symm g) * (W2 (s2.symm g) * + (W1 (s1.symm g) * (G0 g).Xm.Φ 1))) = kernel α + /-- pair 2, idle in stages 0 and 1: ONE block from its start frame to the frame of its + twin after stage 1. -/ + aX2 : ∀ g t, XReach ((G0 g).Xm.Φ (tt (I.tv t))) ((G1 (r1 t g)).Xm.Φ 1) caX + aY2 : ∀ g t, XReach ((G0 g).Xm.Φ 0) ((G1 (r1 t g)).Xm.Φ (1 + tt (I.tv t))) caY + /-- pair 1, idle in stages 3 and 4: ONE block from its frame after stage 2 to the end. -/ + fX1 : ∀ g t, XReach ((G2 (r2 t g)).Xm.Φ 1) (kernel α) cX1 + fY1 : ∀ g t, XReach ((G2 (r2 t g)).Xm.Φ (1 + tt (I.tv t))) (YF g t) cY1 + /-- pair 2 after stage 4: ONE final block. -/ + fX2 : ∀ g t, XReach ((G4 (r4 t g)).Xm.Φ 1) (kernel α) cX2 + fY2 : ∀ g t, XReach ((G4 (r4 t g)).Xm.Φ (1 + tt (I.tv t))) (YF g t) cY2 + term : ∀ g t, kernel α * (G0 g).Xm.Φ (tt (I.tv t)) = shift (u g t) * YF g t + +section Net +variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] + [Fintype Γ] [DecidableEq Γ] +variable (N : GBridge2 H T Sl C α Γ) + +/-- price of the blocks of one unit (one element of `Γ`): five invocations and the idle +climbs of the four bank roles of every class. -/ +def GBridge2.unit (φ : ℕ → ℝ) : ℝ := + 5 * N.I.cost φ + (Fintype.card T : ℝ) * ((N.caX φ + N.caY φ) + + 2 * bcost φ (Fintype.card H - 1) + + (N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ)) + +/-- the bank index of stage 0: the class `(g,t)` belongs to the invocation `g`. -/ +abbrev bid (Γ T : Type) : T → Γ ≃ Γ := fun _ => Equiv.refl Γ + +/-! bank frames of the class `(g,t)`: `xi l`, `yi l` at the START of stage `l`; `xo l`, +`yo l` at the END of stage `l` (only where no next stage follows on that pair). -/ +def GBridge2.xi0 (g : Γ) (t : T) : CMat α := (N.G0 g).Xm.Φ (tt (N.I.tv t)) +def GBridge2.yi0 (g : Γ) (t : T) : CMat α := (N.G0 g).Xm.Φ 0 +def GBridge2.xi1 (g : Γ) (t : T) : CMat α := (N.G1 (N.r1 t g)).Xm.Φ (tt (N.I.tv t)) +def GBridge2.yi1 (g : Γ) (t : T) : CMat α := (N.G1 (N.r1 t g)).Xm.Φ 0 +def GBridge2.xi2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ (tt (N.I.tv t)) +def GBridge2.yi2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ 0 +def GBridge2.xo2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ 1 +def GBridge2.yo2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ (1 + tt (N.I.tv t)) +def GBridge2.xi3 (g : Γ) (t : T) : CMat α := (N.G3 (N.r3 t g)).Xm.Φ (tt (N.I.tv t)) +def GBridge2.yi3 (g : Γ) (t : T) : CMat α := (N.G3 (N.r3 t g)).Xm.Φ 0 +def GBridge2.xi4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ (tt (N.I.tv t)) +def GBridge2.yi4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ 0 +def GBridge2.xo4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ 1 +def GBridge2.yo4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ (1 + tt (N.I.tv t)) + +/-- helper slots after the stages 0, 1, 2, 3, 4 (at the start: `cOne Γ Sl α`). -/ +def GBridge2.Z1 : Γ → Sl → CMat α := fun g _ => (N.G0 g).Xm.Φ 1 +def GBridge2.Z2 : Γ → Sl → CMat α := fun g q => N.W1 (N.s1.symm g) * N.Z1 g q +def GBridge2.Z3 : Γ → Sl → CMat α := fun g q => N.W2 (N.s2.symm g) * N.Z2 g q +def GBridge2.Z4 : Γ → Sl → CMat α := fun g q => N.W3 (N.s3.symm g) * N.Z3 g q +def GBridge2.Z5 : Γ → Sl → CMat α := fun g q => N.W4 (N.s4.symm g) * N.Z4 g q + +/-- frames: at the start; after stage 0; after stage 1; after the first idle climb of pair 2 +(first gate layer here); after stage 2; after the second idle climb of pair 2 (second gate +layer here); after stage 3; after stage 4; after the final climbs (third gate layer here). -/ +def GBridge2.S0 : BRole Γ T Sl C → CMat α := mkB N.xi0 N.yi0 N.xi0 N.yi0 (cOne Γ Sl α) (cOne Γ C α) +def GBridge2.S1 : BRole Γ T Sl C → CMat α := mkB N.xi1 N.yi1 N.xi0 N.yi0 N.Z1 (cOne Γ C α) +def GBridge2.S2 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi0 N.yi0 N.Z2 (cOne Γ C α) +def GBridge2.S3 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi2 N.yi2 N.Z2 (cOne Γ C α) +def GBridge2.S4 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi2 N.yi2 N.Z3 (cOne Γ C α) +def GBridge2.S5 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi3 N.yi3 N.Z3 (cOne Γ C α) +def GBridge2.S6 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi4 N.yi4 N.Z4 (cOne Γ C α) +def GBridge2.S7 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xo4 N.yo4 N.Z5 (cOne Γ C α) +def GBridge2.S8 : BRole Γ T Sl C → CMat α := + mkB (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) (cOne Γ C α) + + +end Net +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.GBridge2.unit diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetStage.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetStage.lean new file mode 100644 index 0000000000..670f4c6a84 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__NetStage.lean @@ -0,0 +1,186 @@ +import Work.GCert.Chain.NetDef + +/-! +# (key: gx-chain) The bridged word B_2 on the generalised engine: the five stages and the idle climbs + +GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/Bridge/Net.lean. The stages are +`GRoute`s (`gistage_fwd` / `gistage_bwd`, `gblift1` / `gblift2`); the idle climbs of the bank +roles are the OLD block routes, unchanged. No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB RF BR GF +noncomputable section + +section Net +variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] + [Fintype Γ] [DecidableEq Γ] +variable (N : GBridge2 H T Sl C α Γ) + +/-- stage 0: forward on pair 1, every helper set starts at the matrix `1`. -/ +lemma GBridge2.stage0 : + GRoute N.S0 N.S1 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + have h := gistage_fwd N.I (bid Γ T) (Equiv.refl Γ) N.G0 (fun _ => 1) (fun g => (N.G0 g).Xm.Φ 1) + (fun d => by rw [N.z0, Matrix.mul_one]) (fun d => by rw [N.z0, Matrix.mul_one]) + (cOne Γ Sl α) (cOne Γ C α) (cOne Γ C α) + have e : bOut N.I.tv (bid Γ T) (fun d => (N.G0 d).Xm) + (fun g (q : Sl) => (N.G0 ((Equiv.refl Γ).symm g)).Xm.Φ 1 * cOne Γ Sl α g q) (cOne Γ C α) + = mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α) := by + funext r + rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩) + · exact N.x01 g t + · exact N.y01 g t + · show (N.G0 g).Xm.Φ 1 * 1 = (N.G0 g).Xm.Φ 1 + rw [Matrix.mul_one] + · rfl + have h' : GRoute (bIn N.I.tv (bid Γ T) (fun d => (N.G0 d).Xm) (cOne Γ Sl α) (cOne Γ C α)) + (mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α)) GF (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + rw [← e] + exact h + exact gblift1 N.xi0 N.yi0 h' + +/-- stage 1: backward on pair 1; the helper set `g` serves the invocation `s1⁻¹ g`. -/ +lemma GBridge2.stage1 : + GRoute N.S1 N.S2 (lift1 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + have h := gistage_bwd N.I N.r1 N.s1 N.G1 N.N1 N.W1 N.hN1 N.hW1 N.Z1 (cOne Γ C α) (cOne Γ C α) + have e : bOut N.I.tv N.r1 (fun d => (N.G1 d).Xm) (fun g q => N.W1 (N.s1.symm g) * N.Z1 g q) (cOne Γ C α) + = mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α) := by + funext r + rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩) + · exact N.x12 g t + · exact N.y12 g t + · rfl + · rfl + have h' : GRoute (bIn N.I.tv N.r1 (fun d => (N.G1 d).Xm) N.Z1 (cOne Γ C α)) + (mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + rw [← e] + exact h + exact gblift1 N.xi0 N.yi0 h' + +/-- stage 2: forward on pair 1. -/ +lemma GBridge2.stage2 : + GRoute N.S3 N.S4 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := + gblift1 N.xi2 N.yi2 + (gistage_fwd N.I N.r2 N.s2 N.G2 N.N2 N.W2 N.hN2 N.hW2 N.Z2 (cOne Γ C α) (cOne Γ C α)) + +/-- stage 3: backward on pair 2. -/ +lemma GBridge2.stage3 : + GRoute N.S5 N.S6 (lift2 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + have h := gistage_bwd N.I N.r3 N.s3 N.G3 N.N3 N.W3 N.hN3 N.hW3 N.Z3 (cOne Γ C α) (cOne Γ C α) + have e : bOut N.I.tv N.r3 (fun d => (N.G3 d).Xm) (fun g q => N.W3 (N.s3.symm g) * N.Z3 g q) (cOne Γ C α) + = mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α) := by + funext r + rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩) + · exact N.x34 g t + · exact N.y34 g t + · rfl + · rfl + have h' : GRoute (bIn N.I.tv N.r3 (fun d => (N.G3 d).Xm) N.Z3 (cOne Γ C α)) + (mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by + rw [← e] + exact h + exact gblift2 N.xo2 N.yo2 h' + +/-- stage 4: forward on pair 2. -/ +lemma GBridge2.stage4 : + GRoute N.S6 N.S7 (lift2 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := + gblift2 N.xo2 N.yo2 + (gistage_fwd N.I N.r4 N.s4 N.G4 N.N4 N.W4 N.hN4 N.hW4 N.Z4 (cOne Γ C α) (cOne Γ C α)) + +/-- first idle climb of pair 2 (stages 0 and 1 in ONE block per role), to the frames of +pair 1 after stage 1. -/ +lemma GBridge2.climbA : + XRoute N.S2 N.S3 id (fun φ => (Fintype.card Γ : ℝ) * + ((Fintype.card T : ℝ) * (N.caX φ + N.caY φ))) := by + have H1 := XRoute.reach_all N.S2 N.S3 + (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ => N.caX) + (fun _ _ => N.caY) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact XReach.refl _ + · exact XReach.refl _ + · show XReach ((N.G0 g).Xm.Φ (tt (N.I.tv t))) ((N.G2 (N.r2 t g)).Xm.Φ (tt (N.I.tv t))) N.caX + rw [← N.x12 g t] + exact N.aX2 g t + · show XReach ((N.G0 g).Xm.Φ 0) ((N.G2 (N.r2 t g)).Xm.Φ 0) N.caY + rw [← N.y12 g t] + exact N.aY2 g t + · exact XReach.refl _ + · exact XReach.refl _) + refine H1.cast (fun φ => ?_) + simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)] + rw [sum_mkB] + simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add] + ring + +/-- second idle climb of pair 2: the window of stage 2, ONE block of rank `card H - 1` per +role, to the frames of pair 1 after stage 2. -/ +lemma GBridge2.climbB : + XRoute N.S4 N.S5 id (fun φ => (Fintype.card Γ : ℝ) * + ((Fintype.card T : ℝ) * (2 * bcost φ (Fintype.card H - 1)))) := by + have H1 := XRoute.reach_all N.S4 N.S5 + (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ φ => bcost φ (Fintype.card H - 1)) + (fun _ _ φ => bcost φ (Fintype.card H - 1)) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact XReach.refl _ + · exact XReach.refl _ + · show XReach ((N.G2 (N.r2 t g)).Xm.Φ (tt (N.I.tv t))) ((N.G3 (N.r3 t g)).Xm.Φ (tt (N.I.tv t))) + (fun φ => bcost φ (Fintype.card H - 1)) + rw [← N.x23 g t] + exact N.I.climbX (N.G2 (N.r2 t g)).Xm t + · show XReach ((N.G2 (N.r2 t g)).Xm.Φ 0) ((N.G3 (N.r3 t g)).Xm.Φ 0) + (fun φ => bcost φ (Fintype.card H - 1)) + rw [← N.y23 g t] + exact N.I.climbY (N.G2 (N.r2 t g)).Xm t + · exact XReach.refl _ + · exact XReach.refl _) + refine H1.cast (fun φ => ?_) + simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)] + rw [sum_mkB] + simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add] + ring + +/-- the final climbs: pair 1 from its frames after stage 2 (ONE block per role for the +stages 3, 4 and the rest), pair 2 from its frames after stage 4; the helper slots are +already at the kernel. -/ +lemma GBridge2.fin : + XRoute N.S7 N.S8 id (fun φ => (Fintype.card Γ : ℝ) * + ((Fintype.card T : ℝ) * ((N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ)))) := by + have H1 := XRoute.reach_all N.S7 N.S8 + (mkB (fun _ _ => N.cX1) (fun _ _ => N.cY1) (fun _ _ => N.cX2) (fun _ _ => N.cY2) + (fun _ _ _ => 0) (fun _ _ _ => 0)) (by + rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) + · exact N.fX1 g t + · exact N.fY1 g t + · exact N.fX2 g t + · exact N.fY2 g t + · show XReach (N.W4 (N.s4.symm g) * (N.W3 (N.s3.symm g) * (N.W2 (N.s2.symm g) * + (N.W1 (N.s1.symm g) * (N.G0 g).Xm.Φ 1)))) (kernel α) (fun _ => 0) + rw [N.sfin g] + exact XReach.refl _ + · exact XReach.refl _) + refine H1.cast (fun φ => ?_) + simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)] + rw [sum_mkB] + simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add] + ring + +end Net +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.GBridge2.stage0 +#print axioms OAI.PowerSaving.GX.GBridge2.stage1 +#print axioms OAI.PowerSaving.GX.GBridge2.stage2 +#print axioms OAI.PowerSaving.GX.GBridge2.stage3 +#print axioms OAI.PowerSaving.GX.GBridge2.stage4 +#print axioms OAI.PowerSaving.GX.GBridge2.climbA +#print axioms OAI.PowerSaving.GX.GBridge2.climbB +#print axioms OAI.PowerSaving.GX.GBridge2.fin diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__Phased.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__Phased.lean new file mode 100644 index 0000000000..e68c42377f --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__Phased.lean @@ -0,0 +1,199 @@ +import Work.Carrier.Phases +import Work.GFrame.Labels.All + +/-! +# (key: gx-chain) The certificate as two halves: `Scal` (scalar half) and `Lab` (label half) + +`Scal T Sl C`: the scalar fields of `CR.Phased` (three facts about ONE matrix +`Mt = MB * scatM (Jr * Cc) * MA`). The scalar identities of the forward and of the backward +word (`fwd_scalar`, `bwd_scalar`) are those of `Work/Carrier/Phases.lean`, word for word. + +`Lab H S`: subspace labels (`GF.SLbl H`), routes of the three phases on the generalised engine +(`GF.GRoute`) through every stage-B frame map (`GF.BXMap H α`), and facts about dimensions at +the two ends. No projector, no `Climb`. + +No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB BR CR GF +noncomputable section + +/-- **The scalar half.** -/ +structure Scal (T Sl C : Type) [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] where + Cc : Matrix C Sl ℚ + Jr : Matrix T C ℚ + MA : Matrix (L3 T Sl) (L3 T Sl) ℚ + MAi : Matrix (L3 T Sl) (L3 T Sl) ℚ + MB : Matrix (L3 T Sl) (L3 T Sl) ℚ + MBi : Matrix (L3 T Sl) (L3 T Sl) ℚ + hAi : MAi * MA = 1 + hBi : MBi * MB = 1 + hx : ∀ t j, (MB * scatM (Jr * Cc) * MA) (lX t) j = if lX t = j then 1 else 0 + hy : ∀ i t, (MB * scatM (Jr * Cc) * MA) i (lY t) = if i = lY t then 1 else 0 + hid : ∀ S t, (MB * scatM (Jr * Cc) * MA) (lY S) (lX t) = if S = t then 1 else 0 + +/-- **The label half.** See `checks/wht26/shared/gcert-interface.md`, section CHAIN. -/ +structure Lab (H : Type) [Fintype H] [DecidableEq H] {T Sl C : Type} [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] (S : Scal T Sl C) where + tv : T → Space H + hunit : ∀ t, dot (tv t) (tv t) = 1 + hgap : ∀ t, ∃ p, tv t p = 0 + lab0 : L3 T Sl → SLbl H + labA : L3 T Sl → SLbl H + labB : L3 T Sl → SLbl H + labF : L3 T Sl → SLbl H + cen : C → SLbl H + z0 : SLbl H + hz0 : z0.dim = 0 + h0x : ∀ t, (lab0 (lX t)).dim = 1 ∧ (lab0 (lX t)).Mem (tv t) + h0y : ∀ t, (lab0 (lY t)).dim = 0 + h0s : ∀ q, (lab0 (lS q)).dim = 0 + hAy : ∀ t, labA (lY t) = z0 + hAc : ∀ k q, S.Cc k q ≠ 0 → cen k = labA (lS q) + hcen : ∀ k, 0 < (cen k).dim + hFx : ∀ t, (labF (lX t)).dim = Fintype.card H + hFs : ∀ q, (labF (lS q)).dim = Fintype.card H + hFy : ∀ t, (labF (lY t)).dim + 1 = Fintype.card H ∧ + ∀ x, (labF (lY t)).Mem x → dot (tv t) x = 0 + rA : List ℕ + rB : List ℕ + rF : List ℕ + pA : ∀ {α : Type} [Fintype α] [DecidableEq α] (X : BXMap H α), + Fintype.card H < Fintype.card α → + GRoute (fun r => X.Φ (lab0 r)) (fun r => X.Φ (labA r)) (actPoint S.MA) + (fun φ => rcost φ rA) ∧ + GRoute (fun r => X.Φ (lab0 r)) (fun r => X.Φ (labA r)) (actPoint S.MAiᵀ) + (fun φ => rcost φ rA) + pB : ∀ {α : Type} [Fintype α] [DecidableEq α] (X : BXMap H α), + Fintype.card H < Fintype.card α → + GRoute (fun r => X.Φ (labA r)) (fun r => X.Φ (labB r)) (actPoint S.MB) + (fun φ => rcost φ rB) ∧ + GRoute (fun r => X.Φ (labA r)) (fun r => X.Φ (labB r)) (actPoint S.MBiᵀ) + (fun φ => rcost φ rB) + pF : ∀ {α : Type} [Fintype α] [DecidableEq α] (X : BXMap H α), + Fintype.card H < Fintype.card α → + GRoute (fun r => X.Φ (labB r)) (fun r => X.Φ (labF r)) id (fun φ => rcost φ rF) + +/-- price of one invocation: the blocks of the three phases and one block per copy. -/ +def Lab.cost {H : Type} [Fintype H] [DecidableEq H] {T Sl C : Type} [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] {S : Scal T Sl C} (L : Lab H S) + (φ : ℕ → ℝ) : ℝ := + rcost φ L.rA + (∑ k, bcost φ (L.cen k).dim) + rcost φ L.rB + rcost φ L.rF + +section Alg +variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T] + [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] + +namespace Scal +variable (K : Scal T Sl C) + +/-- the product of ALL gates of the main phase (scatter with the copies eliminated). -/ +def Mt : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MB * scatM (K.Jr * K.Cc) * K.MA +/-- its inverse. -/ +def Mti : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MAi * scatMi (K.Jr * K.Cc) * K.MBi + + +lemma im : K.Mti * K.Mt = 1 := by + unfold Mti Mt + have hB : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ, K.MBi * (K.MB * X) = X := fun X => by + rw [← Matrix.mul_assoc, K.hBi, Matrix.one_mul] + have hS : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ, + scatMi (K.Jr * K.Cc) * (scatM (K.Jr * K.Cc) * X) = X := fun X => by + rw [← Matrix.mul_assoc, scat_inv, Matrix.one_mul] + rw [Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc, hB, hS, K.hAi] + +lemma mi : K.Mt * K.Mti = 1 := mul_eq_one_comm.mp K.im + +lemma hxm : PX * K.Mt = PX := PX_mul_of (M := K.Mt) K.hx +lemma hmy : K.Mt * PY = PY := mul_PY_of (M := K.Mt) K.hy +lemma hiy : K.Mti * PY = PY := dirt_iy K.hmy K.im + +/-- **the dirt rule, forward**, for the matrices of the circuit. -/ +lemma hF : g8 K.Mti * K.Mt * g1 K.Mt = 1 + PY * K.Mt * PX := by + unfold g8 g1 + exact dirt_fwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi + +/-- **the dirt rule, backward.** -/ +lemma hB : g1i K.Mt * K.Mti * h8 K.Mt = 1 - PY * K.Mt * PX := by + unfold g1i h8 + exact dirt_bwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi + +/-- live part of the forward word before the scatter: `y -= K s`, then phase A. -/ +def f1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MA (actPoint (g1 K.Mt) u) +/-- after the scatter: phase B, then the slot rows of the inverse. -/ +def f2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (g8 K.Mti) (actPoint K.MB u) +/-- the same for the backward word (transposed inverses). -/ +def b1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u) +def b2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ u) + +lemma fwd_live (u : L3 T Sl → ℂ) : + K.f2 (actPoint (scatM (K.Jr * K.Cc)) (K.f1 u)) = actPoint (1 + PY * K.Mt * PX) u := by + rw [← K.hF] + show actPoint (g8 K.Mti) (actPoint K.MB (actPoint (scatM (K.Jr * K.Cc)) + (actPoint K.MA (actPoint (g1 K.Mt) u)))) + = actPoint (g8 K.Mti * (K.MB * scatM (K.Jr * K.Cc) * K.MA) * g1 K.Mt) u + rw [actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul] + +lemma bwd_live (u : L3 T Sl → ℂ) : + K.b2 (actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 u)) + = actPoint ((1 - PY * K.Mt * PX)ᵀ) u := by + rw [← K.hB] + show actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ (actPoint (scatMi (K.Jr * K.Cc))ᵀ + (actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u)))) + = actPoint ((g1i K.Mt * (K.MAi * scatMi (K.Jr * K.Cc) * K.MBi) * h8 K.Mt)ᵀ) u + rw [Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul, + actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul] + +lemma liveAct_glue (φ : (L3 T Sl → ℂ) → (L3 T Sl → ℂ)) (u : L3 T Sl → ℂ) (c : C → ℂ) : + liveAct φ (glue u c) = glue (φ u) c := + congrArg (fun v => glue (φ v) c) (glue_up u c) + +/-- **Scalar identity of the forward word**: `y += x`, slots restored, for ARBITRARY slot +contents (the copies are erased before and after). -/ +theorem fwd_scalar (z : Box T Sl C → ℂ) : + eraseC (liveAct K.f2 (gateS bY bC K.Jr (gateS bC bS K.Cc (liveAct K.f1 (eraseC z))))) + = shearF z := by + have h1 : liveAct K.f1 (eraseC z) = glue (K.f1 (z ∘ upE)) (fun _ => 0) := by + rw [erase_glue z, liveAct_glue] + have hs : gateS bY bC K.Jr (gateS bC bS K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0))) ∘ upE + = actPoint (scatM (K.Jr * K.Cc)) (K.f1 (z ∘ upE)) := by + have h := scat_glue K.Jr K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0)) rfl + rw [glue_up] at h + exact h + rw [h1] + unfold liveAct + rw [erase_of_glue, hs, K.fwd_live, act_shear K.Mt K.hid] + exact glue_shearF z + +/-- **Scalar identity of the backward word**: `x -= y`, slots restored. -/ +theorem bwd_scalar (z : Box T Sl C → ℂ) : + eraseC (liveAct K.b2 (gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (liveAct K.b1 (eraseC z))))) + = shearB z := by + have h1 : liveAct K.b1 (eraseC z) = glue (K.b1 (z ∘ upE)) (fun _ => 0) := by + rw [erase_glue z, liveAct_glue] + have hs : gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (glue (K.b1 (z ∘ upE)) (fun _ => 0))) ∘ upE + = actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 (z ∘ upE)) := by + have h := scat_glueT K.Jr K.Cc (glue (K.b1 (z ∘ upE)) (fun _ => 0)) rfl + rw [glue_up] at h + exact h + rw [h1] + unfold liveAct + rw [erase_of_glue, hs, K.bwd_live, act_shearT K.Mt K.hid] + exact glue_shearB z + +end Scal +end Alg +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.Scal.fwd_scalar +#print axioms OAI.PowerSaving.GX.Scal.bwd_scalar diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__WinB.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__WinB.lean new file mode 100644 index 0000000000..951edde5cf --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Chain__WinB.lean @@ -0,0 +1,132 @@ +import Work.GCert.Chain.Win +import Work.BridgeGeom.Stage + +/-! +# (key: gx-chain) THE WINDOW LIFT (spec statement 3): the windows of the bridged geometry + +For the block frame map `BG.fmapB hB hH d q` of a stage of the bridged word (label space +`LB B H = H ⊕ (B × H)`, old frame of the projector `P`: `PhiB d q P`) the stage-B frame map is +`BXMap.ofConj d q` (`U ↦ d (U ⊕ span q)`, arbitrary subspaces `U` of `H`). At a coordinate +subspace of an orthonormal basis `A` of `H` the two name the SAME subspace of the address space +(`insub_old_new`), so the old frame and the representative of any label of that subspace +differ by a free factor (`frame_core_free`, `rep_change`): `old_to_new`, `new_to_old`. + +`gwinB hB hH d q : GWin H (LB B H)` with `(gwinB hB hH d q).Xm = fmapB hB hH d q` (`rfl`). + +No `sorry`. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving +namespace GX +open Binary Matrix Finset RAM SS CB RF BR BG GF +noncomputable section + +section +variable {B H : Type} [Fintype B] [DecidableEq B] [Fintype H] [DecidableEq H] + +lemma blkT_mulVec (A : OBase H) (z : Space (LB B H)) (k : LB B H) : + ((blkB B A)ᵀ *ᵥ z) k = Sum.elim (fun i => dot (A.v i) (fun j => z (Sum.inl j))) + (fun b => z (Sum.inr b)) k := by + cases k with + | inl i => + simp [blkB, Matrix.fromBlocks_transpose, Matrix.mulVec, dotProduct, Fintype.sum_sum_type, + cmat, dot] + | inr b => + simp [blkB, Matrix.fromBlocks_transpose, Matrix.mulVec, dotProduct, Fintype.sum_sum_type, + Matrix.one_apply] + +/-- coordinates of `y` in the basis `d (A ⊕ std)`. -/ +lemma dot_cols (d : Orth (LB B H)) (A : OBase H) (y : Space (LB B H)) (k : LB B H) : + dot ((ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A))).v k) y + = Sum.elim (fun i => dot (A.v i) (fun j => (d.1ᵀ *ᵥ y) (Sum.inl j))) + (fun b => (d.1ᵀ *ᵥ y) (Sum.inr b)) k := by + rw [← blkT_mulVec A (d.1ᵀ *ᵥ y) k, Matrix.mulVec_mulVec, ← Matrix.transpose_mul] + rfl + +/-- **the old frame and the lifted label name the same subspace of the address space.** -/ +theorem insub_old_new (d : Orth (LB B H)) (q : Finset (B × H)) (A : OBase H) (s : Finset H) + (U : SLbl H) (h : ∀ x, U.Mem x ↔ InSub (coordPerm A) s x) (y : Space (LB B H)) : + InSub (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q) y ↔ + InSub (conjPerm d.1 d.2 (Orth.orth' d) U.G) (U.s.disjSum q) y := by + rw [insub_conj, insub_coordPerm] + have hU := h (fun i => (d.1ᵀ *ᵥ y) (Sum.inl i)) + constructor + · intro hk + refine ⟨hU.mpr ((insub_coordPerm A s _).mpr (fun i hi => ?_)), fun b hb => ?_⟩ + · have h1 := hk (Sum.inl i) (fun hm => hi (Finset.inl_mem_disjSum.mp hm)) + rw [dot_cols] at h1 + exact h1 + · have h1 := hk (Sum.inr b) (fun hm => hb (Finset.inr_mem_disjSum.mp hm)) + rw [dot_cols] at h1 + exact h1 + · rintro ⟨h1, h2⟩ k hk + rw [dot_cols] + cases k with + | inl i => + exact (insub_coordPerm A s _).mp (hU.mp h1) i + (fun hm => hk (Finset.inl_mem_disjSum.mpr hm)) + | inr b => exact h2 b (fun hm => hk (Finset.inr_mem_disjSum.mpr hm)) + +/-- old frame of a coordinate subspace → representative of any label naming it: FREE. -/ +theorem old_to_new (d : Orth (LB B H)) (q : Finset (B × H)) (A : OBase H) (s : Finset H) + (U : SLbl H) (h : ∀ x, U.Mem x ↔ InSub (coordPerm A) s x) : + GReach (PhiB d q (A.mat s)) ((BXMap.ofConj d.1 d.2 (Orth.orth' d) q).Φ U) (fun _ => 0) := by + rw [PhiB_eq] + obtain ⟨N, hN, e⟩ := repChange (LB B H) _ _ _ _ (insub_old_new d q A s U h) + have a : GReach (frame (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A))) (JB s q)) + (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q)) + (fun φ => rcost φ []) := + GReach.ofReach (fmv_old_to_core _ _).reach + have b : GReach + (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q)) + (core (conjPerm d.1 d.2 (Orth.orth' d) U.G) (U.s.disjSum q)) (fun φ => rcost φ []) := + GReach.ofReach (fmv_of_free hN e).reach + exact (a.trans b).cast (fun φ => by simp [rcost]) + +/-- the same, backwards. -/ +theorem new_to_old (d : Orth (LB B H)) (q : Finset (B × H)) (A : OBase H) (s : Finset H) + (U : SLbl H) (h : ∀ x, U.Mem x ↔ InSub (coordPerm A) s x) : + GReach ((BXMap.ofConj d.1 d.2 (Orth.orth' d) q).Φ U) (PhiB d q (A.mat s)) (fun _ => 0) := by + rw [PhiB_eq] + obtain ⟨N, hN, e⟩ := repChange (LB B H) _ _ _ _ + (fun y => (insub_old_new d q A s U h y).symm) + have a : GReach (core (conjPerm d.1 d.2 (Orth.orth' d) U.G) (U.s.disjSum q)) + (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q)) + (fun φ => rcost φ []) := + GReach.ofReach (fmv_of_free hN e).reach + have b : GReach + (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q)) + (frame (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A))) (JB s q)) + (fun φ => rcost φ []) := + GReach.ofReach (fmv_core_to_old _ _).reach + exact (a.trans b).cast (fun φ => by simp [rcost]) + +/-- **The window of a stage of the bridged geometry.** -/ +def gwinB (hB : 1 ≤ Fintype.card B) (hH : 1 ≤ Fintype.card H) (d : Orth (LB B H)) + (q : Finset (B × H)) : GWin H (LB B H) where + Xm := fmapB hB hH d q + X := BXMap.ofConj d.1 d.2 (Orth.orth' d) q + big := by + rw [card_LB] + have := Nat.mul_le_mul_right (Fintype.card H) (show 2 ≤ Fintype.card B + 1 by omega) + omega + toNew := fun A s U h => old_to_new d q A s U h + toOld := fun A s U h => new_to_old d q A s U h + +@[simp] lemma gwinB_Xm (hB : 1 ≤ Fintype.card B) (hH : 1 ≤ Fintype.card H) (d : Orth (LB B H)) + (q : Finset (B × H)) : (gwinB hB hH d q).Xm = fmapB hB hH d q := rfl + +end +end +end GX +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.GX.insub_old_new +#print axioms OAI.PowerSaving.GX.old_to_new +#print axioms OAI.PowerSaving.GX.new_to_old +#print axioms OAI.PowerSaving.GX.gwinB diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean new file mode 100644 index 0000000000..d7ecfebcea --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean @@ -0,0 +1,33 @@ +import Work.GCert.Scalar.Main +import Work.GCert.Data.P193ScalOK +import Work.GCert.Data.B2Gp193 + +/-! +# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `P193`: THE CLOSED THEOREMS +GENERATED by checks/wht26/gx-data/py/gxrate.py. No `sorry`. +`wht_B2Gp193_of_ids` (label half, price, unit list, rate fact, gx-chain's end theorem) applied to the scalar half +`GXD.P193.scalOK : GS.ScalOK GXD.P193.raw` (kernel checks `scal_ok`, `ychk_ok`; soundness: gx-scalar). +-/ + +namespace OAI +namespace PowerSaving +namespace WHT +open BlockWHT + +/-- **Walsh-Hadamard transform in `O(n (log n)^(1 - 74745/10^8))`-type time bounds: whole-block recursion on the bridged +word B_2 with the kernel-checked gcert/1 circuit `P193`.** -/ +theorem wht_main_B2Gp193 : + ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - 74745/(10:ℝ)^8) W := + wht_B2Gp193_of_ids GXD.P193.scalOK.hv GXD.P193.scalOK.hAi GXD.P193.scalOK.hBi GXD.P193.scalOK.hx GXD.P193.scalOK.hy GXD.P193.scalOK.hid + +/-- **Walsh-Hadamard transform in `O(n (log n)^(1 - 7474547/10^10))`-type time bounds: whole-block recursion on the bridged +word B_2 with the kernel-checked gcert/1 circuit `P193`.** -/ +theorem wht_main_B2Gp193x : + ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - 7474547/(10:ℝ)^10) W := + wht_B2Gp193x_of_ids GXD.P193.scalOK.hv GXD.P193.scalOK.hAi GXD.P193.scalOK.hBi GXD.P193.scalOK.hx GXD.P193.scalOK.hy GXD.P193.scalOK.hid + +end WHT +end PowerSaving +end OAI +#print axioms OAI.PowerSaving.WHT.wht_main_B2Gp193 +#print axioms OAI.PowerSaving.WHT.wht_main_B2Gp193x diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__Glue.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__Glue.lean new file mode 100644 index 0000000000..c6cbac38e3 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__Glue.lean @@ -0,0 +1,97 @@ +import Work.GCert.Scalar.Roles +import Work.GCert.Labels.Main + +/-! +# (key: gx-data) Glue between the scalar half (gx-scalar) and the label half (gx-labels) + +For ANY `c : GXD.Raw` (nothing is evaluated here): + +* `roles` gx-scalar's numbering `GS.emb` / `GS.unemb` is gx-labels' `GLab.Roles`; +* `scalOf` gx-chain's record `GX.Scal (Fin v) (Fin R) (Fin ret.length)` from the FIVE scalar + identities of the matrices `GS.MA`, `GS.MAi`, `GS.MB`, `GS.MBi`, `GS.Jrm`, `GS.Ccm`; +* `scalOf_MA .. scalOf_MBi` its gate matrices are the ordered products of the adds of `c`, in + the form asked by `GLab.Valid.lab` (`GLab.mic GS.toCoef`); +* `reg`, `reg_slot`, `reg_mem` copy `k` reads the retained register `(c.ret.getD k (0,0)).1`. + +So a generated instance needs from the scalar side exactly `hAi hBi hx hy hid` (see +`wht__of_ids` in `Work/GCert/Data/.lean`). No `sorry`. +-/ + +namespace GXD +open SSC OAI.PowerSaving OAI.PowerSaving.CR Matrix + +/-- the numbering of the roles: `x_t = t`, `y_t = v + t`, slot `q = 2 v + q` -/ +theorem roles (c : Raw) (hv : 0 < c.v) : GLab.Roles c.v c.R (GS.emb c) (GS.unemb c hv) where + hun := GS.unemb_emb hv + hin := GS.emb_unemb hv + hlt := fun q => by + rcases q with (t | S) | q + · have := t.isLt; show t.val < 2 * c.v + c.R; omega + · have := S.isLt; show c.v + S.val < 2 * c.v + c.R; omega + · have := q.isLt; show 2 * c.v + q.val < 2 * c.v + c.R; omega + eX := fun _ => rfl + eY := fun _ => rfl + eS := fun _ => rfl + +/-- **the scalar half as gx-chain's record**, from the five scalar identities -/ +noncomputable def scalOf (c : Raw) (hv : 0 < c.v) + (hAi : GS.MAi c hv * GS.MA c hv = 1) (hBi : GS.MBi c hv * GS.MB c hv = 1) + (hx : ∀ (t : Fin c.v) (j : GS.Rl c), (GS.MB c hv * scatM (GS.Jrm c * GS.Ccm c) * GS.MA c hv) + (lX t) j = if lX t = j then 1 else 0) + (hy : ∀ (i : GS.Rl c) (t : Fin c.v), (GS.MB c hv * scatM (GS.Jrm c * GS.Ccm c) * GS.MA c hv) + i (lY t) = if i = lY t then 1 else 0) + (hid : ∀ S t : Fin c.v, (GS.MB c hv * scatM (GS.Jrm c * GS.Ccm c) * GS.MA c hv) + (lY S) (lX t) = if S = t then 1 else 0) : + GX.Scal (Fin c.v) (Fin c.R) (Fin c.ret.length) where + Cc := GS.Ccm c + Jr := GS.Jrm c + MA := GS.MA c hv + MAi := GS.MAi c hv + MB := GS.MB c hv + MBi := GS.MBi c hv + hAi := hAi + hBi := hBi + hx := hx + hy := hy + hid := hid + +section +variable (c : Raw) (hv : 0 < c.v) (hAi) (hBi) (hx) (hy) (hid) + +theorem scalOf_MA : (scalOf c hv hAi hBi hx hy hid).MA + = matP (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesA.flatMap Gate.adds)) := rfl + +theorem scalOf_MAi : (scalOf c hv hAi hBi hx hy hid).MAi + = matPinv (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesA.flatMap Gate.adds)) := rfl + +theorem scalOf_MB : (scalOf c hv hAi hBi hx hy hid).MB + = matP (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesB.flatMap Gate.adds)) := rfl + +theorem scalOf_MBi : (scalOf c hv hAi hBi hx hy hid).MBi + = matPinv (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesB.flatMap Gate.adds)) := rfl + +end + +/-- the register read by copy `k` (the slot of the retained total `k`) -/ +def reg (c : Raw) (k : Fin c.ret.length) : Nat := (c.ret.getD k.val (0, 0)).1 + +theorem reg_slot (c : Raw) (hv : 0 < c.v) (hAi) (hBi) (hx) (hy) (hid) (k : Fin c.ret.length) + (q : Fin c.R) (h : (scalOf c hv hAi hBi hx hy hid).Cc k q ≠ 0) : + reg c k = 2 * c.v + q.val := by + have h' : (if 2 * c.v + q.val = (c.ret.getD k.val (0, 0)).1 then (1:ℚ) else 0) ≠ 0 := h + by_cases e : 2 * c.v + q.val = (c.ret.getD k.val (0, 0)).1 + · exact e.symm + · rw [if_neg e] at h'; exact absurd rfl h' + +theorem reg_mem (c : Raw) (k : Fin c.ret.length) : reg c k ∈ c.ret.map Prod.fst := by + unfold reg + rw [List.getD_eq_getElem _ _ k.isLt] + exact List.mem_map.mpr ⟨_, List.getElem_mem k.isLt, rfl⟩ + +end GXD + +#print axioms GXD.roles +#print axioms GXD.scalOf +#print axioms GXD.scalOf_MA +#print axioms GXD.reg_slot +#print axioms GXD.reg_mem diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean new file mode 100644 index 0000000000..9b9925ce26 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean @@ -0,0 +1,55 @@ +import Work.BridgeGeom.B2 +import Work.FoldRate.Group +import Work.GCert.Data.RateB2Gp193 +import Work.GCert.Data.RateB2Gp193x + +/-! +# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `P193`, h = 22: the unit price is the price of the block list + +GENERATED by checks/wht26/gx-data/py/gxrate.py. No `sorry`. Arithmetic only (no certificate is imported). + +One unit: `W = 14692` live roles (`4 v + R`, `v = 1320`, `R = 9412`), label dimension `m = 110`, deficit `D = 3080`. +`invPrice φ` = the price of the blocks of ONE invocation (the kernel-counted list `GXD.P193.hist`, by rank); +`unitPrice φ = 5 * invPrice φ + v * BG.bankPrice h φ = listCost φ BlockAccounting.blocksB2Gp193` (`unit_cost`). +Whole-block `1 - 74745/10^8` (fails at 74746), ten decimals `1 - 7474547/10^10` (fails at 7474550). +-/ + +namespace OAI +namespace PowerSaving +namespace BridgeRate +namespace B2Gp193 +open Finset Binary Matrix RAM CB FoldRate +noncomputable section + +/-- price of the blocks of ONE invocation of `P193` (all roles and scratch copies), by rank -/ +def invPrice (φ : ℕ → ℝ) : ℝ := 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20 + +/-- price of ONE unit of the bridged word B_2 -/ +def unitPrice (φ : ℕ → ℝ) : ℝ := 5 * invPrice φ + ((1320:ℕ):ℝ) * BG.bankPrice 22 φ + +/-- **The unit price is the price of the unit block list.** -/ +theorem unit_cost (φ : ℕ → ℝ) : unitPrice φ = listCost φ BlockAccounting.blocksB2Gp193 := by + unfold unitPrice invPrice + simp only [BG.bankPrice, listCost, BlockAccounting.blocksB2Gp193, List.map_cons, List.map_nil, + List.sum_cons, List.sum_nil] + norm_num [bcost] + try ring + +theorem blocks_x : BlockAccounting.blocksB2Gp193 = BlockAccounting.blocksB2Gp193x := rfl + +theorem unit_cost_x (φ : ℕ → ℝ) : unitPrice φ = listCost φ BlockAccounting.blocksB2Gp193x := by + rw [unit_cost, blocks_x] + +/-- one unit stands for `W * m - D` unit moves -/ +theorem unit_tally : unitPrice (fun r => (r:ℝ)) = ((14692:ℕ):ℝ) * ((110:ℕ):ℝ) - ((3080:ℕ):ℝ) := by + rw [unit_cost] + exact BlockAccounting.blocksB2Gp193_tally + +end +end B2Gp193 +end BridgeRate +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.BridgeRate.B2Gp193.unit_cost +#print axioms OAI.PowerSaving.BridgeRate.B2Gp193.unit_tally diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193Cert.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193Cert.lean new file mode 100644 index 0000000000..3c3250279e --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193Cert.lean @@ -0,0 +1,81 @@ +import Work.GCert.Labels.Main +import Work.GCert.Data.P193Price +import Work.GCert.Data.Gen.P193Tab0 +import Work.GCert.Data.Gen.P193Tab1 +import Work.GCert.Data.Gen.P193Tab2 +import Work.GCert.Data.Gen.P193Tab3 +import Work.GCert.Data.Gen.P193Seg0 +import Work.GCert.Data.Gen.P193Seg1 +import Work.GCert.Data.Gen.P193Seg2 +import Work.GCert.Data.Gen.P193Seg3 +import Work.GCert.Data.Gen.P193Seg4 +import Work.GCert.Data.Gen.P193SegF +import Work.GCert.Data.Gen.P193End +/-! # (key: gx-data) gcert/1 certificate `P193` (h = 22): THE LABEL HALF and its price +GENERATED by checks/wht26/gx-data/py/gxgen.py from gcert1-p11-pr193.json.gz. No `sorry`. +`v = 1320` ports, `R = 9412` slots, 22 scratch copies of rank 20, copy cost `cst = 440`, `N = 262944` unit moves. +* `aux : GLab.Aux`, `valid : GLab.Valid raw aux`: frame table (`tab_ok` composed by `tabK_node`), the replays of phase A + and phase B (`seg_ok` composed by `Run.append`), the final climbs (`segF_ok`, `fin_ok`), the ends (`end_ok`); +* `lab_cost`, `lab_price`: for ANY scalar half `S` the label half `valid.lab S ..` costs + `rcost rA + card C * bcost 20 + rcost rB + rcost rF` = `invPrice φ` = 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20 + (`Gen.P193Hist`, `P193Price`) when there are 22 copies. -/ +set_option maxRecDepth 100000 +namespace GXD.P193 +open GXD OAI.PowerSaving OAI.PowerSaving.CB +theorem raw_h : raw.h = 22 := rfl +theorem raw_v : raw.v = 1320 := rfl +theorem raw_R : raw.R = 9412 := rfl +/-- what the label checks read next to `raw` -/ +noncomputable def aux : GLab.Aux := ⟨par, st0, st2, st5, st6, fins, rA, rB, rF⟩ +theorem fin_ok : GLab.noAdds fins = true := by decide +kernel +theorem cen_ok : (raw.ret.map Prod.fst).all (fun r => Nat.beq (GLab.cdim (par.tab.get (st2.get r))) 20) = true := by + decide +kernel +theorem tab_ok : GLab.tabK par.h par.tab = true := (GLab.tabK_node _ _ _ (GLab.tabK_node _ _ _ tab0_ok tab1_ok) (GLab.tabK_node _ _ _ tab2_ok tab3_ok)) +theorem gatesA_eq : raw.gatesA = seg0 ++ (seg1) := by + show List.flatten [seg0, seg1] = _ + simp only [List.flatten_cons, List.flatten_nil, List.append_nil] +/-- the replay of phase A, composed from its kernel-checked segments -/ +theorem runA : GLab.Run par st0 raw.gatesA st2 rA := by + rw [gatesA_eq] + exact GLab.Run.append (GLab.Run.of_seg seg0_ok) (GLab.Run.of_seg seg1_ok) +theorem gatesB_eq : raw.gatesB = seg2 ++ (seg3 ++ (seg4)) := by + show List.flatten [seg2, seg3, seg4] = _ + simp only [List.flatten_cons, List.flatten_nil, List.append_nil] +/-- the replay of phase B, composed from its kernel-checked segments -/ +theorem runB : GLab.Run par st2 raw.gatesB st5 rB := by + rw [gatesB_eq] + exact GLab.Run.append (GLab.Run.of_seg seg2_ok) (GLab.Run.append (GLab.Run.of_seg seg3_ok) (GLab.Run.of_seg seg4_ok)) +/-- **the kernel-checked label facts** of the certificate -/ +theorem valid : GLab.Valid raw aux where + hn := rfl + tab := tab_ok + runA := runA + runB := runB + runF := GLab.Run.of_seg segF_ok + fin := fin_ok + ends := end_ok +theorem cen_dim (r : Nat) (hr : r ∈ raw.ret.map Prod.fst) : GLab.cdim (par.tab.get (st2.get r)) = 20 := + Nat.eq_of_beq_eq_true (List.all_eq_true.mp cen_ok r hr) +/-- price of the label half over ANY scalar half -/ +theorem lab_cost {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C) + {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)} + (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi) + (reg : C → Nat) (hCc) (hreg) (φ : ℕ → ℝ) : + (valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg).cost φ + = rcost φ rA + (Fintype.card C : ℝ) * bcost φ 20 + rcost φ rB + rcost φ rF := by + obtain ⟨h1, h2, h3, h4⟩ := valid.lab_ranks S Ro cf hMA hMAi hMB hMBi reg hCc hreg + have h5 : ∀ k, bcost φ (((valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg).cen k).dim) = bcost φ 20 := + fun k => by rw [h4 k]; exact congrArg (bcost φ) (cen_dim _ (hreg k)) + rw [GX.Lab.cost, h1, h2, h3, Finset.sum_congr rfl (fun k _ => h5 k), Finset.sum_const, Finset.card_univ, nsmul_eq_mul] + rfl +/-- **the label half costs one invocation price** (`invPrice`, by rank, from the kernel counts) -/ +theorem lab_price {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C) + {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)} + (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi) + (reg : C → Nat) (hCc) (hreg) (hC : Fintype.card C = 22) (φ : ℕ → ℝ) : + (valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg).cost φ = invPrice φ := by + rw [lab_cost S Ro cf hMA hMAi hMB hMBi reg hCc hreg φ, hC, ← inv_cost φ] + norm_num [bcost] +end GXD.P193 +#print axioms GXD.P193.valid +#print axioms GXD.P193.lab_price diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193Price.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193Price.lean new file mode 100644 index 0000000000..eeed011d64 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193Price.lean @@ -0,0 +1,54 @@ +import Work.GCert.Data.Price +import Work.GCert.Data.Gen.P193Hist + +/-! +# (key: gx-data) gcert/1 certificate `P193` (h = 22): the price of one invocation + +GENERATED by checks/wht26/gx-data/py/gxgen.py from gcert1-p11-pr193.json.gz. No `sorry`. +`rA_cost`, `rB_cost`, `rF_cost`: the price of the block rank lists of phase A, phase B and the final climbs +(lists `GXD.P193.rA / rB / rF`, counted by the kernel in `Gen.P193Hist`); +`inv_cost`: with the 22 scratch copies (22 * φ 20) the blocks of ONE invocation cost +`invPrice φ` = 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20; `inv_moves`: they stand for `R h + 2 v (h - 1) + cst = 262944` unit moves. +-/ + +namespace GXD.P193 +open GXD OAI.PowerSaving.CB + +/-- price of the blocks of phase A, from the kernel counts -/ +theorem rA_cost (φ : ℕ → ℝ) : rcost φ rA = 9900 * φ 1 + 500 * φ 2 + 2402 * φ 3 + 82 * φ 4 + 242 * φ 5 + 175 * φ 6 + 88 * φ 7 + 68 * φ 8 + 110 * φ 9 + 132 * φ 10 + 10 * φ 11 + 67 * φ 12 + 3 * φ 13 + 19 * φ 15 := by + rw [GXD.rcost_wsumK φ rA 19 rankBoundA] + simp only [Finset.sum_range_succ, Finset.sum_range_zero, rankA0_ok, rankA1_ok, rankA2_ok, rankA3_ok, rankA4_ok, rankA5_ok, rankA6_ok, rankA7_ok, rankA8_ok, rankA9_ok, rankA10_ok, rankA11_ok, rankA12_ok, rankA13_ok, rankA14_ok, rankA15_ok, rankA16_ok, rankA17_ok, rankA18_ok, rankA19_ok] + norm_num [bcost] + +/-- price of the blocks of phase B, from the kernel counts -/ +theorem rB_cost (φ : ℕ → ℝ) : rcost φ rB = 16638 * φ 1 + 10380 * φ 2 + 4111 * φ 3 + 3476 * φ 4 + 1531 * φ 5 + 1137 * φ 6 + 531 * φ 7 + 1346 * φ 8 + 293 * φ 9 + 629 * φ 10 + 343 * φ 11 + 1067 * φ 12 + 123 * φ 13 + 712 * φ 14 + 1860 * φ 15 + 81 * φ 16 + 39 * φ 17 + 2640 * φ 18 := by + rw [GXD.rcost_wsumK φ rB 19 rankBoundB] + simp only [Finset.sum_range_succ, Finset.sum_range_zero, rankB0_ok, rankB1_ok, rankB2_ok, rankB3_ok, rankB4_ok, rankB5_ok, rankB6_ok, rankB7_ok, rankB8_ok, rankB9_ok, rankB10_ok, rankB11_ok, rankB12_ok, rankB13_ok, rankB14_ok, rankB15_ok, rankB16_ok, rankB17_ok, rankB18_ok, rankB19_ok] + norm_num [bcost] + +/-- price of the blocks of the final climbs, from the kernel counts -/ +theorem rF_cost (φ : ℕ → ℝ) : rcost φ rF = 2640 * φ 1 + 1518 * φ 2 + 3410 * φ 3 + 651 * φ 4 + 177 * φ 5 + 293 * φ 6 + 87 * φ 7 + 117 * φ 8 + 26 * φ 9 + 169 * φ 10 + 1 * φ 11 + 63 * φ 12 + 1 * φ 13 + 30 * φ 14 + 7 * φ 16 + 42 * φ 17 + 13 * φ 18 + 167 * φ 19 := by + rw [GXD.rcost_wsumK φ rF 19 rankBoundF] + simp only [Finset.sum_range_succ, Finset.sum_range_zero, rankF0_ok, rankF1_ok, rankF2_ok, rankF3_ok, rankF4_ok, rankF5_ok, rankF6_ok, rankF7_ok, rankF8_ok, rankF9_ok, rankF10_ok, rankF11_ok, rankF12_ok, rankF13_ok, rankF14_ok, rankF15_ok, rankF16_ok, rankF17_ok, rankF18_ok, rankF19_ok] + norm_num [bcost] + +/-- price of the blocks of ONE invocation (x roles, y roles, slots, scratch copies), by rank -/ +def invPrice (φ : ℕ → ℝ) : ℝ := 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20 + +/-- **Price of ONE invocation** in the shape of `GX.Lab.cost`: phase A, the scratch copies, phase B, final climbs. -/ +theorem inv_cost (φ : ℕ → ℝ) : rcost φ rA + (22 * φ 20) + rcost φ rB + rcost φ rF = invPrice φ := by + rw [rA_cost, rB_cost, rF_cost] + unfold invPrice + ring + +theorem inv_moves : invPrice (fun r => (r:ℝ)) = 9412 * 22 + 2 * 1320 * (22 - 1) + 440 := by + unfold invPrice + norm_num + +theorem inv_count : invPrice (fun _ => (1:ℝ)) = 70169 := by + unfold invPrice + norm_num + +end GXD.P193 +#print axioms GXD.P193.inv_cost +#print axioms GXD.P193.inv_moves diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean new file mode 100644 index 0000000000..3539192ac1 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean @@ -0,0 +1,17 @@ +import Work.GCert.Scalar.Main +import Work.GCert.Data.Gen.P193Scal0 +import Work.GCert.Data.Gen.P193Scal1 +import Work.GCert.Data.Gen.P193YChk +/-! # (key: gx-data) gcert/1 certificate `P193` (h = 22): THE SCALAR HALF `scalOK : GS.ScalOK raw` +GENERATED by checks/wht26/gx-data/py/gxgen.py from gcert1-p11-pr193.json.gz. No `sorry`. +Kernel checks `scal_ok` (2 source batches) and `ychk_ok`. -/ +namespace GXD.P193 +open GXD +theorem scalOK : GS.ScalOK raw where + hv := by decide + hs := fun t ht => + if h0 : t < 660 then ⟨0, 660, Nat.zero_le t, h0, GS.XRun.of_check scal0_ok⟩ else + ⟨660, 660, Nat.le_of_not_lt h0, Nat.lt_of_lt_of_le ht (by decide), GS.XRun.of_check scal1_ok⟩ + hyc := fun t ht => ⟨0, 1320, Nat.zero_le t, Nat.lt_of_lt_of_le ht (by decide), GS.YRun.of_check ychk_ok⟩ +end GXD.P193 +#print axioms GXD.P193.scalOK diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean new file mode 100644 index 0000000000..df563cde73 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean @@ -0,0 +1,327 @@ +import Work.BlockAccounting.Moment +import Work.BlockAccounting.Sharp +import Work.FoldRate.RankTools + +/-! +# (key: gx-data) Rate lemmas for one unit of the bridged word B_2 with the gcert/1 circuit `P193` (h = 22) + +GENERATED by checks/wht26/gx-data/py/gxrate.py with the text generators of checks/wht9/bridge-rate/py +(used read-only). No `sorry`. Arithmetic only. + +One unit: `W = 14692` live roles, label dimension `m = 110`, deficit `D = 3080`. + +* `blocksB2Gp193x` the block list (rank, count) of ONE unit; +* `fold_B2Gp193x` whole-block fact at `1 - 7474547/10^10`, fill parameter `s = 40` (slack 2.37e-12); +* `fold_B2Gp193x_sharp` `W ≤ moment` at `1 - 7474550/10^10`: fails for every table (excess 1.40e-10); +* `foldRank_B2Gp193x` per-rank fact at `1 - 4058344/10^10`, `s = 40` (slack 1.10e-07); +* `foldRank_B2Gp193x_sharp` per-rank: fails for every table at `1 - 4058349/10^10`. + +That a Lean word has `G` times this list as its block histogram is NOT proved here. +-/ + +set_option maxRecDepth 4000 + +namespace OAI.PowerSaving.BlockAccounting +open Finset + +/-- Block list `(rank, count)` of ONE unit of the bridged five-stage word B_2 with the gcert/1 circuit `P193` (h = 22, m = 110): five invocations (ALL blocks of one invocation: x roles, y roles, slots, scratch copies; the kernel-counted list `GXD.P193.hist`), and for each of the v = 1320 ports the idle climbs of its two twins (2 blocks each of rank h-1, 2h-2, 2h+2, 4); R = 9412 slots shared by the five stages. -/ +def blocksB2Gp193x : List (ℕ × ℕ) := [(1, 145890), (2, 61990), (3, 49615), (4, 23685), (5, 9750), (6, 8025), (7, 3530), (8, 7655), (9, 2145), (10, 4650), (11, 1770), (12, 5985), (13, 635), (14, 3710), (15, 9395), (16, 440), (17, 405), (18, 13265), (19, 835), (20, 110), (21, 2640), (42, 2640), (46, 2640)] + +/-- total rank of one unit (= number of unit moves it stands for): `W * m - D` -/ +theorem blocksB2Gp193x_rank : (blocksB2Gp193x.map fun b => b.1 * b.2).sum = 14692 * 110 - 3080 := by + norm_num [blocksB2Gp193x] + +/-- the same as a price: the unit list at the price list `r ↦ r` (shape of `hPid` in `RAM.engine_program_group_perRank`) -/ +theorem blocksB2Gp193x_tally : (blocksB2Gp193x.map fun b => (b.2:ℝ) * ((b.1:ℕ):ℝ)).sum = ((14692:ℕ):ℝ) * ((110:ℕ):ℝ) - ((3080:ℕ):ℝ) := by + norm_num [blocksB2Gp193x] + +theorem blocksB2Gp193x_count : (blocksB2Gp193x.map fun b => b.2).sum = 361405 := by + norm_num [blocksB2Gp193x] + +theorem blocksB2Gp193x_proper : ∀ b ∈ blocksB2Gp193x, 1 ≤ b.1 ∧ b.1 < 110 := by + decide + +/-- **whole-block fact** of `blocksB2Gp193x` at exponent `1 - 7474547/10^10`, fill parameter `s = 40`: the hypothesis `hnum` of `RAM.engine_program_group_list` (u = 110, W = 14692). Relative slack of the rational certificate: 2.37e-12. -/ +theorem fold_B2Gp193x : + ((2:ℝ)^40 - 1) * moment 110 (1 - 7474547/(10:ℝ)^10) blocksB2Gp193x + + ((14692:ℕ):ℝ) * (T 110 (1 - 7474547/(10:ℝ)^10) (110 - 1) 1 + T 110 (1 - 7474547/(10:ℝ)^10) 1 1) + < (2:ℝ)^40 * ((14692:ℕ):ℝ) := by + have hsum : moment 110 (1 - 7474547/(10:ℝ)^10) blocksB2Gp193x + = T 110 (1 - 7474547/(10:ℝ)^10) 1 145890 + (T 110 (1 - 7474547/(10:ℝ)^10) 2 61990 + (T 110 (1 - 7474547/(10:ℝ)^10) 3 49615 + (T 110 (1 - 7474547/(10:ℝ)^10) 4 23685 + (T 110 (1 - 7474547/(10:ℝ)^10) 5 9750 + (T 110 (1 - 7474547/(10:ℝ)^10) 6 8025 + (T 110 (1 - 7474547/(10:ℝ)^10) 7 3530 + (T 110 (1 - 7474547/(10:ℝ)^10) 8 7655 + (T 110 (1 - 7474547/(10:ℝ)^10) 9 2145 + (T 110 (1 - 7474547/(10:ℝ)^10) 10 4650 + (T 110 (1 - 7474547/(10:ℝ)^10) 11 1770 + (T 110 (1 - 7474547/(10:ℝ)^10) 12 5985 + (T 110 (1 - 7474547/(10:ℝ)^10) 13 635 + (T 110 (1 - 7474547/(10:ℝ)^10) 14 3710 + (T 110 (1 - 7474547/(10:ℝ)^10) 15 9395 + (T 110 (1 - 7474547/(10:ℝ)^10) 16 440 + (T 110 (1 - 7474547/(10:ℝ)^10) 17 405 + (T 110 (1 - 7474547/(10:ℝ)^10) 18 13265 + (T 110 (1 - 7474547/(10:ℝ)^10) 19 835 + (T 110 (1 - 7474547/(10:ℝ)^10) 20 110 + (T 110 (1 - 7474547/(10:ℝ)^10) 21 2640 + (T 110 (1 - 7474547/(10:ℝ)^10) 42 2640 + (T 110 (1 - 7474547/(10:ℝ)^10) 46 2640 + 0)))))))))))))))))))))) := rfl + have b0 : T 110 (1 - 7474547/(10:ℝ)^10) 1 145890 ≤ ((16636758090767:ℝ)/12500000000) := + term_le 110 1 145890 (7474547/(10:ℝ)^10) ((2350240182897:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 27 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b1 : T 110 (1 - 7474547/(10:ℝ)^10) 2 61990 ≤ ((22609438945563:ℝ)/20000000000) := + term_le 110 2 61990 (7474547/(10:ℝ)^10) ((2003666592617:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 26 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b2 : T 110 (1 - 7474547/(10:ℝ)^10) 3 49615 ≤ ((135678423202471:ℝ)/100000000000) := + term_le 110 3 49615 (7474547/(10:ℝ)^10) ((1800934038563:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 24 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b3 : T 110 (1 - 7474547/(10:ℝ)^10) 4 23685 ≤ ((43170446048157:ℝ)/50000000000) := + term_le 110 4 23685 (7474547/(10:ℝ)^10) ((1657093002337:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 24 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b4 : T 110 (1 - 7474547/(10:ℝ)^10) 5 9750 ≤ ((44420693582197:ℝ)/100000000000) := + term_le 110 5 9750 (7474547/(10:ℝ)^10) ((38638030667:ℝ)/12500000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 23 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b5 : T 110 (1 - 7474547/(10:ℝ)^10) 6 8025 ≤ ((43867998737293:ℝ)/100000000000) := + term_le 110 6 8025 (7474547/(10:ℝ)^10) ((1454360448283:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 22 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b6 : T 110 (1 - 7474547/(10:ℝ)^10) 7 3530 ≤ ((2250993477097:ℝ)/10000000000) := + term_le 110 7 3530 (7474547/(10:ℝ)^10) ((2754570216739:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 22 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b7 : T 110 (1 - 7474547/(10:ℝ)^10) 8 7655 ≤ ((55781903078171:ℝ)/100000000000) := + term_le 110 8 7655 (7474547/(10:ℝ)^10) ((1310519412057:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 21 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b8 : T 110 (1 - 7474547/(10:ℝ)^10) 9 2145 ≤ ((17582868029823:ℝ)/100000000000) := + term_le 110 9 2145 (7474547/(10:ℝ)^10) ((1251627894229:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 21 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b9 : T 110 (1 - 7474547/(10:ℝ)^10) 10 4650 ≤ ((10587140349809:ℝ)/25000000000) := + term_le 110 10 4650 (7474547/(10:ℝ)^10) ((2997369091:ℝ)/1250000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 21 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b10 : T 110 (1 - 7474547/(10:ℝ)^10) 11 1770 ≤ ((17730489316207:ℝ)/100000000000) := + term_le 110 11 1770 (7474547/(10:ℝ)^10) ((575646273249:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 20 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b11 : T 110 (1 - 7474547/(10:ℝ)^10) 12 5985 ≤ ((65399123107589:ℝ)/100000000000) := + term_le 110 12 5985 (7474547/(10:ℝ)^10) ((1107786858003:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 20 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b12 : T 110 (1 - 7474547/(10:ℝ)^10) 13 635 ≤ ((3758266936203:ℝ)/50000000000) := + term_le 110 13 635 (7474547/(10:ℝ)^10) ((533882752083:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 20 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b13 : T 110 (1 - 7474547/(10:ℝ)^10) 14 3710 ≤ ((47290992642379:ℝ)/100000000000) := + term_le 110 14 3710 (7474547/(10:ℝ)^10) ((2061423036179:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b14 : T 110 (1 - 7474547/(10:ℝ)^10) 15 9395 ≤ ((128304571925681:ℝ)/100000000000) := + term_le 110 15 9395 (7474547/(10:ℝ)^10) ((498107541173:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b15 : T 110 (1 - 7474547/(10:ℝ)^10) 16 440 ≤ ((6409229123791:ℝ)/100000000000) := + term_le 110 16 440 (7474547/(10:ℝ)^10) ((963945821777:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b16 : T 110 (1 - 7474547/(10:ℝ)^10) 17 405 ≤ ((1253566561349:ℝ)/20000000000) := + term_le 110 17 405 (7474547/(10:ℝ)^10) ((933633510869:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b17 : T 110 (1 - 7474547/(10:ℝ)^10) 18 13265 ≤ ((217357516651669:ℝ)/100000000000) := + term_le 110 18 13265 (7474547/(10:ℝ)^10) ((905054303949:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b18 : T 110 (1 - 7474547/(10:ℝ)^10) 19 835 ≤ ((14441670418763:ℝ)/100000000000) := + term_le 110 19 835 (7474547/(10:ℝ)^10) ((1756041386627:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b19 : T 110 (1 - 7474547/(10:ℝ)^10) 20 110 ≤ ((2002550068509:ℝ)/100000000000) := + term_le 110 20 110 (7474547/(10:ℝ)^10) ((21309351153:ℝ)/12500000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b20 : T 110 (1 - 7474547/(10:ℝ)^10) 21 2640 ≤ ((25231210703361:ℝ)/50000000000) := + term_le 110 21 2640 (7474547/(10:ℝ)^10) ((165595792807:ℝ)/100000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b21 : T 110 (1 - 7474547/(10:ℝ)^10) 42 2640 ≤ ((100872567578631:ℝ)/100000000000) := + term_le 110 42 2640 (7474547/(10:ℝ)^10) ((962810747511:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 15 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b22 : T 110 (1 - 7474547/(10:ℝ)^10) 46 2640 ≤ ((22094393345401:ℝ)/20000000000) := + term_le 110 46 2640 (7474547/(10:ℝ)^10) ((174367793861:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 14 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b23 : T 110 (1 - 7474547/(10:ℝ)^10) 109 1 ≤ ((198183171:ℝ)/200000000) := + term_le 110 109 1 (7474547/(10:ℝ)^10) ((1826496713:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 5 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have b24 : T 110 (1 - 7474547/(10:ℝ)^10) 1 1 ≤ ((456145263:ℝ)/50000000000) := + term_le 110 1 1 (7474547/(10:ℝ)^10) ((2350240182897:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 27 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + have e1 : ((2:ℝ)^40 - 1) = 1099511627775 := by norm_num + have e2 : (2:ℝ)^40 = 1099511627776 := by norm_num + have e3 : ((14692:ℕ):ℝ) = 14692 := by norm_num + have e4 : T 110 (1 - 7474547/(10:ℝ)^10) (110 - 1) 1 = T 110 (1 - 7474547/(10:ℝ)^10) 109 1 := rfl + rw [e1, e2, e3, e4, hsum] + linarith + +/-- **sharpness**: at exponent `1 - 7474550/10^10` the moment of one unit is at least its number of live roles `14692`, so the moment criterion fails for EVERY table made of copies of the unit and idle roles (`FoldRate.block_fails`). Relative excess of the rational certificate: 1.40e-10. -/ +theorem fold_B2Gp193x_sharp : ((14692:ℕ):ℝ) ≤ moment 110 (1 - 7474550/(10:ℝ)^10) blocksB2Gp193x := by + have hsum : moment 110 (1 - 7474550/(10:ℝ)^10) blocksB2Gp193x + = T 110 (1 - 7474550/(10:ℝ)^10) 1 145890 + (T 110 (1 - 7474550/(10:ℝ)^10) 2 61990 + (T 110 (1 - 7474550/(10:ℝ)^10) 3 49615 + (T 110 (1 - 7474550/(10:ℝ)^10) 4 23685 + (T 110 (1 - 7474550/(10:ℝ)^10) 5 9750 + (T 110 (1 - 7474550/(10:ℝ)^10) 6 8025 + (T 110 (1 - 7474550/(10:ℝ)^10) 7 3530 + (T 110 (1 - 7474550/(10:ℝ)^10) 8 7655 + (T 110 (1 - 7474550/(10:ℝ)^10) 9 2145 + (T 110 (1 - 7474550/(10:ℝ)^10) 10 4650 + (T 110 (1 - 7474550/(10:ℝ)^10) 11 1770 + (T 110 (1 - 7474550/(10:ℝ)^10) 12 5985 + (T 110 (1 - 7474550/(10:ℝ)^10) 13 635 + (T 110 (1 - 7474550/(10:ℝ)^10) 14 3710 + (T 110 (1 - 7474550/(10:ℝ)^10) 15 9395 + (T 110 (1 - 7474550/(10:ℝ)^10) 16 440 + (T 110 (1 - 7474550/(10:ℝ)^10) 17 405 + (T 110 (1 - 7474550/(10:ℝ)^10) 18 13265 + (T 110 (1 - 7474550/(10:ℝ)^10) 19 835 + (T 110 (1 - 7474550/(10:ℝ)^10) 20 110 + (T 110 (1 - 7474550/(10:ℝ)^10) 21 2640 + (T 110 (1 - 7474550/(10:ℝ)^10) 42 2640 + (T 110 (1 - 7474550/(10:ℝ)^10) 46 2640 + 0)))))))))))))))))))))) := rfl + have c0 : ((665470322971:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 1 145890 := + term_ge 110 1 145890 (7474550/(10:ℝ)^10) ((4700480365791:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 4 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c1 : ((565235973477:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 2 61990 := + term_ge 110 2 61990 (7474550/(10:ℝ)^10) ((4007333185231:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 4 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c2 : ((1356784232023:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 3 49615 := + term_ge 110 3 49615 (7474550/(10:ℝ)^10) ((3601868077123:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c3 : ((431704460547:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 4 23685 := + term_ge 110 4 23685 (7474550/(10:ℝ)^10) ((3314186004671:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c4 : ((44420693593:ℝ)/100000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 5 9750 := + term_ge 110 5 9750 (7474550/(10:ℝ)^10) ((3091042453357:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c5 : ((87735997501:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 6 8025 := + term_ge 110 6 8025 (7474550/(10:ℝ)^10) ((2908720896563:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c6 : ((112549673893:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 7 3530 := + term_ge 110 7 3530 (7474550/(10:ℝ)^10) ((86080319273:ℝ)/31250000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c7 : ((557819030987:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 8 7655 := + term_ge 110 8 7655 (7474550/(10:ℝ)^10) ((2621038824111:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c8 : ((87914340183:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 9 2145 := + term_ge 110 9 2145 (7474550/(10:ℝ)^10) ((500651157691:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c9 : ((423485614161:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 10 4650 := + term_ge 110 10 4650 (7474550/(10:ℝ)^10) ((2397895272797:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c10 : ((88652446617:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 11 1770 := + term_ge 110 11 1770 (7474550/(10:ℝ)^10) ((2302585092993:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c11 : ((130798246269:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 12 5985 := + term_ge 110 12 5985 (7474550/(10:ℝ)^10) ((2215573716003:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c12 : ((15033067751:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 13 635 := + term_ge 110 13 635 (7474550/(10:ℝ)^10) ((2135531008329:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c13 : ((23645496331:ℝ)/50000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 14 3710 := + term_ge 110 14 3710 (7474550/(10:ℝ)^10) ((128838939761:ℝ)/62500000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c14 : ((320761429947:ℝ)/250000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 15 9395 := + term_ge 110 15 9395 (7474550/(10:ℝ)^10) ((1992430164689:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 12 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c15 : ((1001442051:ℝ)/15625000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 16 440 := + term_ge 110 16 440 (7474550/(10:ℝ)^10) ((1927891643551:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c16 : ((62678328093:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 17 405 := + term_ge 110 17 405 (7474550/(10:ℝ)^10) ((373453404347:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c17 : ((1086787583699:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 18 13265 := + term_ge 110 18 13265 (7474550/(10:ℝ)^10) ((362021721579:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c18 : ((28883340849:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 19 835 := + term_ge 110 19 835 (7474550/(10:ℝ)^10) ((13719073333:ℝ)/7812500000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c19 : ((20025500693:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 20 110 := + term_ge 110 20 110 (7474550/(10:ℝ)^10) ((1704748092237:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c20 : ((63078026783:ℝ)/125000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 21 2640 := + term_ge 110 21 2640 (7474550/(10:ℝ)^10) ((1655957928067:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c21 : ((126090709507:ℝ)/125000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 42 2640 := + term_ge 110 42 2640 (7474550/(10:ℝ)^10) ((240702686877:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 1 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have c22 : ((552359833771:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 46 2640 := + term_ge 110 46 2640 (7474550/(10:ℝ)^10) ((435919484651:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 1 11 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) + have e3 : ((14692:ℕ):ℝ) = 14692 := by norm_num + rw [e3, hsum] + linarith + +/-- **per-rank fact** at exponent `1 - 4058344/10^10`, fill parameter `s = 40`: the hypothesis `hnum` of `RAM.engine_program_group_perRank` (u = 110, W = 14692, D = 3080). Relative slack: 1.10e-07. -/ +theorem foldRank_B2Gp193x : + (2:ℝ)^40 * ((14692:ℕ):ℝ) * (((110:ℕ):ℝ) - ((110:ℕ):ℝ)^(1 - 4058344/(10:ℝ)^10)) < ((2:ℝ)^40 - 1) * ((3080:ℕ):ℝ) := + FoldRate.rank_fact3 110 14692 3080 40 (4058344/(10:ℝ)^10) ((2350240182897:ℝ)/500000000000) (by norm_num) (by norm_num) (by norm_num) + (log_le_of_le_expsum _ _ 27 (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + +/-- **per-rank sharpness**: at exponent `1 - 4058349/10^10` the deficit of one unit is at most `W (m - m^z)`, so the per-rank inequality fails for EVERY table made of copies of the unit and idle roles (`FoldRate.rank_fails`). -/ +theorem foldRank_B2Gp193x_sharp : + ((3080:ℕ):ℝ) ≤ ((14692:ℕ):ℝ) * (((110:ℕ):ℝ) - ((110:ℕ):ℝ)^(1 - 4058349/(10:ℝ)^10)) := + FoldRate.rank_fact_sharp 110 14692 3080 (4058349/(10:ℝ)^10) ((4700480365791:ℝ)/1000000000000) (by norm_num) (by norm_num) (by norm_num) + (le_log_of_expbound _ _ 4 10 (by norm_num) (by norm_num) (by norm_num) + (by norm_num [Finset.sum_range_succ, Nat.factorial])) + (by norm_num) (by norm_num) + +end OAI.PowerSaving.BlockAccounting + +#print axioms OAI.PowerSaving.BlockAccounting.fold_B2Gp193x +#print axioms OAI.PowerSaving.BlockAccounting.fold_B2Gp193x_sharp +#print axioms OAI.PowerSaving.BlockAccounting.foldRank_B2Gp193x +#print axioms OAI.PowerSaving.BlockAccounting.foldRank_B2Gp193x_sharp diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__Raw.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__Raw.lean new file mode 100644 index 0000000000..499fa67e88 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__Raw.lean @@ -0,0 +1,93 @@ +/-! +# (key: gx-data) Raw Lean form of a `gcert/1` file (EXTENDED-CERT-SPEC section 1) + +Core Lean only (no Mathlib): the generated data modules import nothing else, so they elaborate +cheaply. Field order and meaning are those of the JSON; `pairs` and `cuts` are DERIVED by the +generator (`checks/wht26/gx-data/py/gxgen.py`) and must be tested by the label check. + +Registers: `x_t = t`, `y_t = v + t`, slot `k = 2 v + k`. Vectors are bit masks. +-/ + +namespace GXD + +/-- the rational `(-1)^neg * num / den` (`den > 0`) -/ +structure Co where + neg : Bool + num : Nat + den : Nat + deriving DecidableEq, Repr + +/-- a gate: every register named (extras too) first moves to `frame`; then the adds happen in order -/ +inductive Gate where + /-- `tgt += co * src` for each `(tgt, co)` -/ + | out (frame src : Nat) (tgts : List (Nat × Co)) (extra : List Nat) + /-- `tgt += co * src` for each `(src, co)` -/ + | inn (frame tgt : Nat) (srcs : List (Nat × Co)) + deriving Repr + +/-- the scatter `y_t += sum over k of coef(t, k) * total k` -/ +inductive Scat where + /-- star rule: `inside` if coordinate `k` is in port `t`, else `outside` -/ + | star (inside outside : Co) + /-- explicit rows: row `t` = `[(total k, coef)]` -/ + | table (rows : List (List (Nat × Co))) + deriving Repr + +/-- a `gcert/1` file -/ +structure Raw where + h : Nat + v : Nat + R : Nat + N : Nat + cst : Nat + /-- `v` masks: the unit vectors of the ports -/ + ports : List Nat + /-- CHUNKED frame table (`flatten` = the table); an entry is a reduced echelon basis, leading bits descending -/ + frames : List (List (List Nat)) + /-- frame id per register at the start -/ + start : List Nat + /-- frame id per register at the end -/ + final : List Nat + /-- position `k` = retained total `k`: (slot REGISTER, frame id) -/ + ret : List (Nat × Nat) + scat : Scat + /-- CHUNKED gates before the scatter -/ + A : List (List Gate) + /-- CHUNKED gates after the scatter -/ + B : List (List Gate) + /-- CHUNKED, derived: the distinct climbs `(a, b)`, `a ≠ b`, in order of first use -/ + pairs : List (List (Nat × Nat)) + /-- derived: replay cut points `(gates of A ++ B before the cut, frame id per register there)` -/ + cuts : List (Nat × List Nat) + /-- slots that start off the zero frame `(slot REGISTER, frame id)` (rule 2); EMPTY on the path -/ + ext : List (Nat × Nat) + /-- derived (questions.md Q1): per port `h` masks, the first is the port, pairwise dots = delta + (an orthonormal basis of `F_2^h` through the port) -/ + obase : List (List Nat) + +/-- registers of a gate, in the order they are moved -/ +def Gate.regs : Gate → List Nat + | .out _ s ts ex => s :: (ts.map Prod.fst ++ ex) + | .inn _ t ss => t :: ss.map Prod.fst + +/-- frame of a gate -/ +def Gate.frame : Gate → Nat + | .out f _ _ _ => f + | .inn f _ _ => f + +/-- single adds `(tgt, src, co)` of a gate, in order -/ +def Gate.adds : Gate → List (Nat × Nat × Co) + | .out _ s ts _ => ts.map fun p => (p.1, s, p.2) + | .inn _ t ss => ss.map fun p => (t, p.1, p.2) + +/-- the frame table -/ +def Raw.frameTab (c : Raw) : List (List Nat) := c.frames.flatten + +/-- all gates of phase A / phase B -/ +def Raw.gatesA (c : Raw) : List Gate := c.A.flatten +def Raw.gatesB (c : Raw) : List Gate := c.B.flatten + +/-- number of registers -/ +def Raw.nreg (c : Raw) : Nat := 2 * c.v + c.R + +end GXD diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean new file mode 100644 index 0000000000..0392e1f05b --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean @@ -0,0 +1,38 @@ +import Work.GCert.Data.B2Gp193Main + +/-! +Comparator solution `Work.GCert.Data.SolutionB2Gp193x` (not part of github.com/openai/math). + +The whole proof is in `Work.GCert.Data.B2Gp193Main` (and its imports), which ends with + + theorem wht_main_B2Gp193x : + ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - 7474547/(10:ℝ)^10) W + +This module adds nothing mathematical: it defines `WHT.WHTTimeBoundsAt` with the text of the +challenge (the same definition as `BlockWHT.WHTTimeBoundsAt` of `Work.Block.WHT`) and restates +the theorem under the name the challenge uses. Its proof is that single term. +`WHTProgram`, `wht`, `whtInKind`, `whtOutKind` come from `WHTCheck.Solution`. +-/ + +namespace OAI +namespace PowerSaving +namespace WHT +open Filter Asymptotics + +/-- `W = O(2^k (k+1)^β)` and `W = o(2^k k)` (`WHTTimeBounds` is the case `β = 1 - 2/10^11`). -/ +def WHTTimeBoundsAt (β : ℝ) (W : ℕ → ℕ) : Prop := + let W' := fun k : ℕ => (W k : ℝ) + W' =O[atTop] (fun k : ℕ => (2:ℝ)^k * ((k:ℝ)+1)^β) ∧ + W' =o[atTop] (fun k : ℕ => (2:ℝ)^k * (k:ℝ)) + +/-- **Walsh-Hadamard transform with the improved exponent**: the statement `WHTGoal` of +`WHTCheck/Challenge.lean` with `1 - 2/10^11` replaced by `1 - 7474547/10^10`. -/ +theorem wht_main_block_B2Gp193x : + ∃ solve W, WHTProgram solve W ∧ WHTTimeBoundsAt (1 - 7474547/(10:ℝ)^10) W := + wht_main_B2Gp193x + +end WHT +end PowerSaving +end OAI + +#print axioms OAI.PowerSaving.WHT.wht_main_block_B2Gp193x diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Adapt.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Adapt.lean new file mode 100644 index 0000000000..24e261ad08 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Adapt.lean @@ -0,0 +1,183 @@ +import Work.GFrame.Engine.Letters + +/-! +# GFrame engine, part 2: the free adapter letters in the RAM (agent key: eng-ram) + +One free adapter on the array of one role is ONE linear pass, no recursive call: + +* `Gear.permuted` the address of `x ↦ G.π x` (columnwise), O(1) per entry with the XOR table + (as `Gear.shifted`, through `Gear.body_additive` of `Work/Block/Address.lean`); +* `Gear.phase_row` the `f`-bit row `(z·x_j + c)_j` of an entry, O(1); +* `GRig` a rig together with the table of `lum` over `Bits f` (`i^popcount`); +* `Rig.perm_cost`, `GRig.phase_cost` the two letters, work = length of the role table. +-/ + +set_option linter.unusedSectionVars false + +namespace OAI +namespace PowerSaving.GF +open Binary Matrix Finset +noncomputable section +section +variable {α : Type*} [Fintype α] [DecidableEq α] + +/-- An address permutation acts on an array with `f` columns by moving the address. -/ +lemma act_perm (G : APerm α) (f : ℕ) (x : Sky α f → ℂ) (j : Sky α f) : + matAct f (permMat G) x j = x (fun k => G.π (j k)) := by + change ∑ i, digitProd (fun _ : Fin f => permMat G) j i * x i = _ + have h (i : Sky α f) : digitProd (fun _ : Fin f => permMat G) j i = + if (fun k => G.π (j k)) = i then 1 else 0 := + prod_delta (κ:=fun _ : Fin f => Space α) _ _ + simp_rw [h] + simp [ite_mul] + +/-- A diagonal phase acts by the scalar `lum` of the row of phase bits. -/ +lemma act_phase (z : Space α) (c : F) (f : ℕ) (x : Sky α f → ℂ) (j : Sky α f) : + matAct f (phaseMat z c) x j = lum (fun k : Fin f => dot z (j k) + c) * x j := by + unfold matAct phaseMat + rw [digitProd_diagonal, Matrix.mulVec_diagonal] + rfl + +end +end +end PowerSaving.GF + +namespace PowerSaving.RAM +open Binary Ty Finset GF +noncomputable section +universe U V +variable {ι : Type U} + +namespace Gear +variable {m : Bool} {B f : ι → ℕ} {s t : Ty} {x : ι → T s} +variable (g : Gear m B f s x) +include g +variable {α : Type*} [Fintype α] [DecidableEq α] (X : Layout α) (r : ℕ) + +/-- Address moved by an additive bijection (across the `f` columns). -/ +lemma permuted (G : APerm α) (v : ∀ i, Grid α r (f i)) + (h : Knows m B s x (fun i => (book X r (f i)).loc (v i))) : + Knows m B s x (fun i => (book X r (f i)).loc ((v i).1, + fun j => G.π ((v i).2 j))) := by + obtain ⟨hg, hh⟩ := g.grid_fetch X r (v:=v) h + let K (i : ι) : Grid α r (f i) := ((v i).1, fun j => G.π ((v i).2 j)) + apply g.grid_combine X r (v:=K) hg (g.body_set X (fun i => (K i).2) _) + intro a + exact g.body_additive X (fun y => G.π y a) (fun p q => by rw [G.add]; rfl) + (fun i => (v i).2) hh + +/-- The row of phase bits `(z·x_j + c)_j` of an address. -/ +lemma phase_row (z : Space α) (c : F) (v : ∀ i, Grid α r (f i)) + (h : Knows m B s x (fun i => (book X r (f i)).loc (v i))) : + Knows m B s x (fun i => (bits (f i)).loc (fun l : Fin (f i) => dot z ((v i).2 l) + c)) := by + obtain ⟨_, hh⟩ := g.grid_fetch X r (v:=v) h + have hv := g.body_additive X (fun y => dot z y) (fun p q => dot_add_right z p q) + (fun i => (v i).2) hh + rcases bit_cases c with hc|hc + · refine hv.cong (fun _ => rfl) (fun i => ?_) + simp [hc] + · let q i : Bits (f i) × Bits (f i) := (fun l => dot z ((v i).2 l), fun _ => 1) + have hf : Knows m B s x (fun i => (bits (f i)).loc (q i).2) := + (g.power.minus (Can.wc 1) g.bound).cong (fun _ => rfl) (fun i => by + change 2^f i-1=(bits _).loc (fun _ => 1) + rw [← bits_lit] + simp) + refine (g.addr_xor q hv hf).cong (fun _ => rfl) (fun i => ?_) + apply congr_arg (bits (f i)).loc + ext k; simp [q,hc] + +end Gear + +/-- A rig together with the table of `lum` over `Bits f` (the fourth roots `i^popcount`). -/ +structure GRig {α ρ : Type*} (m : Bool) (B f : ι → ℕ) (s : Ty) (x : ι → T s) + (cl : Paint) (X : Layout α) (R : Layout ρ) (r : ℕ) (v : ∀ i, Grain α ρ r (f i)) : Prop where + rig : Rig m B f s x cl X R r v + lumT : Has m B s (a sc) x fun i => (bits (f i)).tape (fun y : Bits (f i) => lum y) + +section +variable {α ρ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ] +variable {m : Bool} {B f : ι → ℕ} {s t : Ty} {x : ι → T s} + {cl : Paint} {X : Layout α} {R : Layout ρ} {r : ℕ} {v : ∀ i, Grain α ρ r (f i)} + +omit [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ] in +lemma GRig.gather (h : GRig m B f s x cl X R r v) {κ : Type V} (j : κ → ι) (y : κ → T t) + (g : Has m (fun k => B (j k)) t s y (fun k => x (j k))) : + GRig m (fun k => B (j k)) (fun k => f (j k)) t y cl X R r (fun k => v (j k)) := + ⟨h.rig.gather j y g, g.then_do (h.lumT.reindex j)⟩ + +lemma Rig.ferry_perm (h : Rig m B f s x cl X R r v) (l : ρ) (G : APerm α) + (q : ∀ i, Beach α ρ r (f i)) + (g : Knows m B s x fun i => (fleet X R r (f i)).loc (q i)) : + Has m B s (c cl) x fun i => ferryM r (f i) l (permMat G) (v i) (q i) := by + obtain ⟨hl,hk⟩ := h.addrs q g + have hb := h.read q hl hk + let t i : Beach α ρ r (f i) := ((q i).1,(q i).2.1,fun j => G.π ((q i).2.2 j)) + have hh := h.read t hl (h.gear.permuted X r G _ hk) + apply Can.cases_code (q:=fun i => (q i).1) R hl + intro p + by_cases he : p=l + · refine (hh.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl) + (fun i => ?_) + have hql : (q i.val).1 = l := i.property.trans he + change v i.val ((q i.val).1,(q i.val).2.1,fun j => G.π ((q i.val).2.2 j)) = _ + simp only [ferryM, roleAct, hql, if_true, act_perm] + · refine (hb.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl) + (fun i => ?_) + have H : (q i.val).1≠l := fun hh => he (i.property.symm.trans hh) + simp only [ferryM, roleAct, H, ite_false, Prod.mk.eta] + +/-- **Free address permutation**: one pass over the role table. -/ +lemma Rig.perm_cost (h : Rig m B f s x cl X R r v) (l : ρ) (G : APerm α) : + Can m B s (a (c cl)) x + (fun i => (fleet X R r (f i)).tape (ferryM r (f i) l (permMat G) (v i))) + (fun i => (fleet X R r (f i)).n) := by + refine h.chart _ ?_ + let S := Σ i, Beach α ρ r (f i) + have hg := h.gather (fun d : S => d.1) + (fun d => ((x d.1, (fleet X R r (f d.1)).loc d.2):T (p s w))) (t:=p s w) Can.first + exact hg.ferry_perm l G (fun d => d.2) Can.second + +lemma GRig.ferry_phase (h : GRig m B f s x cl X R r v) (l : ρ) (z : Space α) (c' : F) + (q : ∀ i, Beach α ρ r (f i)) + (g : Knows m B s x fun i => (fleet X R r (f i)).loc (q i)) : + Has m B s (c cl) x fun i => ferryM r (f i) l (phaseMat z c') (v i) (q i) := by + obtain ⟨hl,hk⟩ := h.rig.addrs q g + have hb := h.rig.read q hl hk + have hrow := h.rig.gear.phase_row X r z c' (fun i => (q i).2) hk + have hs : Has m B s sc x + (fun i => lum (fun k : Fin (f i) => dot z ((q i).2.2 k) + c')) := + Can.fetch (t:=sc) (L:=fun i => bits (f i)) (v:=fun i (y : Bits (f i)) => lum y) + (i:=fun i => (fun k : Fin (f i) => dot z ((q i).2.2 k) + c')) h.lumT hrow + have hh := hs.mul hb + apply Can.cases_code (q:=fun i => (q i).1) R hl + intro p + by_cases he : p=l + · refine (hh.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl) + (fun i => ?_) + have hql : (q i.val).1 = l := i.property.trans he + change lum (fun k : Fin (f i.val) => dot z ((q i.val).2.2 k) + c') * + v i.val ((q i.val).1,(q i.val).2.1,(q i.val).2.2) = _ + simp only [ferryM, roleAct, hql, if_true, act_phase] + · refine (hb.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl) + (fun i => ?_) + have H : (q i.val).1≠l := fun hh => he (i.property.symm.trans hh) + simp only [ferryM, roleAct, H, ite_false, Prod.mk.eta] + +/-- **Free diagonal phase**: one pass over the role table (one table look-up per entry). -/ +lemma GRig.phase_cost (h : GRig m B f s x cl X R r v) (l : ρ) (z : Space α) (c' : F) : + Can m B s (a (c cl)) x + (fun i => (fleet X R r (f i)).tape (ferryM r (f i) l (phaseMat z c') (v i))) + (fun i => (fleet X R r (f i)).n) := by + refine h.rig.chart _ ?_ + let S := Σ i, Beach α ρ r (f i) + have hg := h.gather (fun d : S => d.1) + (fun d => ((x d.1, (fleet X R r (f d.1)).loc d.2):T (p s w))) (t:=p s w) Can.first + exact hg.ferry_phase l z c' (fun d => d.2) Can.second + +end +end +end PowerSaving.RAM +end OAI + +#print axioms OAI.PowerSaving.RAM.Rig.perm_cost +#print axioms OAI.PowerSaving.RAM.GRig.phase_cost diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Letters.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Letters.lean new file mode 100644 index 0000000000..2eca6e713b --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Letters.lean @@ -0,0 +1,199 @@ +import Work.Block.Bridge + +/-! +# GFrame engine, part 1: the generalised word language (agent key: eng-ram) + +Letters = the letters of `BWord` (`localM`, `shift`, `block`) plus TWO FREE ADAPTER letters on +one role: + +* `perm l G` address permutation by an additive bijection `G` of the label space + (matrix `permMat G`, `(permMat G) x y = 1` iff `G.π x = y`); +* `phase l z c` the diagonal phase `i^[z·x + c]` IN THE STANDARD BASIS + (matrix `phaseMat z c = diagonal (fun x => tint (dot z x + c))`). + +Products of `phaseMat` are exactly the diagonal matrices `i^Q(x)`, `Q` a `ℤ/4`-valued quadratic +form plus a constant; with the old free letter `shift` the free letters generate every affine +address permutation times such a phase. A "twisted block" is NOT a primitive letter: it is the +word `free adapters ++ [block] ++ free adapters` (see `Free.lean`, `gstep_of_factor`). + +The semantics is `gwalk` (no unit-move word behind it: the new letters are not Walsh-diagonal). +Old words embed by `GWord.ofB`, and `gwalk f (ofB w) = walk f w.flat`. +-/ + +set_option linter.unusedSectionVars false + +namespace OAI +namespace PowerSaving.GF +open Binary Matrix Finset +noncomputable section + +/-- An additive bijection of the label space. -/ +structure APerm (α : Type*) where + π : Space α ≃ Space α + add : ∀ x y, π (x + y) = π x + π y + +section Mats +variable {α : Type*} [Fintype α] [DecidableEq α] + +/-- Permutation matrix of an additive bijection: `(permMat G *ᵥ v) x = v (G.π x)`. -/ +def permMat (G : APerm α) : CMat α := fun x y => if G.π x = y then 1 else 0 + +/-- Elementary diagonal phase in the standard basis: `i^[z·x + c]`. -/ +def phaseMat (z : Space α) (c : F) : CMat α := Matrix.diagonal fun x => tint (dot z x + c) + +end Mats + +/-- Letters of the generalised word language. -/ +inductive GMove (α ρ : Type*) + | old (mv : BMove α ρ) + | perm (l : ρ) (G : APerm α) + | phase (l : ρ) (z : Space α) (c : F) + +abbrev GWord (α ρ : Type*) := List (GMove α ρ) + +namespace GMove +variable {α ρ : Type*} + +/-- The letter of `BWord` with the same price (free adapters ↦ a free letter). -/ +def shadow : GMove α ρ → BMove α ρ + | .old mv => mv + | .perm _ _ => .localM 0 + | .phase _ _ _ => .localM 0 + +/-- Rank paid by a letter (`none` = free). -/ +def rk? (mv : GMove α ρ) : Option ℕ := mv.shadow.rk? +def costR (φ : ℕ → ℝ) (mv : GMove α ρ) : ℝ := mv.shadow.costR φ +def Proper (u : ℕ) (mv : GMove α ρ) : Prop := mv.shadow.Proper u + +@[simp] lemma costR_perm (φ : ℕ → ℝ) (l : ρ) (G : APerm α) : + (GMove.perm l G : GMove α ρ).costR φ = 0 := rfl +@[simp] lemma costR_phase (φ : ℕ → ℝ) (l : ρ) (z : Space α) (c : F) : + (GMove.phase l z c : GMove α ρ).costR φ = 0 := rfl +@[simp] lemma costR_old (φ : ℕ → ℝ) (mv : BMove α ρ) : (GMove.old mv).costR φ = mv.costR φ := rfl +lemma proper_perm (u : ℕ) (l : ρ) (G : APerm α) : (GMove.perm l G : GMove α ρ).Proper u := + trivial +lemma proper_phase (u : ℕ) (l : ρ) (z : Space α) (c : F) : + (GMove.phase l z c : GMove α ρ).Proper u := trivial +@[simp] lemma proper_old (u : ℕ) (mv : BMove α ρ) : (GMove.old mv).Proper u ↔ mv.Proper u := + Iff.rfl + +end GMove + +namespace GWord +variable {α ρ : Type*} + +def shadow (w : GWord α ρ) : BWord α ρ := w.map GMove.shadow +def costR (φ : ℕ → ℝ) (w : GWord α ρ) : ℝ := (w.map (GMove.costR φ)).sum +def Proper (u : ℕ) (w : GWord α ρ) : Prop := ∀ mv ∈ w, mv.Proper u +def hist (w : GWord α ρ) (r : ℕ) : ℕ := w.shadow.hist r +/-- Natural-number cost when a block of rank `rk` costs `g rk`. -/ +def cost (g : ℕ → ℕ) (w : GWord α ρ) : ℕ := w.shadow.cost g +/-- An old block word as a generalised word. -/ +def ofB (w : BWord α ρ) : GWord α ρ := w.map GMove.old + +@[simp] lemma shadow_nil : shadow ([] : GWord α ρ) = [] := rfl +@[simp] lemma shadow_cons (mv : GMove α ρ) (w : GWord α ρ) : + shadow (mv :: w) = mv.shadow :: shadow w := rfl +@[simp] lemma shadow_ofB (w : BWord α ρ) : shadow (ofB w) = w := by + simp [shadow, ofB, List.map_map, Function.comp_def, GMove.shadow] + +lemma costR_shadow (φ : ℕ → ℝ) (w : GWord α ρ) : costR φ w = BWord.costR φ w.shadow := by + unfold costR BWord.costR shadow + rw [List.map_map] + rfl +@[simp] lemma costR_nil (φ : ℕ → ℝ) : costR φ ([] : GWord α ρ) = 0 := rfl +lemma costR_cons (φ : ℕ → ℝ) (mv : GMove α ρ) (w : GWord α ρ) : + costR φ (mv :: w) = mv.costR φ + costR φ w := by simp [costR] +lemma costR_append (φ : ℕ → ℝ) (w w' : GWord α ρ) : + costR φ (w ++ w') = costR φ w + costR φ w' := by simp [costR] +lemma costR_ofB (φ : ℕ → ℝ) (w : BWord α ρ) : costR φ (ofB w) = BWord.costR φ w := by + rw [costR_shadow, shadow_ofB] +@[simp] lemma cost_nil (g : ℕ → ℕ) : cost g ([] : GWord α ρ) = 0 := rfl +lemma cost_cons (g : ℕ → ℕ) (mv : GMove α ρ) (w : GWord α ρ) : + cost g (mv :: w) = mv.shadow.cost g + cost g w := BWord.cost_cons g _ _ + +lemma Proper.shadow {u : ℕ} {w : GWord α ρ} (h : Proper u w) : BWord.Proper u w.shadow := by + intro mv hmv + obtain ⟨g, hg, rfl⟩ := List.mem_map.mp hmv + exact h g hg +lemma Proper.tail {u : ℕ} {mv : GMove α ρ} {w : GWord α ρ} (h : Proper u (mv :: w)) : + Proper u w := fun x hx => h x (List.mem_cons_of_mem _ hx) +lemma Proper.head {u : ℕ} {mv : GMove α ρ} {w : GWord α ρ} (h : Proper u (mv :: w)) : + mv.Proper u := h mv List.mem_cons_self +lemma Proper.append {u : ℕ} {w w' : GWord α ρ} (h : Proper u w) (h' : Proper u w') : + Proper u (w ++ w') := by + intro mv hmv + rcases List.mem_append.mp hmv with h1|h1 + · exact h mv h1 + · exact h' mv h1 +lemma proper_nil (u : ℕ) : Proper u ([] : GWord α ρ) := fun _ h => by simp at h +lemma proper_ofB {u : ℕ} {w : BWord α ρ} (h : w.Proper u) : Proper u (ofB w) := by + intro mv hmv + obtain ⟨g, hg, rfl⟩ := List.mem_map.mp hmv + exact h g hg + +lemma costR_eq_sum (u : ℕ) (φ : ℕ → ℝ) (w : GWord α ρ) (hw : Proper u w) : + costR φ w = ∑ r : Fin u, (w.hist r:ℝ) * φ r := by + rw [costR_shadow]; exact BWord.costR_eq_sum u φ _ hw.shadow +lemma cost_eq_sum (u : ℕ) (g : ℕ → ℕ) (w : GWord α ρ) (hw : Proper u w) : + cost g w = ∑ r : Fin u, hist w r * g r := BWord.cost_eq_sum u g _ hw.shadow +lemma hist_zero (u : ℕ) (w : GWord α ρ) (hw : Proper u w) : hist w 0 = 0 := + BWord.hist_zero u _ hw.shadow +lemma hist_large (u : ℕ) (w : GWord α ρ) (hw : Proper u w) (r : ℕ) (hr : u ≤ r) : + hist w r = 0 := BWord.hist_large u _ hw.shadow r hr + +end GWord + +section Sem +variable {α ρ σ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ] + +/-- The action of one letter. -/ +def GMove.act (f : ℕ) : GMove α ρ → Data α ρ f → Data α ρ f + | .old mv => mv.act f + | .perm l G => roleAct f l (permMat G) + | .phase l z c => roleAct f l (phaseMat z c) + +/-- **Semantics of a generalised word.** -/ +def gwalk (f : ℕ) (w : GWord α ρ) (x : Data α ρ f) : Data α ρ f := + w.foldl (fun y mv => mv.act f y) x + +@[simp] lemma gwalk_nil (f : ℕ) (x : Data α ρ f) : gwalk f ([] : GWord α ρ) x = x := rfl +lemma gwalk_cons (f : ℕ) (mv : GMove α ρ) (w : GWord α ρ) (x : Data α ρ f) : + gwalk f (mv :: w) x = gwalk f w (mv.act f x) := rfl +lemma gwalk_append (f : ℕ) (p q : GWord α ρ) (x : Data α ρ f) : + gwalk f (p ++ q) x = gwalk f q (gwalk f p x) := by simp [gwalk] + +/-- Old words keep their meaning. -/ +lemma gwalk_ofB (f : ℕ) (w : BWord α ρ) (x : Data α ρ f) : + gwalk f (GWord.ofB w) x = walk f w.flat x := by + induction w generalizing x with + | nil => rfl + | cons mv w ih => + rw [walk_flat_cons, ← ih] + rfl + +/-- The scratch certificate for the generalised language (cf. `RAM.LiveKernel`). -/ +def GLiveKernel (e : σ → ρ) (w : GWord α ρ) : Prop := + ∀ (f : ℕ) (v : Data α ρ f), (∀ s, (∀ l, e l ≠ s) → v s = 0) → + ∀ l, gwalk f w v (e l) = matAct f (kernel α) (v (e l)) + +/-- Every old certificate is a generalised certificate. -/ +theorem GLiveKernel.ofB {e : σ → ρ} {w : BWord α ρ} (h : RAM.LiveKernel e w.flat) : + GLiveKernel e (GWord.ofB w) := by + intro f v hv l + rw [gwalk_ofB] + exact h f v hv l + +/-- One role goes from matrix `M` to `N` by a generalised word (cf. `XStep`). -/ +def GStep (l : ρ) (M N : CMat α) (c : (ℕ → ℝ) → ℝ) : Prop := + ∃ (w : GWord α ρ) (U : CMat α), w.Proper (Fintype.card α) ∧ (∀ φ, w.costR φ = c φ) ∧ + U * M = N ∧ ∀ (f : ℕ) (x : Data α ρ f), gwalk f w x = roleAct f l U x + +end Sem +end +end PowerSaving.GF +end OAI + +#print axioms OAI.PowerSaving.GF.gwalk_ofB +#print axioms OAI.PowerSaving.GF.GLiveKernel.ofB +#print axioms OAI.PowerSaving.GF.GWord.costR_eq_sum diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Level.lean b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Level.lean new file mode 100644 index 0000000000..6bd7cfa0c3 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/Work__GFrame__Engine__Level.lean @@ -0,0 +1,163 @@ +import Work.GFrame.Engine.Sweep +import WHTCheck.Solution + +/-! +# GFrame engine, part 4: one level of the recursion for generalised words (agent key: eng-ram) + +Copy of `amplifyB` (`Work/Block/Recursion.lean`) with two changes: + +* the word is a `GWord` with certificate `GLiveKernel e w` (semantics `gwalk`); +* after the XOR table the level also builds the table of `lum` over `Bits f` + (`WHT.lum_tables`, doubling construction, `O(2^f)` work), which the free diagonal phases + read; it is part of the environment of the sweep (`GRig`). + +Cost: the same `2^k` linear term and the same recursive calls as `amplifyB`. +-/ + +set_option linter.unusedSectionVars false + +namespace OAI +namespace PowerSaving.RAM +open Binary Matrix Ty Finset GF +noncomputable section + +section +universe U +variable {α ρ σ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ] + [Fintype σ] [DecidableEq σ] +variable {ι : Type U} {m : Bool} + {k f : ι → ℕ} {cl : Paint} {b : Bank (hatch cl) ι} + {v : ∀ i, Sim σ (k i)→ℂ} + +/-- **One level of the recursion for a generalised word** (replaces `amplifyB`). -/ +lemma gamplify (m : Bool) (b : Bank (hatch cl) ι) (v : ∀ i, Sim σ (k i) → ℂ) + (X : Layout α) (R : Layout ρ) (Q : Layout σ) (e : σ → ρ) (he : Function.Injective e) + (r : ℕ) (hr : 3 ≤ X.n) (hq : ∀ i, X.n*f i+r=k i) (hd : r < X.n) (hQ : 0 rk * f i) (τ rk) Q) + (hB : Small b.B (fun i => 2^k i)) + (w : GWord α ρ) (hw : GLiveKernel e w) (hP : w.Proper X.n) : + Able m b (simIn cl) (bundle cl) (fun i => loaded cl Q (k i) (v i)) + (fun i => (sim Q (k i)).tape (whole (k i) (v i))) + (fun i => 2^k i) (fun i => w.cost (fun rk => n rk i * τ rk i)) := by + let W i := 2^k i + let s := simIn cl + let eQ (i : ι) := ebb X r (f i) (k i) (hq i) (v i) + let zQ (i : ι) : Grain α σ r (f i) := + fun j => chords r (f i) (fun l => eQ i (j.1,l)) j.2 + let Z (i : ι) : Grain α ρ r (f i) := crossAct r (f i) (embM e) (zQ i) + let O (i : ι) : Grain α σ r (f i) := + crossAct r (f i) (extM e) (gcoast r (f i) w (Z i)) + let szQ i := (fleet X Q r (f i)).n + let szR i := (fleet X R r (f i)).n + have hszQ : Dom W szQ := by + refine ⟨Q.n+1, fun i => ?_⟩ + unfold W szQ fleet + rw [Layout.pair_len, book_len,hq] + simp only [mul_add,add_mul,mul_one,one_mul] + gcongr <;> simp + have hszR : Dom W szR := by + refine ⟨R.n+1, fun i => ?_⟩ + unfold W szR fleet + rw [Layout.pair_len, book_len,hq] + simp only [mul_add,add_mul,mul_one,one_mul] + gcongr <;> simp + have hn (i : ι) : 0 < W i := pow_pos (by decide) _ + have hf (i) : f i ≤ k i := by + have g := hq i + have h := Nat.le_mul_of_pos_left (f i) (show 0 < X.n by omega) + omega + have hy (i) : k i ≤ W i := Nat.lt_two_pow_self.le + have sd := hB.mono hy + have sq := sd.mono hf + let p2 i := 2^f i + have hm : Small b.B p2 := + hB.mono (fun i => Nat.pow_le_pow_right (by omega) (hf i)) + have hcube (i) : p2 i ^ 3 ≤ W i := by + unfold p2 W + rw [← pow_mul] + apply Nat.pow_le_pow_right (by omega) + have hh : 3*f i ≤ X.n*f i := Nat.mul_le_mul_right _ hr + have hg := hq i + rw [mul_comm] + omega + let env (i : ι) := loaded cl Q (k i) (v i) + -- construct fields (f,p2) + have hmf : Can m b.B s Ty.w env f W := by + have ht : Can m b.B s Ty.w env k W := Can.first + refine (ht.dvdW (Can.wc X.n) sd).cong (fun _ => rfl) ?_ + intro i + rw [← hq,mul_comm X.n] + exact Layout.div_join hd + have hmp : Can m b.B s (Ty.p Ty.w Ty.w) env (fun i => (f i,p2 i)) W := + hmf.pair (hmf.wTwo sq hm (Dom.of_le (fun i => (hf i).trans (hy i)))) + let T1 := Ty.p s (Ty.p Ty.w Ty.w) + let Y (i : ι) : T T1 := (env i,f i,p2 i) + have hmx : Can m b.B T1 (Ty.a Ty.w) Y (fun i => xorTab (f i)) W := + (Can.xor_prepare (m:=m) (f:=f) (B:=b.B) (s:=T1) (x:=Y) + (Can.fst Can.second) (Can.snd Can.second) sq hm).mono hcube + let T2 := Ty.p T1 (Ty.a Ty.w) + let Y' (i : ι) : T T2 := (Y i,xorTab (f i)) + -- NEW: the table of `lum` over `Bits f` + have hE2 : Has m b.B T2 s Y' env := Can.fst Can.first + have hml : Can m b.B T2 (Ty.a sc) Y' + (fun i => (bits (f i)).tape (fun y : Bits (f i) => lum y)) W := + (WHT.lum_tables (k:=f) (Can.fst (Can.snd Can.first)) sq hm hE2.snd.fst).1.mono + (fun i => Nat.pow_le_pow_right (by omega) (hf i)) + let T2' := Ty.p T2 (Ty.a sc) + let Y'' (i : ι) : T T2' := (Y' i,(bits (f i)).tape (fun y : Bits (f i) => lum y)) + -- the input, read as a table on the slots + have hG0 : Rig m b.B f T2 Y' cl X Q r eQ := by + refine ⟨⟨?_,?_,Can.second,hm⟩,?_,?_⟩ + · exact Can.snd (Can.snd Can.first) + · exact Can.fst (Can.snd Can.first) + · exact hE2.snd.fst + · exact hE2.snd.snd.cong (fun _ => rfl) (fun i => (ebb_eq ..).symm) + have hG : Rig m b.B f T2' Y'' cl X Q r eQ := hG0.gather id Y'' (t:=T2') Can.first + -- leftover bits, still on the slots + have HH : Can m b.B T2' (bundle cl) Y'' (fun i => (fleet X Q r (f i)).tape (zQ i)) W := + hG.dots_cost.weaken hszQ + let T3 := Ty.p T2' (bundle cl) + let A (i) : T T3 := (Y'' i,(fleet X Q r (f i)).tape (zQ i)) + have Hg := hG.gather id A (t:=T3) Can.first + have G : Rig m b.B f T3 A cl X Q r zQ := ⟨Hg.gear,Hg.imag,Can.second⟩ + -- embed the slots into the table of roles; scratch roles are created as zero + have HE : Can m b.B T3 (bundle cl) A (fun i => (fleet X R r (f i)).tape (Z i)) W := + (G.cross_point R (embM e)).weaken hszR + let T4 := Ty.p T3 (bundle cl) + let A4 (i) : T T4 := (A i,(fleet X R r (f i)).tape (Z i)) + have Hg4 := G.gather id A4 (t:=T4) Can.first + have hL4 : Has m b.B T4 (Ty.a sc) A4 + (fun i => (bits (f i)).tape (fun y : Bits (f i) => lum y)) := + Can.snd (Can.fst Can.first) + have G4 : GRig m b.B f T4 A4 cl X R r Z := ⟨⟨Hg4.gear,Hg4.imag,Can.second⟩,hL4⟩ + -- the GENERALISED word on the table of roles + have Hy := G4.gsweep n τ Q hQ hk + (fun rk h1 h2 i => by rw [book_len,hq]; exact hc rk h1 h2 i) w hP + -- extract the slots; scratch roles are dropped + let T5 := Ty.p T4 (bundle cl) + let A5 (i) : T T5 := (A4 i,(fleet X R r (f i)).tape (gcoast r (f i) w (Z i))) + have Hg5 := G4.rig.gather id A5 (t:=T5) Can.first + have G5 : Rig m b.B f T5 A5 cl X R r (fun i => gcoast r (f i) w (Z i)) := + ⟨Hg5.gear,Hg5.imag,Can.second⟩ + have HX : Can m b.B T5 (bundle cl) A5 (fun i => (fleet X Q r (f i)).tape (O i)) W := + (G5.cross_point Q (extM e)).weaken hszQ + have heq (i) : O i = ebb X r (f i) (k i) (hq i) (whole (k i) (v i)) := + gamplifies X r (f i) (k i) (hq i) e he w hw (v i) + have ha (i) : (fleet X Q r (f i)).tape (O i) = + (sim Q (k i)).tape (whole (k i) (v i)) := by + rw [heq]; exact ebb_eq .. + have Ho := ((Hy.weaken hszR (fun _ => le_rfl)).save_keep + (Able.of_can (b:=b) HX)).cong (fun _ => rfl) ha + have Hx := (Able.of_can (b:=b) hmp).save_keep + ((Able.of_can (b:=b) hmx).save_keep ((Able.of_can (b:=b) hml).save_keep + ((Able.of_can (b:=b) HH).save_keep ((Able.of_can (b:=b) HE).save_keep Ho)))) + simpa only [Nat.zero_add,Nat.add_zero] using Hx + +end +end +end PowerSaving.RAM +end OAI + +#print axioms OAI.PowerSaving.RAM.gamplify diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__ORIGIN.md b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__ORIGIN.md new file mode 100644 index 0000000000..e28b355d0a --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__ORIGIN.md @@ -0,0 +1,64 @@ +# tools/gx/ and tools/certificate/gcert1-p11-pr193.json.gz: where they come from + +Everything in this directory is Python 3 (standard library only). No Lean is run by these scripts. +"The outside repository" means github.com/CrocSwap/integer-mult-bounds (Apache-2.0, `LICENSE` and `NOTICE` +of that repository; `NOTICE`: "A sharper exponent for integer multiplication, Copyright 2026 Douglas Colkitt"). + +## The scripts (this project) + +| file | what it is | +|---|---| +| `gxcore.py` | loads a certificate of format gcert/1, brings it to normal form, and is a Python mirror of both Lean checks (labels and scalars); cut points and label data for the generator | +| `gxunit.py` | split plan of the kernel evaluations, the unit of the word, expected exponents | +| `gxbind.py` | the texts of the generated Lean modules (check, histogram, price, certificate and final modules) | +| `gxgen.py` | certificate -> Lean data modules and one module per kernel evaluation | +| `gxrate.py` | rate lemmas, unit price list, final theorem modules, comparator challenge / solution / configuration | +| `gxdry.py`, `gxmut.py` | dry run (mirror, sizes, plan, exponents); mutants that the mirror must reject | +| `gxconv.py` | converts the earlier format gcert/0 to gcert/1 when a step scales its target (the in-place pair step of #193) | +| `gsmirror.py` | a second, separately written Python mirror of the scalar check `GS.scalCheck` / `GS.yCheck`; `gxbind.py` calls it for the check parameters | +| `gx.py`, `gxrun.py` | the reference checker of the format (`gx.check1`: labels and exact scalar identity) and its command line | +| `gxpaths.py`, `regen_check.py`, `refcheck.py` | written for this repository: relative paths; regenerate and compare; `gx.check1` on a gcert/1 file | + +The scripts are the working copies with path edits only (absolute paths of the author's machine replaced through +`gxpaths.py`; templates read from `tools/gen/templates/`; `foldlib.py` / `foldgen.py` taken from `tools/gen/`). +The sentences "Generated by checks/..." that the generators print into Lean files name the scripts by their +working paths; they are kept so that the generated modules come out byte for byte. +No script of the outside repository is included, imported or executed. + +## The certificate `tools/certificate/gcert1-p11-pr193.json.gz` + +sha256 of the file: `15251deb0c11e0504d7b297c9786818888a93f08b92b03c6c122784b70488633` +(gzip stores a time stamp; sha256 of the uncompressed JSON: `c64d2a95431c3cf24452de7391cce780bd71521da44ce23923ebb235f6c7962c`). + +It is ONE explicit program (numbered registers, every addition, every frame) for the helper circuit at +h = 22: 1,320 ports, 9,412 slots, 262,944 unit moves. **The design of that circuit is not this project's.** It is the +"paired-cube" circuit of the outside repository as of pull request #193 (which builds on #191, #184, #168 and #144). +The outside repository publishes the design as generator scripts, frozen data files and a ledger; it has no file +of this kind. This file was produced by this project's own code, which + +1. loaded the following files of the outside repository **as data** (JSON; nothing was executed): + - at pull request #168, commit `4a3c769e5c5430e7114c4d3e099ff34664677f17`: + `references/paired-cube/sources/local_L1.json`, + `references/paired-cube/sources/tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json`, + `references/paired-cube/sources/pmod_J0_full_6.0666810e-4.json`, + `references/paired-cube/sources/qmod_climb3u_best.json` (the local modules), + `references/paired-cube/selected-module/matching-arcs.json` (carrier arcs), + `references/paired-cube/physical/frames.json` (moved frames); + - at pull request #193, commit `187e1010ac8b259af8e9b5166f68b64bc27b4b47`: + `research/source-assisted-v4/data/physical-pairs.json` (the hand-over pairs); +2. rebuilt the circuit from them step by step, following the algorithms of the outside scripts (read as text), + and checked the result exactly (every published count reproduced; the scalar identity as an identity of + linear maps with arbitrary contents of the helper registers); +3. converted it to gcert/1 with `gxconv.py` (tested by `gx.check1`). + +Step 1 and step 2 (the rebuild scripts and the outside inputs they read) are NOT part of this repository; +the inputs can be fetched from the outside repository at the two commits above. +What this repository proves about the file does not depend on how it was made: the Lean kernel checks the +generated modules (labels, scalar identity, price), and `regen_check.py` shows that those modules are exactly +what `gxgen.py` / `gxrate.py` produce from this file. + +## Check it yourself + + python3 tools/gx/regen_check.py # regenerates the Lean modules and compares + python3 tools/gx/gxdry.py tools/certificate/gcert1-p11-pr193.json.gz # Python mirror of both checks + python3 tools/gx/refcheck.py # the reference checker gx.check1 on the certificate diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxbind.py b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxbind.py new file mode 100644 index 0000000000..fcb625619c --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxbind.py @@ -0,0 +1,438 @@ +"""gxbind.py: THE BINDING of the generator to the Lean terms of gx-scalar / gx-labels / +gx-chain. Everything that names somebody else's definition is in this file (tables SCAL, LAB, CHAIN).""" +import os, re, subprocess +import gxpaths +ROOT = gxpaths.OUT +GSM = gxpaths.HERE + '/gsmirror.py' + +# gx-scalar (gcert-interface.md, SCALAR v1): GS.scalCheck c p lo n, GS.yCheck c p lo n, p : GS.SPar = +SCAL = dict(imports=['Work.GCert.Scalar.Seg'], scal='GS.scalCheck raw ⟨%(d)d, %(sw)d, %(ux)d, %(us)d, %(uy)d⟩ %(lo)d %(n)d', + ychk='GS.yCheck raw ⟨%(d)d, %(sw)d, %(ux)d, %(us)d, %(uy)d⟩ 0 %(v)d') +# gx-labels (gcert-interface.md LABELS v1 + questions.md N1): namespace GLab; frame = code, table and states = SSC.Trie, +# GLab.Par = . `end` = GLab.endK of Labels/EndDef.lean (questions.md N2; no obaseK: Q1 closed). +LAB = dict(imports=['Work.GCert.Labels.Def'], end_imports=['Work.GCert.Labels.EndDef'], par='⟨%(h)d, %(n)d, %(d)d, %(K)d, tab⟩', ns='GLab', + tab='GLab.tabK %(h)d %(sub)s', seg='GLab.segK par %(gs)s st%(i)d st%(j)d %(rs)s', + end='GLab.endK par %(v)d ports (raw.ret.map Prod.fst) st0 st%(a)d st%(f)d', noadds='GLab.noAdds fins') +# gx-chain (CHAIN v1): GX.wht_of_halves; the terms below are PLACEHOLDERS until Labels/Main and Scalar/Main compile +CHAIN = dict(imports=['Work.GCert.Chain.End', 'Work.GCert.Labels.Main'], scal='(GS.ScalOK.scal GXD.%(c)s.scalOK)', + lab='(GXD.%(c)s.valid.lab GXD.%(c)s.scalOK)', lab_cost='GXD.%(c)s.lab_cost') + + +def code(h, basis): + """gx-labels' code of a reduced echelon basis: dim + 2^8 M + 2^(8+h) P (M = pivot mask, vector of pivot p at digit p)""" + M = P = 0 + for b in basis: + p = b.bit_length() - 1 + M |= 1 << p + P |= b << (h * p) + return len(basis) + (M << 8) + (P << (8 + h)) + + +MOD = '''import %(imports)s +/-! Kernel check (%(doc)s) of the gcert/1 certificate `GXD.%(name)s.raw`. Source: %(src)s. +Generated by checks/wht26/gx-data/py/gxgen.py (do not edit). -/ +set_option maxRecDepth 100000 +namespace GXD.%(name)s +open GXD +theorem %(thm)s : %(stmt)s := by decide +kernel +end GXD.%(name)s +#print axioms GXD.%(name)s.%(thm)s +''' + + +def spar(D, path): + """scalar parameters: from gx-scalar's mirror when it exists, else from my own replay (sw is then a GUESS)""" + import gxcore as gc + n, d = 2 * D['v'] + D['R'], 0 + while (1 << d) < n: + d += 1 + dg = gc.digits(D) + S = dict(d=d, sw=16, ux=dg['x'][0], us=dg['s'][0], uy=dg['y'][0], source='gx-data replay; sw = 16 is a GUESS', digits=dg) + if os.path.exists(GSM): + try: + o = subprocess.run(['python3', '-B', GSM, path], capture_output=True, text=True, timeout=1800).stdout + m = re.search(r'SPar[^\d]*(\d+)[, ]+(\d+)[, ]+(\d+)[, ]+(\d+)[, ]+(\d+)', o) + if m: + S.update(zip(('d', 'sw', 'ux', 'us', 'uy'), map(int, m.groups())), source='gsmirror.py') + except Exception as e: + S['source'] += ' (gsmirror.py failed: %r)' % (e,) + return S + + +def write_checks(D, M, name, src, out, L, S, plan): + base = out[len(ROOT):].replace('/', '.') + raw = '%s.Gen.%s' % (base, name) + h, v, n = D['h'], D['v'], 2 * D['v'] + D['R'] + res = [] + + def mod(suffix, imports, thm, stmt, doc): + m = '%s.Gen.%s%s' % (base, name, suffix) + open('%s/Gen/%s%s.lean' % (out, name, suffix), 'w').write(MOD % dict( + imports='\nimport '.join(imports), doc=doc, name=name, src=src, thm=thm, stmt=stmt)) + res.append((m, thm, stmt)) + for i, (lo, hi) in enumerate(M['scal']['batches']): + mod('Scal%d' % i, SCAL['imports'] + [raw], 'scal%d_ok' % i, SCAL['scal'] % dict(S, lo=lo, n=hi - lo) + ' = true', + 'scalar identity E3 E5 for the sources %d .. %d' % (lo, hi - 1)) + mod('YChk', SCAL['imports'] + [raw], 'ychk_ok', SCAL['ychk'] % dict(S, v=v) + ' = true', 'the y -> y adds, E4') + par = '%s.Gen.%sPar' % (base, name) + open('%s/Gen/%sPar.lean' % (out, name), 'w').write( + 'import %s\nimport %s\n/-! Parameters `GLab.Par` of the label check of `GXD.%s.raw` (h, registers, trie depth, K, frame table).\n' + 'Generated by checks/wht26/gx-data/py/gxgen.py (do not edit). -/\nnamespace GXD.%s\nopen GXD\n' + 'noncomputable def par : GLab.Par := %s\nend GXD.%s\n' + % (LAB['imports'][0], raw, name, name, LAB['par'] % dict(h=h, n=n, d=L['d'], K=L['K']), name)) + paths = [format(i, '0%db' % L['ts']) for i in range(1 << L['ts'])] if L['ts'] else [''] + for i, pth in enumerate(paths): + mod('Tab%d' % i, LAB['imports'] + [raw], 'tab%d_ok' % i, + LAB['tab'] % dict(h=h, sub='tab_n' + pth if pth else 'tab') + ' = true', + 'E0: subtrie %d of %d of the frame table (codes of reduced echelon bases)' % (i, len(paths))) + for i in range(L['nseg']): + mod('Seg%d' % i, [par], 'seg%d_ok' % i, + LAB['seg'] % dict(gs='seg%d' % i, i=i, j=i + 1, rs='rs%d' % i) + ' = true', + 'E1: replay of the gates %d .. %d (phase %s) and their block ranks' % ( + M['cuts'][i], M['cuts'][i + 1] - 1, 'A' if i < L['nsegA'] else 'B')) + mod('SegF', [par], 'segF_ok', LAB['seg'] % dict(gs='fins', i=L['nseg'], j=L['nseg'] + 1, rs='rF') + ' = true', + 'E1: the final climbs and their block ranks') + if LAB['end']: + mod('End', LAB['end_imports'] + [par], 'end_ok', LAB['end'] % dict(v=v, a=L['nsegA'], f=L['nseg'] + 1) + ' = true', + 'E1 E2: frames at the start, at the scatter, at the end') + return res + + +def nest(xs): + return ' ++ ('.join(xs) + ')' * (len(xs) - 1) if xs else '[]' + + +def write_hist(D, M, name, src, out, L): + """kernel counts of the block ranks on the EXPLICIT rank lists rA, rB, rF (no replay)""" + base = out[len(ROOT):].replace('/', '.') + h, rs = D['h'], M['lab']['rs'] + ph = {'A': sum(rs[:L['nsegA']], []), 'B': sum(rs[L['nsegA']:], []), 'F': M['lab']['rF']} + top = max(max(x) if x else 0 for x in ph.values()) + cop = [M['dim'][f] for _, f in D['ret']] + tot = {r: sum(x.count(r) for x in ph.values()) + cop.count(r) for r in range(1, max(top, max(cop)) + 1)} + assert {r: c for r, c in tot.items() if c} == M['hist'], 'histogram of the rank lists differs from the mirror' + lines = ['/-- block ranks of phase A / phase B, in order (the lists of the replay segments) -/', + 'noncomputable def rA : List Nat := ' + nest(['rs%d' % i for i in range(L['nsegA'])]), + 'noncomputable def rB : List Nat := ' + nest(['rs%d' % i for i in range(L['nsegA'], L['nseg'])])] + names = [] + for X, lst in sorted(ph.items()): + l = 'r' + X + lines.append('theorem rankBound%s : wsumK (SSC.wGt %d) %s 0 = 0 := by decide +kernel' % (X, top, l)) + lines += ['theorem rank%s%d_ok : wsumK (SSC.wEq %d) %s 0 = %d := by decide +kernel' % (X, r, r, l, lst.count(r)) + for r in range(top + 1)] + names += ['rankBound' + X] + ['rank%s%d_ok' % (X, r) for r in range(top + 1)] + txt = ('import Work.GCert.Data.Count\nimport %s.Gen.%s\n' % (base, name) + + '/-! Kernel count of the BLOCK RANKS of the gcert/1 certificate `GXD.%s.raw` (h = %d) on the explicit rank lists\n' + '`rA`, `rB`, `rF` (phase A, phase B, final climbs): `wsumK (SSC.wEq r) l 0` blocks of rank `r`, none above %d\n' + '(`GXD.sum_of_wsumK` turns a count into `(l.map wt).sum` = `SSC.wsum wt l`).\n' + 'The %d scratch copies (ranks %s) are not in these lists. Whole invocation, by rank: %s.\n' + 'Source: %s. Generated by checks/wht26/gx-data/py/gxgen.py (do not edit). -/\n' + 'set_option maxRecDepth 100000\nnamespace GXD.%s\nopen GXD\n' % ( + name, h, top, len(cop), sorted(set(cop)), M['hist'], src, name) + '\n'.join(lines) + + '\nend GXD.%s\n' % name + '\n'.join('#print axioms GXD.%s.%s' % (name, x) for x in names[:2]) + '\n') + open('%s/Gen/%sHist.lean' % (out, name), 'w').write(txt) + return '%s.Gen.%sHist' % (base, name) + + +def write_price(D, M, name, src, out, L): + """Mathlib module: the price of the rank lists rA, rB, rF from the kernel counts, and of one whole invocation""" + base = out[len(ROOT):].replace('/', '.') + rs = M['lab']['rs'] + ph = {'A': sum(rs[:L['nsegA']], []), 'B': sum(rs[L['nsegA']:], []), 'F': M['lab']['rF']} + top = max(max(x) if x else 0 for x in ph.values()) + cop = [M['dim'][f] for _, f in D['ret']] + tm = lambda cnt: ' + '.join('%d * φ %d' % (n, r) for r, n in sorted(cnt.items()) if n) or '0' + thms = '' + for X in 'ABF': + cnt = {r: ph[X].count(r) for r in range(1, top + 1)} + thms += ''' +/-- price of the blocks of %s, from the kernel counts -/ +theorem r%s_cost (φ : ℕ → ℝ) : rcost φ r%s = %s := by + rw [GXD.rcost_wsumK φ r%s %d rankBound%s] + simp only [Finset.sum_range_succ, Finset.sum_range_zero, %s] + norm_num [bcost] +''' % ({'A': 'phase A', 'B': 'phase B', 'F': 'the final climbs'}[X], X, X, tm(cnt), X, top, X, + ', '.join('rank%s%d_ok' % (X, r) for r in range(top + 1))) + copies = tm({r: cop.count(r) for r in set(cop)}) + txt = '''import Work.GCert.Data.Price +import %(base)s.Gen.%(name)sHist + +/-! +# (key: gx-data) gcert/1 certificate `%(name)s` (h = %(h)d): the price of one invocation + +GENERATED by checks/wht26/gx-data/py/gxgen.py from %(src)s. No `sorry`. +`rA_cost`, `rB_cost`, `rF_cost`: the price of the block rank lists of phase A, phase B and the final climbs +(lists `GXD.%(name)s.rA / rB / rF`, counted by the kernel in `Gen.%(name)sHist`); +`inv_cost`: with the %(nc)d scratch copies (%(copies)s) the blocks of ONE invocation cost +`invPrice φ` = %(terms)s; `inv_moves`: they stand for `R h + 2 v (h - 1) + cst = %(N)d` unit moves. +-/ + +namespace GXD.%(name)s +open GXD OAI.PowerSaving.CB +%(thms)s +/-- price of the blocks of ONE invocation (x roles, y roles, slots, scratch copies), by rank -/ +def invPrice (φ : ℕ → ℝ) : ℝ := %(terms)s + +/-- **Price of ONE invocation** in the shape of `GX.Lab.cost`: phase A, the scratch copies, phase B, final climbs. -/ +theorem inv_cost (φ : ℕ → ℝ) : rcost φ rA + (%(copies)s) + rcost φ rB + rcost φ rF = invPrice φ := by + rw [rA_cost, rB_cost, rF_cost] + unfold invPrice + ring + +theorem inv_moves : invPrice (fun r => (r:ℝ)) = %(R)d * %(h)d + 2 * %(v)d * (%(h)d - 1) + %(cst)d := by + unfold invPrice + norm_num + +theorem inv_count : invPrice (fun _ => (1:ℝ)) = %(count)d := by + unfold invPrice + norm_num + +end GXD.%(name)s +#print axioms GXD.%(name)s.inv_cost +#print axioms GXD.%(name)s.inv_moves +''' % dict(base=base, name=name, h=D['h'], src=src, nc=len(cop), copies=copies, terms=tm(M['hist']), N=D['N'], thms=thms, + R=D['R'], v=D['v'], cst=D['cst'], count=sum(M['hist'].values())) + open('%s/%sPrice.lean' % (out, name), 'w').write(txt) + return '%s.%sPrice' % (base, name) + + +def write_cert(D, M, name, src, out, chk, L): + """Cert: THE LABEL HALF (aux, valid : GLab.Valid raw aux, lab_cost, lab_price) -- compiles without the scalar half; + ScalOK: the scalar half `scalOK : GS.ScalOK raw` (needs Work/GCert/Scalar/Main).""" + base = out[len(ROOT):].replace('/', '.') + bs = M['scal']['batches'] + hs = '' + for i, (lo, hi) in enumerate(bs): + lo_pf = 'Nat.zero_le t' if i == 0 else 'Nat.le_of_not_lt h%d' % (i - 1) + if i + 1 < len(bs): + hs += ' if h%d : t < %d then ⟨%d, %d, %s, h%d, GS.XRun.of_check scal%d_ok⟩ else\n' % (i, hi, lo, hi - lo, lo_pf, i, i) + else: + hs += ' ⟨%d, %d, %s, Nat.lt_of_lt_of_le ht (by decide), GS.XRun.of_check scal%d_ok⟩' % (lo, hi - lo, lo_pf, i) + terms = ' + '.join('%d * φ %d' % (n, r) for r, n in M['hist'].items()) + info = dict(base=base, name=name, h=D['h'], src=src, v=D['v'], R=D['R'], nc=len(D['ret']), cst=D['cst'], N=D['N'], + nb=len(bs), terms=terms, hs=hs) + txt = '''import Work.GCert.Scalar.Main +import %(imps)s +/-! # (key: gx-data) gcert/1 certificate `%(name)s` (h = %(h)d): THE SCALAR HALF `scalOK : GS.ScalOK raw` +GENERATED by checks/wht26/gx-data/py/gxgen.py from %(src)s. No `sorry`. +Kernel checks `scal_ok` (%(nb)d source batches) and `ychk_ok`. -/ +namespace GXD.%(name)s +open GXD +theorem scalOK : GS.ScalOK raw where + hv := by decide + hs := fun t ht => +%(hs)s + hyc := fun t ht => ⟨0, %(v)d, Nat.zero_le t, Nat.lt_of_lt_of_le ht (by decide), GS.YRun.of_check ychk_ok⟩ +end GXD.%(name)s +#print axioms GXD.%(name)s.scalOK +''' % dict(info, imps='\nimport '.join(c[0] for c in chk if 'Scal' in c[0] or 'YChk' in c[0])) + open('%s/%sScalOK.lean' % (out, name), 'w').write(txt) + # ---- the label half + nA, ns, ts = L['nsegA'], L['nseg'], L['ts'] + cop = sorted(set(M['dim'][f] for _, f in D['ret'])) + assert len(cop) == 1, 'scratch copies of several ranks %s: lab_cost of gxbind.py assumes one' % cop + + def tabpf(path): + if len(path) == ts: + return 'tab%d_ok' % (int(path, 2) if path else 0) + return '(GLab.tabK_node _ _ _ %s %s)' % (tabpf(path + '0'), tabpf(path + '1')) + + def runpf(a, b): + return 'GLab.Run.of_seg seg%d_ok' % a if a + 1 == b else 'GLab.Run.append (GLab.Run.of_seg seg%d_ok) (%s)' % (a, runpf(a + 1, b)) + + def run(X, a, b): + segs = ['seg%d' % i for i in range(a, b)] + return '''theorem gates%(X)s_eq : raw.gates%(X)s = %(cat)s := by + show List.flatten %(lst)s = _ + simp only [List.flatten_cons, List.flatten_nil, List.append_nil] +/-- the replay of phase %(X)s, composed from its kernel-checked segments -/ +theorem run%(X)s : GLab.Run par st%(a)d raw.gates%(X)s st%(b)d r%(X)s := by + rw [gates%(X)s_eq] + exact %(pf)s +''' % dict(X=X, cat=nest(segs), lst=ll_(segs), a=a, b=b, pf=runpf(a, b)) + lab = 'valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg' + txt = '''import Work.GCert.Labels.Main +import %(base)s.%(name)sPrice +import %(imps)s +/-! # (key: gx-data) gcert/1 certificate `%(name)s` (h = %(h)d): THE LABEL HALF and its price +GENERATED by checks/wht26/gx-data/py/gxgen.py from %(src)s. No `sorry`. +`v = %(v)d` ports, `R = %(R)d` slots, %(nc)d scratch copies of rank %(c)d, copy cost `cst = %(cst)d`, `N = %(N)d` unit moves. +* `aux : GLab.Aux`, `valid : GLab.Valid raw aux`: frame table (`tab_ok` composed by `tabK_node`), the replays of phase A + and phase B (`seg_ok` composed by `Run.append`), the final climbs (`segF_ok`, `fin_ok`), the ends (`end_ok`); +* `lab_cost`, `lab_price`: for ANY scalar half `S` the label half `valid.lab S ..` costs + `rcost rA + card C * bcost %(c)d + rcost rB + rcost rF` = `invPrice φ` = %(terms)s + (`Gen.%(name)sHist`, `%(name)sPrice`) when there are %(nc)d copies. -/ +set_option maxRecDepth 100000 +namespace GXD.%(name)s +open GXD OAI.PowerSaving OAI.PowerSaving.CB +theorem raw_h : raw.h = %(h)d := rfl +theorem raw_v : raw.v = %(v)d := rfl +theorem raw_R : raw.R = %(R)d := rfl +/-- what the label checks read next to `raw` -/ +noncomputable def aux : GLab.Aux := ⟨par, st0, st%(a)d, st%(n)d, st%(f)d, fins, rA, rB, rF⟩ +theorem fin_ok : GLab.noAdds fins = true := by decide +kernel +theorem cen_ok : (raw.ret.map Prod.fst).all (fun r => Nat.beq (GLab.cdim (par.tab.get (st%(a)d.get r))) %(c)d) = true := by + decide +kernel +theorem tab_ok : GLab.tabK par.h par.tab = true := %(tabpf)s +%(runA)s%(runB)s/-- **the kernel-checked label facts** of the certificate -/ +theorem valid : GLab.Valid raw aux where + hn := rfl + tab := tab_ok + runA := runA + runB := runB + runF := GLab.Run.of_seg segF_ok + fin := fin_ok + ends := end_ok +theorem cen_dim (r : Nat) (hr : r ∈ raw.ret.map Prod.fst) : GLab.cdim (par.tab.get (st%(a)d.get r)) = %(c)d := + Nat.eq_of_beq_eq_true (List.all_eq_true.mp cen_ok r hr) +/-- price of the label half over ANY scalar half -/ +theorem lab_cost {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C) + {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)} + (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi) + (reg : C → Nat) (hCc) (hreg) (φ : ℕ → ℝ) : + (%(lab)s).cost φ + = rcost φ rA + (Fintype.card C : ℝ) * bcost φ %(c)d + rcost φ rB + rcost φ rF := by + obtain ⟨h1, h2, h3, h4⟩ := valid.lab_ranks S Ro cf hMA hMAi hMB hMBi reg hCc hreg + have h5 : ∀ k, bcost φ (((%(lab)s).cen k).dim) = bcost φ %(c)d := + fun k => by rw [h4 k]; exact congrArg (bcost φ) (cen_dim _ (hreg k)) + rw [GX.Lab.cost, h1, h2, h3, Finset.sum_congr rfl (fun k _ => h5 k), Finset.sum_const, Finset.card_univ, nsmul_eq_mul] + rfl +/-- **the label half costs one invocation price** (`invPrice`, by rank, from the kernel counts) -/ +theorem lab_price {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C) + {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)} + (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi) + (reg : C → Nat) (hCc) (hreg) (hC : Fintype.card C = %(nc)d) (φ : ℕ → ℝ) : + (%(lab)s).cost φ = invPrice φ := by + rw [lab_cost S Ro cf hMA hMAi hMB hMBi reg hCc hreg φ, hC, ← inv_cost φ] + norm_num [bcost] +end GXD.%(name)s +#print axioms GXD.%(name)s.valid +#print axioms GXD.%(name)s.lab_price +''' % dict(info, imps='\nimport '.join(c[0] for c in chk if 'Scal' not in c[0] and 'YChk' not in c[0]), c=cop[0], a=nA, n=ns, + f=ns + 1, tabpf=tabpf(''), runA=run('A', 0, nA), runB=run('B', nA, ns), lab=lab) + open('%s/%sCert.lean' % (out, name), 'w').write(txt) + return ['%s.%sCert' % (base, name), '%s.%sScalOK' % (base, name)] + + +def ll_(xs): + return '[' + ', '.join(xs) + ']' + + +def final_module(name, cname, d, dx, g, out): + """.lean: the final theorems OVER ANY SCALAR HALF (compiles as soon as Cert does): label half, price, + unit list, rate fact and gx-chain's `GX.wht_of_halves` are all closed; what is left is `S : GX.Scal` with its + gate matrices = the ordered products of the adds of `raw` (gx-scalar).""" + h, v, R, W, m, nc = g['h'], g['v'], g['R'], d['W'], d['m'], g['nc'] + X = 'GXD.%s' % cname + args = 'S Ro cf hMA hMAi hMB hMBi reg hCc hreg' + + def thm(nm, rate, A, B, s, ux): + return f''' +/-- **Walsh-Hadamard transform, whole-block recursion on the bridged word B_2 with the gcert/1 circuit `{cname}`**, +over any scalar half of `{X}.raw`: the statement `WHTGoal` of `WHTCheck/Challenge.lean` with `1 - 2/10^11` +replaced by `1 - {A}/10^{B}`. -/ +theorem {nm} (Ro : GLab.Roles {X}.raw.v {X}.raw.R emb un) + (hMA : S.MA = SSC.matP un (GLab.mic cf ({X}.raw.gatesA.flatMap GXD.Gate.adds))) + (hMAi : S.MAi = SSC.matPinv un (GLab.mic cf ({X}.raw.gatesA.flatMap GXD.Gate.adds))) + (hMB : S.MB = SSC.matP un (GLab.mic cf ({X}.raw.gatesB.flatMap GXD.Gate.adds))) + (hMBi : S.MBi = SSC.matPinv un (GLab.mic cf ({X}.raw.gatesB.flatMap GXD.Gate.adds))) + (reg : C → Nat) (hCc : ∀ k (q : Fin {X}.raw.R), S.Cc k q ≠ 0 → reg k = 2 * {X}.raw.v + q.val) + (hreg : ∀ k, reg k ∈ {X}.raw.ret.map Prod.fst) (hC : Fintype.card C = {nc}) : + ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - {A}/(10:ℝ)^{B}) W := + GX.wht_of_halves S ({X}.valid.lab {args}) + (by rw [Fintype.card_fin]; decide) {m} {W} (by rw [Fintype.card_fin]; rfl) + (by rw [Fintype.card_fin, Fintype.card_fin]; rfl) (by norm_num) BlockAccounting.blocks{rate} + (fun φ => by + rw [{X}.lab_price {args} hC φ, Fintype.card_fin, Fintype.card_fin] + exact BridgeRate.{name}.unit_cost{ux} φ) + {s} (1 - {A}/(10:ℝ)^{B}) (by norm_num) (by norm_num) BlockAccounting.fold_{rate} + +/-- the same from the FIVE scalar identities of gx-scalar's matrices of `{X}.raw` (`Work/GCert/Data/Glue.lean`) -/ +theorem {nm.replace('_of_scal', '_of_ids')} (hv : 0 < {X}.raw.v) + (hAi : GS.MAi {X}.raw hv * GS.MA {X}.raw hv = 1) (hBi : GS.MBi {X}.raw hv * GS.MB {X}.raw hv = 1) + (hx : ∀ (t : Fin {X}.raw.v) (j : GS.Rl {X}.raw), (GS.MB {X}.raw hv * CR.scatM (GS.Jrm {X}.raw * GS.Ccm {X}.raw) + * GS.MA {X}.raw hv) (CR.lX t) j = if CR.lX t = j then 1 else 0) + (hy : ∀ (i : GS.Rl {X}.raw) (t : Fin {X}.raw.v), (GS.MB {X}.raw hv * CR.scatM (GS.Jrm {X}.raw * GS.Ccm {X}.raw) + * GS.MA {X}.raw hv) i (CR.lY t) = if i = CR.lY t then 1 else 0) + (hid : ∀ S t : Fin {X}.raw.v, (GS.MB {X}.raw hv * CR.scatM (GS.Jrm {X}.raw * GS.Ccm {X}.raw) + * GS.MA {X}.raw hv) (CR.lY S) (CR.lX t) = if S = t then 1 else 0) : + ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - {A}/(10:ℝ)^{B}) W := + {nm} (GXD.scalOf {X}.raw hv hAi hBi hx hy hid) GS.toCoef (GXD.roles {X}.raw hv) + (GXD.scalOf_MA _ hv hAi hBi hx hy hid) (GXD.scalOf_MAi _ hv hAi hBi hx hy hid) + (GXD.scalOf_MB _ hv hAi hBi hx hy hid) (GXD.scalOf_MBi _ hv hAi hBi hx hy hid) + (GXD.reg {X}.raw) (GXD.reg_slot {X}.raw hv hAi hBi hx hy hid) (GXD.reg_mem {X}.raw) + (by rw [Fintype.card_fin]; decide) +''' + txt = f'''import Work.GCert.Chain.End +import Work.GCert.Data.Glue +import Work.GCert.Data.{cname}Cert +import Work.GCert.Data.Inst{name} + +/-! +# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `{cname}` (h = {h}, m = {m}): final theorems over a scalar half +GENERATED by checks/wht26/gx-data/py/gxrate.py. No `sorry`. +`GX.wht_of_halves` (gx-chain) on the label half `{X}.valid` (kernel checks of `Work/GCert/Data/Gen/{cname}*`), its +price `{X}.lab_price`, the unit list `BlockAccounting.blocks{name}` (`W = {W}` live roles) and the rate facts +`fold_{name}` (`1 - {d['A']}/10^{d['B']}`), `fold_{name}x` (`1 - {dx['A']}/10^{dx['B']}`). +The ONLY hypothesis left is the scalar half: `S : GX.Scal (Fin {v}) (Fin {R}) C` whose gate matrices are the ordered +products of the adds of `raw` (`hMA .. hMBi`), with `{nc}` copies read from the retained registers (`reg`, `hCc`, `hreg`, `hC`); +`wht_{name}_of_ids`, `wht_{name}x_of_ids`: the same from the five identities `hAi hBi hx hy hid` of gx-scalar's matrices +`GS.MA GS.MAi GS.MB GS.MBi GS.Jrm GS.Ccm` of `raw` (the fields of the planned `GS.ScalOK`). +-/ + +namespace OAI +namespace PowerSaving +namespace WHT +open BlockWHT + +variable {{C : Type}} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin {X}.raw.v) (Fin {X}.raw.R) C) + {{emb : CR.L3 (Fin {X}.raw.v) (Fin {X}.raw.R) → Nat}} {{un : Nat → CR.L3 (Fin {X}.raw.v) (Fin {X}.raw.R)}} + (cf : GXD.Co → SSC.Coef) +''' + thm('wht_' + name + '_of_scal', name, d['A'], d['B'], d['s'], '') + thm('wht_' + name + 'x_of_scal', name + 'x', dx['A'], dx['B'], dx['s'], '_x') + f''' +end WHT +end PowerSaving +end OAI +#print axioms OAI.PowerSaving.WHT.wht_{name}_of_scal +#print axioms OAI.PowerSaving.WHT.wht_{name}x_of_scal +#print axioms OAI.PowerSaving.WHT.wht_{name}_of_ids +#print axioms OAI.PowerSaving.WHT.wht_{name}x_of_ids +''' + open(out + '%s.lean' % name, 'w').write(txt) + + +def main_module(name, cname, d, dx, out): + """Main.lean: the CLOSED theorems (what the comparator solution cites). WRITTEN against gx-scalar's planned + `GS.ScalOK` (fields / theorems hv hAi hBi hx hy hid, SCALAR v2); compiles when Work/GCert/Scalar/Main.lean exists.""" + K = 'GXD.%s.scalOK' % cname + + def thm(sfx, A, B): + return f''' +/-- **Walsh-Hadamard transform in `O(n (log n)^(1 - {A}/10^{B}))`-type time bounds: whole-block recursion on the bridged +word B_2 with the kernel-checked gcert/1 circuit `{cname}`.** -/ +theorem wht_main_{name}{sfx} : + ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - {A}/(10:ℝ)^{B}) W := + wht_{name}{sfx}_of_ids {K}.hv {K}.hAi {K}.hBi {K}.hx {K}.hy {K}.hid +''' + txt = f'''import Work.GCert.Scalar.Main +import Work.GCert.Data.{cname}ScalOK +import Work.GCert.Data.{name} + +/-! +# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `{cname}`: THE CLOSED THEOREMS +GENERATED by checks/wht26/gx-data/py/gxrate.py. No `sorry`. +`wht_{name}_of_ids` (label half, price, unit list, rate fact, gx-chain's end theorem) applied to the scalar half +`{K} : GS.ScalOK GXD.{cname}.raw` (kernel checks `scal_ok`, `ychk_ok`; soundness: gx-scalar). +-/ + +namespace OAI +namespace PowerSaving +namespace WHT +open BlockWHT +''' + thm('', d['A'], d['B']) + thm('x', dx['A'], dx['B']) + f''' +end WHT +end PowerSaving +end OAI +#print axioms OAI.PowerSaving.WHT.wht_main_{name} +#print axioms OAI.PowerSaving.WHT.wht_main_{name}x +''' + open(out + '%sMain.lean' % name, 'w').write(txt) diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxconv.py b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxconv.py new file mode 100644 index 0000000000..49ebb65973 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxconv.py @@ -0,0 +1,131 @@ +"""gxconv.py: gcert/0 -> gcert/1 for files whose in-place pair op SCALES its target +(`op dest ctrl ca cb`: dest := ca dest + cb ctrl with ca not in {1}; #193 has ca = -2), questions.md D2. +COPY of `convert` of gx.py with ONE change: a scaling is not a gate, it changes the +UNIT in which the register is stored. wanted(r) = lam[r] * stored(r), lam = 1 at the start; + op dest ctrl ca cb : lam[dest] *= ca ; stored(dest) += (cb lam[ctrl] / lam[dest]) stored(ctrl) + any other add t += k s : stored(t) += (k lam[s] / lam[t]) stored(s) + scatter y_t += coef(t, k) total_k : coefficient coef(t, k) * lam[slot of total k] (a TABLE if some lam != 1) +Only slots may be scaled (tested: lam = 1 on every x and y register, always), so the x / y rows and columns of +the product (E5: hx, hy, hid) are those of the unscaled circuit; frames, blocks and N are untouched. +The result is tested by the reference checker `gx.check1` (labels and scalars) before it is written. + +usage: gxconv.py out= [noext] +""" +import sys, time +from fractions import Fraction as Fr +import gxpaths +import gx +import gxrun +need = gx.need + + +def convert(c0): + need(c0["format"] == "gcert/0", "input is not gcert/0") + v, h, R = c0["v"], c0["h"], c0["R"] + ev = c0["events"] + cen = [n for n, e in enumerate(ev) if e[0] == "centre"] + need(cen == list(range(cen[0], cen[0] + len(cen))), "centres are not contiguous: no single scatter cut") + A, B, ret = [], [], [] + neg, openq, lam, nsc = set(), {}, {}, 0 + L = lambda r: lam.get(r, Fr(1)) + + def co(t, s, a, b=1): # stored coefficient of the wanted add t += (a/b) s + w = Fr(a, b) * L(s) / L(t) + return [w.numerator, w.denominator] + + def o(f, s, tg, ex=()): + return ["out", f, s, [[t] + co(t, s, a, b) for t, a, b in tg], list(ex)] + + def i(f, t, sr): + return ["in", f, t, [[s] + co(t, s, a, b) for s, a, b in sr]] + scat_lam = None + for n, e in enumerate(ev): + k = e[0] + out = A if n < cen[0] else B + if k in ("read0", "undo", "uninj"): + continue + if k == "inj": + need(e[2] not in neg, "inj reads a negated x register") + out.append(o(e[3], e[2], [[e[1], 1, 1]])) + elif k == "op": + if e[3] != 1: + need(e[3] != 0 and e[1] >= 2 * v, "op scales a register that is not a slot (or by 0)") + lam[e[1]] = L(e[1]) * e[3] + nsc += 1 + out.append(o(e[5], e[2], [[e[1], e[4], 1]])) + elif k == "centre": + ret.append([e[2], e[1], e[3]]) + if scat_lam is None: + scat_lam = dict(lam) + elif k == "gread": + hit = set(t for t, _, _ in e[3]) + out.append(o(e[2], e[1], [[v + t, a, b] for t, a, b in e[3]], [v + t for t in e[4] if t not in hit])) + elif k == "rread": + out.append(o(e[2], e[1], [[v + t, a, b] for t, a, b in e[3]])) + elif k == "ysh": + out.append(o(e[4], e[2], [[e[1], e[3], 1]])) + elif k == "yw": + out.append(o(e[4], e[2], [[e[1], e[3][0], e[3][1]]])) + elif k == "kd": + out.append(o(e[3], e[2], [[e[1], -1 if e[2] in neg else 1, 1]])) + elif k == "ktr": + q = tuple(e[1]) + need(e[2] == gx.KJ, "ktr matrix is not 1 - J/2") + x1, rest = q[0], list(q[1:]) + if q not in openq: + out.append(i(e[3], x1, [[r, 1, 1] for r in rest])) + out.append(o(e[3], x1, [[r, -1, 2] for r in rest])) + out.append(i(e[3], x1, [[r, 1, 1] for r in rest])) + openq[q] = n + neg.add(x1) + else: + out.append(i(e[3], x1, [[r, -1, 1] for r in rest])) + out.append(o(e[3], x1, [[r, 1, 2] for r in rest])) + out.append(i(e[3], x1, [[r, -1, 1] for r in rest])) + del openq[q] + neg.discard(x1) + else: + raise AssertionError("unknown event kind %r" % k) + need(all(r >= 2 * v for r, x in lam.items() if x != 1), "an x or y register is scaled") + need(not openq and not neg, "a K block is not closed") + ret = sorted(ret) + need([z[0] for z in ret] == list(range(len(ret))), "retained totals are not 0 .. n-1") + sin, sout = Fr(*c0["scat"]["inside"]), Fr(*c0["scat"]["outside"]) + sl = [(scat_lam or {}).get(z[1], Fr(1)) for z in ret] + if all(x == 1 for x in sl): + scat = dict(inside=c0["scat"]["inside"], outside=c0["scat"]["outside"]) + else: + need(len(ret) == h, "star rule with a number of totals other than h") + scat = dict(table=[[[kk, (w * sl[kk]).numerator, (w * sl[kk]).denominator] + for kk in range(h) for w in [sin if c0["ports"][t] >> kk & 1 else sout]] for t in range(v)]) + ext = sorted([z[3], z[1]] for z in c0["gauge"] if z[5]) + c1 = dict(format="gcert/1", derived_from=c0["name"] + " (gcert/0; %d scalings as units, gxconv.py)" % nsc, p=c0["p"], + h=h, v=v, R=R, cst=c0["cst"], N=c0["N"], ports=c0["ports"], frames=c0["frames"], start=c0["start"], + final=c0["final"], ext=ext, ret=ret, scat=scat, A=A, B=B, blocks=c0["blocks"]) + return c1, nsc, sorted(set(str(x) for x in lam.values())), sum(1 for x in sl if x != 1) + + +if __name__ == "__main__": + t0 = time.time() + opt = dict(a.split("=", 1) for a in sys.argv[2:] if "=" in a) + c0 = gxrun.load(sys.argv[1]) + c1, nsc, lams, nsl = convert(c0) + c1["two_hosts"] = sorted(2 * c0["v"] + i for i, hs in enumerate(c0["hosts"]) if len(hs) > 1) + if "noext" in sys.argv and c1["ext"]: # as gxrun.py: exterior gauges become first blocks + for r, f in c1["ext"]: + d = str(len(c1["frames"][f])) + c1["start"][r] = 0 + c1["blocks"]["s"][d] = c1["blocks"]["s"].get(d, 0) + 1 + c1["N"] += int(d) + c1["blocks"]["s"] = {k: c1["blocks"]["s"][k] for k in sorted(c1["blocks"]["s"], key=int)} + c1["derived_from"] += ", exterior gauges turned into first blocks" + c1["ext"] = [] + st = {} + gx.check1(c1, scalar=True, stats=st) + print("ACCEPTED by gx.check1: %s h=%d v=%d R=%d N=%d gates A=%d B=%d ext=%d; scalings %d, final units %s, scaled totals %d, " + "scatter %s, stats %s (%.1fs)" % (c0["name"], c1["h"], c1["v"], c1["R"], c1["N"], len(c1["A"]), len(c1["B"]), + len(c1["ext"]), nsc, lams, nsl, "table" if "table" in c1["scat"] else "star", st, + time.time() - t0)) + if "out" in opt: + gxrun.dump(c1, opt["out"]) + print("written", opt["out"]) diff --git a/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxcore.py b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxcore.py new file mode 100644 index 0000000000..3e3f0a3076 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/inputs/complex/sources/tools__gx__gxcore.py @@ -0,0 +1,375 @@ +"""gxcore.py: gcert/1 JSON -> normal data D, PYTHON MIRROR of the two Lean checks +(label side: E0 frames, E1 pairs, E1/E2/E6 replay in segments; scalar side: E3 kinds, E4 bracket, E5 replay +in source batches, E7), derived data (distinct climbs, cut states, block histogram) and kernel sizes. +Own code; the reference is gx.py (check1), not imported. Python 3, standard library only.""" +import json, gzip +from collections import Counter +from math import gcd + + +class Reject(Exception): + pass + + +def need(c, msg): + if not c: + raise Reject(msg) + + +def load(path): + c = json.load(gzip.open(path, "rt") if path.endswith(".gz") else open(path)) + need(c.get("format") == "gcert/1", "E0: format is not gcert/1") + return c + + +def co(num, den): + need(den > 0, "E0: denominator") + return (num < 0, abs(num), den) + + +def normal(c): + """the Lean-facing data: gates as tuples, coefficients (neg, num, den), ret by total""" + def gate(g): + if g[0] == "out": + return ("out", g[1], g[2], [(t, co(a, b)) for t, a, b in g[3]], list(g[4])) + need(g[0] == "in", "E0: unknown gate kind %r" % (g[0],)) + return ("in", g[1], g[2], [(s, co(a, b)) for s, a, b in g[3]]) + ret = sorted(c["ret"]) + need([k for k, _, _ in ret] == list(range(len(ret))), "E2: retained totals numbered 0..") + sc = c["scat"] + scat = ("table", [[(k, co(a, b)) for k, a, b in row] for row in sc["table"]]) if "table" in sc else \ + ("star", co(*sc["inside"]), co(*sc["outside"])) + return dict(h=c["h"], v=c["v"], R=c["R"], N=c["N"], cst=c["cst"], ports=list(c["ports"]), + frames=[list(f) for f in c["frames"]], start=list(c["start"]), final=list(c["final"]), + ext=[tuple(e) for e in c.get("ext", [])], ret=[(s, f) for _, s, f in ret], scat=scat, + A=[gate(g) for g in c["A"]], B=[gate(g) for g in c["B"]], + blocks={k: {int(r): n for r, n in c["blocks"][k].items()} for k in "xysc"}) + + +def regs(g): + return [g[2]] + [t for t, _ in g[3]] + (g[4] if g[0] == "out" else []) + + +def adds(g): + """single adds (tgt, src, coef) in order""" + return [(t, g[2], k) for t, k in g[3]] if g[0] == "out" else [(g[2], s, k) for s, k in g[3]] + + +def reduce(u, basis): + for b in basis: + if u >> (b.bit_length() - 1) & 1: + u ^= b + return u + + +def echelon_ok(f): + piv = [b.bit_length() - 1 for b in f] + if any(b == 0 for b in f) or any(piv[i] <= piv[i + 1] for i in range(len(piv) - 1)): + return False + return all(not (b >> p & 1) for i, b in enumerate(f) for j, p in enumerate(piv) if i != j) + + +def par(x): + return bin(x).count("1") & 1 + + +def cls(D, r): + return "x" if r < D["v"] else "y" if r < 2 * D["v"] else "s" + + +# ---- label side ------------------------------------------------------------------------------------------ +def chk_frames(D): + """E0: ports; frame table reduced echelon, no repeats, zero first, full second""" + h, v, fr = D["h"], D["v"], D["frames"] + p = D["ports"] + need(len(p) == v == len(set(p)) and all(par(q) and 0 < q < 1 << h for q in p), "E0: ports") + need(len(fr) >= 2 and fr[0] == [] and fr[1] == [1 << i for i in range(h - 1, -1, -1)], "E0: frames 0 and 1") + need(all(echelon_ok(f) and all(b < 1 << h for b in f) for f in fr), "E0: frame table not reduced echelon") + need(len(set(map(tuple, fr))) == len(fr), "E0: frame table has repeats") + return [len(f) for f in fr] + + +def derive(D): + """distinct climbs in order of first use (NOT a check: plain replay of the frame ids)""" + cur, seen, pairs = list(D["start"]), set(), [] + + def go(r, f): + o = cur[r] + if o != f: + if (o, f) not in seen: + seen.add((o, f)) + pairs.append((o, f)) + cur[r] = f + for g in D["A"] + D["B"]: + for r in regs(g): + go(r, g[1]) + for r, f in enumerate(D["final"]): + go(r, f) + return pairs + + +def chk_pairs(D, dim, pairs): + """E1, level 2: every listed pair is a nested pair of the table; returns the mask-operation count""" + fr, n, work = D["frames"], len(D["frames"]), 0 + for a, b in pairs: + need(a < n and b < n and dim[a] < dim[b], "E1: pair (%d, %d) does not climb" % (a, b)) + need(all(reduce(u, fr[b]) == 0 for u in fr[a]), "E1: pair (%d, %d) is not nested" % (a, b)) + work += dim[a] * dim[b] + return work + + +def cut_points(D, gmax, wmax=210000): + """replay segments: each phase (A, B) is cut into k pieces of equal gate count, k = the least number with at + most `gmax` gates AND at most `wmax` work per piece on average (work = sum over gates of the dimension of the + gate's frame; measured at h = 22: about 1 ms of kernel time per unit, NOTES.md). + Cut points = gate counts in A ++ B (always 0, |A| and the end).""" + nA, nB = len(D["A"]), len(D["B"]) + cuts = [0] + for off, gs in ((0, D["A"]), (nA, D["B"])): + n = len(gs) + k = max(1, -(-n // gmax), -(-sum(len(D["frames"][g[1]]) for g in gs) // wmax)) + cuts += [off + (n * i) // k for i in range(1, k + 1)] + return sorted(set(cuts)) + + +def label_data(D, dim, cuts): + """for gx-labels (gcert-interface.md LABELS v1): ordered block ranks per replay segment, the final climbs as + gates without adds (same final frame, at most 32 registers, only registers that still climb), their ranks, + the largest number of registers named by one gate. Plain replay (the mirror has accepted the file).""" + cur, G = list(D["start"]), D["A"] + D["B"] + rs, most = [], 0 + for a, b in zip(cuts, cuts[1:]): + r = [] + for g in G[a:b]: + most = max(most, len(regs(g))) + for x in regs(g): + if cur[x] != g[1]: + r.append(dim[g[1]] - dim[cur[x]]) + cur[x] = g[1] + rs.append(r) + by = {} + for x, f in enumerate(D["final"]): + if cur[x] != f: + by.setdefault(f, []).append(x) + fins, rF = [], [] + for f in sorted(by): + for i in range(0, len(by[f]), 32): + xs = by[f][i:i + 32] + fins.append(("out", f, xs[0], [], xs[1:])) + rF += [dim[f] - dim[cur[x]] for x in xs] + return dict(rs=rs, fins=fins, rF=rF, most=max(most, 32 if fins else 0)) + + +def chk_replay(D, dim, pairs, cuts): + """E1 (labels), E2 (scatter cut), E6 (price), in the segments given by `cuts`. + Returns (states at the cuts, histogram by class, visits per segment).""" + h, v, R, fr = D["h"], D["v"], D["R"], D["frames"] + n = 2 * v + R + start, final, ext = D["start"], D["final"], dict(D["ext"]) + need(len(start) == len(final) == n, "E0: start / final") + for t in range(v): + need(fr[start[t]] == [D["ports"][t]] and final[t] == 1, "E1: x start / final") + f = fr[final[v + t]] + need(start[v + t] == 0 and len(f) == h - 1 and all(par(u & D["ports"][t]) == 0 for u in f), "E1: y start / final") + for r in range(2 * v, n): + need(start[r] == ext.get(r, 0) and final[r] == 1, "E1: slot start / final") + ok = set(pairs) + need(len(ok) == len(pairs), "E1: pair list has repeats") + cur = list(start) + H = {k: Counter() for k in "xysc"} + + def climb(r, f, where): + o = cur[r] + if o != f: + need((o, f) in ok, "E1: climb (%d, %d) is not a tested pair (%s, register %d)" % (o, f, where, r)) + H[cls(D, r)][dim[f] - dim[o]] += 1 + seg[dim[f] - dim[o]] += 1 + cur[r] = f + seg, segs = Counter(), [] + G, nA = D["A"] + D["B"], len(D["A"]) + states, visits = [], [] + need(cuts[0] == 0 and cuts[-1] == len(G) and cuts == sorted(cuts), "cut points") + for a, b in zip(cuts, cuts[1:]): + states.append(list(cur)) + vs = 0 + for i in range(a, b): + if i == nA: + scatter_cut(D, cur, dim, H, seg) + g = G[i] + rs = regs(g) + need(all(0 <= r < n for r in rs) and len(set(rs)) == len(rs), "E3: registers of gate %d" % i) + need(i >= nA or all(r < v or r >= 2 * v for r in rs), "E2: y role named in phase A (gate %d)" % i) + vs += len(rs) + for r in rs: + climb(r, g[1], "gate %d" % i) + visits.append(vs) + if b < len(G): + segs.append(seg) + seg = Counter() + if nA == len(G): + scatter_cut(D, cur, dim, H, seg) + states.append(list(cur)) + for r in range(n): + climb(r, final[r], "final") + segs.append(seg) + Hb = {k: dict(sorted(w.items())) for k, w in H.items()} + need(Hb == D["blocks"], "E6: block histogram differs from `blocks`") + N = sum(r * c for w in H.values() for r, c in w.items()) + need(N == D["N"] and N + sum(dim[f] for f in ext.values()) == R * h + 2 * v * (h - 1) + D["cst"], "E6: N") + need(sum(r * c for r, c in H["c"].items()) == D["cst"], "E6: cst") + return states, Hb, visits, [[w[r] for r in range(h + 1)] for w in segs] + + +def scatter_cut(D, cur, dim, H, seg): + v = D["v"] + if D["scat"][0] == "table": + need(len(D["scat"][1]) == v and all(k < len(D["ret"]) for row in D["scat"][1] for k, _ in row), "E2: scatter table") + else: + need(len(D["ret"]) == D["h"], "E2: one retained total per coordinate (star rule)") + for s, f in D["ret"]: + need(s >= 2 * v and cur[s] == f, "E2: retained slot %d is not at its frame" % s) + H["c"][dim[f]] += 1 + seg[dim[f]] += 1 + need(all(cur[v + t] == 0 for t in range(v)), "E2: a y role left frame 0 before the scatter") + + +# ---- scalar side ----------------------------------------------------------------------------------------- +SC = 6 # all scalar entries are kept as integers in units of 1/6 +KINDS = {("x", "s"), ("s", "s"), ("s", "y"), ("x", "y"), ("x", "x"), ("y", "y")} + + +def chk_shape(D): + """E3 kinds, E4 bracket""" + ysum, piv, ytg = Counter(), set(), set() + for ph in "AB": + for i, g in enumerate(D[ph]): + for t, s, (ng, a, b) in adds(g): + kd = (cls(D, s), cls(D, t)) + need(kd in KINDS and t != s, "E3: gate kind %s -> %s (gate %s%d)" % (kd + (ph, i))) + need(ph == "B" or kd in {("x", "s"), ("s", "s")}, "E3: bank gate before the scatter (gate A%d)" % i) + if kd == ("y", "y"): + need(SC % b == 0, "E4: denominator beyond sixths") + ysum[s, t] += (-a if ng else a) * (SC // b) + piv.add(s) + ytg.add(t) + need(not (piv & ytg), "E4: a pivot is the target of a gate reading y") + need(all(x == 0 for x in ysum.values()), "E4: y-reads of a (pivot, target) pair do not cancel") + return len(ysum) + + +def scat_row(D, t): + sc = D["scat"] + if sc[0] == "table": + return sc[1][t] + return [(k, sc[1] if D["ports"][t] >> k & 1 else sc[2]) for k in range(D["h"])] + + + +def replay(D, init, count=None): + """exact replay through A, scatter, B of sparse vectors with entries in sixths; `init` = register -> {key: 6}""" + v = D["v"] + reg = [dict(init.get(r, ())) for r in range(2 * v + D["R"])] + + def axpy(t, src, k): + ng, a, b = k + d = reg[t] + for key, u in src.items(): + w = u * a + need(w % b == 0, "E5: an entry needs a denominator beyond sixths (register %d)" % t) + w = w // b + w = d.get(key, 0) + (-w if ng else w) + if count is not None: + count[key] += 1 + if w: + d[key] = w + else: + d.pop(key, None) + + def run(gates): + for g in gates: + if g[0] == "out": + src = dict(reg[g[2]]) + for t, k in g[3]: + axpy(t, src, k) + else: + for s, k in g[3]: + axpy(g[2], reg[s], k) + run(D["A"]) + afterA = sum(len(d) for d in reg) + tot = [dict(reg[s]) for s, _ in D["ret"]] + for t in range(v): + for k, cf in scat_row(D, t): + if tot[k]: + axpy(v + t, tot[k], cf) + run(D["B"]) + return reg, afterA + + +def chk_scalar(D, batches=1): + """E5 (+ E7 when `ext` is non-empty). Returns stats: denominators and largest numerators per class in the + units of the spec (x halves, y sixths, slots integers), updates per source batch, entries after A.""" + v = D["v"] + count = Counter() + reg, afterA = replay(D, {t: {t: SC} for t in range(v)}, count) + need(all(reg[t] == {t: SC} for t in range(v)), "E5: an x role is not restored (X3)") + need(all(reg[v + t] == {t: SC} for t in range(v)), "E5: a target does not receive exactly its source (hid)") + if D["ext"]: + rg, _ = replay(D, {r: {r: SC} for r, _ in D["ext"]}) + need(all(not rg[v + t] for t in range(v)), "E7: the dirt of an exterior-gauged slot reaches a y role") + step = -(-v // batches) + per = [sum(count[k] for k in range(lo, min(lo + step, v))) for lo in range(0, v, step)] + return dict(updates=sum(count.values()), per_batch=per, afterA=afterA, + batches=[(lo, min(lo + step, v)) for lo in range(0, v, step)]) + + +def digits(D): + """largest |entry| met and denominators per class, by a second replay that watches every write (slow path + kept separate from the check): returns {class: (lcm of denominators, largest |numerator| in units 1/den)}""" + v = D["v"] + den = {"x": 1, "y": 1, "s": 1} + mx = {"x": 0, "y": 0, "s": 0} + reg = [({r: SC} if r < v else {}) for r in range(2 * v + D["R"])] + + def axpy(t, src, k): + ng, a, b = k + d, kk = reg[t], cls(D, t) + for key, u in src.items(): + w = d.get(key, 0) + (-(u * a // b) if ng else u * a // b) + if w: + d[key] = w + q = SC // gcd(w, SC) + if den[kk] % q: + den[kk] = den[kk] * q // gcd(den[kk], q) + if abs(w) > mx[kk]: + mx[kk] = abs(w) + else: + d.pop(key, None) + for g in D["A"]: + for t, s, k in adds(g): + axpy(t, reg[s], k) + tot = [dict(reg[s]) for s, _ in D["ret"]] + for t in range(v): + for k, cf in scat_row(D, t): + axpy(v + t, tot[k], cf) + for g in D["B"]: + for t, s, k in adds(g): + axpy(t, reg[s], k) + return {k: (den[k], mx[k] * den[k] // SC if (mx[k] * den[k]) % SC == 0 else "%d/6" % mx[k]) for k in "xys"} + + +def mirror(D, gmax=12000, batches=1, pairs=None, cuts=None): + """both checks; returns everything the generator and the dry run need""" + dim = chk_frames(D) + pairs = derive(D) if pairs is None else pairs + work = chk_pairs(D, dim, pairs) + cuts = cut_points(D, gmax) if cuts is None else cuts + states, H, visits, segH = chk_replay(D, dim, pairs, cuts) + ysh = chk_shape(D) + S = chk_scalar(D, batches) + hist = Counter() + for k in "xysc": + for r, n in H[k].items(): + hist[r] += n + return dict(dim=dim, pairs=pairs, pair_work=work, cuts=cuts, states=states, H=H, visits=visits, ysh=ysh, + scal=S, hist=dict(sorted(hist.items())), segH=segH, + lab=label_data(D, dim, cuts)) diff --git a/research/multicut-delayed-inverse-checkpoint/kernel-candidates.json.gz b/research/multicut-delayed-inverse-checkpoint/kernel-candidates.json.gz new file mode 100644 index 0000000000000000000000000000000000000000..ca1360f111e32a699ae687d979dec1d473083377 GIT binary patch literal 59015 zcma&O2|U!>`v+dPT}oysl7unVFqR}mC0j-+dqPH(vQ&s{Q^_)seVHLiGO}l1hZ_@> zkr=xWiV&mh4F2acL%H{R|G)qL-q$_zndNiN=bYzx-tYJGoa6EZZr=d?^B2pHZi^5> z6ZiJv#Vs>Rh{00hV7Tx(dWVuKehaHUa5?Uu~=I0$I^dVRMMwv`LHY0 z?B(p3iQxw9qJo9gX?O#6ASJ<@KE#KejZ;}#9%{o*u{mOAztQ()uS86w>ypwjgDqYa z9}KgX@BHjle!^Lyn;JoGL(VjN+F6qLOpms8rIVL-dK9YUO*FczluayqWT#IY^vEum zPanZ%7lLcc0~m{H$d~A?846X@m12tbuHxo7WHyCZ4@P1^CB&f zm|pqx@q$$4*UHL!5A@4dp7hdx)~cM-dO-cGMSUM|y|xW& zyw_qycLjW=pt2Gog@dRNrCs}V$r1vXrbhv`cLb0UhcTzs$&pq9NEfIVKowiZSMtA@m50!Aa1V`1CXAu7bMp zfJEzYDa=KqXwbQfS`-}hEmvT0xQV@!%M3^YdpWWHI^k6jNtw?Wz*M*vr^4!BX>vN!jL6oz9)u;W=hJNv+{?d(a?>{c(* zq}h`cpv$D9sxz$Y1COEA#V_LUxgv2xV*!d!qf24}?ky!9;6&+A2hX4jUIolAT#kjI zykhR`MVV241Pq^$x9F+(`!YNHh#=;iW{~W4Xg=+m;{q-iJ2Ecw;-M96G=yu5uJqDox05;s*;3f&!7#-FL$t0zn(!~xEOJZIp77Tr9Ba1?ZdrE|tM?x@WASKm}vb;0BOW=zz~xU7?1P-ARNUj=3}xj5oVM z@WA0q?KRZU-5iSSdQ##T^91b(0pifb9yPR)zM4AuX+o@&qs5i109#jnXhy>u*j33Q z846JQ5?OPN^ds(-2JSuX%QGH!WgbBQ1kRWW5vL_srUIz-L(+&Es|u;iRl7HV8Osc zpn;Rp1c*VF9!i)a>DJ>yn2QP(g2;jnUewr{&wU6hL%Q%Aa4|}dQc)j{7ec^ zo+2PT9Q7-D_d=R$4Vkh>X#*VYY)7`GK=Zz~I_n>*PfJILdg;s~TTB;0)(1FbG zJ@fVAMP5x$0vz^xgmf1i{@8(Q8Wd^)g>(kp5}71GJZo@{9JwN?|4e%r_}@ny zel}&Wu4^A&93v)RLoyef;|^UviQf=c-fIAY2SgJO2zDbeUZb zjvAfb|7KPf3U&rGwld5CTJRRtr?>X_%McB%veM!4Dap5X9ci9YJqE3oU;bD-=GVV+G2 zY#~cbhp+h!L@*5R0T*x-b2{$gP)s9pAXh7@8rLpR`RRgXcd~)qB}-&Mv%tuUVcci` z-VaBORS6*Fr~-pOE`9{`VoQLdXu9<=MN-m6-Sx6YwB&pSJVHj^?VC=BkVqyv4VDGKD^8w&&-1NSr;N%_2r#l%FM2DsG2< zld)QP;CY@LnR<{DWlb3i5Dm0eql`M)UL;Rl{$N6W`GOLFpVXWDHnx+(iQ)y;wE+NA;(%-6~>8}Ic!x%&v0f^E;1I-OQ3?+yuMX|&41dzMW zpcQnpATl5@0gFnB|MJt0Vr?N{k^GXhJa%AZm^It5NKx z?i~B^b9tK@di%Fm0iva1!k7>Lm{hZig9c?jb^h1PmchqdN+1|jGdyMn<@aJ6l3zPO zZp;97CcrSMcpVxuxK3bDQh;cBSA&&3FCbtCTmT8!`0^#PGdDXG_}nV_e`co^Zec9h ziC0DZtxfXhVDSBrQ@tikSJa2moXH#hFWy&rjL+KM(8ca_^f2IMJs0~ENK9ImgOdR9 zk|DWVIR>eQZb*KrhBnn-$1BJ^AT0S?ecUJa2W~UMb1|T%J_Crd6J1PY_TT#K2D_fJ zBbbh?={nYrWU@Y!SE`P)Q(bVtS6OomKPhcM9fz%<@C3@eOe#vWwe{^jrg#zE3%+7< z7lR(N0`Q~&yz!&_kG!a|W|>(S%AOkh7+)xPM}TOqPxkbZ)PbN_F%X|6K~zK2R!xJC z1)~KS8-7}RM8@A*Lmgmon^s_?XJhLChsh@l(KM0>IvoP8gcQfb-e!*|eHviPBo%;X ztE4h5!441nXciH`B^X^WyZa<1=l5)^9;=bRk5%Zi!21_vG=R0Ns#4Yj8PgY%y3PXE zy8WB(WAA!ups80~$@xrvj0_kqi*#j&KYmubpC2_e1c=25uZ=m2*os40ncSnn;i|mc zKluUhJRu)qv?4*!g&@kEC-)Fa6Of}IXo;GvFdy>gc(Q(%DY(cV(8k0bD?<=XW6Ap5 zFumY#JW?Bzwh<>f6o$VQ{1Eh@8RR;+jT0rlNekSX^twkg2(=fMqyS5jh7EkwXIaE= z`ehyD3x4P>zB-_0P`XAazObt;8gH-?*z<(^v?!fb6Wv|F4TA^zYN5My!Xb)`z-N63 z!_P8jqJ{aG?i9)eX4CP7Hb*9(*HCZQ4C3Djp7Hl=zCR&@IdT0I4T9VNd{cZH18%X| zjT?UKWsg(bj?;o8i0{Mjucg5!wlJr=qMKk5cMyY1B_Jv}2gC5WcHs9lLDngxP3(C( zp^sMk=Ca~PxeG#Tr5RKOR;tGMQ=|F&m{a^UjM2lTInnX6HI=IIf>~kK@e5J%w?lD&Fltz za0&))2Fe9y_3j~=xk#&q2lMh6vp>I`*FN48tlA+|Rw$knhHs6=*Dh_PX<#0MS>%IR z97bb0KS$#oBFXxSFzW6iva_rfT7t1w-|mxi(UeBzYkga`;7(yUT6u66xNxX`)wN4N z`H&Cht3$R>hpxH9@gJk{yTb9!I-z)^2BxVBL9DaG0e!a-rv>`oRY)ddkT|W4pg)T7 zly{YGpvJft%bI&Z>+xqM%tZn7aeQ4=<)BmG@+%oyD^Cr~!^0@H0U1|8l!4+Y%*X*; 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/multicut-delayed-inverse-checkpoint/loader/MANIFEST.json b/research/multicut-delayed-inverse-checkpoint/loader/MANIFEST.json new file mode 100644 index 0000000000..c9b7512456 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/MANIFEST.json @@ -0,0 +1,60 @@ +{ + "excluded_outputs": [ + "check_development_equivalence.py", + "equivalence_result.json", + "preparation_result.json", + "raw_ledger_result.json", + "qualification_result.json" + ], + "files": { + "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "NOTICE.md": "ee632728cc4d89565292d288f8289f483ac794b5d02cc6e42c2b2c43c43ff1d8", + "README.md": "4c36794b0f702ed521c0a7eaeacb1b7cc258760d59193cc8eb29be590e709dcd", + "SOURCE_PINS.json": "5a4fa413ea0ecc74050359329e90521723151e421cc0c19f774b92bd608a3ac9", + "expected_ledger.json": "4910004287e016f7f2db10f0a54eed24bf1912aa148b347b3cb4795fa3ac63f5", + "expected_semantics.json": "b7de3178436c824b6be6d99842270a82e1821b8008dda66d8b18519ad347da2d", + "raw_ledger.py": "89ba53114488a77a12b941db2787850c2c41612ad0332c47699a011c0c596b32", + "source_loader.py": "e5e19ee6d6899fce8273d0f06a49caf7cd8bfe940a2a9a103dbc9d3a62b6b291", + "test_public.py": "2103c953b37cf447d1a2bbb86c26f96fa5569a3ab018567e93884c12ebde8a2d" + }, + "schema": "five-stage-focused-loader-package/1", + "scope": "Clean source preparation and raw path ledger only; 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It preserves the original source files, headers and notices in the supplied source roots. + +- eumemic: the current PR210 operation frames, joint gauges, donor assignments, source borrowing and retimed source471 schedule at df95878d11190518e45ef9717c9ee05011f88ace; original OpenAI Codex assistance disclosure retained. +- Chafik Boukhalfa (chafreaky): the PR200 physical bit construction and terminal interfaces, carried by the consumed PR202 base at 8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d. +- Rohan Arun, Dugongue, evmckinney9, icekylinx, James Chang, eumemic, Douglas Colkitt, OpenAI, and preceding contributors retain the attribution and assistance disclosures recorded in the supplied original notices. + +The new contribution here is the explicit public-root binding, source-derived terminal records, exact semantic comparison and independent raw-path recount. It does not claim authorship of the inherited helper, frames, compiler or theorem interfaces, nor review or endorsement by the named contributors. + +Assisted with ChatGPT. diff --git a/research/multicut-delayed-inverse-checkpoint/loader/README.md b/research/multicut-delayed-inverse-checkpoint/loader/README.md new file mode 100644 index 0000000000..0bc303ed81 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/README.md @@ -0,0 +1,48 @@ +# Minimal source471 preparation and raw-ledger adapter + +This focused adapter consumes the current public PR210 source and pinned base-bit files directly. It prepares the actual retimed helper context and recounts its physical paths. It does not run the all-column scalar proof, prime suite, five-stage global lowering, or outer assembly. + +## Inputs + +`SOURCE_PINS.json` lists 22 exact public files, including original licences and notices: + +- eumemic/integer-mult-bounds at `df95878d11190518e45ef9717c9ee05011f88ace` +- CrocSwap/integer-mult-bounds at `8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d` + +Supply those source roots with their original relative paths. The supplied `source_inputs/` contains just the listed files for this qualification. Normal pinned upstream checkouts can be used instead. No network access or implicit downloading is performed by the adapter. Python 3.11 or newer with assertions enabled is required; no third-party Python package is used by this focused loader. + +## Targeted verification + +From this directory: + + python3 -B test_public.py --pr210-root source_inputs/pr210 --base-bit-root source_inputs/base_bit + +To print the prepared-source fingerprint: + + python3 -B source_loader.py --pr210-root source_inputs/pr210 --base-bit-root source_inputs/base_bit + +To print the independently rebuilt raw ledger: + + python3 -B raw_ledger.py --pr210-root source_inputs/pr210 --base-bit-root source_inputs/base_bit + +These commands do not alter inputs or write result files. Redirect output to a new external path if desired. The tests distinguish executable checks from the compact deterministic fixtures `expected_semantics.json` and `expected_ledger.json`. + +## Programmatic interface + +`source_loader.prepare(pr210_root, base_bit_root)` returns the source preparation namespace. It contains actual `W`, `C`, `schedule`, `borrow`, `terminal`, `regs`, `adj`, `at`, `kpair`, `deliveries`, selections and `SOURCE_TEXT` for the pinned scalar source. The payload replay function is deliberately not loaded or invoked. + +The unmodified public joint candidate body is loaded with its base-bit root explicitly rebound. The unmodified current source471 preparation body is then executed, except for the source's historical input-list lines and terminal receipt lookup. Terminal records are reconstructed from pinned source root data and the pinned sink selectors. All input hashes are checked before importing source code. + +`raw_ledger.reconstruct(context)` recounts internal, copied-center, original-source and target paths, including actual compensation and terminal chronology. It pays all copied-center calls and independent exterior completions. It neither reads nor rescales a packed historical profile. + +The semantic fingerprint binds exact basis rows, original-source/root frames, operation frames, gauge order/read times, aliases, partner mix/delivery chains, retiming, terminal records, all borrowed and independent dirty roles, and target responses. Dynamically allocated frame numbers differ from older preparation code that registered unused historical frames; exact basis/chronology equality is the relevant comparison. No rank-only substitution is made. + +## Qualification and exclusions + +The one-time development comparison is `check_development_equivalence.py`. It intentionally loads the earlier audited preparation once to compare exact semantics. It is not a runtime dependency and is excluded from the minimal loader package. `equivalence_result.json`, `preparation_result.json`, `raw_ledger_result.json`, and test logs are qualification outputs, not proof inputs. + +The minimal loader package consists of source_loader.py, raw_ledger.py, test_public.py, SOURCE_PINS.json, expected_semantics.json, expected_ledger.json, README.md, NOTICE.md, LICENSE and MANIFEST.json, plus the supplied source inputs or equivalent externally supplied pinned roots. Do not include historical search code, whole archived checkouts, or qualification receipts in that runtime closure. + +A successful focused run establishes only the preparation/ledger scope above. A complete portable five-stage candidate still needs its global lowering, mathematical witness coverage, finite bills, conditional contract binding and exact assembly connected to these actual source objects. + +Assisted with ChatGPT. diff --git a/research/multicut-delayed-inverse-checkpoint/loader/SOURCE_PINS.json b/research/multicut-delayed-inverse-checkpoint/loader/SOURCE_PINS.json new file mode 100644 index 0000000000..c3e9ef5269 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/SOURCE_PINS.json @@ -0,0 +1,37 @@ +{ + "base_bit": { + "commit": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d", + "files": { + "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "NOTICE": "c9850418622aea8d1382a230306c940947b5da818f4854b4b272ed60cf239f6c", + "research/paired-cube-diagonal-bit-168/bit/base_word.py": "9fe3a81b7561d2b09c086f63fac4626e88a32369ef8bc5579c94878de1218a89", + "research/paired-cube-diagonal-bit-168/bit/word.py": 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"03e2888758f0a95abfab5582bf77d435c484c8f4d1737eeb7a8ab9752fccda7a", + "root_bases": "747757d16a1e9b5112a296bff94754ac43e047126849b50599be3ded16d1e36d", + "schedule": "73294d81e5d32745f6df91ebe874153a5791e3c774eae6f833b4ca919f255e67", + "source_bases": "ff2b49b40d245a593561561457ec9ce2b16b739b2e4c277ce96898dba42c2c62", + "source_loans": "8d42978f946bd21279dc7c84f8f360974702e3d8e63e7278a484ff40805ab216", + "source_selection": "bd9e2b029487ba5b9a794c7a1721c92b8e9c18fc39d8903d3c2df0dbf5ba28d8", + "target_response": "f7302a72deaebd599cfc671eab4690a6866c547cefd41dd71f313a10e27f4aaf", + "terminal": "c8edf8b469bc4886f8af4ad7535cb8f35452fd3ff2e3236e93aa6b44877d359d" + }, + "independent_dirty_roles": 16643, + "local_independent_columns": 20163, + "logical_roles": 18908, + "retimed_operations": 168, + "schema": "source471-exact-semantic-fingerprint/1", + "source_head": "df95878d11190518e45ef9717c9ee05011f88ace", + "source_loans": 471, + "terminal_sinks": 34, + "v": 1760 +} diff --git a/research/multicut-delayed-inverse-checkpoint/loader/raw_ledger.py b/research/multicut-delayed-inverse-checkpoint/loader/raw_ledger.py new file mode 100644 index 0000000000..c5e44b07fe --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/raw_ledger.py @@ -0,0 +1,127 @@ +#!/usr/bin/env python3 +"""Fresh raw source471 path ledger from the clean public loader, no receipts. + +Reconstructs actual retimed source, target, auxiliary and copied-center paths. +No packed histogram, historical search, scalar replay or network input is used. +Assisted with ChatGPT. Retain supplied source licences and notices. +""" +from collections import Counter, defaultdict +from pathlib import Path +import argparse, hashlib, json, sys +import source_loader +sys.dont_write_bytecode = True +if not __debug__: raise SystemExit('refusing optimized Python') + + +def clean(hist): + return {str(k):v for k,v in sorted(hist.items()) if v} + + +def reconstruct(context): + w, c = context['W'], context['C'] + kp = {e['passive']:e for e in w.k['entries']} + borrow = context['borrow']; omitted = context['omitted'] + recipient = {d:b for b,d in w.pairs} + roots = defaultdict(list) + for j, role in enumerate(w.w['rootroles']): + roots[role].append(w.w['root_frame'][j]) + def chain(role): + out = [w.opframe[i] for i in w.role_ops[role] if i not in omitted]+roots[role] + if role in recipient: + r=recipient[role] + out += [w.gauge[r]['frame']]+[w.opframe[i] for i in w.role_ops[r] if i not in omitted]+roots[r] + return out+[w.w['full_frame']] + inclusion_cache={} + def included(a,b): + if (a,b) not in inclusion_cache: inclusion_cache[a,b]=c.sub(a,b) + return inclusion_cache[a,b] + def hist(start,path): + prev=start;rank=0 if start is None else c.dimf[start];result=Counter() + for frame in path: + assert prev is None or included(prev,frame), ('nonnested actual path',prev,frame) + delta=c.dimf[frame]-rank;assert delta>=0 + if delta:result[delta]+=1 + prev=frame;rank=c.dimf[frame] + return result + start={r:w.w['source_frame'][n] for n,r in w.source.items()} + start.update({r:g['frame'] for r,g in w.gauge.items()}) + internal=Counter() + for r in context['regs']: + hh=hist(start.get(r),chain(r)) + if r in w.source.values(): hh[1]+=1 + internal.update(hh) + copies=Counter(c.dimf[w.w['root_frame'][j]] for j,r in enumerate(w.g['roots']) if r['kind']=='center') + by_source={r['source']:r for r in context['selection']+context['gauge_selection']} + source=Counter();source_paths=[] + for n,e in kp.items(): + source.update(hist(e['carrier_chain'][0],e['carrier_chain'][1:])) + if n not in by_source: + source.update(hist(e['passive_chain'][0],e['passive_chain'][1:]));continue + r=by_source[n];role=r['role'];path=chain(role) + if 'early' in r: + assert all(i>=r['early'] for i in w.role_ops[role] if i not in omitted) + path=[e['mix_frame']]+path + else: + f=w.register(r['gauge_basis']) if 'gauge_basis' in r else w.gauge[role]['frame'] + path=[e['mix_frame'],f]+path + hh=hist(w.w['source_frame'][n],path);source.update(hh) + source_paths.append(dict(source=n,role=role,rank_path=[1]+[c.dimf[f] for f in path],histogram=clean(hh))) + # Exact target chronology from the source preparation, including compensated + # gauge reads, terminal writes/posts, side-root reads and partner deliveries. + target=[[] for _ in range(w.v)] + def read(t,frame): target[t].append(frame) + for j,i in enumerate(w.rest): + for role in context['at'][j]: + for t in context['adj'][role]: read(t,w.gauge[role]['frame']) + if i in context['bywrite']: + e=context['bywrite'][i];read(e['pivot'],w.opframe[i]) + if i in context['afterwrite']: + e=context['afterwrite'][i] + for t in e['targets']:read(t,e['root_frame']) + for role in context['at'][len(w.rest)]: + for t in context['adj'][role]:read(t,w.gauge[role]['frame']) + for j,r in enumerate(w.g['roots']): + if r['kind']=='side' and j not in context['deletedroots']: + for t in r['targets']:read(t,w.w['root_frame'][j]) + for e in context['deliveries'][j]: + for t in e['receivers']:read(t,e['deliver_frame']) + targets=Counter() + for t,path in enumerate(target): + assert path + assert all(all(w.module.dot(c.cov[t],row)==0 for row in c.B[f]) for f in set(path)) + targets.update(hist(None,path));gap=w.h-1-c.dimf[path[-1]];assert gap>=0 + if gap:targets[gap]+=1 + entrances=Counter(g['dim'] for r,g in w.gauge.items() if r not in w.donor and r not in borrow) + helper=internal+copies+source+targets + R=len(context['regs']);v=w.v;h=w.h + assert copies=={22:24} and entrances=={20:2200,18:13,12:18,13:48} + assert len(source_paths)==len(borrow)==471 + assert sum(r*n for r,n in source.items())==sum(r*n for r,n in targets.items())==v*(h-1) + assert sum(r*n for r,n in helper.items())==h*R+2*v*(h-1)+sum(r*n for r,n in copies.items())-sum(r*n for r,n in entrances.items()) + idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)}) + five=Counter({r:5*n for r,n in helper.items()});five.update(idle) + for a,n in entrances.items():five[5*a]+=n + W=4*v+R;mass=sum(r*n for r,n in five.items()) + assert (sum(helper.values()),sum(five.values()),mass,5*h*W-mass,max(five))==(95789,495304,2837560,4400,100) + return dict(schema='source471-clean-raw-ledger/1',source_head='df95878d11190518e45ef9717c9ee05011f88ace', + h=h,v=v,physical_R=R,source_aliases=len(borrow), + terminal_sinks=len(context['terminal']),physical_donor_recipient_identifications=len(w.pairs), + physical_internal_excluding_center_copies=clean(internal), + paid_center_copy_histogram=clean(copies),physical_source_histogram=clean(source), + physical_target_histogram=clean(targets),one_stage_helper_histogram_including_copies=clean(helper), + auxiliary_entrance_rank_histogram=clean(entrances), + auxiliary_entrance_count=sum(entrances.values()),helper_rank_mass=sum(r*n for r,n in helper.items()), + five_stage_profile=dict(m=5*h,W=W,histogram=clean(five),idle_histogram=clean(idle),calls=sum(five.values()),rank_mass=mass,deficit=5*h*W-mass,maxchild=max(five)), + exact_nested_path_pairs_checked=len(inclusion_cache), + source_paths=source_paths, + scope='Fresh exact retimed physical-path/rank ledger; no scalar all-column replay or five-stage global-word admission.') + + +def main(): + ap=argparse.ArgumentParser(description=__doc__) + ap.add_argument('--pr210-root',type=Path,required=True) + ap.add_argument('--base-bit-root',type=Path,required=True) + args=ap.parse_args() + print(json.dumps(reconstruct(source_loader.prepare(args.pr210_root,args.base_bit_root)),sort_keys=True,indent=2)) + +if __name__=='__main__':main() diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/LICENSE b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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Its Apache-2.0 +license is retained at upstream/LICENSE. The exact source URLs and hashes +are recorded in upstream/manifest.json. Proposed modifications are supplied +as separate files under patches/; upstream source files remain unchanged. + +The original local research note, scripts, certificates, patches, and +documentation were prepared by Douglas Colkitt with assistance from OpenAI +Codex. The community release incorporates work by the contributors identified +in the notices below; it is not solely Douglas Colkitt's work. +This project is not an official OpenAI release or an OpenAI endorsement. + +Related contributions (not imported in this checkpoint): +Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset +motif, finite-alphabet transfer and a different compressed producer: +https://github.com/CrocSwap/integer-mult-bounds/pull/7 +The present checkpoint claims no priority for that motif. Related compressed +complex-network work was submitted by eumemic in PR #3: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +The pending contributions and their stronger bounds require their own review; +this notice does not represent adoption or verification of their claims. + +Additional community work, including parallel and superseded submissions, is +acknowledged in CONTRIBUTORS.md, with pinned submission metadata in +docs/research/contribution-snapshot.json. These acknowledgements do not imply +joint authorship of the checkpoint, incorporation of pending code, or verification +of submitted mathematical claims. Original source-specific notices remain in force. + +Community release (accepted by conditional maintainer audit): +The following original contributor notices apply to the imported PR #39 chain. +The preceding non-import statements describe the preserved 2^-30 checkpoint. + +Ternary five-subset construction and common-four-point resynthesis +Copyright 2026 Zhihao Chen (jacklightChen) + +This is Zhihao Chen's first submission of this specific construction to this +repository. Research, proof development, code and verification were prepared +with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by +the contributor. This is not independent review or an OpenAI endorsement. +Future research building on this contribution should explicitly acknowledge +Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and +retain the preceding contributors' attribution. This notice records research +provenance; the repository's Apache-2.0 license remains unchanged. + +Dependencies retained from eumemic's Claude-assisted PRs #3 and #5: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +https://github.com/CrocSwap/integer-mult-bounds/pull/5 +PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893 +These include complex_circuit.py, complex_network.py, fast_gaussian.py, +their generators, construction notes and certificates. The Gaussian input +scaling proof is corrected in this derivative; see the modification notice below. +Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank +sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6. +Their results and ideas are credited separately from this new construction. + +Bulk recursion and controlled tensor basis, 8 October 2026 +Copyright 2026 icekylinx + +The new projector batching, controlled simultaneous basis, mixed-width tape +recursions, whole-residual complex recursion, and dependency-path precision +guard in the batched-23 proof package are contributed by icekylinx, developed +with substantial OpenAI GPT-6 Astra assistance. Review corrections and +repository integration were prepared with OpenAI Codex assistance. They build on the retained +PR #7 finite producers and the preceding contributions credited above. +The new package retains the repository's Apache-2.0 license and the original +authors' attribution. It is not an official OpenAI release or endorsement. + +Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian +proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2). +The P=34p precision budget and asymptotic exponent are unchanged. New batching +checks are integrated into make verify. Existing attribution and licenses +are retained. + +Partial-swap extension, 8 October 2026 +Copyright 2026 icekylinx + +The partial-swap framing, binary retained-total triple construction, +dependency-preserving positive labels and carrier reuse, unequal (25,23,57) +common basis, improved complex batching, and exact assembly are contributed +by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance. +This extension directly inherits PR #10, commit +62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters +modify the retained paired-exclusion and shared-point implementations credited +above. The original licenses, attribution, and Gaussian correction are retained. + +Endpoint-gauge extension, 8 October 2026 +Copyright 2026 icekylinx + +The complementary auxiliary endpoint gauges, (32,30,40) controlled basis, +general nondegenerate binary phase residuals, retained-center complex producer, +updated path guard and exact assembly are contributed by icekylinx with +substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits +f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable +adapters retain the paired-exclusion and shared-point contributions credited +above. Original licenses, notices and the Gaussian correction are retained. + +Structured-basis and semantic-bulk extension, 8 October 2026 +Copyright 2026 icekylinx + +The selected (32,30,36) bit construction, A1/A5 block refinements, fixed +middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic +complex centers, complete data-front batching and circuit-specific assembly +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505. + +The semantic/bulk compatibility argument, literal scalar charge, product row +stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23, +commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints +are adopted from the same contributor's PR #21, +commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame +work and the earlier nested-basis contributions. + +Completed-child semantic precision, arbitrary-coordinate routing, phase-cell +Gaussian inversion and bulk tape transfer are credited to the RaD project +(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through +PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof +sources retain their separate CC0 license and original NOTICE under +references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI +Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra +assistance. These are source attributions, not claims about this session's +runtime model. Related contributions by Rohan Arun and Aurel Prosz remain +acknowledged in the imported notices. No exclusive priority or additional +license condition is asserted. + +Copied retained-center extension, 8 October 2026 +Copyright 2026 icekylinx + +The copied retained-center schedule in both orientations, its complex-phase +transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. This extension inherits PR #32 at +0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work. + +The two-stage topology and paid copied-stream endpoint correction are due +to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab, +through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition, +9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil +Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a. + +The contiguous corner method, incidence factorization and exact partition +certificates are due to Rohan Arun (PR #31), method commit +02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate +specializes Dominik Scholz's parameterization (PR #33), +a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof +snapshots are preserved in references/copied-centers. Historical OpenAI +Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to +the respective imported sources. Existing PR #21/#23 and RaD attribution, +licenses and the Gaussian correction remain in force. + +Copied centers with reversed (23,25) data corners, 2026-10-08 +Rohan Arun with substantial OpenAI Codex assistance supplies the composition, +exact dimension23 controls and cost/assembly integration. James Chang's PR34 +supplies the reversed boundary family; icekylinx's PR36 supplies copied retained +centers and the selected complex graph. The balanced assembly remains attributed +to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan +Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx, +eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses. +This is not a claim of formal verification, new foundational rank identities, +global optimality or measured runtime speedup. + +Fixed middle basis with copied reversed corners (2026-10-08): +Rohan Arun, with substantial OpenAI Codex assistance. Composition and +simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34, +Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures, +eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed. + +Changed-graph composition with copied centers and fixed bases, 8 October 2026 +Prepared with OpenAI Codex assistance. +The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J +projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument, +Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and +Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods, +Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc) +PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40 +reversed integration, modular data tests and both-fixed baseline; +RaD (hipotures) PR41 changed scalar DAGs, alternating point order and +independent role compiler; RaD balanced routing/bulk interfaces, and all +retained predecessors. +Complete attribution and the conditional proof boundary are recorded in +research/copied-fixed/PROOF.md; imported originals and source pins are retained. +Original Apache-2.0 and CC0 notices remain applicable. + +Community follow-up integration (PRs #49, #45 and #26), 8 October 2026: +Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out +orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction +composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and +reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD / +hipotures's alternating pair order and independent physical role compiler (#41). +All predecessor attribution and conditional proof interfaces remain in force. + +Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic +proofs, finite tensor execution, profile precision accounting and mixed-center +reference checks (#45), with substantial OpenAI Codex assistance. See +research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution. +Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic +frame/movement/guard lemmas and independent paired-circuit checks (#26). Original +source notices and historical snapshots are preserved. Neither formal package +claims a full formal proof of the multiplication algorithm or a new kappa. + +Joint frame compilation of skip-prefix strips + +The local compiler and exact finite witnesses were prepared by eumemic with +substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic +Claude assistance supplied the unmodified PR #53 skip-prefix producer. +The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating +Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang, +Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt, +OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors +credited in the retained sources. See references/frame-compiler/pr48/NOTICE +and references/frame-compiler/pr53/NOTICE for their original attribution. +No authorship or review by those predecessors is claimed for this compiler. + +Joint dual-skip/frame-compiler composition (2026-10-08) +--------------------------------------------------- +Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip +scalar graph; original LICENSE and NOTICE are retained under +references/frame-compiler/pr55. eumemic with substantial OpenAI Codex +assistance: PR57 joint reversible frame compiler, serialized physical-word +replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex +assistance: this composition, independent integration and exact certificate. +Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all +predecessor authors retain the notices above and in pinned source files. +No original copyright or Apache-2.0 notice is removed. + +Ranked reclamation after the PR58 joint-dual composition (2026-10-08) +---------------------------------------------------------------- +Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical +slots by descending current frame rank, retaining slot ID as the tie-breaker; +complete replay, exact profiles, comparison and adversarial controls. +The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57 +compiler. Its original shared compiler remains byte-identical. The PR55 +producer and all preceding attribution remain unchanged. The complete PR58 +comparison certificate and its original source/proof provenance are retained +under references/frame-compiler/pr58. + +Skip-prefix strip producer, 8 October 2026 +By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5) +assistance. The skip-prefix leave-one-out strip layout in +research/skip-strips/skip_graph.py is new. Everything else reuses Chafik +Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification +code, exact data recovery and support-order sorting, together with: +- Rohan Arun's PR #44 weighted carrier matching optimizer; +- RaD (hipotures) PR #41 alternating point order; +- every predecessor credited above. +Complete attribution and the conditional proof boundary are recorded in +research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain +applicable. + +Local split-pair skip-prefix research continuation (2026-10-08) +Prepared with substantial OpenAI Codex assistance at the user's request. +New contribution: singleton/pair recursive partition refinement, finite summand +order and moment search, and verified composition with inherited paid clones. +Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips. +Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone +specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51 +whole-chain/source-partition cloning and earlier producer/interfaces. +Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all +stated historical AI assistance. All predecessor notices remain applicable. + +Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones): +Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain +source-partition clone operation is adapted from RaD / hipotures, PR #51, +with original envelopes retained here. The scalar base is Avi Eisenberg's +PR #53 skip-prefix layout, prepared with Anthropic Claude assistance. +All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain +in force. This extension is provided under the repository's Apache-2.0 terms. + +Community review checkpoint #50–#61, 8 October 2026 +Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex. +Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude. +Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance) +and dual-suffix strips (#55, Anthropic Claude assistance). +Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex. +Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked +reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame +composition (#56), with Anthropic Claude and stated Codex validation assistance. +RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones +(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone +composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter +refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with +his disclosed multi-tool and Codex assistance. +Parallel #50/#52 records are reviewed and credited without replacing the earlier +climbed-48 implementation; their exact commit history is retained. The separate +#51 witness is retained as research; its source-graph/negative-basis data proof +has not received the same full replay as #56's consumed compiler machinery. +See docs/research/community-round2-review.md for precise validation boundaries. +All earlier licenses, contributions and notices remain applicable. + +Interval strips and core-aware pair assembly + +The interval leave-one-out strips and the core-aware group-pair assembly in +research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic +Claude assistance. They build on PR #53's skip-prefix pipeline and its +inherited PR #48 sources. The stacked witness in research/pair-assembly/frame +uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier +are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and +verify_skip_frame.py. Original notices are retained. + +Stopped product-ring extension, 9 October 2026 +Copyright 2026 icekylinx + +The opposite-bank generic projector factorization, componentwise atom-ring +adapters, stopped recurrence and outer ordinary-interchange conversion, +aligned pair-star producer with rational centers, exact denominator-21 +grid and selected (23,23)/(24,24) composition are contributed by icekylinx +with substantial OpenAI GPT-6 Astra and Codex assistance. This extension +builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is +integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37. + +The retained ordinary leaf uses the preceding projector, corner and +copied-center proofs. The selected two-stage topology retains Aurel Prosz +(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly +retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching, +original source notices and each imported license remain unchanged. + +Partial-gauge extension, 9 October 2026 +Copyright 2026 icekylinx + +The pair-first cube assembly, compatible backward frame enlargement, +partial source-gauge schedule and stopped whole-projector composition are +contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834. + +The physical deferred bit word, reflected event accounting, fan-order +clarification and signed two-stage endpoint are adopted from Zhihao Chen +(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with +Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen +witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38, +with the original Claude disclosure retained. Cyclic interval and core-aware +pair primitives build on Avi Eisenberg (ikeboy), PR #62, +ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance. +Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and +precision credits remain in force. Imported source notices and licenses +are preserved under references/partial-gauge/; see SOURCES.json. + +Three-stage cover extension, 9 October 2026 +Copyright 2026 icekylinx + +The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry, +weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring +fallback, uniform batching, internal row borrowing and exact composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This +repository integration uses Codex assistance and extends +PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86. + +The selected local complex DAG and support/label replay are adopted from +Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf, +with its original Apache-2.0 terms and Claude assistance disclosure. +The unchanged bit physical word retains Zhihao Chen (jacklightChen), +PR #97, and Swapnil Jain's underlying witness and assistance notices. +The prior opposite-bank, stopped ordinary-leaf, copied-center, topology, +semantic/bulk, precision and analytic credits remain applicable. +Original source notices and license bytes are preserved in +references/three-stage-cover/pr117/ and references/partial-gauge/. + +Paired-cube and shared-core extension, 9 October 2026 +Copyright 2026 icekylinx + +The paired-coordinate triple-cube identity, signed overlap channels, +original-source involution, coordinate-star centers, concrete frame/carrier +compiler, chronological source gauges, PR97 gauge-subset selection, +orthogonal cross-stage routing and exact supplier/assembly integration +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. +Repository integration uses Codex assistance and extends PR #130 at +6a9970a530119174507904e23592fd59ede19a5d. + +The substantial completed-core workspace-sharing principle is credited to +an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported +OpenAI Codex assistance. No source code, partition or serialized witness +from that PR is imported. The new routing and paired-motif integration +are described separately and claim no exclusive priority for broad sharing, +paired-coordinate, Fourier or carrier-matching ideas. + +Selected coarse queries restrict the retained eumemic PR #117 positive DAG; +its original Apache-2.0 and Claude assistance notices remain unchanged. +The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil +Jain's frozen witness, notices and historical assistance disclosures. +Earlier generalized Clifford, three-stage, uniform weighted recursion, +ordinary-leaf, semantic/bulk, precision and analytic credits remain in force. +Source-specific license bytes are preserved under references/. + +Source-assisted common-frame extension (2026-10-09) +Copyright 2026 icekylinx. Apache-2.0. +Research construction and local certificates: OpenAI GPT-6 Astra assistance. +Repository integration and independent complex scalar replay: Codex assistance. +Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee; +its byte-preserved sources, licenses and original Claude assistance disclosure +are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz. +Additional inherited contributions are recorded in research/source-assisted/CREDITS.md. +The new work concerns source-assisted invertible common-frame lifts, controlled +affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for +broad register reuse, paired cubes, frame theory or ordinary linear elimination. diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/base_word.py b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/base_word.py new file mode 100644 index 0000000000..044f278fbe --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/base_word.py @@ -0,0 +1,202 @@ +#!/usr/bin/env python3 +"""Reusable independent formal checks for a physical partner-pair bit word. + +The supplied frozen inputs are source-pinned. This never runs their producer. +The Z check is of the defining integer decoder; only its F2 reduction is I. +Prepared with OpenAI Codex assistance; Apache-2.0. +""" +from collections import Counter, defaultdict +import importlib.util +import json +from pathlib import Path +import sys +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +BASE = HERE.parent / 'references/pr168-v4/research/paired-cube-bit' +P = 12 +if sys.flags.optimize: raise SystemExit('refusing -O') + + +def need(ok, msg): + if not ok: + raise ValueError(msg) + +class Candidate: + def exact_frames(self): + C = self.C + C.decoder(); C.geometry() + for i in self.changed_frames: + f = self.opframe[i]; x = self.ops[i][2]; bits = C.sup[x] + need(C.nondeg(f), 'replacement operation frame nondegenerate') + while bits: + low = bits&-bits; s=low.bit_length()-1; bits-=low + need(C.in_frame(C.chi[s],f), 'value span in replacement operation frame') + for b, d in self.pairs: + need(C.sub(self.endframe[d], self.gauge[b]['frame']), 'exact Q donor-to-birth containment') + need(C.nondeg(self.endframe[d]) and C.nondeg(self.gauge[b]['frame']), 'handoff endpoints nondegenerate') + + def register(self, rows): + C = self.C + B,_ = self.module.reduce_rows(rows,self.h if hasattr(self,'h') else C.h) + B = tuple(sorted(map(tuple,B),key=lambda r:next(i for i,x in enumerate(r) if x))) + if not hasattr(self,'frame_ids'): + self.frame_ids = {} + if B in self.frame_ids: return self.frame_ids[B] + f=max(C.B)+1; A,_=self.module.kernel(B,C.h) + C.B[f],C.A[f],C.dimf[f]=B,A,len(B);self.frame_ids[B]=f + return f + + + def row(self): + C,h,v=self.C,self.h,self.v + _,baseline_Y,src,_=C.chains() + seq=defaultdict(list) + for i,(a,b,x) in enumerate(self.ops): + seq[a].append(self.opframe[i]);seq[b].append(self.opframe[i]) + for j,s in enumerate(self.w['rootroles']):seq[s].append(self.w['root_frame'][j]) + start={b:self.w['source_frame'][x] for x,b in self.source.items()} + start.update({b:z['frame'] for b,z in self.gauge.items()}) + recipient={d:b for b,d in self.pairs} + H=Counter() + for s in range(self.R): + if s in self.donor:continue + chain=list(seq[s]) + if s in recipient: + b=recipient[s];chain += [self.gauge[b]['frame']]+seq[b] + chain.append(self.w['full_frame']) + prev=start.get(s);dp=0 if prev is None else C.dimf[prev] + if s in self.source.values():H[1]+=1 + for f in chain: + need(prev is None or C.sub(prev,f),'physical role chain nested') + d=C.dimf[f] + if d>dp:H[d-dp]+=1 + prev,dp=f,d + for j,r in enumerate(self.g['roots']): + if r['kind']=='center':H[C.dimf[self.w['root_frame'][j]]]+=1 + Y=Counter();current=[None]*v + def read(t,f): + old=current[t];d=0 if old is None else C.dimf[old] + need(old is None or C.sub(old,f),'actual target frame chronology') + if C.dimf[f]>d:Y[C.dimf[f]-d]+=1 + current[t]=f + for b in sorted(self.order,key=lambda b:self.readtime[b]): + for t in self.gauge[b]['targets']:read(t,self.gauge[b]['frame']) + deliveries=defaultdict(list) + for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e) + for j,r in enumerate(self.g['roots']): + if r['kind']!='center': + for t in r['targets']:read(t,self.w['root_frame'][j]) + for e in deliveries[j]: + for t in e['receivers']:read(t,e['deliver_frame']) + for t,f in enumerate(current): + need(f is not None and all(self.module.dot(C.cov[t],b)==0 for b in C.B[f]),'final target cap') + gap=h-1-C.dimf[f] + if gap:Y[gap]+=1 + need(sum(r*n for r,n in Y.items())==v*(h-1),'target rank mass') + hist=Counter() + for part in (H,Y,src): + for r,n in part.items(): + if r and n:hist[r]+=3*n + for b,z in self.gauge.items(): + if b not in self.donor:hist[3*z['dim']]+=1 + hist[2]+=2*v + hist={r:n for r,n in sorted(hist.items()) if n} + need(all(00 for r,n in hist.items()),'positive proper child bins') + R=self.R-len(self.pairs);W=2*v+R;mass=sum(r*n for r,n in hist.items());m=3*h + need(W*m-mass==C.prof['deficit_per_vertex'],'reuse preserves telescoping deficit') + return dict(h=h,v=v,R=R,virtual_R=self.R,reused_registers=len(self.pairs),W_per_vertex=W,m=m, + loss=C.prof['loss'],rank_per_vertex=mass,deficit_per_vertex=W*m-mass, + child_histogram=hist,maxchild=max(hist),selected_roles=len(self.gauge)-len(self.pairs), + selected_rank_histogram=dict(Counter(z['dim'] for b,z in self.gauge.items() if b not in self.donor)), + changed_operation_frames=len(self.changed_frames), + frames_not_contained_in_original=sum(not C.sub(self.opframe[i],self.original_opframe[i]) for i in self.changed_frames), + physical_target_histogram=dict(Y),physical_internal_histogram=dict(H), + foreign_producer_replays=0) + + def adjoint(self): + adj = [dict() for _ in range(self.R)] + for r,s in zip(self.g['roots'], self.w['rootroles']): + for t in r['targets']: adj[s][t] = adj[s].get(t,0)+1 + for a,b,_ in reversed(self.ops): + for t,c in adj[a].items(): adj[b][t] = adj[b].get(t,0)+c + return adj + + def formal(self, ring, tamper=None): + """All source, target and dirty-register columns; no random sampling.""" + v, phys = self.v, self.phys + regs = sorted(set(phys.values())); index = {r:2*v+j for j,r in enumerate(regs)} + if ring == 2: + unit = lambda i: 1<=0) + need(closure==set(self.phase1),'phase one is exactly centre closure') + content=[0]*self.R + for n,s in self.source.items():content[s]=1<0,'gauge dimension and complete response support') + src=Counter();members=Counter();labels=list(map(set,self.g['labels'])) + for e in self.k['entries']: + c,d,f,cap=e['carrier'],e['passive'],e['mix_frame'],e['deliver_frame'] + need(len(labels[c]&labels[d])==1,'orthogonal partner pair') + need(C.dimf[f]==2 and C.in_frame(C.chi[c],f) and C.in_frame(C.chi[d],f),'partner mix span') + need(C.sub(f,cap) and all(self.module.dot(C.cov[t],u)==0 for t in e['receivers'] for u in C.B[cap]),'partner mix delivery cap') + need(e['carrier_chain']==[w['source_frame'][c],f,cap,w['full_frame']] and e['passive_chain']==[w['source_frame'][d],f,w['full_frame']],'exact partner chains') + root=self.g['roots'][e['deliver_after_root']] + need(root['kind']=='side' and sorted(e['receivers'])==sorted(root['targets']),'partner delivery at own receiver root') + for chain in (e['carrier_chain'],e['passive_chain']): + for a,b in zip(chain,chain[1:]): + need(C.sub(a,b),'nested source chronology');src[C.dimf[b]-C.dimf[a]]+=1 + members[c]+=1;members[d]+=1 + need(all(members[s]==1 for s in range(v)),'every source in one partner pair') + return Counter(),Counter(),src,self.R diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py new file mode 100644 index 0000000000..69d535a6c2 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py @@ -0,0 +1,465 @@ +#!/usr/bin/env python3 +"""Standalone checker for subset-port bit words with broadcast roots (paired-cube bit word with partner-pair +source mixing). Standard library only; independent of the generator (own exact integer linear algebra, own +register simulation). Refuses to run under python -O. + +Inputs (plain JSON written by paired_cube_bit_word.py): graph_pP.json, word_pP.json, frames_pP.json, +kchron_pP.json, profile_pP.json. Checks: + 1. exact frames: every frame record (integer annihilator rows 'a' or integer basis rows 'b') is completed to + the other representation by exact integer elimination; A.B^T = 0 exactly and rank A + rank B = h; + 2. the mod-2 decoder: supports of all additions are disjoint; for every target the deliveries (star roots, + side roots, partner singles, partner-pair sums) XOR to {T}; stars read are exactly the c in T and the + side part is exactly sum_{|S cap T|=1} x_S; + 3. geometry: source frames are the lines ; side root frames lie in every receiver's cap + ker(3 chi_T - 1) and equal the common cap of the group; centre frames are the star spans; the frame of + every addition contains its operands' frames; every root node's frame lies in its root frame; + 4. every role chain (start -> each op frame -> root frame -> full) is nested, including gauged starts + (sigma inside the first op frame, gauged roles untouched in phase one); every target chain (gauge reads + ascending, side root caps in root order, partner-pair delivery at the level-2 cap, final cap) is nested; + the partner-pair chronology: mix frame span{chi_c,chi_d} with H0-Gram I_2, contained in the receivers' + level-2 cap, carrier chain [, mix, cap, full] and passive chain [, mix, full] nested; + 5. G = I - J/9 nondegenerate on every frame used (exact integer determinant test); + 6. literal F2 replay with arbitrary dirty scratch: y = y0 + x, scratch and sources restored; + 7. independent ledger recount from the role chains equals the saved child histogram (PR #144 bit ledger); + 8. mutation controls (must all be REJECTED): broken partner mix, delivery outside the shared cap (full + frame), delivery below the level (face cap without span{chi_c,chi_d}), dropped star, non-nested role chain. +Usage: python3 check_paired_cube_bit.py --dir out --p 12 +""" +import argparse, json, random, sys +from collections import Counter, defaultdict +from math import gcd +from operator import mul +from pathlib import Path + +if sys.flags.optimize: + raise SystemExit('refusing to run under python -O: assertions are part of the check') + + +class Fail(Exception): + pass + + +def need(cond, msg): + if not cond: + raise Fail(msg) + + +# ------------------------------------------------------------------ exact integer linear algebra +def prim(v): + g = 0 + for x in v: + g = gcd(g, abs(x)) + if g > 1: + v = [x // g for x in v] + f = next((x for x in v if x), 0) + return tuple(-x for x in v) if f < 0 else tuple(v) + + +def reduce_rows(rows, h): + """Integer reduced echelon form: returns (rows, pivot columns); each pivot column is zero outside its row.""" + rows = [list(r) for r in rows] + out, pivs = [], [] + for r in rows: + for pr, pc in zip(out, pivs): + if r[pc]: + a, b = pr[pc], r[pc] + r = [x * a - y * b for x, y in zip(r, pr)] + r = list(prim(r)) + c = next((j for j in range(h) if r[j]), None) + if c is None: + continue + for i, (pr, pc) in enumerate(zip(out, pivs)): + if pr[c]: + a, b = r[c], pr[c] + out[i] = list(prim([x * a - y * b for x, y in zip(pr, r)])) + out.append(r); pivs.append(c) + return out, pivs + + +def kernel(rows, h): + red, pivs = reduce_rows(rows, h) + ps = set(pivs) + basis = [] + for f in range(h): + if f in ps: + continue + D = 1 + for r, c in zip(red, pivs): + if r[f]: + D = D * abs(r[c]) // gcd(D, abs(r[c])) + v = [0] * h + v[f] = D + for r, c in zip(red, pivs): + if r[f]: + v[c] = -r[f] * D // r[c] + basis.append(prim(v)) + return basis, len(red) + + +def rank(rows, h): + return len(reduce_rows(rows, h)[0]) + + +def dot(a, b): + return sum(map(mul, a, b)) + + +# ------------------------------------------------------------------ checker +class Checker: + def __init__(self, d, p, mutate=None): + self.mut = mutate + rd = lambda name: json.loads((d / ('%s_p%d.json' % (name, p))).read_text()) + self.g, self.w, self.fr, self.k, self.prof = rd('graph'), rd('word'), rd('frames'), rd('kchron'), rd('profile') + self.h, self.v = self.g['h'], self.g['v'] + if mutate == 'dropped_star': + j = next(i for i, r in enumerate(self.g['roots']) if r['kind'] == 'center') + self.g['roots'][j]['targets'] = [] + if mutate == 'broken_mix': + self.k['entries'][0]['skip_mix'] = True + if mutate == 'outside_cap': + # raise the mixed carrier past the receivers' shared cap (to the full frame) and deliver there + e = self.k['entries'][0] + e['deliver_frame'] = self.w['full_frame'] + e['carrier_chain'][2] = e['deliver_frame'] + if mutate == 'below_level': + # deliver at the receivers' level-1 (face) cap, which does not contain span{chi_c, chi_d} + e = self.k['entries'][0] + face = next(j for j, r in enumerate(self.g['roots']) if r['kind'] == 'side' and set(e['receivers']) < set(r['targets'])) + e['deliver_frame'] = self.w['root_frame'][face] + e['carrier_chain'][2] = e['deliver_frame'] + if mutate == 'non_nested': + # move one addition's frame down to the frame of its smallest operand's source line + x = next(int(k) for k, f in sorted(self.w['node_frame'].items(), key=lambda kv: int(kv[0])) if int(k) >= self.v and str(int(k)) not in set(map(str, self.w['plain']))) + self.w['node_frame'][str(x)] = self.w['source_frame'][0] + + # ---------------- frames + def frames(self): + h = self.h + self.A, self.B, self.dimf = {}, {}, {} + for key, rec in self.fr['frames'].items(): + f = int(key) + if 'a' in rec: + A = [tuple(r) for r in rec['a']] + need(rank(A, h) == len(A), 'independent annihilator rows %d' % f) + B, _ = kernel(A, h) + else: + B = [tuple(r) for r in rec['b']] + need(rank(B, h) == len(B), 'independent basis rows %d' % f) + A, _ = kernel(B, h) + need(len(A) + len(B) == h and len(B) == rec['dim'], 'dimensions of frame %d' % f) + need(all(dot(a, b) == 0 for a in A for b in B), 'A.B = 0 for frame %d' % f) + self.A[f], self.B[f], self.dimf[f] = A, B, len(B) + self.cc = {} + return len(self.A) + + def sub(self, f1, f2): + """frame f1 contained in frame f2 (exact)""" + if f1 == f2: + return True + key = (f1, f2) + r = self.cc.get(key) + if r is None: + B1, A2 = self.B[f1], self.A[f2] + r = len(B1) <= len(self.B[f2]) and all(dot(a, b) == 0 for a in A2 for b in B1) + self.cc[key] = r + return r + + def in_frame(self, vec, f): + return all(dot(a, vec) == 0 for a in self.A[f]) + + def nondeg(self, f): + h = self.h + B, A = self.B[f], self.A[f] + if not B or len(B) == h: + return True + if len(B) <= len(A): + s = [sum(b) for b in B] + M = [[9 * dot(B[i], B[j]) - s[i] * s[j] for j in range(len(B))] for i in range(len(B))] + return rank(M, len(B)) == len(B) + s = [sum(a) for a in A] + M = [[(9 - h) * dot(A[i], A[j]) + s[i] * s[j] for j in range(len(A))] for i in range(len(A))] + return rank(M, len(A)) == len(A) + + # ---------------- decoder + def decoder(self): + g, v = self.g, self.v + args = g['args'] + lab = [frozenset(l) for l in g['labels']] + sup = [1 << i for i in range(v)] + [0] * (len(args) - v) + for x in range(v, len(args)): + a, b = args[x] + need(not sup[a] & sup[b], 'support-disjoint addition %d' % x) + sup[x] = sup[a] | sup[b] + self.sup = sup + acc, stars, side = [0] * v, [set() for _ in range(v)], [0] * v + for r in g['roots']: + need(r.get('coefficient', 1) == 1, 'mod-2 unit coefficients') + for t in r['targets']: + acc[t] ^= sup[r['node']] + if r['kind'] == 'center': + stars[t].add(r['coordinate']) + else: + side[t] ^= sup[r['node']] + for e in self.k['entries']: + for t in e['receivers']: + acc[t] ^= (1 << e['carrier']) | (1 << e['passive']) + side[t] ^= (1 << e['carrier']) | (1 << e['passive']) + for t in range(v): + need(acc[t] == 1 << t, 'decoder identity at target %d' % t) + need(stars[t] == set(lab[t]), 'stars read by target %d' % t) + orth = sum(1 << s for s in range(v) if len(lab[s] & lab[t]) == 1) + need(side[t] == orth, 'side part = sum over |S cap T| = 1 at target %d' % t) + for r in g['roots']: + if r['kind'] == 'center': + c = r['coordinate'] + need(sup[r['node']] == sum(1 << s for s in range(v) if c in lab[s]), 'star support %d' % c) + + # ---------------- geometry and chains + def geometry(self): + g, w, h, v = self.g, self.w, self.h, self.v + lab = g['labels'] + chi = [tuple(1 if j in l else 0 for j in range(h)) for l in lab] + cov = [tuple((3 if j in l else 0) - 1 for j in range(h)) for l in lab] + nf = {int(k): f for k, f in w['node_frame'].items()} + rf = w['root_frame'] + full = w['full_frame'] + need(self.dimf[full] == h, 'full frame') + for s in range(v): + f = w['source_frame'][s] + need(self.dimf[f] == 1 and self.in_frame(chi[s], f), 'source line ') + args = g['args'] + for x, f in nf.items(): + if x < v: + need(self.in_frame(chi[x], f), 'leaf label inside its frame') + else: + for y in args[x]: + need(self.sub(nf[y], f), 'operand frame inside node frame (%d <- %d)' % (x, y)) + loss = 0 + for j, r in enumerate(g['roots']): + f = rf[j] + need(self.sub(nf[r['node']], f), 'root node frame inside root frame %d' % j) + if r['kind'] == 'center': + c = r['coordinate'] + S = [chi[s] for s in range(v) if c in lab[s]] + need(all(self.in_frame(x, f) for x in S) and rank(S, h) == self.dimf[f], 'centre frame = star span') + loss += self.dimf[f] + else: + need(all(dot(cov[t], b) == 0 for t in r['targets'] for b in self.B[f]), 'root frame inside caps') + need(self.dimf[f] == h - rank([cov[t] for t in r['targets']], h), 'root frame = common cap') + need(loss == self.prof['loss'], 'copied-centre loss') + self.chi, self.cov, self.nf, self.rf = chi, cov, nf, rf + + def chains(self): + """Role chains of the physical word and target chains; returns the recounted child histogram.""" + g, w, h, v = self.g, self.w, self.h, self.v + nf, rf, full = self.nf, self.rf, w['full_frame'] + ops, rootroles = w['ops'], w['rootroles'] + R = 1 + max(max(a, b) for a, b, _ in ops) + R = max(R, 1 + max(rootroles), 1 + max(w['sources'].values())) + need(R == self.prof['R'], 'role count') + start = {} + for leaf, s in w['sources'].items(): + start[s] = w['source_frame'][int(leaf)] + phase1 = set(w['phase1']) + touched_p1 = set() + for i in phase1: + touched_p1.update(ops[i][:2]) + # phase one = closure of the centre roots' last ops under per-role precedence (recomputed) + prev_op, pred = {}, [] + for i, (a, b, _) in enumerate(ops): + pred.append((prev_op.get(a, -1), prev_op.get(b, -1))) + prev_op[a] = prev_op[b] = i + stack = [prev_op[s] for r, s in zip(g['roots'], rootroles) if r['kind'] == 'center' and s in prev_op] + closure = set() + while stack: + i = stack.pop() + if i not in closure: + closure.add(i); stack.extend(j for j in pred[i] if j >= 0) + need(closure == phase1, 'phase one is the centre closure') + # conservative response support of every role (union incidence through the XOR word) + co = [0] * R + for j, (r, s) in enumerate(zip(g['roots'], rootroles)): + for t in r['targets']: + co[s] |= 1 << t + for a, b, _ in reversed(ops): + co[b] |= co[a] + gauge_of = {} + for z in w['gauges']: + need(sum(1 << t for t in z['targets']) == co[z['role']], 'gauge targets = response support of its role') + s = z['role'] + need(s not in start, 'gauged role is not a source role') + need(s not in touched_p1, 'gauged role untouched in phase one') + need(self.dimf[z['frame']] == z['dim'] and z['dim'] > 0, 'gauge dimension') + need(self.nondeg(z['frame']), 'gauge frame nondegenerate') + start[s] = z['frame']; gauge_of[s] = z + seq = defaultdict(list) + for i, (a, b, x) in enumerate(ops): + f = nf[x] + seq[a].append(f); seq[b].append(f) + for j, s in enumerate(rootroles): + seq[s].append(rf[j]) + H = Counter() + for s in range(R): + chain = [start.get(s)] + seq[s] + [full] + prev = chain[0] + dprev = 0 if prev is None else self.dimf[prev] + if prev is not None and s not in gauge_of: + H[1] += 1 # source role: injection frame (entrance child of width 1) + for f in chain[1:]: + if prev is not None: + need(self.sub(prev, f), 'nested role chain (role %d)' % s) + d = self.dimf[f] + need(d >= dprev, 'ascending role chain (role %d)' % s) + if d > dprev: + H[d - dprev] += 1 + prev, dprev = f, d + for j, r in enumerate(g['roots']): + if r['kind'] == 'center': + H[self.dimf[rf[j]]] += 1 # copied-centre transform + # partner-pair chronology + src = Counter() + lab = [set(l) for l in g['labels']] + for e in self.k['entries']: + c, d = e['carrier'], e['passive'] + need(len(lab[c] & lab[d]) == 1, 'carrier and passive orthogonal (H0-Gram I_2)') + mf, cf = e['mix_frame'], e['deliver_frame'] + need(self.dimf[mf] == 2 and self.in_frame(self.chi[c], mf) and self.in_frame(self.chi[d], mf), 'mix frame = span{chi_c, chi_d}') + need(self.nondeg(mf), 'mix frame nondegenerate') + need(self.sub(mf, cf), 'mix frame inside the receivers shared cap') + need(all(dot(self.cov[t], b) == 0 for t in e['receivers'] for b in self.B[cf]), 'delivery frame inside receiver caps') + need(e['carrier_chain'][0] == w['source_frame'][c] and e['passive_chain'][0] == w['source_frame'][d], 'chains start at source lines') + need(e['carrier_chain'][1:] == [mf, cf, w['full_frame']] and e['passive_chain'][1:] == [mf, w['full_frame']], 'chain shape') + for ch in (e['carrier_chain'], e['passive_chain']): + for f1, f2 in zip(ch, ch[1:]): + need(self.sub(f1, f2), 'nested source-register chain') + for f1, f2 in zip(ch, ch[1:]): + src[self.dimf[f2] - self.dimf[f1]] += 1 + # target chains: gauge reads (time order = reverse selection order), side caps in root order, + # partner-pair delivery at the level-2 cap right after its root, final cap + events = defaultdict(list) + for z in reversed(w['gauges']): + for t in z['targets']: + events[t].append(z['frame']) + deliveries = defaultdict(list) + for e in self.k['entries']: + deliveries[e['deliver_after_root']].append(e) + for j, r in enumerate(g['roots']): + if r['kind'] == 'center': + continue + for t in r['targets']: + events[t].append(rf[j]) + for e in deliveries.get(j, []): + need(sorted(e['receivers']) == sorted(r['targets']), 'delivery after the receivers own level-2 root') + for t in e['receivers']: + events[t].append(e['deliver_frame']) + Y = Counter() + cap = {} + for t in range(v): + capf = None + prev, dprev = None, 0 + for f in events[t]: + if prev is not None: + need(self.sub(prev, f), 'nested target chain (target %d)' % t) + need(all(dot(self.cov[t], b) == 0 for b in self.B[f]), 'target frame inside cap (target %d)' % t) + if self.dimf[f] > dprev: + Y[self.dimf[f] - dprev] += 1 + prev, dprev = f, self.dimf[f] + need(dprev <= h - 1, 'target chain below the cap') + Y[h - 1 - dprev] += 1 if h - 1 - dprev else 0 + # every source register is in exactly one mixing pair + members = Counter() + for e in self.k['entries']: + members[e['carrier']] += 1; members[e['passive']] += 1 + need(all(members[s] == 1 for s in range(v)), 'each source in exactly one partner pair') + for f in set(self.nf.values()) | set(self.rf): + need(self.nondeg(f), 'G-nondegenerate frame %d' % f) + return H, Y, src, R + + # ---------------- replay + def replay(self, seeds=3): + g, w, v = self.g, self.w, self.v + ops, rootroles = w['ops'], w['rootroles'] + R = self.prof['R'] + srcmap = {int(k): s for k, s in w['sources'].items()} + roots = g['roots'] + for seed in range(seeds): + rng = random.Random(1000 + seed) + x = [rng.getrandbits(1) for _ in range(v)] + y0 = [rng.getrandbits(1) for _ in range(v)] + z0 = [rng.getrandbits(1) for _ in range(R)] + X, Yv, z = x[:], y0[:], z0[:] + # y <- y - J M z (old-value response of the dirty scratch) + zd = z0[:] + for a, b, _ in ops: + zd[a] ^= zd[b] + for j, r in enumerate(roots): + for t in r['targets']: + Yv[t] ^= zd[rootroles[j]] + for leaf, s in srcmap.items(): # z <- z + V x (at the rank-one source lines) + z[s] ^= X[leaf] + for a, b, _ in ops: # z <- M z + z[a] ^= z[b] + for j, r in enumerate(roots): # y <- y + J z + for t in r['targets']: + Yv[t] ^= z[rootroles[j]] + for e in self.k['entries']: # partner-pair sums on the source registers + if not e.get('skip_mix'): + X[e['carrier']] ^= X[e['passive']] + for t in e['receivers']: + Yv[t] ^= X[e['carrier']] + for e in self.k['entries']: # undo at the full frame + if not e.get('skip_mix'): + X[e['carrier']] ^= X[e['passive']] + for a, b, _ in reversed(ops): # z <- M^-1 z + z[a] ^= z[b] + for leaf, s in srcmap.items(): # z <- z - V x + z[s] ^= X[leaf] + need(all(Yv[t] == y0[t] ^ x[t] for t in range(v)), 'replay: y = y0 + x (seed %d)' % seed) + need(z == z0 and X == x, 'replay: scratch and sources restored (seed %d)' % seed) + + def run(self): + nfr = self.frames() + self.decoder() + self.geometry() + H, Y, src, R = self.chains() + self.replay() + h, v = self.h, self.v + m, W = 3 * h, 2 * v + R + C = Counter() + for part in (H, Y, src): + for r, c in part.items(): + if r and c: + C[r] += 3 * c + for z in self.w['gauges']: + C[3 * z['dim']] += 1 + C[2] += 2 * v + saved = {int(k): c for k, c in self.prof['child_histogram'].items()} + need(dict(C) == saved, 'independent recount equals the saved child histogram') + mass = sum(r * c for r, c in C.items()) + need(W == self.prof['W_per_vertex'] and W * m - mass == 2 * v - 3 * self.prof['loss'] == self.prof['deficit_per_vertex'], + 'W = 2v + R and deficit 2v - 3 ell') + return dict(frames=nfr, roles=R, W=W, m=m, deficit=W * m - mass, children=sum(C.values())) + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--dir', type=Path, default=Path(__file__).resolve().parent / 'out') + ap.add_argument('--p', type=int, default=12) + ap.add_argument('--no-controls', action='store_true') + a = ap.parse_args() + res = Checker(a.dir, a.p).run() + print('PASS p=%d: %s' % (a.p, res)) + if a.no_controls: + return + for m in ('broken_mix', 'outside_cap', 'below_level', 'dropped_star', 'non_nested'): + try: + Checker(a.dir, a.p, mutate=m).run() + except Fail as e: + print('control %-13s REJECTED: %s' % (m, e)) + continue + raise SystemExit('FAIL: control %s was accepted' % m) + print('PASS: all 5 mutation controls rejected') + + +if __name__ == '__main__': + main() diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz new file mode 100644 index 0000000000000000000000000000000000000000..adc471b1324673f3fb5fd965b53060d38da9ed91 GIT binary patch literal 528291 zcmagF2{_bi|36;Il2IsQBw5mCFt$jMHIyhaV;$L->|2;4WEo^BEo0w1p+VWQFB$tz zB1^UyOO`T3viVzjVGm#PkwXRaA#Q=_h0qj>#Nu<^5+1;+)n=$d*-5r

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48],[18211,25849],[25850,25851],[18221,25852],[25853,25854],[25855,25856],[25857,25858],[25860,25861],[25859,25862],[25863,25864],[3679,3919],[4129,4309],[4669,4729],[4759,4999],[5539,5659],[5749,5809],[5839,6049],[7459,7579],[7669,7729],[7759,7879],[7969,8029],[8059,8149],[18595,25866],[18656,25867],[25868,25869],[18829,25870],[25871,25872],[18540,18543],[25873,25874],[25875,25876],[18526,25877],[18580,18665],[25878,25879],[25880,25881],[18631,25882],[18653,25883],[25884,25885],[18663,25886],[25887,25888],[25889,25890],[25891,25892],[25894,25895],[25893,25896],[25897,25898]],"decoder":"mod 2: Y_T += sum of root nodes whose targets contain T (star roots: T reads star(c) for c in T; side roots; partner singles) + x_carrier + x_passive for each partner_mix entry receiving T; equals x_T = sum_{c in T} star(c) + sum_{|S cap T|=1} 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE new file mode 100644 index 0000000000..d08cb66ac7 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE @@ -0,0 +1,431 @@ +A sharper exponent for integer multiplication +Copyright 2026 Douglas Colkitt + +This project includes unmodified source files from OpenAI's manuscript +"Integer multiplication below n log n" and its accompanying repository: +https://github.com/openai/math +Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a + +The original manuscript identifies OpenAI as its author. Its Apache-2.0 +license is retained at upstream/LICENSE. The exact source URLs and hashes +are recorded in upstream/manifest.json. Proposed modifications are supplied +as separate files under patches/; upstream source files remain unchanged. + +The original local research note, scripts, certificates, patches, and +documentation were prepared by Douglas Colkitt with assistance from OpenAI +Codex. The community release incorporates work by the contributors identified +in the notices below; it is not solely Douglas Colkitt's work. +This project is not an official OpenAI release or an OpenAI endorsement. + +Related contributions (not imported in this checkpoint): +Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset +motif, finite-alphabet transfer and a different compressed producer: +https://github.com/CrocSwap/integer-mult-bounds/pull/7 +The present checkpoint claims no priority for that motif. Related compressed +complex-network work was submitted by eumemic in PR #3: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +The pending contributions and their stronger bounds require their own review; +this notice does not represent adoption or verification of their claims. + +Additional community work, including parallel and superseded submissions, is +acknowledged in CONTRIBUTORS.md, with pinned submission metadata in +docs/research/contribution-snapshot.json. These acknowledgements do not imply +joint authorship of the checkpoint, incorporation of pending code, or verification +of submitted mathematical claims. Original source-specific notices remain in force. + +Community release (accepted by conditional maintainer audit): +The following original contributor notices apply to the imported PR #39 chain. +The preceding non-import statements describe the preserved 2^-30 checkpoint. + +Ternary five-subset construction and common-four-point resynthesis +Copyright 2026 Zhihao Chen (jacklightChen) + +This is Zhihao Chen's first submission of this specific construction to this +repository. Research, proof development, code and verification were prepared +with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by +the contributor. This is not independent review or an OpenAI endorsement. +Future research building on this contribution should explicitly acknowledge +Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and +retain the preceding contributors' attribution. This notice records research +provenance; the repository's Apache-2.0 license remains unchanged. + +Dependencies retained from eumemic's Claude-assisted PRs #3 and #5: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +https://github.com/CrocSwap/integer-mult-bounds/pull/5 +PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893 +These include complex_circuit.py, complex_network.py, fast_gaussian.py, +their generators, construction notes and certificates. The Gaussian input +scaling proof is corrected in this derivative; see the modification notice below. +Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank +sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6. +Their results and ideas are credited separately from this new construction. + +Bulk recursion and controlled tensor basis, 8 October 2026 +Copyright 2026 icekylinx + +The new projector batching, controlled simultaneous basis, mixed-width tape +recursions, whole-residual complex recursion, and dependency-path precision +guard in the batched-23 proof package are contributed by icekylinx, developed +with substantial OpenAI GPT-6 Astra assistance. Review corrections and +repository integration were prepared with OpenAI Codex assistance. They build on the retained +PR #7 finite producers and the preceding contributions credited above. +The new package retains the repository's Apache-2.0 license and the original +authors' attribution. It is not an official OpenAI release or endorsement. + +Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian +proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2). +The P=34p precision budget and asymptotic exponent are unchanged. New batching +checks are integrated into make verify. Existing attribution and licenses +are retained. + +Partial-swap extension, 8 October 2026 +Copyright 2026 icekylinx + +The partial-swap framing, binary retained-total triple construction, +dependency-preserving positive labels and carrier reuse, unequal (25,23,57) +common basis, improved complex batching, and exact assembly are contributed +by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance. +This extension directly inherits PR #10, commit +62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters +modify the retained paired-exclusion and shared-point implementations credited +above. The original licenses, attribution, and Gaussian correction are retained. + +Endpoint-gauge extension, 8 October 2026 +Copyright 2026 icekylinx + +The complementary auxiliary endpoint gauges, (32,30,40) controlled basis, +general nondegenerate binary phase residuals, retained-center complex producer, +updated path guard and exact assembly are contributed by icekylinx with +substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits +f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable +adapters retain the paired-exclusion and shared-point contributions credited +above. Original licenses, notices and the Gaussian correction are retained. + +Structured-basis and semantic-bulk extension, 8 October 2026 +Copyright 2026 icekylinx + +The selected (32,30,36) bit construction, A1/A5 block refinements, fixed +middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic +complex centers, complete data-front batching and circuit-specific assembly +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505. + +The semantic/bulk compatibility argument, literal scalar charge, product row +stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23, +commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints +are adopted from the same contributor's PR #21, +commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame +work and the earlier nested-basis contributions. + +Completed-child semantic precision, arbitrary-coordinate routing, phase-cell +Gaussian inversion and bulk tape transfer are credited to the RaD project +(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through +PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof +sources retain their separate CC0 license and original NOTICE under +references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI +Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra +assistance. These are source attributions, not claims about this session's +runtime model. Related contributions by Rohan Arun and Aurel Prosz remain +acknowledged in the imported notices. No exclusive priority or additional +license condition is asserted. + +Copied retained-center extension, 8 October 2026 +Copyright 2026 icekylinx + +The copied retained-center schedule in both orientations, its complex-phase +transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. This extension inherits PR #32 at +0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work. + +The two-stage topology and paid copied-stream endpoint correction are due +to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab, +through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition, +9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil +Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a. + +The contiguous corner method, incidence factorization and exact partition +certificates are due to Rohan Arun (PR #31), method commit +02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate +specializes Dominik Scholz's parameterization (PR #33), +a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof +snapshots are preserved in references/copied-centers. Historical OpenAI +Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to +the respective imported sources. Existing PR #21/#23 and RaD attribution, +licenses and the Gaussian correction remain in force. + +Copied centers with reversed (23,25) data corners, 2026-10-08 +Rohan Arun with substantial OpenAI Codex assistance supplies the composition, +exact dimension23 controls and cost/assembly integration. James Chang's PR34 +supplies the reversed boundary family; icekylinx's PR36 supplies copied retained +centers and the selected complex graph. The balanced assembly remains attributed +to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan +Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx, +eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses. +This is not a claim of formal verification, new foundational rank identities, +global optimality or measured runtime speedup. + +Fixed middle basis with copied reversed corners (2026-10-08): +Rohan Arun, with substantial OpenAI Codex assistance. Composition and +simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34, +Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures, +eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed. + +Changed-graph composition with copied centers and fixed bases, 8 October 2026 +Prepared with OpenAI Codex assistance. +The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J +projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument, +Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and +Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods, +Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc) +PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40 +reversed integration, modular data tests and both-fixed baseline; +RaD (hipotures) PR41 changed scalar DAGs, alternating point order and +independent role compiler; RaD balanced routing/bulk interfaces, and all +retained predecessors. +Complete attribution and the conditional proof boundary are recorded in +research/copied-fixed/PROOF.md; imported originals and source pins are retained. +Original Apache-2.0 and CC0 notices remain applicable. + +Community follow-up integration (PRs #49, #45 and #26), 8 October 2026: +Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out +orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction +composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and +reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD / +hipotures's alternating pair order and independent physical role compiler (#41). +All predecessor attribution and conditional proof interfaces remain in force. + +Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic +proofs, finite tensor execution, profile precision accounting and mixed-center +reference checks (#45), with substantial OpenAI Codex assistance. See +research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution. +Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic +frame/movement/guard lemmas and independent paired-circuit checks (#26). Original +source notices and historical snapshots are preserved. Neither formal package +claims a full formal proof of the multiplication algorithm or a new kappa. + +Joint frame compilation of skip-prefix strips + +The local compiler and exact finite witnesses were prepared by eumemic with +substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic +Claude assistance supplied the unmodified PR #53 skip-prefix producer. +The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating +Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang, +Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt, +OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors +credited in the retained sources. See references/frame-compiler/pr48/NOTICE +and references/frame-compiler/pr53/NOTICE for their original attribution. +No authorship or review by those predecessors is claimed for this compiler. + +Joint dual-skip/frame-compiler composition (2026-10-08) +--------------------------------------------------- +Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip +scalar graph; original LICENSE and NOTICE are retained under +references/frame-compiler/pr55. eumemic with substantial OpenAI Codex +assistance: PR57 joint reversible frame compiler, serialized physical-word +replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex +assistance: this composition, independent integration and exact certificate. +Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all +predecessor authors retain the notices above and in pinned source files. +No original copyright or Apache-2.0 notice is removed. + +Ranked reclamation after the PR58 joint-dual composition (2026-10-08) +---------------------------------------------------------------- +Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical +slots by descending current frame rank, retaining slot ID as the tie-breaker; +complete replay, exact profiles, comparison and adversarial controls. +The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57 +compiler. Its original shared compiler remains byte-identical. The PR55 +producer and all preceding attribution remain unchanged. The complete PR58 +comparison certificate and its original source/proof provenance are retained +under references/frame-compiler/pr58. + +Skip-prefix strip producer, 8 October 2026 +By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5) +assistance. The skip-prefix leave-one-out strip layout in +research/skip-strips/skip_graph.py is new. Everything else reuses Chafik +Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification +code, exact data recovery and support-order sorting, together with: +- Rohan Arun's PR #44 weighted carrier matching optimizer; +- RaD (hipotures) PR #41 alternating point order; +- every predecessor credited above. +Complete attribution and the conditional proof boundary are recorded in +research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain +applicable. + +Local split-pair skip-prefix research continuation (2026-10-08) +Prepared with substantial OpenAI Codex assistance at the user's request. +New contribution: singleton/pair recursive partition refinement, finite summand +order and moment search, and verified composition with inherited paid clones. +Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips. +Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone +specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51 +whole-chain/source-partition cloning and earlier producer/interfaces. +Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all +stated historical AI assistance. All predecessor notices remain applicable. + +Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones): +Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain +source-partition clone operation is adapted from RaD / hipotures, PR #51, +with original envelopes retained here. The scalar base is Avi Eisenberg's +PR #53 skip-prefix layout, prepared with Anthropic Claude assistance. +All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain +in force. This extension is provided under the repository's Apache-2.0 terms. + +Community review checkpoint #50–#61, 8 October 2026 +Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex. +Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude. +Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance) +and dual-suffix strips (#55, Anthropic Claude assistance). +Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex. +Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked +reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame +composition (#56), with Anthropic Claude and stated Codex validation assistance. +RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones +(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone +composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter +refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with +his disclosed multi-tool and Codex assistance. +Parallel #50/#52 records are reviewed and credited without replacing the earlier +climbed-48 implementation; their exact commit history is retained. The separate +#51 witness is retained as research; its source-graph/negative-basis data proof +has not received the same full replay as #56's consumed compiler machinery. +See docs/research/community-round2-review.md for precise validation boundaries. +All earlier licenses, contributions and notices remain applicable. + +Interval strips and core-aware pair assembly + +The interval leave-one-out strips and the core-aware group-pair assembly in +research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic +Claude assistance. They build on PR #53's skip-prefix pipeline and its +inherited PR #48 sources. The stacked witness in research/pair-assembly/frame +uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier +are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and +verify_skip_frame.py. Original notices are retained. + +Stopped product-ring extension, 9 October 2026 +Copyright 2026 icekylinx + +The opposite-bank generic projector factorization, componentwise atom-ring +adapters, stopped recurrence and outer ordinary-interchange conversion, +aligned pair-star producer with rational centers, exact denominator-21 +grid and selected (23,23)/(24,24) composition are contributed by icekylinx +with substantial OpenAI GPT-6 Astra and Codex assistance. This extension +builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is +integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37. + +The retained ordinary leaf uses the preceding projector, corner and +copied-center proofs. The selected two-stage topology retains Aurel Prosz +(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly +retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching, +original source notices and each imported license remain unchanged. + +Partial-gauge extension, 9 October 2026 +Copyright 2026 icekylinx + +The pair-first cube assembly, compatible backward frame enlargement, +partial source-gauge schedule and stopped whole-projector composition are +contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834. + +The physical deferred bit word, reflected event accounting, fan-order +clarification and signed two-stage endpoint are adopted from Zhihao Chen +(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with +Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen +witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38, +with the original Claude disclosure retained. Cyclic interval and core-aware +pair primitives build on Avi Eisenberg (ikeboy), PR #62, +ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance. +Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and +precision credits remain in force. Imported source notices and licenses +are preserved under references/partial-gauge/; see SOURCES.json. + +Three-stage cover extension, 9 October 2026 +Copyright 2026 icekylinx + +The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry, +weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring +fallback, uniform batching, internal row borrowing and exact composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This +repository integration uses Codex assistance and extends +PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86. + +The selected local complex DAG and support/label replay are adopted from +Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf, +with its original Apache-2.0 terms and Claude assistance disclosure. +The unchanged bit physical word retains Zhihao Chen (jacklightChen), +PR #97, and Swapnil Jain's underlying witness and assistance notices. +The prior opposite-bank, stopped ordinary-leaf, copied-center, topology, +semantic/bulk, precision and analytic credits remain applicable. +Original source notices and license bytes are preserved in +references/three-stage-cover/pr117/ and references/partial-gauge/. + +Paired-cube and shared-core extension, 9 October 2026 +Copyright 2026 icekylinx + +The paired-coordinate triple-cube identity, signed overlap channels, +original-source involution, coordinate-star centers, concrete frame/carrier +compiler, chronological source gauges, PR97 gauge-subset selection, +orthogonal cross-stage routing and exact supplier/assembly integration +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. +Repository integration uses Codex assistance and extends PR #130 at +6a9970a530119174507904e23592fd59ede19a5d. + +The substantial completed-core workspace-sharing principle is credited to +an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported +OpenAI Codex assistance. No source code, partition or serialized witness +from that PR is imported. The new routing and paired-motif integration +are described separately and claim no exclusive priority for broad sharing, +paired-coordinate, Fourier or carrier-matching ideas. + +Selected coarse queries restrict the retained eumemic PR #117 positive DAG; +its original Apache-2.0 and Claude assistance notices remain unchanged. +The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil +Jain's frozen witness, notices and historical assistance disclosures. +Earlier generalized Clifford, three-stage, uniform weighted recursion, +ordinary-leaf, semantic/bulk, precision and analytic credits remain in force. +Source-specific license bytes are preserved under references/. + +Source-assisted common-frame extension (2026-10-09) +Copyright 2026 icekylinx. Apache-2.0. +Research construction and local certificates: OpenAI GPT-6 Astra assistance. +Repository integration and independent complex scalar replay: Codex assistance. +Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee; +its byte-preserved sources, licenses and original Claude assistance disclosure +are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz. +Additional inherited contributions are recorded in research/source-assisted/CREDITS.md. +The new work concerns source-assisted invertible common-frame lifts, controlled +affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for +broad register reuse, paired cubes, frame theory or ordinary linear elimination. + +Fourth finite bootstrap refinement (2026-10-09) +Prepared by eumemic with substantial OpenAI Codex assistance. +This arithmetic refinement builds on PR207 at +cd14825023b75af4f5919a30e5e6d548b6ade5bc by adding one finite ordinary-leaf +level. The 302 frame overrides, completed banks, scalar words and upstream +assistance disclosures are inherited, not newly attributed to this work. +The Apache-2.0 license and all preceding notices remain applicable. + +PR211 plateau frame refinement (2026-10-09) +Prepared by eumemic with substantial OpenAI Codex assistance. +The additional seven equal-frame component moves build directly on Rohan +Arun's PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347, +prepared with Anthropic Claude assistance. Its original notice is preserved +byte-for-byte in search/PR211-NOTICE. The offline certificate package and +independent stage-private bank admission derive from Dugongue's PR207 at +cd14825023b75af4f5919a30e5e6d548b6ade5bc. Original contributor notices, +licenses, source pins and all upstream AI disclosures remain applicable. + +The final refinement also includes seeded bidirectional cascade search after +the seven initial component moves. Exact canonical subspace comparison +records 274 changed operations beyond PR211. Search, integration and replay +were prepared with substantial OpenAI Codex assistance. + +Shared rank-18 gauges and two donor substitutions (2026-10-09) +Prepared by eumemic with substantial OpenAI Codex assistance. +The new actual gauge/read chronology, gauged-donor handoff, shared target +transitions, rank6 residual banks and donor substitutions extend the same +PR200/PR211 word and PR207/PR197 completed-bank framework. All original +licenses, notices, authorship and upstream AI disclosures remain in force. + +Nested-gauge extension: three rank-19 physical entrances compose with thirteen shared rank-18 entrances; all target transitions and mixed4/5/6/24 banks are explicit. Prepared by eumemic with substantial OpenAI Codex assistance. 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+"""Actual joint entrance gauges on the frozen relaxed PR211 frame word. +Experimental adapter, prepared with OpenAI Codex assistance; Apache-2.0. +Pinned supplier files are never modified. +""" +from pathlib import Path +from copy import deepcopy +from collections import defaultdict +import sys,json,hashlib +sys.dont_write_bytecode=True +D=Path(__file__).resolve().parent +PACKAGE=D.parent;ROOT=PACKAGE.parents[1] +P=ROOT/'research/paired-cube-diagonal-bit-168' +sys.path.insert(0,str(P/'bit')) +from word import Candidate as Original +class Candidate(Original): + def __init__(self): + super().__init__() + self.original_w=deepcopy(self.w) + pairs=json.loads((D/'pairs.json').read_text()) + assert len(pairs)==1760 and set(b for b,d in pairs)==set(self.donor) + assert len(set(d for b,d in pairs))==1760 + old=dict(self.pairs);changed=[(b,d) for b,d in pairs if old[b]!=d] + assert changed==[(18892,8292),(18894,8293)] + assert all(d not in self.source.values() and d not in self.gauge and d not in self.rootroles for b,d in changed) + assert all(self.last[d]=0 + if d:target_hist[d]+=1 + target_frames[t]=f;target_paths[t].append(f) + X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[] + def track(value): + nonlocal top + if major:top=max(top,value) + return value + def value(s): + s=W.phys[s];return x[borrow[s]]if s in borrow else z[s] + def assign(s,val): + s=W.phys[s] + if s in borrow:x[borrow[s]]=track(val) + else:z[s]=track(val) + responses=W.adjoint() if mutation=='old_adjoint' else adj + def read(s): + if mutation=='omit_gauge_compensation' and s==new_gauge_selection[-1]['role']:return + for t,c in responses[s].items(): + if s in W.gauge:target_move(t,W.gauge[s]['frame']) + else:assert target_frames[t]is None,'initial correction leaves target atD0' + y[t]=track(add(y[t],-direction*c,value(s))) + def gate(i,sign): + a,b,_=W.ops[i];assign(a,add(value(a),sign,value(b))) + for s in range(W.R): + if s not in W.gauge and s not in borrow and s not in removed:read(s) + for n,s in W.source.items():assign(s,add(value(s),1,x[n])) + for r in gauge_selection: + if mutation!='omit_gauge_mix' or r is not new_gauge_selection[-1]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']])) + def forward(i): + if i not in omitted:gate(i,1);done.append(i) + for k in schedule['center_prefix_events']: + event=schedule['events'][k];i=event['op'] + if event['kind']=='early_mix': + r=early[i];a,b=r['partner'],r['source'] + if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b])) + else:forward(i) + for r,s in zip(W.g['roots'],W.w['rootroles']): + if r['kind']=='center': + for t in r['targets']: + assert all(f is None for f in target_frames),'copied centers scatter before any nonzero target movement' + center_reads.append((s,t));y[t]=track(add(y[t],direction,value(s))) + for e in terminal: + assert all(target_frames[t]is None for t in e['targets']),'terminal setup afterD0center scatter' + for t in e['targets']: + if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']])) + for j,i in enumerate(W.rest): + for s in at[j]: + if mutation!='late_phase1_gauge' or s not in new_phase_roles:read(s) + if mutation=='late_phase1_gauge'and j==168: + for s in sorted(new_phase_roles):read(s) + if i in bywrite: + e=bywrite[i];a,b,n=W.ops[i];target_move(e['pivot'],W.opframe[i]);y[e['pivot']]=track(add(y[e['pivot']],direction,value(b))) + else:forward(i) + if i in afterwrite: + e=afterwrite[i] + for t in e['targets']: + target_move(t,e['root_frame']) + if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']])) + for s in at[len(W.rest)]:read(s) + for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])): + if r['kind']=='side' and j not in deletedroots: + for t in r['targets']:target_move(t,W.w['root_frame'][j]);y[t]=track(add(y[t],direction,value(s))) + for e in deliveries[j]: + a,b=e['carrier'],e['passive'] + if b not in by_source:x[a]=track(add(x[a],1,x[b])) + for t in e['receivers']:target_move(t,e['deliver_frame']);y[t]=track(add(y[t],direction,x[a])) + for e in W.k['entries']: + if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']])) + if mutation=='undo_before_restore': + for e in W.k['entries']: + if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']])) + for i in reversed(done):gate(i,-1) + if mutation!='undo_before_restore': + for e in W.k['entries']: + if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']])) + for n,s in W.source.items():assign(s,add(value(s),-1,x[n])) + if binary:want=[Y0[t]^X0[t]for t in range(v)] + else: + want=Y0[:];cache={} + for root in W.g['roots']: + n=root['node'] + if n not in cache: + mask=C.sup[n];val=0 + while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1]) + cache[n]=val + for t in root['targets']:want[t]=add(want[t],direction,cache[n]) + for e in W.k['entries']: + val=add(X0[e['carrier']],1,X0[e['passive']]) + for t in e['receivers']:want[t]=add(want[t],direction,val) + # Finish every local target at its codimension-one cap, then reverse the exact + # annihilator stream. All copied-center scatters lie before these movements, + # hence atD0 forward and atD1 after the complementary movements reverse. + reflected=Counter() + for t,chain in enumerate(target_paths): + assert chain + d=W.h-1-C.dimf[chain[-1]];assert d>=0 + if d:target_hist[d]+=1 + previous=[C.cov[t]];dimension=1 + for f in reversed(chain): + assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f]) + basis=C.A[f];step=len(basis)-dimension;assert step>=0 + if step:reflected[step]+=1 + previous=basis;dimension=len(basis) + if W.h>dimension:reflected[W.h-dimension]+=1 + assert reflected==target_hist + assert len(center_reads)==sum(len(r['targets'])for r in W.g['roots']if r['kind']=='center') + assert sum(k*n for k,n in target_hist.items())==W.v*(W.h-1) + if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0))) + assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10]) + assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10]) + assert z==Z0,('dirty restoration',mode) + return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True,copied_center_target_reads=len(center_reads),all_forward_centers_at_D0=True,all_reflected_centers_at_D1=True,actual_target_histogram={str(k):n for k,n in sorted(target_hist.items())},actual_reflected_target_histogram={str(k):n for k,n in sorted(reflected.items())}) +if __name__=='__main__': + start=time.monotonic();out=dict(status='PASS_SCALAR_SOURCE471_RETIMED_PHASE1_GAUGES',selection=selection,gauge_selection=gauge_selection,new_gauge_selection=new_gauge_selection,geometry_and_paid_admission=False) + out['F2']=replay();print('PASS F2',out['F2'],'seconds',time.monotonic()-start,flush=True) + bound=replay('bound');bits=8*((bound.bit_length()+2+7)//8);out['bound']=bound;out['bits']=bits;out['terminal_sinks']=len(terminal);print('bound',bound,'bits',bits,flush=True) + out['integer']=[] + if True: + for direction in(1,-1):out['integer'].append(replay('Z',direction,bits));gc.collect();print('PASS integer',direction,'seconds',time.monotonic()-start,flush=True) + out['controls']={} + for mutation in ['omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation','late_phase1_gauge']: + try:replay(mutation=mutation) + except AssertionError as error:out['controls'][mutation]=dict(status='REJECTED',error=str(error)) + else:raise AssertionError('Invalid mutation accepted: '+mutation) + assert (1<2*bound + assert input_pins=={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths} + out['input_pins']=input_pins + (HERE/'replay.json').write_text(json.dumps(out,indent=2)+'\n') + print('PASS actual471 source word, both integer signs, all columns and six controls',flush=True) diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/schedule.json b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/schedule.json new file mode 100644 index 0000000000..51c2c8df5a --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/schedule.json @@ -0,0 +1 @@ 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8584, + "source": 98, + "partner": 97, + "mix_frame": 126131, + "targets": [ + 208, + 209, + 210, + 211 + ] + }, + { + "role": 9424, + "source": 458, + "partner": 457, + "mix_frame": 125321, + "targets": [ + 568, + 569, + 570, + 571 + ] + } +] diff --git a/research/multicut-delayed-inverse-checkpoint/loader/source_loader.py b/research/multicut-delayed-inverse-checkpoint/loader/source_loader.py new file mode 100644 index 0000000000..5665769ab2 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/source_loader.py @@ -0,0 +1,207 @@ +#!/usr/bin/env python3 +"""Source-bound current PR210 preparation with no historical search imports. + +This is a focused preparation adapter, not a full scalar or supplier verifier. +The pinned public joint and source471 preparation bodies are retained literally; +only the explicit base source root and terminal-selector source are rebound. +Original source-specific notices remain beside the supplied public inputs. +New path adapter and semantic fingerprints: assisted with ChatGPT. +""" +from collections import Counter, defaultdict +from copy import deepcopy +from pathlib import Path +import argparse +import contextlib +import gzip +import hashlib +import importlib.util +import io +import json +import sys + +sys.dont_write_bytecode = True +if not __debug__: + raise SystemExit('refusing optimized Python: assertions are required') +HERE = Path(__file__).resolve().parent +BASE_PACKAGE = 'research/paired-cube-diagonal-bit-168' +PR210_PACKAGE = 'research/coordinated-crossover-pr200' + + +def sha(data): + return hashlib.sha256(data).hexdigest() + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(',', ':')).encode() + + +def check_sources(pr210_root, base_bit_root): + """Validate all inputs before importing any source implementation.""" + roots = {'pr210': Path(pr210_root).resolve(), 'base_bit': Path(base_bit_root).resolve()} + pins = json.loads((HERE/'SOURCE_PINS.json').read_text()) + checked = {} + for group, root in roots.items(): + checked[group] = {} + for relative, expected in pins[group]['files'].items(): + path = (root/relative).resolve() + if not path.is_relative_to(root): + raise ValueError('source path escapes supplied root') + got = sha(path.read_bytes()) + if got != expected: + raise ValueError('source hash mismatch: '+group+'/'+relative) + checked[group][relative] = got + return roots, pins, checked + + +def import_file(name, path): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +def terminal_records(word, selectors): + """Derive terminal data from pinned roots and selectors, never a receipt.""" + writes = defaultdict(list) + for i, (a, b, node) in enumerate(word.ops): + writes[a].append(i) + result = [] + for selected in selectors: + root = selected['root']; pivot = selected['pivot'] + role = word.w['rootroles'][root] + entry = word.g['roots'][root] + assert entry['kind'] == 'side' and pivot in entry['targets'] + assert role not in word.source.values() and role not in word.gauge + result.append(dict(root=root, pivot=pivot, role=role, + targets=entry['targets'], root_frame=word.w['root_frame'][root], + writes=writes[role])) + assert len(result) == 34 + assert len({r['role'] for r in result}) == 34 + return result + + +def prepare(pr210_root, base_bit_root): + roots, pins, checked = check_sources(pr210_root, base_bit_root) + p = roots['base_bit']/BASE_PACKAGE + package = roots['pr210']/PR210_PACKAGE + base = import_file('five_stage_public_base_word', p/'bit/word.py') + joint_path = package/'joint/joint_word.py' + joint = joint_path.read_text() + old = "P=ROOT/'research/paired-cube-diagonal-bit-168'\nsys.path.insert(0,str(P/'bit'))\nfrom word import Candidate as Original" + assert joint.count(old) == 1, 'joint import boundary changed' + joint = joint.replace(old, 'P=__base_package__\nOriginal=__base_candidate__', 1) + js = {'__name__': 'five_stage_public_joint', '__file__': str(joint_path), + '__base_package__': p, '__base_candidate__': base.Candidate} + exec(compile(joint, str(joint_path)+' [explicit base-root binding]', 'exec'), js) + word = js['Candidate']() + c = word.C + # These are decoder/geometry checks, not all-column payload replay. + c.decoder(); c.geometry() + word.changed_frames = [i for i, (a, b) in enumerate(zip(word.original_opframe, word.opframe)) if a != b] + word.endframe = {r: word.opframe[ii[-1]] for r, ii in word.role_ops.items()} + selection = json.loads((package/'borrow/selection.json').read_text()) + gauges = json.loads((package/'gaugeb/selection.json').read_text()) + current = json.loads((package/'newg/selection.json').read_text()) + if isinstance(current, dict): + current = current['selection'] + terminal = terminal_records(word, json.loads((p/'selected/bit/sinks.json').read_text())) + source_path = package/'newg/replay.py' + source = source_path.read_text() + start = source.index('schedule=json.loads') + stop = source.index('\ndef replay(') + preparation = source[start:stop] + # Original input_paths names an output-only joint replay receipt. All actual + # code/data inputs have already been checked against SOURCE_PINS.json. + lines = preparation.splitlines() + removed = [line for line in lines if line.startswith(('input_paths=', 'input_pins='))] + assert len(removed) == 2, 'source input-list boundary changed' + preparation = '\n'.join(line for line in lines if line not in removed)+'\n' + old = "terminal=json.loads((D/'joint-replay.json').read_text())['selected'];" + assert preparation.count(old) == 1, 'terminal source boundary changed' + preparation = preparation.replace(old, 'terminal=__terminal_records__;', 1) + namespace = {'__name__': 'five_stage_public_source471', '__file__': str(source_path), + 'Path': Path, 'Counter': Counter, 'defaultdict': defaultdict, + 'json': json, 'sys': sys, 'hashlib': hashlib, + 'HERE': package/'newg', 'D': package/'joint', + 'W': word, 'C': c, 'selection': selection, + 'new_gauge_selection': current, 'gauge_selection': gauges+current, + '__terminal_records__': terminal} + with contextlib.redirect_stdout(io.StringIO()): + exec(compile(preparation, str(source_path)+' [preparation only; terminal selectors rebound]', 'exec'), namespace) + assert 'replay' not in namespace, 'payload replay must not be executed or loaded by preparation' + namespace['SOURCE_TEXT'] = source + namespace['source_pins'] = checked + namespace['source_revisions'] = {key: pins[key]['commit'] for key in roots} + namespace['source_preparation_sha256'] = sha(source[start:stop].encode()) + namespace['loader_scope'] = 'Pinned source preparation and exact geometry only; no all-column payload replay, five-stage supplier admission, or final exponent.' + return namespace + + +def semantic_fingerprint(context): + """Exact-basis semantics; generated frame IDs need not match legacy order.""" + w, c = context['W'], context['C'] + basis_cache = {} + def basis(f): + if f not in basis_cache: + basis_cache[f] = sha(canonical([list(r) for r in c.B[f]])) + return basis_cache[f] + def framed(entry): + out = dict(entry) + for key in ('frame', 'root_frame', 'mix_frame', 'deliver_frame'): + if key in out: + out[key] = basis(out[key]) + for key in ('carrier_chain', 'passive_chain'): + if key in out: + out[key] = [basis(f) for f in out[key]] + return out + parts = dict( + partner_chronology=[framed(e) for e in w.k['entries']], + partner_lookup=[(n,framed(e)) for n,e in sorted(context['kpair'].items())], + graph=w.g, + operations=w.ops, + operation_bases=[basis(f) for f in w.opframe], + source_bases=[basis(f) for f in w.w['source_frame']], + root_bases=[basis(f) for f in w.w['root_frame']], + full_basis=basis(w.w['full_frame']), + gauge_records=[framed(w.gauge[r]) for r in sorted(w.gauge)], + gauge_read_order=w.order, + gauge_readtimes=sorted(w.readtime.items()), + donor_recipient_pairs=w.pairs, + physical_role_map=sorted(w.phys.items()), + phase1=w.phase1, + rest=w.rest, + schedule=context['schedule'], + source_loans=sorted(context['borrow'].items()), + terminal=[framed(e) for e in context['terminal']], + dirty_roles=context['regs'], + target_response=context['adj'], + live_gauge_reads=sorted(context['at'].items()), + source_selection=context['selection'], + gauge_selection=context['gauge_selection'], + ) + hashes = {key: sha(canonical(value)) for key, value in parts.items()} + return dict(schema='source471-exact-semantic-fingerprint/1', + hashes=hashes, combined_sha256=sha(canonical(hashes)), + source_head='df95878d11190518e45ef9717c9ee05011f88ace', + h=w.h, v=w.v, logical_roles=w.R, + independent_dirty_roles=len(context['regs']), + local_independent_columns=2*w.v+len(context['regs']), + source_loans=len(context['borrow']), + terminal_sinks=len(context['terminal']), + retimed_operations=len(context['deferred']), + formal_frame_id_equality_required=False) + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--pr210-root', type=Path, required=True) + ap.add_argument('--base-bit-root', type=Path, required=True) + args = ap.parse_args() + ctx = prepare(args.pr210_root, args.base_bit_root) + print(json.dumps(dict(status='PASS_PINNED_PUBLIC_SOURCE_PREPARATION', + scope=ctx['loader_scope'], + fingerprint=semantic_fingerprint(ctx), + source_revisions=ctx['source_revisions']), indent=2, sort_keys=True)) + +if __name__ == '__main__': + main() diff --git a/research/multicut-delayed-inverse-checkpoint/loader/test_public.py b/research/multicut-delayed-inverse-checkpoint/loader/test_public.py new file mode 100644 index 0000000000..255fce4f7d --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/loader/test_public.py @@ -0,0 +1,88 @@ +#!/usr/bin/env python3 +"""Targeted preparation/ledger tests. No all-column scalar replay is invoked.""" +import argparse +import json +from pathlib import Path +import subprocess +import sys +import unittest +from unittest.mock import patch +import source_loader +import raw_ledger + +HERE=Path(__file__).resolve().parent +PR210=HERE/'source_inputs/pr210' +BASE=HERE/'source_inputs/base_bit' + +class IntegrityControls(unittest.TestCase): + def test_all_public_input_hashes(self): + _,_,pins=source_loader.check_sources(PR210,BASE) + self.assertEqual(sum(map(len,pins.values())),22) + def test_corrupt_input_rejected_before_import(self): + target=(PR210/source_loader.PR210_PACKAGE/'frames/opframe-bases.json').resolve() + original=Path.read_bytes + def corrupt(path): + data=original(path) + return data+b'\n' if path.resolve()==target else data + with patch.object(Path,'read_bytes',corrupt): + with self.assertRaisesRegex(ValueError,'source hash mismatch'): + source_loader.check_sources(PR210,BASE) + def test_missing_input_rejected(self): + target=(PR210/source_loader.PR210_PACKAGE/'newg/schedule.json').resolve() + original=Path.read_bytes + def missing(path): + if path.resolve()==target:raise FileNotFoundError(str(path)) + return original(path) + with patch.object(Path,'read_bytes',missing): + with self.assertRaises(FileNotFoundError):source_loader.check_sources(PR210,BASE) + def test_optimized_execution_rejected(self): + proc=subprocess.run([sys.executable,'-O','-B',str(HERE/'source_loader.py'),'--help'],capture_output=True,text=True) + self.assertNotEqual(proc.returncode,0) + self.assertIn('refusing optimized',proc.stderr) + +class SemanticControls(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.context=source_loader.prepare(PR210,BASE) + cls.fingerprint=source_loader.semantic_fingerprint(cls.context) + cls.expected=json.loads((HERE/'expected_semantics.json').read_text()) + cls.ledger=raw_ledger.reconstruct(cls.context) + def test_exact_semantic_fixture(self): + self.assertEqual(self.fingerprint,self.expected) + def test_no_scalar_replay_loaded(self): + self.assertNotIn('replay',self.context) + self.assertIn('SOURCE_TEXT',self.context) + def test_wrong_gauge_order_detected(self): + w=self.context['W'];before=w.order[:] + try: + w.order[0],w.order[1]=w.order[1],w.order[0] + self.assertNotEqual(source_loader.semantic_fingerprint(self.context)['hashes']['gauge_read_order'],self.expected['hashes']['gauge_read_order']) + finally:w.order[:]=before + def test_wrong_partner_delivery_detected(self): + e=self.context['W'].k['entries'][0];before=e['deliver_after_root'] + try: + e['deliver_after_root']=before+1 + changed=source_loader.semantic_fingerprint(self.context)['hashes'] + self.assertNotEqual(changed['partner_chronology'],self.expected['hashes']['partner_chronology']) + self.assertNotEqual(changed['partner_lookup'],self.expected['hashes']['partner_lookup']) + finally:e['deliver_after_root']=before + def test_raw_ledger_fixture(self): + expected=json.loads((HERE/'expected_ledger.json').read_text()) + for key,value in expected.items():self.assertEqual(self.ledger[key],value,key) + def test_all_center_and_completion_calls_paid(self): + self.assertEqual(self.ledger['paid_center_copy_histogram'],{'22':24}) + profile=self.ledger['five_stage_profile'] + self.assertEqual((profile['calls'],profile['rank_mass'],profile['deficit'],profile['maxchild']),(495304,2837560,4400,100)) + self.assertEqual({r:profile['histogram'][r] for r in ['60','65','90','100']},{'60':18,'65':48,'90':13,'100':2200}) + +if __name__=='__main__': + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--pr210-root',type=Path,default=PR210) + parser.add_argument('--base-bit-root',type=Path,default=BASE) + parser.add_argument('--pins-only',action='store_true') + args=parser.parse_args();PR210=args.pr210_root;BASE=args.base_bit_root + suite=unittest.TestSuite() + suite.addTests(unittest.defaultTestLoader.loadTestsFromTestCase(IntegrityControls)) + if not args.pins_only:suite.addTests(unittest.defaultTestLoader.loadTestsFromTestCase(SemanticControls)) + result=unittest.TextTestRunner(verbosity=2).run(suite) + raise SystemExit(not result.wasSuccessful()) diff --git a/research/multicut-delayed-inverse-checkpoint/math_check.py b/research/multicut-delayed-inverse-checkpoint/math_check.py new file mode 100644 index 0000000000..5beb34c701 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/math_check.py @@ -0,0 +1,57 @@ +"""Source-bound banked five-stage moments and all47outerconstraints. +Prepared with substantial OpenAI Codex assistance; Apache-2.0. +Uses PR234 five-stage architecture and complex baseline, our527 helper and +PR197-derived actual five-stage entrance banks. Inputs must be live verifier outputs. +""" +from pathlib import Path +from fractions import Fraction as Q +from collections import Counter +import importlib.util +HERE=Path(__file__).resolve().parent + +def load(name): + p=HERE/(name+'.py');s=importlib.util.spec_from_file_location('banked527_'+name,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m + +def serial(x): + if isinstance(x,Q):return str(x) + if isinstance(x,dict):return {str(k):serial(v)for k,v in x.items()} + if isinstance(x,(list,tuple)):return list(map(serial,x)) + return x + +def hist(x):return Counter({int(k):v for k,v in x.items()if v}) + +def run(raw,complex_histogram,banks,global_result): + cost=load('moment');other=load('base_two_moment');outer=load('outer') + assert raw['source_aliases']==527 and raw['physical_R']==16587 and raw['h']==24 and raw['v']==1760 + H=hist(raw['five_stage_profile']['histogram']);assert H==hist(global_result['paid_histogram']) + assert global_result['paid_rank_mass']==sum(k*n for k,n in H.items())==2830840 + gauges=hist(raw['auxiliary_entrance_rank_histogram']);assert gauges=={1: 369, 2: 41, 3: 3, 4: 19, 5: 31, 6: 1, 8: 1, 9: 20, 10: 2, 11: 14, 12: 30, 13: 55, 14: 8, 15: 4, 17: 7, 18: 22, 20: 2200} + assert banks['assignments']==3317400 and banks['source527_roles']==16587 and banks['physical_replicas']==40 + assert banks['literal_stock']==867045 and banks['banks_total']==585445 and banks['charts']==770 + for a,n in gauges.items(): + assert H[5*a]>=n;H[5*a]-=n + if not H[5*a]:del H[5*a] + assert len(H)>0 and min(H.values())>0 and max(H)==50 + literal={k:n*40 for k,n in H.items()};literal_mass=sum(k*n for k,n in literal.items());literal_W=banks['literal_stock'] + assert 120*literal_W-literal_mass==40*4400 + # Retain the complete literal40-replica integer moment and literal stock. + normalized=dict(literal);W=literal_W;m=120;mass=literal_mass + bp=dict(m=m,W=W,histogram=normalized,calls=sum(normalized.values()),rank_mass=mass,deficit=m*W-mass,maxchild=max(normalized),normalization=40,physical_replicas=40,literal_stock=literal_W,literal_children=sum(literal.values()),literal_rank_mass=literal_mass) + root=cost.certify(normalized,m,W,True);c=Q(int(Q(root['lower'])*10**18),10**18);bm=cost.moment(normalized,m,W,c,True);nextbm=cost.moment(normalized,m,W,c+Q(1,10**18),True);assert bm[1]<10 + q=bit*(1-2*eta);minimum=(1-eta)*q/(1+q);ticks=minimum*10**18;k=Q((ticks.numerator-1)//ticks.denominator,10**18) + assembly=outer.assembly(bit,b,bridge,k,eta=eta,beta=beta);assert len(assembly['strict_constraints'])==47 and len(assembly['margins'])==7 and min(assembly['strict_constraints'].values())>0 + try:outer.assembly(bit,b,bridge,k+Q(1,10**18),eta=eta,beta=beta) + except AssertionError:pass + else:raise AssertionError('Adjacentkappaadmitted') + return serial(dict(status='PASS_BANKED_FIVE_STAGE527_EXACT_MOMENTS_AND_OUTER47',mathematics=dict(bit_profile=bp,complex_profile=cp,bit_coarse=c,complex_coarse=b,bit_moment_interval=bm,complex_moment_interval=cm,next_bit_grid_excluded=nextbm,independent_base_two_pass=True,ordinary_bootstrap_chain=chain,finite_bridge=bridge,assembly=assembly,kappa=k,kappa_decimal=cost.decimal(k),kappa_scientific=format(float(k),'.15e'),binding='bit'),adjacent_grid_point_rejected=True)) diff --git a/research/multicut-delayed-inverse-checkpoint/moment.py b/research/multicut-delayed-inverse-checkpoint/moment.py new file mode 100644 index 0000000000..1e43aaeaca --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/moment.py @@ -0,0 +1,114 @@ +"""Independent prospective ordinary five-stage cost, never supplier admission. + +Consumes a raw one-invocation ledger made from the actual source/internal/target +paths. No packed profile is used as an input. Missing physical/finite/compiler +obligations must remain explicit in the accompanying report. +""" +from collections import Counter +from decimal import Decimal, localcontext, ROUND_FLOOR +from fractions import Fraction as F +from functools import lru_cache +from math import factorial +import hashlib, json, sys +from pathlib import Path + +DYAD = 1 << 224 + +def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD) +def ceil(x): return -floor(-x) + +@lru_cache(None) +def logarithm(x): + """Mercator bounds after binary reduction; ln(2) via two Mercator sums. + + For 1 <= y < 2, log(y) is an alternating series in z=y-1. + We instead reduce to 1 <= y < 3/2, and bound log(3/2) and + log(2) by the same alternating series at z=1/2 and z=1/3, + using log(2)=log(3/2)+log(4/3). All sums are exact rationals. + """ + assert x >= 1 + k = 0 + while x >= 2: x /= 2; k += 1 + j = int(x >= F(3, 2)) + if j: x /= F(3, 2) + def mercator(y): + z=y-1 + assert 0 <= z <= F(1,2) + # An even partial sum is below log; the next term bounds its tail. + t=z; s=F(0) + for n in range(1, 225): + s += (t if n & 1 else -t)/n + t *= z + return s, s+t/225 + a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3)) + return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f) + +def exponential(l,u): + assert 0 <= l <= u < F(1,10) + def poly(x): + s=t=F(1) + for j in range(1,25): t=t*x/j; s+=t + return s,t + lo,_=poly(l);hi,last=poly(u) + return floor(lo),ceil(hi+last*u/25/(1-u/26)) + +def moment(H,m,W,a,bad=False): + lo=hi=F(0) + for r,n in H.items(): + assert 0 < r < m and n > 0 + l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u) + p=F(r*n,m*W);lo+=p*e;hi+=p*f + if bad: + # Existing envelope shape, only conditionally portable to the new cover. + l,u=logarithm(F(m));e,f=exponential(a*l,a*u) + p=F(32*m*sum(H.values()),10**16*W) + lo+=p*e;hi+=p*f + return floor(lo),ceil(hi) + +def decimal(x): + with localcontext() as c: + c.prec=70 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def certify(H,m,W,bad=False): + with localcontext() as c: + c.prec=70 + terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()] + fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0) + lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003') + for _ in range(210): + a=(lo+hi)/2 + if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a + else:hi=a + q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR)) + lo=F(q,10**24);hi=lo+F(1,10**24) + lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad) + assert lv[1]<10 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0} + assert min(margins.values())==minimum + assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2 + assert 1-eps*(1+c)==eta-eps*eta/4 + return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma, + eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam, + lambda_prime=lp,alpha_squared_power=r,delta=delta, + C0=bridge['semantic']['C0'],C1=1,kappa=kappa), + strict_constraints=slacks,margins=margins,minimum_margin=minimum, + absorption_gap=minimum-kappa, + recurrence=dict(internal=internal,leaf=leaf,reservations=1-c)) diff --git a/research/multicut-delayed-inverse-checkpoint/parity_transform.py b/research/multicut-delayed-inverse-checkpoint/parity_transform.py new file mode 100644 index 0000000000..11c218eafa --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/parity_transform.py @@ -0,0 +1,143 @@ +"""F2 payload identity elision and exact nested address-MOVE fusion. +The inherited signed source decoder is retained as producer evidence. +All odd-prime address projectors and routes remain unchanged. +Prepared with substantial OpenAI Codex assistance; Apache-2.0. +""" +from array import array +from collections import Counter +from pathlib import Path +import gzip,hashlib,json,time +if not __debug__:raise SystemExit('assertions required') + +def transform(records): + out = array('i') + pending = {} + chains = [] + removed = Counter() + hist = Counter() + merged = 0 + maxchain = 0 + uses = Counter() + + def flush(a): + nonlocal merged, maxchain + if a in pending: + old, new, rank, count, first, last = pending.pop(a) + if rank: + hist[rank] += 1 + out.extend((0, a, old, new, rank, 0)) + merged += count - 1 + maxchain = max(maxchain, count) + if count > 1: + chains.append(dict(stream=a, old=old, new=new, rank=rank, moves=count, first=first, last=last)) + for index in range(0, len(records), 6): + op, a, b, c, f, z = records[index:index + 6] + if op == 1 and c % 2 == 0: + removed[c] += 1 + continue + if op == 0: + assert z == 0 + if a in pending: + old, prev, rank, count, first, last = pending[a] + assert prev == b, ('disconnected movechain', a, prev, b) + pending[a] = (old, c, rank + f, count + 1, first, index // 6) + else: + pending[a] = (b, c, f, 1, index // 6, index // 6) + continue + assert op in (1, 2, 3) + flush(a) + flush(b) + if op == 1: + uses[a] += 1 + uses[b] += 1 + elif op == 2: + hist[z] += 1 + uses[a] += 1 + uses[b] += 1 + out.extend((op, a, b, c, f, z)) + for a in sorted(pending): + flush(a) + return (out, dict(removed_even_adds=dict(removed), removed_even_add_count=sum(removed.values()), merged_move_count=merged, max_chain=maxchain, fused_chains=len(chains), paid_histogram=dict(sorted(hist.items())), paid_calls=sum(hist.values()), paid_rank_mass=sum((r * n for r, n in hist.items())), surviving_additions=sum((out[k] == 1 for k in range(0, len(out), 6))), output_records=len(out) // 6, unused_streams=[i for i in range(20107) if not uses[i]], stream_use_counts=dict(uses)), chains) + +def run(producer,output_dir=None): + begun=time.monotonic();W,C=producer['W'],producer['C'];v=W.v;n=20107 + old=producer['records'];out,stats,chains=transform(old);oldmeta=producer['physical'] + assert stats['removed_even_add_count']==1783196 and stats['surviving_additions']==786621 + assert stats['paid_rank_mass']==434502 and stats['merged_move_count']==3 + initial=producer['initial_state'];ZERO,FULL=producer['ZERO'],producer['FULL'] + # Independent required-frame census: ignore all original MOVE instructions. + # Use only surviving gate operands, copy sources and mathematical endpoints. + needs={i:[] for i in initial};copies=Counter();copy=None + for k in range(0,len(old),6): + op,a,b,c,f,z=old[k:k+6] + if op==1 and c%2: + needs[a].append(f) + if b!=n:needs[b].append(f) + else:assert copy is not None + elif op==2: + assert copy is None;copy=(a,b,c);needs[a].append(c);copies[z]+=1 + elif op==3:assert copy==(a,b,c);copy=None + assert copy is None and copies=={22:24} + final={i:FULL for i in initial} + for t in range(v):final[v+t]=W.register(W.module.kernel([C.cov[t]],W.h)[0]) + census=Counter(copies);per_role={};pairs=set() + for i,oldf in initial.items(): + H=Counter() + for f in needs[i]+[final[i]]: + assert C.sub(oldf,f);rank=C.dimf[f]-C.dimf[oldf] + if rank:H[rank]+=1 + pairs.add((oldf,f));oldf=f + per_role[i]=H;census.update(H) + assert census==Counter({int(r):c for r,c in stats['paid_histogram'].items()}) + # Read back the actual annihilator rows as a separate reflected ledger. + # A(new) annihilating B(old) states new^perp <= old^perp exactly. + reflected_pairs=0;endpoint_frames={f for pair in pairs for f in pair};nondeg_bases=set() + for f in endpoint_frames: + assert len(C.B[f])==C.dimf[f] and len(C.A[f])+C.dimf[f]==24 + key=tuple(map(tuple,C.B[f])) + if key not in nondeg_bases: + assert C.nondeg(f),'singular required endpoint frame' + nondeg_bases.add(key) + for oldf,newf in sorted(pairs): + assert len(C.A[newf])<=len(C.A[oldf]) + assert all(W.module.dot(a,b)==0 for a in C.A[newf]for b in C.B[oldf]) + reflected_pairs+=1 + # Validate the actual transformed instruction stream against its own frames. + state=dict(initial);copy=None;used=set(initial.values());hist=Counter();cat=Counter();coeff=Counter();centers=[];count=0 + scalar=hashlib.sha256();tagged=hashlib.sha256();category_names=oldmeta['category_names'] + def event(d,row):d.update(json.dumps(row,separators=(',',':')).encode());d.update(b'\n') + for k in range(0,len(out),6): + op,a,b,c,f,z=out[k:k+6] + if op==0: + assert state[a]==b and C.sub(b,c) and C.dimf[c]-C.dimf[b]==f + # Complement reflection reverses containment and preserves the gap. + assert (24-C.dimf[b])-(24-C.dimf[c])==f + state[a]=c;used.update((b,c)) + if f:hist[f]+=1 + elif op==1: + assert c%2 and state[a]==state[b]==f and a!=b + kind=category_names[z];source=copy['source']if b==n else b + event(scalar,[a,source,c]);event(tagged,[a,b,c,f,kind]);count+=1;cat[kind]+=1;coeff[abs(c)]+=1;used.add(f) + elif op==2: + assert copy is None and b==n and state[a]==c and f==ZERO and z==22 + state[b]=f;hist[z]+=1;copy=dict(source=a,frame=c,rank=z,temporary=b,first_event=count,copy_output_frame=f) + else: + assert op==3 and copy is not None and (a,b,c)==(copy['source'],copy['temporary'],copy['frame']) and state[b]==f==ZERO + copy.update(after_event=count,scatter_reads=count-copy['first_event']);assert copy['scatter_reads']==220 + centers.append(copy);copy=None;del state[b] + assert copy is None and state==final and hist==census + assert count==786621 and coeff=={1:785301,3:1320} and len(centers)==24 + delta=Counter(hist);delta.subtract(Counter({int(r):c for r,c in oldmeta['paid_histogram'].items()}));delta={r:c for r,c in delta.items()if c} + assert delta=={8:-3,12:-3,20:3} + sourceH=Counter();targetH=Counter();internalH=Counter(copies) + for i,H in per_role.items():(sourceH if i0,'row reserve gap') + need(2*(dim*(dim-1)+3*dim+dim+3*dim*(dim-1)//2+dim+1)+2*dim0,'factor shape';product*=p**exponent + assert product==abs(determinant),'factor identity' + +def run(context,physical,progress=lambda text:None): + started=time.monotonic();w,c=context['W'],context['C'] + p=HERE/'prime_witnesses.py';spec=importlib.util.spec_from_file_location('source527_exact_prime_validator',p);module=importlib.util.module_from_spec(spec);spec.loader.exec_module(module) + required=set(w.opframe)|set(w.w['source_frame'])|set(w.w['root_frame'])|{w.w['full_frame'],w.register([])} + required.update(g['frame']for g in w.gauge.values()) + for e in w.k['entries']:required.update((e['mix_frame'],e['deliver_frame'])) + for t in range(w.v):required.add(w.register(w.module.kernel([c.cov[t]],w.h)[0])) + required.update(g['frame']for g in context['agg']+context['rankgroups']+context['echelon']) + required.update(r['mix_frame']for r in context['pair_selection']) + used=set();groups={} + for row in physical['used_frames']: + f=row['frame_id'];assert f not in used;used.add(f) + h=basis_hash(c.B[f]);assert h==row['basis_sha256'] and len(c.B[f])==c.dimf[f]==row['dimension'],'fresh physical basis identity' + groups.setdefault(h,f) + # Retiming may remove a producer frame from the executed word. Retain its + # determinant obligation as well as every fresh physically consumed basis. + actual_basis_hashes=set(groups) + for f in required:groups.setdefault(basis_hash(c.B[f]),f) + assert actual_basis_hashes<=set(groups) + assert {basis_hash(c.B[f])for f in required}<=set(groups),'base and added transform coverage' + determinants={};largest=0;prime_factors=set();witnesses={} + for j,(h,f)in enumerate(sorted(groups.items())): + B=c.B[f];s=list(map(sum,B)) + gram=[[9*sum(a*b for a,b in zip(x,y))-s[i]*s[k]for k,y in enumerate(B)]for i,x in enumerate(B)] + det=module.det(gram);powers,residual=module.factor_witness(det) + row=dict(dimension=len(B),determinant=det,residual=residual,factors={p:e for p,e in powers.items()if e}) + validate_record(row,len(B),det,module.PRIMES);prime_factors.update(map(int,row['factors']));witnesses[h]=row + determinants[h]=det;largest=max(largest,abs(det).bit_length()) + if j and j%5000==0:progress('Validated '+str(j)+' exact basis determinants') + ranks=Counter();bundle_bases=set() + for role,g in w.gauge.items(): + if role in w.donor or role in context['borrow']:continue + a=g['dim'];h=basis_hash(c.B[g['frame']]);d=determinants[h] + assert 0<5*a<120 and d!=0;ranks[a]+=1;bundle_bases.add(h) + assert abs(d**5)==abs(d)**5 + #770 actual chart IDs include repeated exact bases; prime coverage deduplicates bytes. + assert ranks=={1: 369, 2: 41, 3: 3, 4: 19, 5: 31, 6: 1, 8: 1, 9: 20, 10: 2, 11: 14, 12: 30, 13: 55, 14: 8, 15: 4, 17: 7, 18: 22, 20: 2200}and len(bundle_bases)==459 + assert 2*120**3*10**16<2**80 + from copy import deepcopy + controls=[] + def reject(name,call): + try:call() + except AssertionError:controls.append(name) + else:raise AssertionError('invalid witness accepted: '+name) + first=next(iter(groups));row=witnesses[first];det=determinants[first] + missing=dict(witnesses);missing.pop(first) + def coverage(actual,wanted):assert set(actual)==set(wanted),'complete current basis coverage' + reject('missing current basis',lambda:coverage(groups,missing)) + changed=deepcopy(row);changed['determinant']=det+1;reject('incorrect determinant',lambda:validate_record(changed,row['dimension'],det,module.PRIMES)) + changed_factor=deepcopy(row);changed_factor['residual']+=1;reject('incorrect factor identity',lambda:validate_record(changed_factor,row['dimension'],det,module.PRIMES)) + singular=deepcopy(row);singular['determinant']=0;reject('singular basis',lambda:validate_record(singular,row['dimension'],module.det([[1,2],[1,2]]),module.PRIMES)) + huge=deepcopy(row);huge['residual']=2**80;reject('residual at prime lower bound',lambda:validate_record(huge,row['dimension'],det,module.PRIMES)) + noninteger=deepcopy(row);noninteger['residual']=float(row['residual']);reject('noninteger residual',lambda:validate_record(noninteger,row['dimension'],det,module.PRIMES)) + return dict(status='PASS_FRESH_ALL_ACTUAL527_BASIS_DETERMINANTS_AND_FIVEFOLD_BUNDLES',unique_bases=len(groups),current_frame_ids=len(used),basis_inventory_sha256=hashlib.sha256(json.dumps(sorted(groups),separators=(',',':')).encode()).hexdigest(),maximum_determinant_bits=largest,prime_factors=sorted(prime_factors),all_remaining_factors_below_2_power_80=True,independent_entrances=sum(ranks.values()),bundled_unique_bases=len(bundle_bases),entrance_rank_counts=dict(ranks),controls=controls,physical_inventory_bound=True,seconds=time.monotonic()-started,scope='Every actual physical basis is freshly evaluated over exact integers; all527 source frames, target transformations and caps are covered. Five disjoint copies inherit nondegeneracy; compiler and prime supply remain theorem dependencies.') diff --git a/research/multicut-delayed-inverse-checkpoint/prime_witnesses.py b/research/multicut-delayed-inverse-checkpoint/prime_witnesses.py new file mode 100644 index 0000000000..7f16d93030 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/prime_witnesses.py @@ -0,0 +1,99 @@ +"""Exact finite prime exclusions for the selected new rational bit frames. + +For integer basis B and G=I-J/9, det(9 B G B^t) is a nonzero integer. +Excluding its prime divisors and 2,3,5 makes the reduced frame nondegenerate +over every odd local ring Z/q^w. Projector denominators divide this determinant +times fixed powers of 3. Retained frames retain their old exclusion witnesses. +The selected q is an arbitrarily large fixed prime chosen after this finite +set; this is not verification at a preselected numeric q. +""" +from pathlib import Path +import argparse,gzip,hashlib,json,sys + +HERE=Path(__file__).resolve().parent + +def det(A): + A=[list(r) for r in A];n=len(A) + if not n:return 1 + sign=1;previous=1 + for k in range(n-1): + pivot=next((i for i in range(k,n) if A[i][k]),None) + if pivot is None:return 0 + if pivot!=k:A[k],A[pivot]=A[pivot],A[k];sign=-sign + p=A[k][k] + for i in range(k+1,n): + for j in range(k+1,n): + v=A[i][j]*p-A[i][k]*A[k][j] + assert v%previous==0 + A[i][j]=v//previous + A[i][k]=0 + previous=p + return sign*A[-1][-1] + +PRIMES=(2,3,5,7,11,13,17,19,23,29,31) + +def factor_witness(determinant): + assert type(determinant) is int and determinant != 0 + residual=abs(determinant);powers={} + for prime in PRIMES: + power=0 + while residual%prime==0:residual//=prime;power+=1 + powers[str(prime)]=power + validate_factor(determinant,powers,residual) + return powers,residual + +def validate_factor(determinant,powers,residual): + assert type(determinant) is int and determinant != 0 + assert set(powers)==set(map(str,PRIMES)) + assert all(type(x) is int and x>=0 for x in powers.values()) + assert type(residual) is int and 02^80 before the input widths +n,w are chosen. All represented determinant and route exclusions lie in +{2,3,5,7}. The primitive constant C may depend on q, the fixed source recipe, +the finite bootstrap level and the retained primitive implementations. It +does not depend on n,w, ancestor values or unrelated spectator widths except +through logical volume V. + +The three byte-pinned public sources supplied here are +`three-stage-cover-bit.tex`, `three-stage-cover-rows.tex`, and +`copied-centers-lemma.tex`. Their exact-tape, completed ordinary-leaf and +complete-stream hypotheses are retained. All high matrix fibers are batched; +prefix descriptor preparation has fixed polynomial cost in w and is paid +over the complete q^w target fiber. Unrelated spectators use ripple counters. +Fresh child role fields never enter the ancestor weight descriptor. Finite +low-type tables, initialization and sequential scans are charged in C. +There is no unit-cost random-access oracle or new numerical Turing-machine +constant requirement. + +## Historical source471 finite recipe + +Let m=120, W=23,683, E=495,304, v=1,760, R=16,643, d=m^2 and +J=20v+10R+4v=208,670. The last 4v=7,040 families pay the explicit final +complete-stream exchange; they are not equal-frame payload shears. + +The check derives from the fresh word 12,858,690 weighted additions or +21,787,770 unit additions, including all six bridge families. Its +cancellation-free prefix lift is +64*37,623^3*9,430,154^2=303,094,329,841,208,644,962,256,113,408<2^98. +No old defining-integer endpoint is imposed on the F2 supplier. + +The actual rational 48-dimensional route is reconstructed, eliminated and +inversely replayed: 195 additions, 44 swaps and 27 scalings, with denominator +set {1,2,3,6}. A missing addition fails reconstruction. Coordinate +permutations and embedding give the other ports and stages. + +Per route, pay at most 23,707*|GL_120(F_q)| low-role transpositions and +16*(d+1)^2 high affine factors. Per good edge, pay at most 8m^2+8 elementary +normalizer wrappers. The same compiler pays all exterior completions. +There are 120 copied-center episodes. Their recursive rank-22 calls and +their scalar reads are already included in E and the scalar count; their +copy/erase and parking passes are separately paid by the complete-stream +primitive, using one additional work stream. + +For a bad edge with N=2m=240, the exact inherited Gaussian fallback uses at +most 6N(N-1)+3N+6(N-1)=346,314<32m^2=460,800 children. Keep the entire +larger allowance in the moment. Coefficient preparation uses only unit +pivot inversions. Allocate 128N^3 ring operations per edge per low class: +four 8N^3 elimination/inversion allocations and sixteen 2N^3 matrix products +give 64N^3; the remaining 64N^3 pays extraction, copying, generic tests, +factor output and the bad-branch elimination. Matrix bit costs and descriptor +arithmetic are paid by the inherited primitive theorem. Invertibility testing +of each candidate low matrix additionally gets 16m^3 operations. + +Overcharging every finite recipe item over every low matrix gives + + B(q) <= [unit+J*16(d+1)^2+J*(W+24)+E*(8m^2+8) + +E*128(2m)^3+16m^3+120+E+1]*q^14400 + = 1,568,901,757,016,837*q^14400 < q^14401. + +These are explicit ring/recipe counts, not single-step machine costs. All +fixed-tape, table, setup and selector constants stay in C_primitive, giving +the normalized toll C*n+C*w^(1-a_old) after increasing one finite C. +No recursive child is absorbed into this nonrecursive constant. + +## Weighted induction and rows + +Set c=704197288590873/10^18 and tau=1-c. Exact interval recomputation, +adding the entire 10^-16*32m^2 fallback envelope per edge, proves + + 1-mu_tau > 774/10^21, + 1-mu_1 = 107421874442783/69383789062500000 > 0. + +For the retained recurrence +T(n,w)<=sum_r c_r*T(r*floor(n/120),w)+C*n+C*w^(1-a_old), +take A=C*(1+1/(1-mu_1)+1/(1-mu_tau)). Induction bounds T by +A*(n+n^tau*w^(1-a_old)): the linear and stopped gaps separately pay the +two tolls; the leaves follow A>=C. Floors only decrease the child sizes. + +The maximum ratio is 5/6, so four levels halve. The depth bound +4*ceil(log2(max(2,n))) changes stock, not the one-level moment. Paying all +24 temporary roles gives per-level log_q stock below 14400w+1. Add the +inherited old-leaf reserve. The new contribution is at most +57,604w*ceil(log2(max(2,n))). Borrow and restore these complete high digits; +successive residuals and the two-call ordinary wrapper reuse the reserve. +One outer row padding factor is below two; binary-radix padding costs less +than q^2 in logical volume. The temporary roles never enter the denominator W. + +## Finite ordinary bootstrap and cutoff + +Starting with completed ordinary a0=384599/10^10, set beta=c and repeat +a_(j+1)=(1-c)c+c*a_j exactly three times. The finite result is + + a3=704197288358392787581160456097840009769165126434639147437106139267359/10^72. + +At each stage, the paid exponents are 1-c, 1-a_(j+1), c with a logarithm, +and c(1-a_j). The exact positive gaps are c-a_(j+1), +1-a_(j+1)-c, 1-a_(j+1)-c(1-a_j), and 1-c (for row fitting). + +Choose a finite integer C>=1 dominating that stage's already fixed primitive, +recurrence, wrapper, row, setup, padding and previous ordinary constants. +Let delta be the minimum positive gap and + + L=ceil(max(1,36/delta^2,2*(4+ceil(log2 C))/delta)), e0=2^L. + +For e>=e0, C(1+log2 e)e^(-delta)<=1/16. Indeed for t=log2 e, +log2(1+t)<=3sqrt(t)<=delta*t/2 and log2 C+4<=delta*t/2. +Thus every lower-order toll is covered and the borrowed high part fits. +Below e0 use the previously completed ordinary algorithm. A finite next +constant is C_large+C_old*2^(L*(a_(j+1)-a_j)). This choice is sequential, +not circular, and uniform in the input and complete spectators. + +This proves the finite-accounting and ordinary-bootstrap step at the same +conditional theorem level as the retained public baseline. It does not claim +an independent proof of those inherited primitives, a measured universal +machine constant, or execution at every input width. diff --git a/research/multicut-delayed-inverse-checkpoint/proof/INTEGRATED_PROOF.md b/research/multicut-delayed-inverse-checkpoint/proof/INTEGRATED_PROOF.md new file mode 100644 index 0000000000..ce8aa9567d --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/proof/INTEGRATED_PROOF.md @@ -0,0 +1,91 @@ +# Historical PR234 five-stage composition + +This document preserves the predecessor source471 five-stage composition. +Its source parameters, numerical ledgers, hashes and exponent are historical. +The current construction and its mandatory executable checks are described in +[PROOF.md](../PROOF.md). Its current pinned source closure is recorded in +[PR259-PROVENANCE.md](../PR259-PROVENANCE.md) and +[loader/SOURCE_PINS.json](../loader/SOURCE_PINS.json). + +## 1. Meaning and evidence boundary + +The argument is a conjunction of constructive source theorems and finite +source-bound checks. A numerical moment or a PASS string does not substitute +for source correctness. The current verifier reconstructs the selected word, +geometry, scalar controls, primes, banks, complex supplier, arithmetic and +finite bill, while preserving the stated inherited theorem dependencies. + +## 3. Local bit helper and global lowering + +The source is PR210 df95878d11190518e45ef9717c9ee05011f88ace. The independently recorded local scalar word has 2,569,626 weighted events. Its projection is fixed by SHA256 e74e43692685ad5f716b41b363acb8b443c2c90096634a55f0e3728980713944. Both literal forward and signed inverse pass all 20,163 F2 columns, including arbitrary targets and 16,643 independent dirty helpers. The 471 borrowed sources are source aliases, never extra independent dirty coordinates. + +The tagged physical word has digest 614f19da4288021cd5c10a575ec7759c069962a6030bf30edb047573ed76edf6 and realizes the telescope Dout M Din^-1. Its input frames are X:Pq, Y:0, Z:sigma and output frames X:H, Y:H-Pq, Z:H. Thus a completed invocation has no data/helper cross blocks and each independent helper residual is D_(H-sigma). This is a full arbitrary-input operator identity, not just zero-dirt testing. + +The five-stage compiler maps every local live index through the stage owner injection, maps every local temporary to external family 23,683, and applies the stage's data or helper cover bijection. The active source/target pairs are (X1,Y1), (Y1,X1), (X1,Y1), (Y2,X2), (X2,Y2). The unused pair stays parked. A data route is right multiplication by the exact rational rho_i,S; a helper route is the whole-block exchange tau_i. Both are bijections of the eligible GL cover. They are charged address movements, not dense payload arithmetic. + +Cover expansion is strictly phase-major: complete every cover class in stage i before starting the next routed stage or boundary phase. The same rule applies separately to idle, bridge, completion and terminal-exchange phases. Running all five stages for one class before proceeding to another class would be incorrect because rho and tau refer to other classes. The callable stage and boundary iterators specify the phases; a flattened one-class-per-phase template is not a class-major execution schedule. The admission contract rejects that alternative loop ordering. + +For backward stages the compiler reverses the complete local event order, negates scalar coefficients, and replaces local P by H-P. A copied-center block is lowered afresh: copy the original at H-U, transform the copy to H, perform reverse signed scatter, and erase it. It is forbidden to invert an erased temporary. Every original center also retains its complementary live climb. Copy/live index injectivity is checked for all stage orientations. + +The global lowering artifact must emit bridges, idle children, independent gauge completions, and the final signed bank exchange. That exchange is an explicitly paid complete-stream role permutation, not a same-frame shear. A stage-descriptor string alone does not meet this obligation. + +## 4. Rational five-window geometry and prime eligibility + +Take H=Q^24, G=I-J/9 and a triple line q*. Its projector has denominator 6; coordinate permutations supply the port isometries. The two-block involution rho_i,S fixes the two port lines and exchanges their complementary spaces. With Q_i the direct sum of the first i copied complements, the frame is D_[g rho_i,S(P direct_sum Q_i)rho_i,S^-1 g^-1]. The four elementary two-block projector identities imply all eight bank boundary equalities. Rational complements are I-P. + +The exact rational partial swap is D_P=[[I-P,P],[P,I-P]]. It is involutory. Every same-frame scalar gate telescopes exactly, all idle transitions are nested actual projectors, and D_I D_Pq=D_(I-Pq) gives the terminal all-data address endpoint after the scalar exchange. No Pauli or Fourier phase claim is imported from the complex compiler into this bit proof. + +For each independent physical helper, the five routed residuals occupy different H_i. Their supported projectors have pairwise zero products. Their product is D_(I-sum_i sigma_i), so one paid child D_(sum_i sigma_i) completes D_I. This is one correction for each helper, not a packing of unrelated dirty rows. The 2,279 independent entrances have ranks {12:18,13:48,18:13,20:2200}; the 31 source-owned gauge entrances receive no extra completion. + +All 26,750 used physical frame IDs (26,380 distinct bases) have exact determinant witnesses. Their factors lie in {2,3,5,7}; the 231 distinct gauge-bundle bases use {2,3,5}. Window-map denominators also use {2,3,5}. Hence every retained q>2^80 preserves these ranks. General GL charts use gPg^-1, not a recomputed Euclidean orthogonal projection. + +The weighted generic normalizer allows every proper rank, including 60,65,90,100 in dimension120. Its cleared pivot-polynomial degree is at most m r(r+1)/2 < m^3. Thus the bad fraction is at most2m^3/q<10^-16. Each bad edge retains the full invertible local-ring residual and is paid 32m^2=460,800 rank-one fallback children. No field elimination at nonunit pivots is substituted. + +## 5. Exact bit ledger and ordinary finite recurrence + +One invocation has 95,789 positive-rank children and rank mass435,846. Its24 rank22 center copies are included once, as are each original center's live climbs. With H this histogram, + +n5 = 5H + 3520(delta4+delta23+delta46+delta50) + +18delta60+48delta65+13delta90+2200delta100. + +Thus m=120, W=23,683, calls495,304, mass2,837,560, deficit4,400. Work streams do not enlarge the live moment denominator. Five invocations retain all120 rank22 centers. + +The counted finite-bit adapter pays the literal scalar word, six bridge families, weighted good-edge wrappers, bad-class residuals, copy/read/erase, class routing, selector arithmetic, and fixed-tape movement. Its retained primitive theorem fixes the eligible prime and finite source recipe before varying width, so its finite constant may depend on that prime and recipe. It cannot depend on width, ancestor payload values or unrelated spectators except through complete volume. This is the correct quantifier order; an invented numerical Turing-machine constant is unnecessary. + +The largest child ratio is5/6. Three levels do not halve and four do: (5/6)^4=625/1296<1/2. The same one-level weighted moment induction remains valid; the factor4 is used only in depth/descriptor/borrowed-row bounds. Even paying24 simultaneous temporaries gives physical stock23,707 per GL cover vertex and log_q Wphysical(w)<14400w+1. The resulting internal log_q row charge is at most4ceil(log2(max(2,n)))(14400w+1), plus the old-leaf reserve. High digits are borrowed and restored internally; this is not a new external q-adic group factor. + +The defining-integer lift need not equal the older supplier's decoder endpoint. A safe cancellation-free prefix bound is64*37623^3*9430154^2<2^98. It uses neither signed integer restoration nor commutation between stage-dependent decoder channels. F2 payload exactness does not require complex floating precision. + +Let c=704197288590873/10^18. The freshly enclosed full bit moment, including all bad fallback charges, has a strictly positive gap. Start from the retained completed ordinary leaf a0=384599/10^10. Three finite levels with atom exponent beta=c give + +a_(j+1)=(1-c)c+c a_j, +a3=.000704197288358392787581160456097840009769165126434639147437106139267359. + +At every level a_j32mB^2. The complex half-shrink requirement is preserved:2*46<110. + +The physical row overcharge16,171 remains below20,161. Degree1,000,000 and suffix slope4,000,000 retain exact row gap23,971,789/25. These constants come from the changed stock and counted program, not from simply replacing72 by110 in the old guard. + +## 7. Fresh outer47 arithmetic and completion boundary + +Use complex saving747454944651775/10^18, strictly inside its freshly re-enclosed moment. The bit saving is the exact finite a3 above. Both moment gaps exceed7.7e-19. With eta=10^-12 and stopping fraction10^-9, the unchanged retained balanced assembly produces47 positive strict constraints and7 margins. + +The isolated candidate is kappa=703701743496697/10^18, or.000703701743496697. The minimum outer margin exceeds it by approximately7.3899e-19. This is the predecessor arithmetic value, not a current frontier comparison. Exact rational gaps, rather than printed decimals, are the evidence. + +These historical arguments supply the inherited five-stage, geometry, complex, +finite and outer interfaces. Current execution status, retained numerical pins +and the exact expected certificate are documented in [README.md](../README.md). +The current checks do not independently reprove the inherited all-size +compiler, common-chart, restored-row, routing, prime-supply, recovery and +complex-symbolic/analytic theorems. diff --git a/research/multicut-delayed-inverse-checkpoint/proof/PARITY-CONTRACT.md b/research/multicut-delayed-inverse-checkpoint/proof/PARITY-CONTRACT.md new file mode 100644 index 0000000000..fad6150b1f --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/proof/PARITY-CONTRACT.md @@ -0,0 +1,45 @@ +# Parity-even payload gate removal: inherited contract audit + +Conclusion: removing an even-coefficient pointwise payload ADD is valid for the bit supplier. The odd-prime address arithmetic remains unchanged. A new paid profile still requires the actual filtered word and full frame schedule; this conclusion alone does not certify a new kappa value. + +## Primary source evidence + +The bundled, unmodified primary section [`primary/03-motifs.tex`](primary/03-motifs.tex) states the following. Its exact source, attribution, and file hashes are recorded in [`primary/PROVENANCE.md`](primary/PROVENANCE.md). + +* Line50: in the bit construction scalars lie in F2. +* Lines149–177: replace a scalar by an array in R^Omega, with R the scalar ring of the network. Common R-linear frames give the exact operator `D_out S_Omega D_in^-1`. +* Around line223: the scalar domain need not equal the label field. Bit labels are rational subspaces even though the payload scalars are binary. +* Lines483–493: rational-address frames are F2-linear permutations of array entries; the bit network uses pointwise XOR gates. + +The byte-pinned [`copied-centers-lemma.tex`](copied-centers-lemma.tex) in this package opens with its scalar restoration identity explicitly over F2. Its copy contract preserves the complete role stream, controls and spectators; each copy's frame change and erasure remain paid. + +The retained PR234 [`INTEGRATED_PROOF.md`](INTEGRATED_PROOF.md), line56, explicitly says: "The defining-integer lift need not equal the older supplier's decoder endpoint" and the majorant "uses neither signed integer restoration nor commutation between stage-dependent decoder channels." The package's signed integer decoder checks are stronger supplier evidence; the inherited bit interface does not impose their particular integer endpoint on a newly synthesized F2-equivalent word. + +The retained weighted bit theorem, [`three-stage-cover-bit.tex`](three-stage-cover-bit.tex), uses the odd-prime local ring Z/q^w for address fields and weighted address interchanges. Its rational projectors, unit-pivot tests, route matrices and fallback coefficients are address geometry. They are not the F2 payload ADD coefficients. + +## Operator proof + +Fix the finite complete address set Omega and let one stream be a vector in F2^Omega. A payload gate is `x_a <- x_a + (c mod2) x_b`. For even c this is the identity on every input, including arbitrary correlated dirty streams and spectators. Removing any number of these identity gates leaves the complete scalar operator S_Omega unchanged. Negative odd coefficients and coefficient3 also reduce to1, though the simplest experimental witness can retain their original signed integer representatives. + +Retain each original input/output frame and every remaining gate's original common frame. The common-frame identity then gives the same `D_out S_Omega D_in^-1` for the filtered word. In particular, every independent helper still has its required partial-swap residual and the existing completed-bank proof still applies. No integer identity of the new lift is needed to conclude this F2 operator identity. + +## Safe frame compression + +Deleting an ADD removes its operand-frame alignment requirement. For a given stream, collect successive MOVE operations between surviving uses. A use includes any surviving ADD, COPY read/write, ERASE or required phase/endpoint boundary. No use may be crossed merely because another stream is idle. + +Suppose the original local frames form a nested chain U0 <= U1 <= ... <= Uk. Their exact address changes telescope to `D_Uk D_U0^-1`. Their orthogonal increments are disjoint commuting projectors, so this is one allowed residual of rank dim(Uk)-dim(U0). In the reflected orientation the chain is the reversed complementary chain; the same endpoint difference and rank follow exactly. The ambient stage background cancels. Rebuild both ledgers from the actual surviving uses and actual basis inclusions; do not infer this from summed ranks alone. + +Keep every physical input/output family and its endpoint, all stage boundaries, all data routes/bridges, and every copied-center lifetime. For source527 the120 center copies in the five-stage construction have odd scatter coefficients, so parity filtering does not itself remove those episodes. The shared work stream cannot be used before COPY or after ERASE. A COPY is a barrier even if its original register has no intervening ADD. + +## Required finite admission + +1. Filter actual emitted ADD records by the literal coefficient modulo2; retain all odd-coefficient records in their original order. Check all20107 formal F2 source, target and dirty columns against the admitted shear. The expected local event counts before any further resynthesis are1,783,196 removed and786,621 retained. +2. Generate the actual complemented inverse schedule, with fresh reverse copies as before, and verify every formal F2 column again. Do not reverse an erasure. +3. Compute cancellation-free prefix bounds for the new literal signed forward/inverse projections. Recompute the five-stage bound and finite bill. Failure to reproduce the previous integer decoder is not itself a contract failure; a missing F2 dirty restoration is. +4. Regenerate exact monotone forward/reflected physical paths and the complete paid histogram. Every fused residual's actual endpoints must be checked. Keep all remaining recursive children, copied-center calls, fallback allowance, data idle edges and stock. +5. Rebind the completed banks and final global address lowering to the new word and tags. Their actual rational address projectors, odd-q routes and normalizers must not be reduced modulo2. +6. Rerun actual basis/prime coverage, finite ordinary-leaf assembly, both rational moment engines and all47 outer inequalities before claiming a result. + +For simple monotone MOVE fusion the paid rank mass and physical stock remain unchanged; combining positive increments can improve the concave moment while reducing the number of children. Any different deficit or stock must be explained by a separate exact endpoint change. The displayed bound is still conditional on the retained all-size compiler, selectors, common ancestor charts, restored rows, routing, prime and analytic interfaces. + +Prepared with OpenAI Codex assistance. This audit is independent of the physical parity-filter/fusion implementation owned by the complex research lane. diff --git a/research/multicut-delayed-inverse-checkpoint/proof/copied-centers-lemma.tex b/research/multicut-delayed-inverse-checkpoint/proof/copied-centers-lemma.tex new file mode 100644 index 0000000000..8933aaf508 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/proof/copied-centers-lemma.tex @@ -0,0 +1,103 @@ +\subsection{Scalar, frame and temporary-stream contracts} +The retained source-copy, reversible mixer and scatter maps satisfy +$JLV=I$. Over $\mathbb F_2$ the chronological word +\[ + L,\ J,\ L^{-1},\ V,\ L,\ J,\ L^{-1},\ V +\] +adds the input to the target and restores every arbitrary auxiliary. +Over Gaussian dyadics use its corresponding old-value subtraction and +new-value addition. A designated retained output has a distinct terminal +output-use carrier: a matching may feed that carrier, but cannot merge +two designated terminals or append a later middle-producer consumer. +Its remaining uses before cleanup are its scatter reads, all of which +leave its scalar value unchanged. + +Within one tensor invocation suppress the fixed future line and put +\[ + D_0=B\otimes F_h,\quad D_1=A\otimes F_h,\quad + A=B\mathbin{\perp}P,\quad D_U=D_0\mathbin{\perp}(P\otimes U), +\] +where $P$ and the suppressed future factor are one-dimensional and +$U$ is a nondegenerate retained-center space of dimension $r$. +Thus a local change of rank $d$ has ambient rank $d$. +Write $\mathcal F_Q$ for the address frame: it is the rational partial +swap $D_Q$ in the bit construction, and the binary phase $C_Q$ in the +complex construction. Changing a copy from frame $Q_1$ to $Q_2$ means +applying $\mathcal F_{Q_2}\mathcal F_{Q_1}^{-1}$ to its physical stream. +Orthogonal backgrounds cancel in this quotient. + +Copies use Paureel's retained copy/transform/read/discard interface +\cite{paureel,pr29}. A copy has precisely the original complete role +stream, including all control and spectator ranges, and volume $V/W$. +It is processed while inactive original streams are parked on the fixed +depth-first stack tapes. Copying, pointwise reads and erasure are linear +in that stream's volume. Sequential processing needs one additional +fixed work stream, not an independently varying row coordinate. + +\begin{lemma}[Copied retained-center schedule] +In either invocation orientation a designated retained center's +physical rank profile can be changed from $z^r+z^h$ to +$z^r+z^{h-r}$, with the same scalar action, original role endpoints, +and arbitrary-scratch restoration. The rank-$r$ transform on the copy +is included in the recursive call list. +\end{lemma} +\begin{proof} +In the forward orientation the original center has scalar value $w$ +at frame $D_U$ after the middle mixer. Its old sequence was +$D_U\to D_0\to D_1$: scatter at $D_0$, then cleanup at $D_1$, +with ranks $r,h$. +Keep the original at $D_U$, copy its physical stream, and transform +only the copy to $D_0$, paying rank $r$. Supply every scatter read +from that transformed copy while its data targets remain at their +unchanged common frame $D_0$, then erase the temporary. The reads +are exactly those of $w$ in the old schedule. Since they do not change +$w$, move the original directly from $D_U$ to $D_1$ before its first +cleanup incidence; this costs $h-r$. Every subsequent original gate +therefore receives the same scalar value at its required frame. + +In the reverse complementary orientation the old sequence was +$D_0\to D_1\to D_{U^\perp}$, with ranks $h,r$ and scatter at $D_1$. +After the preceding all-$D_0$ operations, advance the original directly +to $D_{U^\perp}=D_0\mathbin{\perp}(P\otimes U^\perp)$, paying +$h-r$. Copy it and transform the copy to $D_1$, paying $r$. +Use that copy for all early scatter reads, then erase it. The original +already has the exact frame and unchanged scalar value required by the +following inverse mixer. Terminal output-use separation ensures no +intervening producer consumer has been bypassed. + +For partial swaps either direction of the rank-$r$ change is the same +involution. For complex phases the forward copy uses $C_U^{-1}$, +the direct original growth uses $C_{U^\perp}$, and the reverse copy +uses $C_FC_{U^\perp}^{-1}=C_U$. These are the same charged ranks, +implemented by the retained forward/inverse residual interface. +All tensor backgrounds cancel as stated. Every original scalar +incidence, final source-copy cancellation and auxiliary source/sink +operator is unchanged. Thus the scalar identity and all endpoints +remain exact on arbitrary initial auxiliary values. +\end{proof} + +For the selected centers $r=h-1$, and there are $h$ distinct retained +output uses per invocation. Their old histogram has rank mass +$hR_h+2\ell_h$, where $\ell_h=h(h-1)$. The new histogram is exactly +\[ + H'_{h,1}=H_{h,1}+h,\qquad H'_{h,h}=H_{h,h}-h, + \qquad H'_{h,r}=H_{h,r}\quad(r\ne1,h), +\] +and hence $\sum_r rH'_{h,r}=hR_h+\ell_h$. +Every old rank-$(h-1)$ residual remains as a paid transform of a copy; +its projector and ordinary bit pivot profile are unchanged. Only the +full-rank original transition becomes the rank-one complement. +The physical role count $R_h$ is unchanged. + +For two factors, put $N=v_av_b$, $B_a=v_bR_a$, $B_b=v_aR_b$, +$W=2N+B_a+B_b$ and $L=v_b\ell_a+v_a\ell_b$. +All exterior, data-front and endpoint-correction calls remain in place. +The resulting complete rank mass is +\[ + s=Wm-N+L,\qquad Wm-s=N-L. +\] +This follows by subtracting exactly $L$ from the retained two-stage +physical mass $Wm-N+2L$. The $N$ rank-one endpoint-copy calls are +separate and remain fully charged. Because center copies retain the +same complete role-volume fraction $1/W$, the existing complete-row, +integer-width remainder and fixed-tape recurrence proofs apply. diff --git a/research/multicut-delayed-inverse-checkpoint/proof/primary/03-motifs.tex b/research/multicut-delayed-inverse-checkpoint/proof/primary/03-motifs.tex new file mode 100644 index 0000000000..cdb0ba527a --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/proof/primary/03-motifs.tex @@ -0,0 +1,700 @@ +\section{Two finite networks with a rank saving} +\label{sec:finite-networks} + +We construct two fixed linear networks that exchange two banks of values +while restoring every auxiliary value. When the values are replaced by +arrays, a common representation on all wires at a gate lets that gate act +pointwise without disturbing the representation. We explain this invariant +before constructing the subspace labels that make changes of representation +cheap. The first network uses bit values and rational subspaces; the second +uses Gaussian dyadic values and binary subspaces. The scalar domain specifies +the values combined at a gate, while the subspace labels are used to construct operators +on each wire's address-indexed array. + +Put +\[ + h=100,\qquad v=\binom h3=161700,\qquad + N=v^3=4227952113000000,\qquad m=h^3=1000000. +\] +Let $\mathcal T$ be the set of three-element subsets of $[h]$. +There are two data banks $X_a,Y_a$, indexed by +$a=(a_1,a_2,a_3)\in\mathcal T^3$. +A wire carries one scalar. A gate is a fixed linear map on a specified +set of wires; untouched wires retain their values. The vertices on a +wire are its source, the gates that touch it in time order, and its sink; +its edges join consecutive vertices. We call the name of a wire its +\emph{role}, so that we can distinguish a fixed place in the network +from the value currently carried there. + +\subsection{Scalar operations and restored auxiliary values} + +There are three stages. At stage $j$, fix the other two coordinates and +operate on the $v$ values in each bank obtained by varying coordinate $j$. Thus +there are $v^2$ separate invocations at each stage. +Each such invocation has its own auxiliary wires, distinct from all +other invocations, including those in other stages. There are +\[ + I=3v^2=78440670000 +\] +invocations. In a forward invocation, write $x_T$ for the data used as +controls and $y_S$ for the data to be updated; $S,T\in\mathcal T$ are +their varying $j$th coordinates. These are the logical source and +target banks of the invocation. For each ordered pair $(S,T)$ that is +neighboring under the rule below, there is an auxiliary wire $A_{ST}$. +There are also central wires $C_i$, $i\in[h]$, and in the second +construction a wire $C_\ast$. We call all these auxiliary wires +\emph{scratch wires}; their initial values may be arbitrary. + +The following maps specify the construction. +\begin{itemize} +\item In the bit construction, scalars lie in $\mathbb F_2$, and +$S,T$ are neighbors when $|S\cap T|=1$. +The copy map adds $x_T$ to $A_{ST}$ for each neighbor pair. +The side injection adds $\sum_{T:\,|S\cap T|=1}A_{ST}$ to $y_S$. +The gather adds $\sum_{T\ni i}x_T$ to $C_i$, and the scatter adds +$\sum_{i\in S}C_i$ to $y_S$. +\item In the complex construction, scalars lie in +$\mathbb Z[i,1/2]$, and neighbors are distinct triples with even +intersection. Copy is unchanged. Side injection adds +\[ + -\sum_{T\text{ neighboring }S}\frac{|S\cap T|-1}{2}A_{ST} +\] +to $y_S$. Gather adds $\sum_{T\ni i}x_T$ to $C_i$ and +$\sum_Tx_T$ to $C_\ast$. Scatter adds +$(\sum_{i\in S}C_i-C_\ast)/2$ to $y_S$. +\end{itemize} +Each update leaves its control wires unchanged. The first two rows of +the following schedule remove the effect of the old scratch values; +the next four introduce the source values and add their effect to the +target; the last two restore the scratch values. +This cancellation of arbitrary initial scratch values is a linear +instance of the restoration construction for transparent computation in +Buhrman, Cleve, Kouck\'y, Loff and Speelman~\cite[Section~3]{BuhrmanCleveKouckyLoffSpeelman2014}. +The forward schedule is +\begin{equation} +\begin{array}{c|l|l} +\text{row}&\text{operation}&\text{gate grouping}\\ \hline +0&\text{subtract side injection}&\text{one gate per target}\\ +1&\text{subtract scatter}&\text{one gate on targets and center}\\ +2&\text{copy}&\text{one gate per source}\\ +3&\text{gather}&\text{one gate on sources and center}\\ +4&\text{scatter}&\text{one gate on targets and center}\\ +5&\text{side injection}&\text{one gate per target}\\ +6&\text{undo gather}&\text{one gate on sources and center}\\ +7&\text{undo copy}&\text{one gate per source}. +\end{array} +\label{eq:motif-schedule} +\end{equation} + +\begin{lemma}\label{lem:scalar-motifs} +For arbitrary initial values of its auxiliary wires, a forward +invocation changes $y_S$ to $y_S+x_S$ and leaves every other value +unchanged. Performing a forward invocation from $X$ to $Y$ at stage 1, +the inverse forward schedule from $Y$ to $X$ at stage 2, and a forward +invocation from $X$ to $Y$ at stage 3 sends +\[ + (X_a,Y_a)\longmapsto(-Y_a,X_a) +\] +and restores every auxiliary value. In the bit construction this is the +permutation exchanging $X_a$ and $Y_a$. +\end{lemma} + +\begin{proof} +Write $\mathcal V$ for copy, $\mathcal G$ for gather, +$\mathcal J$ for side injection and $\mathcal R$ for scatter. +If the initial side and central vectors are $a,c$, the first two rows +subtract $\mathcal J a+\mathcal R c$ from $y$. Rows 2 and 3 change +them to $a+\mathcal Vx,c+\mathcal Gx$. Rows 4 and 5 therefore add +$\mathcal R(c+\mathcal Gx)+\mathcal J(a+\mathcal Vx)$. +Rows 6 and 7 restore $c,a$. The net target change is +$(\mathcal J\mathcal V+\mathcal R\mathcal G)x$, independently of +$a,c$. + +In the bit case, the central coefficient from $x_T$ to $y_S$ is +$|S\cap T|\bmod2$. The side contribution cancels exactly the +intersection-one cases; distinct triples with intersection zero or two +already have coefficient zero, and the self coefficient is one. +In the complex case, the central coefficient is $(|S\cap T|-1)/2$. +The side cancels it for distinct triples with intersection zero or two; +intersection one contributes zero, and the self coefficient is one. +Thus both net maps are $y\gets y+x$. + +The inverse schedule with logical source $Y$ and target $X$ effects +$X\gets X-Y$. Starting from $(x,y)$, the three stages give +$(x,y+x)$, then $(-y,y+x)$, then $(-y,x)$. +Each invocation restores its own auxiliary wires. +\end{proof} + +A fixed triple has +\[ + z_{\rm b}=3\binom{97}{2}=13968,\qquad + z_{\rm c}=\binom{97}{3}+3\cdot97=147731 +\] +neighbors in the two constructions, respectively. The second count +separates intersection zero and intersection two. Ordered pairs must +be counted: there are $vz$ side wires in each invocation, not $vz/2$. +With $c_{\rm b}=100$ and $c_{\rm c}=101$ central wires per invocation, +the total wire counts are +\begin{align} + W_{\rm b}&=2N+I(vz_{\rm b}+c_{\rm b}) + =177176569091445000000,\label{eq:bit-wire-count}\\ + W_{\rm c}&=2N+I(vz_{\rm c}+c_{\rm c}) + =1873807244643542670000.\label{eq:complex-wire-count} +\end{align} +All these numbers are fixed, independently of any eventual input length. + +\subsection{From scalar values to arrays} + +We record the representation principle before choosing the labels. Let +$R$ be the scalar ring of either network and let $\Omega$ be a finite address +set common to all wires. Replace each scalar by an array in $R^\Omega$, and +apply each scalar gate pointwise at every address. Write $\mathsf S_\Omega$ +for the resulting network on the vector of wire arrays. + +At each source, gate and sink $v$, assign an invertible $R$-linear operator +$D_v$ on $R^\Omega$, called its \emph{frame}. A frame acts on the values +indexed by addresses within one wire's array and may mix them; a gate instead +combines wire values at each fixed address. All incoming and outgoing +incidences of one gate use the same $D_v$. On an edge from $u$ to $v$, apply +$D_vD_u^{-1}$ to that wire's array. Define the operators on all input and +output roles by +\[ + D_{\rm in}=\bigoplus_w D_{\mathrm{source}(w)},\qquad + D_{\rm out}=\bigoplus_w D_{\mathrm{sink}(w)}. +\] +For a supplied physical input $f$, define its logical interpretation to be +$g=D_{\rm in}^{-1}f$; this definition requires no physical preprocessing. +At a vertex $v$, the invariant is that the stored array equals $D_v$ applied +to the logical array there. It holds at the sources by definition. On an +edge whose logical array is $\varphi$, the edge operator changes $D_u\varphi$ +to $D_v\varphi$. At a gate with logical input arrays $\varphi_j$, +$R$-linearity gives +$\sum_j c_jD_v\varphi_j=D_v(\sum_j c_j\varphi_j)$ for each output combination, with +$c_j\in R$. Thus the common frame commutes with the pointwise gate, and +the invariant continues to the sinks. The implemented operator is therefore +\begin{equation} + D_{\rm out}\,\mathsf S_\Omega\,D_{\rm in}^{-1}. + \label{eq:common-frame-identity} +\end{equation} +In particular, if the scalar network is a signed role permutation that +sends input role $w$ to output role $\rho(w)$ with scalar sign +$\varepsilon_w$, that route applies +$\varepsilon_wD_{\mathrm{sink}(\rho(w))} +D_{\mathrm{source}(w)}^{-1}$ to its input array. All roles, including +the auxiliary roles, enter this identity. Once the scalar routing is fixed, +the endpoint frames determine the array operator on each route; the +intermediate frames can then be chosen to make each edge change inexpensive. + +Figure~\ref{fig:frame-compatibility} summarizes this invariant. +\begin{figure}[ht] +\centering +\input{figures/proof-map.tex} +\caption{An edge changes the representation of one role's logical array +$g_w$. At the next gate $G$, all $t$ input arrays have the common frame +$D_v$, so the pointwise linear gate commutes with that frame. The same +$D_v$ acts on each output component.} +\label{fig:frame-compatibility} +\end{figure} + +\subsection{Subspaces attached to the gates} + +The scalar network already has the required signed exchange. We now +choose its intermediate labels to make the changes of frame inexpensive. +A label is a nondegenerate subspace of a fixed bilinear space. On every +edge one label will contain the other. For such an edge, the +\emph{orthogonal residual} is the orthogonal complement of the smaller label inside +the larger one; its dimension is the absolute change in label dimension. +The saving will come from keeping the total dimension lost on decreasing +edges small relative to $N$. + +For the bit network, let $F=\mathbb Q^h$ with bilinear form +\[ + \langle x,y\rangle=x^{\mathsf T}(I-J/9)y, +\] +where $J$ is the all-one matrix. This form has eigenvalues $1$ and +$1-h/9=-91/9$, so it is nondegenerate. For a triple $T$ let $t_T$ be +its indicator vector. Then +\[ + \langle t_S,t_T\rangle=|S\cap T|-1, + \qquad \langle t_T,t_T\rangle=2. +\] +For the complex network, take instead $F=\mathbb F_2^h$ with the +ordinary dot product; here $t_T\cdot t_T=1$. +Thus every triple line is nondegenerate, and neighboring triple lines +are orthogonal, in the label field belonging to either construction. +The scalar domain need not equal the label field. +These degree-one intersection formulas specialize the construction +described by Alon~\cite[Section~3, remark after Theorem~1.1]{Alon1998} +to the prime $2$, with the ground-set size changed to $h=100$. + +The ambient label space is $\mathcal F=F^{\otimes3}$ with its tensor +bilinear form. It has dimension $m$ and is nondegenerate. Write +\[ + u_a=t_{a_1}\otimes t_{a_2}\otimes t_{a_3},\qquad U_a=\langle u_a\rangle. +\] +The prescribed terminal labels are +\[ +\begin{array}{c|cc} +\text{role}&\text{source label}&\text{sink label}\\ \hline +X_a&U_a&\mathcal F\\ +Y_a&0&U_a^\perp\\ +\text{each scratch role}&0&\mathcal F. +\end{array} +\] +Write $W$ for the wire count in the construction under consideration. +The source dimensions sum to $N$ and the sink dimensions to $Wm-N$. +Thus the signed dimension changes will sum to $Wm-2N$, because the +contributions at gates cancel. If decreasing edges lose a total +dimension $L$, the sum of absolute changes is $Wm-2N+2L$. +We will obtain $L1$. +The same argument gives a unit vector in $Q^\perp$ when it is nonzero. +A triple complement, and the complement of two neighboring triple +lines, contain coordinate units outside supports of sizes at most +three and six, respectively. Full tensor spaces contain coordinate +units as well. In each nonzero tensor summand, tensoring these chosen +norm-one vectors gives a norm-one vector; placing it in that orthogonal +summand gives one in the whole residual. Each residual is nondegenerate +by the decompositions above. Thus every nonzero residual is nonalternating. + +For completeness, every nondegenerate nonalternating binary symmetric +space has an orthonormal basis. Split off unit lines until the +remaining space is alternating. A nonzero nondegenerate alternating +space splits into planes with bases $a,b$ satisfying +$a\cdot a=b\cdot b=0$, $a\cdot b=1$; this follows by choosing a +nonzero $a$, choosing $b$ with $a\cdot b=1$, and taking the orthogonal +complement of their nondegenerate plane. Retain one unit vector $w$ +from the lines already split off. A plane orthogonal to $w$ can be +absorbed into that line: the three vectors +\[ + w+a,\qquad w+b,\qquad w+a+b +\] +are independent, pairwise orthogonal, and have norm one. +Use one of these new unit vectors in place of $w$ for the next plane; +the remaining planes are still orthogonal to it. Repeating this +operation absorbs every alternating plane. +\end{proof} + +The loss ratios have the exact values +\[ + \frac{L_{\rm b}}{N}=\frac{100}{539},\qquad + \frac{L_{\rm c}}{N}=\frac{101}{539}. +\] +In particular, both are smaller than $1/2$. We now turn the dimension +count into the two interfaces needed later. + +\subsection{The rational matrix interface} + +For a nondegenerate rational subspace $U\subset\mathcal F$, write +$P_U$ for the projection onto $U$ along $U^\perp$. Assign $P_U$ to +each gate or sink with label $U$. Assign the source matrix $-P_{U_a}$ +to $X_a$ and zero to every other source. +These are fixed rational $m\times m$ matrices; write $M_v$ for the matrix +at vertex $v$. Define +$\rho(X_a)=Y_a$, $\rho(Y_a)=X_a$, and let $\rho$ fix every auxiliary +wire. + +In the tape application, two address chunks are written as vectors $H,D$ +of $m$ fields. For a matrix $M$, the frame $\Phi_M$ moves the array entry +at $(H,D)$ to $(H+MD,D)$, leaving the other address fields unchanged. +The fields use a radix coprime to the fixed denominators, and addition is +modulo their common range. Hence $\Phi_M$ is an invertible +$\mathbb F_2$-linear permutation of array entries, and +$\Phi_{M'}\Phi_M=\Phi_{M'+M}$. Using $D_v=\Phi_{M_v}$ +in \eqref{eq:common-frame-identity}, the endpoint matrix difference on each +routed role is exactly the shear applied to that role. The following identity +makes it the same shear $H\gets H+D$ on every role; +Section~\ref{sec:tape-interchange} turns that shear into a field interchange +and implements an edge change with one smaller interchange per unit of its +rational rank. + +\begin{proposition}\label{prop:bit-motif-interface} +The bit network has $W_{\rm b}$ wires, uses only pointwise XOR gates, +and sends its input vector to the permutation $\rho$ of that vector. +Its rational matrices are common to all incidences of a gate and obey +\[ + M_{\mathrm{sink}(\rho(w))}-M_{\mathrm{source}(w)}=I_m + \quad\text{for every input role }w. +\] +The sum of the rational ranks of all edge differences is +\begin{align} + s_{\rm b} + &=\sum_{e}\operatorname{rank}_{\mathbb Q} + (M_{\mathrm{head}(e)}-M_{\mathrm{tail}(e)})\\ + &=W_{\rm b}m-N+2L_{\rm b} + =1771765690887858612870000000$. + +For an input $X_a$ the endpoint difference is +$P_{U_a^\perp}-(-P_{U_a})=I_m$. Every other role has zero source +matrix and identity sink matrix. This proves the interface. +\end{proof} + +\subsection{The binary phase interface} + +Use the complex scalar network and binary labels. For a binary vector +$x$, let $\operatorname{wt}(x)$ denote its integer Hamming weight, and +write $[b]\in\{0,1\}$ for the integer representative of $b\in\mathbb F_2$. +For a label $U$, let +\[ + q_U(x)=\operatorname{wt}(P_Ux)\pmod4. +\] +The projection here is over $\mathbb F_2$. +Let +\[ + H=\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix},\quad + C=H\begin{pmatrix}1&0\\0&i\end{pmatrix}H + =aI+bX,\qquad a=\frac{1+i}{2},\quad b=\frac{1-i}{2}, +\] +where $X$ interchanges the two coordinates. On functions of +$x\in\mathbb F_2^m$ define +\[ + \mathcal C_U=H^{\otimes m}\operatorname{diag}_x(i^{q_U(x)})H^{\otimes m}. +\] +The use of $H$ in this definition is an identity of matrices. Each entry +of $\mathcal C_U$ and its inverse is a Gaussian integer divided by $2^m$, +so both preserve arrays over $\mathbb Z[i,1/2]$. The kernels implemented +below have Gaussian dyadic coefficients. + +To apply the scalar network to these functions, replace the scalar on +each wire $w$ by an array +$f_w:\mathbb F_2^m\to\mathbb Z[i,1/2]$. At each address, a gate +applies its scalar linear map to the values on the wires it touches. +The frame in \eqref{eq:common-frame-identity} is $D_v=\mathcal C_U$ at a +vertex with label $U$. On an edge from label $U$ to label $V$, apply +$\mathcal C_V\mathcal C_U^{-1}$ to that wire's array. +All frames are diagonalized by the same matrix $H^{\otimes m}$. +For an integer $k\ge1$, an address for $k$ columns is +$(x^{(1)},\ldots,x^{(k)})\in(\mathbb F_2^m)^k$; every frame and edge +operator is then the tensor product of its one-column version over +these $k$ coordinates. Each scalar gate is still applied once at each +joint address, to the vector of values on the roles it touches. + +\begin{proposition}\label{prop:complex-motif-interface} +For each edge of the complex network, write $U,V$ for its tail and +head labels. Its phase-frame difference +$\mathcal C_V\mathcal C_U^{-1}$ is a product of one translation +kernel $aI+bX_v$, or its inverse, per vector in an orthonormal basis +of the edge residual. Here $(X_vf)(x)=f(x+v)$. +The sum of these basis lengths is +\[ + s_{\rm c}=W_{\rm c}m-2N+2L_{\rm c} + =18738072446366712673080000000$. + +For an edge $e$, write $\mathcal B_e$ for its residual basis and +$K_{e,v}$ for the forward or inverse translation kernel associated +with $v\in\mathcal B_e$. Taking tensor products preserves their +product, so on $k$ columns +\[ + \bigl(\mathcal C_V\mathcal C_U^{-1}\bigr)^{\otimes k} + =\prod_{v\in\mathcal B_e}K_{e,v}^{\otimes k}. +\] +Thus each residual vector contributes one factor acting on all $k$ +columns, and there are $s_{\rm c}$ such factors over the network. + +Let $\rho$ denote the underlying permutation of the complex roles: it +exchanges $X_a,Y_a$ and fixes each scratch role. +By Lemma~\ref{lem:scalar-motifs}, the pointwise scalar network sends an input array at +$X_a$ to $Y_a$ with sign $+1$, an input at $Y_a$ to $X_a$ with sign +$-1$, and each scratch input to its own output with sign $+1$. +Applying \eqref{eq:common-frame-identity} with $D_v=\mathcal C_{U_v}$, +where $U_v$ is the label at vertex $v$, shows that an input role $w$ reaches +output role $\rho(w)$ with that sign and with operator +$\mathcal C_{U_{\mathrm{sink}(\rho(w))}} + \mathcal C_{U_{\mathrm{source}(w)}}^{-1}$. +For $k$ columns, use $D_v=\mathcal C_{U_v}^{\otimes k}$ on the joint +address set in the same identity. The route operator is then the $k$th +tensor power of the displayed address operator, multiplied by its scalar +route sign once. +The zero label has frame $I$, while the full label has frame +$C^{\otimes m}$ because $q_{\mathcal F}(x)=\operatorname{wt}(x)$. +Every pair except $X_a\to Y_a$ has these zero and full labels. +For the remaining pair, write $u=u_a$. Its binary norm is one and +its weight is $3^3=27$. Orthogonal weight additivity yields +\begin{align*} + q_{U_a^\perp}(x)-q_{U_a}(x) + &=\operatorname{wt}(x)-2\operatorname{wt}(u)[u\cdot x]\\ + &=\operatorname{wt}(x)+2[u\cdot x]\pmod4. +\end{align*} +Since $2[u\cdot x]\equiv2\sum_{j\in\operatorname{supp}(u)}x_j +\pmod4$, the operator on $X_a\to Y_a$ is the tensor of $C^{-1}$ on the +27 coordinates of $\operatorname{supp}(u)$ and $C$ elsewhere. +Put $Z=\operatorname{diag}(1,-1)$. Direct multiplication gives +\[ + C=iZC^{-1}Z,\qquad C^{-1}=-iZCZ. +\] +For $k$ columns, let $Z_{a,k}$ be the tensor product of $Z$ on +those 27 coordinates in every column and the identity elsewhere. +Apply $Z_{a,k}$ before the network at $X_a$, and apply +$i^{27k}Z_{a,k}$ after the network at $Y_a$; the latter scalar is +applied once to the whole role. There are $27k$ inverse factors, so +the identity $C=iZC^{-1}Z$ turns each of them into $C$. The operator +on this pair is therefore $C^{\otimes mk}$ as well. +Only the physical $X$ outputs still carry the minus sign of the +scalar exchange. Negate those outputs and send output role $\rho(w)$ +back to role $w$. Every role, including every scratch role, now has +the required forward tensor product. +\end{proof} + +\subsection{Explicit rational bounds for the two recurrences} + +The relative rank deficits are +\[ + \eta_{\rm b}=\frac{W_{\rm b}m-s_{\rm b}}{W_{\rm b}m} + =\frac{339}{22587335000000},\qquad + \eta_{\rm c}=\frac{W_{\rm c}m-s_{\rm c}}{W_{\rm c}m} + =\frac{73}{19906842167500}. +\] +For both recurrences we may use the fixed rational exponent +\begin{equation} + \tau=\sigma=1-2^{-50}. + \label{eq:explicit-motif-exponents} +\end{equation} +Both exact deficits displayed above are greater than $20\cdot2^{-50}$. +Since $m<2^{20}$ and $\log 2<1$, we have $\log m<20$, whence +\[ + m^{1-2^{-50}} + =m\exp(-2^{-50}\log m) + >m(1-20\cdot2^{-50})>m(1-\eta) +\] +for either $\eta=\eta_{\rm b}$ or $\eta=\eta_{\rm c}$. +Consequently +\[ + \frac{s_{\rm b}}{W_{\rm b}}2^{80}$ avoiding all fixed rational denominators and rank witnesses. +At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its +right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart +\[ + g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q), + \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2}, +\] +using fixed representatives for $g_0$. Right multiplication permutes $g_0$ +and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed +dimensional rational linear part factors into fixed scalings, ordered +shifts and a constant number of old width-$O(w)$ interchanges. Thus each +fixed port routing costs $O(Vw^{1-a_{\rm old}})$. + +Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$. +The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into +low role type, high matrix and unused stock. Batch \emph{all} high matrices +for each low type in one stream. A frame edge is one child on this whole +batch, of volume $V/W_0$, and routing is executed once per fixed operation +type. Consequently neither the number of tapes nor the number of child +programs grows with $w$. High cardinality cancels in the normalized moment, +but its $m^2(w-1)$ digits are charged in the row reserve. + +For each fixed projector require the leading minors of sizes $1$ through +$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero: +similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the +fixed rank is preserved modulo $q$. Their total degree is at most $m^3$. +The polynomial zero bound and conditioning on invertibility (a factor at +most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type. +No simultaneous union bound over edge types is needed, since their moments +are averaged separately. Good classes use the whole-rank child; bad +classes use the full $32m^2$-child fallback just proved. + +Let $H_r$ be the three-stage histogram described above and +$W=2v+3R_{\rm loc}=90140$. The retained ledger gives +\[ + \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell, + \qquad \max\{r:H_r>0\}=66. +\] +For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is +\[ + \mu_\tau\le + \sum_r\frac{H_r}{W}(r/m)^\tau+ + b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1. +\] +The entire fallback is added; no displaced good contribution is subtracted. +The exact script \path{scripts/three_stage_cover_network.py} and certificate +\path{certificates/three-stage-cover-network.json}, from the retained axis +\path{certificates/three-stage-cover-bit-input.json}, give +$1-\mu_\tau>6.3397366396\cdot10^{-7}$. +The certificate uses the 32-term positive atanh logarithm series with its +geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to +the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper +bounds, not floating-point root estimates. Since every child ratio is at +most one and $\tau<1$, this same strict bound implies $\mu_1<1$. + +With $T(n,w)$ denoting time divided by logical volume, the fixed low-type +profile $\{c_r\}$, including fallback classes, satisfies +\[ + T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}} + \quad(n\ge m),\qquad + T(n,w)\le Cw^{1-a_{\rm old}}\quad(na_{\rm bit}$ and $\beta<1-a_{\rm bit}$ make both +the atom-adapter and borrowing terms lower order. The remainder cost +$O(Vw^{1-a_{\rm old}})$ is also covered. Thus the ordinary radix-$q$ +interchange has cost $O(Ve^{1-a_{\rm bit}})$, without an externally supplied +superpolynomial stock of rows. + +For an original binary chunk range $[2^e]$, let +$t=\lceil e/\log_2q\rceil$ and embed it numerically into $[q^t]$. +Pad both exchanged ranges while retaining every spectator. The total +logical-volume increase is below $q^2$. Execute the completed ordinary +radix-$q$ interchange, then remove the padding. Its endpoint preserves the +rectangular genuine set $[2^e]\times[2^e]$; intermediate maps may mix genuine +and padded addresses, which does not affect this endpoint justification. +As $q$ is fixed and $t=\Theta(e)$, the binary interchange has the same +saving exponent. All schedules use a fixed number of finite-alphabet +tapes; low residue tables and port types are fixed independently of $w$. + +The conclusion retains the original local-producer, ordered-affine, +ordinary-leaf and multiplication-framework hypotheses. It adds no free +unit-scaling, dense chart transformation, or external row-stock primitive. diff --git a/research/multicut-delayed-inverse-checkpoint/raw_ledger.py b/research/multicut-delayed-inverse-checkpoint/raw_ledger.py new file mode 100644 index 0000000000..d0941cfd54 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/raw_ledger.py @@ -0,0 +1,243 @@ +"""Independent fresh source527 physical path census, including both reflections. +Prepared with substantial OpenAI Codex assistance; Apache-2.0. +No saved receipt, search script, or historical profile is consumed. +""" +from collections import Counter,defaultdict +from types import SimpleNamespace + +def run(context): + m=SimpleNamespace(**context);W=m.W;C=m.C;chosen=m.pair_selection;fresh=m.fresh;pair_byalias=m.pair_byalias;pair_bydirty=m.pair_bydirty + compact=lambda h:{str(k):v for k,v in sorted(h.items())if v} + zero=W.register([]);compact=lambda h:{str(k):v for k,v in sorted(h.items())if v} + # Exact raw event sequence, with every actual extra source included. + events=[] + for j,op in enumerate(W.rest): + for s in m.at[j]:events.append(dict(kind='gauge',ref=s,frame=W.gauge[s]['frame'],targets=list(m.adj[s]))) + if op in m.bywrite:events.append(dict(kind='terminalwrite',ref=op,frame=W.opframe[op],targets=[m.bywrite[op]['pivot']])) + if op in m.afterwrite:events.append(dict(kind='terminalread',ref=op,frame=m.afterwrite[op]['root_frame'],targets=m.afterwrite[op]['targets'])) + for s in m.at[len(W.rest)]:events.append(dict(kind='gauge',ref=s,frame=W.gauge[s]['frame'],targets=list(m.adj[s]))) + endgauge=len(events) + for j,r in enumerate(W.g['roots']): + if r['kind']=='side'and j not in m.deletedroots:events.append(dict(kind='root',ref=j,frame=W.w['root_frame'][j],targets=r['targets'])) + for e in m.deliveries[j]:events.append(dict(kind='K',ref=e['passive'],frame=e['deliver_frame'],targets=e['receivers'])) + for t in range(W.v):events.append(dict(kind='endpoint',ref=t,frame=W.register(W.module.kernel([C.cov[t]],W.h)[0]),targets=[t])) + + def histogram(t,path): + h=Counter();rev=Counter();assert path[0]==zero and C.dimf[path[-1]]==W.h-1 + for f in path:assert C.nondeg(f)and all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]),('cap/nondeg',t,f) + for a,b in zip(path,path[1:]): + assert C.sub(a,b),('forward inclusion',t,a,b) + d=C.dimf[b]-C.dimf[a];assert d>=0 + if d:h[d]+=1 + for a,b in zip(reversed(path),list(reversed(path))[1:]): + assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]),('reflected inclusion',t,a,b) + d=len(C.A[b])-len(C.A[a]);assert d>=0 + if d:rev[d]+=1 + assert h==rev and sum(k*v for k,v in h.items())==W.h-1 + return h + + def paths(agg,readframes,echelon,rankgroups): + rankowner={t:k for k,g in enumerate(rankgroups)for t in g['targets']} + out=[[zero]for _ in range(W.v)];owner={t:k for k,g in enumerate(agg)for t in g['targets']};roles=defaultdict(list) + for k,g in enumerate(agg): + for s in g['roles']:roles[s].append(k) + active=set(range(len(agg)));rankactive=set(range(len(rankgroups)));echactive=set(range(len(echelon)));seen=[set()for _ in echelon];restores=[] + for j,e in enumerate(events): + if j>=endgauge:assert not active and not rankactive and not echactive + s=e['ref'];T=set(e['targets']) + if e['kind']=='gauge': + if s in pair_byalias:continue + if s in fresh: + for t,c in fresh[s]['transformed_response'].items(): + if c:out[int(t)].append(W.gauge[s]['frame']) + continue + skip=set() + for k,g in enumerate(echelon): + if k not in echactive:continue + if s not in g['roles']: + if T&set(g['targets']): + assert seen[k]==g['roles'];restores.append(dict(group=k,before_role=s,event=j)) + for t in g['targets']:out[t].append(g['frame']) + echactive.remove(k) + continue + seen[k].add(s);skip.update(g['targets']) + for t,a in g['response'].items(): + if a.get(s,0):out[t].append(W.gauge[s]['frame']) + for k in sorted({rankowner[t]for t in T if t in rankowner}): + if k in rankactive and s not in rankgroups[k]['roles']: + for t in rankgroups[k]['targets']:out[t].append(rankgroups[k]['frame']) + rankactive.remove(k) + for k in sorted({owner[t]for t in T if t in owner}): + if k in active and s not in agg[k]['roles']: + for t in agg[k]['targets']:out[t].append(agg[k]['frame']) + active.remove(k) + for k in roles[s]: + assert k in active;g=agg[k];out[g['pivot']].append(readframes[s,k]);skip.update(g['targets']) + for t in e['targets']: + if t in skip:continue + rk=rankowner.get(t) + if rk in rankactive and t in rankgroups[rk]['dependent']:continue + out[t].append(e['frame']) + else: + assert not(T&{t for k in active for t in agg[k]['targets']}) + assert not(T&{t for k in echactive for t in echelon[k]['targets']}) + for t in e['targets']:out[t].append(e['frame']) + H=Counter() + for t,p in enumerate(out):H.update(histogram(t,p)) + return out,H,restores + + newpaths,target,restores=paths(m.agg,m.read_frames,m.echelon,m.rankgroups) + # Recover raw literal adjoint independently, respecting all omitted copies. + adj=[{}for _ in range(W.R)] + for root,role in zip(W.g['roots'],W.w['rootroles']): + for t in root['targets']:adj[role][t]=adj[role].get(t,0)+1 + for i in reversed(W.phase1+W.rest): + if i in m.omitted:continue + a,b,_=W.ops[i] + for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c + assert adj==m.adj + oldused={r[k]for r in m.selection+m.gauge_selection if r['role']not in pair_byalias for k in('source','partner')} + assert len({r[k]for r in chosen for k in('source','partner')})==62 and not oldused&{r[k]for r in chosen for k in('source','partner')} + for r in chosen: + b=r['aliased_role'];response=Counter(adj[b]) + for a,c in r['retained_dirty_coefficients'].items(): + for t,v in adj[a].items():response[t]+=c*v + assert not any(response.values()) + # Complete physical source and auxiliary census, including all alias handoffs. + recipient={d:b for b,d in W.pairs};roots=defaultdict(list) + for j,s in enumerate(W.w['rootroles']):roots[s].append(W.w['root_frame'][j]) + source_role={s:n for n,s in W.source.items()};plainborrow={r['role']:r for r in m.selection};gauges={r['role']:r for r in m.gauge_selection} + assert len(plainborrow)==440 and len(gauges)==87 and len(m.borrow)==527 + seq={s:[('op',i,W.opframe[i])for i in W.role_ops[s]]+[('root',j,f)for j,f in enumerate(roots[s])]for s in range(W.R)} + def physical_path(s): + a=list(seq[s]) + if s in pair_bydirty:a=[('paired-source-injection',s,pair_bydirty[s]['mix_frame'])]+a + if s in recipient: + b=recipient[s];a+=[('alias',b,W.gauge[b]['frame'])]+seq[b] + return a+[('full',s,W.w['full_frame'])] + count=0 + + def census_hist(start,events): + nonlocal count + count+=1;prev=start;dim=C.dimf[start]if start is not None else 0;out=Counter();chain=[]if start is None else[start] + for _,_,f in events: + assert C.nondeg(f)and(prev is None or C.sub(prev,f));d=C.dimf[f]-dim;assert d>=0 + if d:out[d]+=1 + prev=f;dim=C.dimf[f];chain.append(f) + assert dim==24;reflected=Counter() + for a,b in zip(reversed(chain),list(reversed(chain))[1:]): + assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);d=len(C.A[b])-len(C.A[a]);assert d>=0 + if d:reflected[d]+=1 + if start is None: + d=24-len(C.A[chain[0]]) + if d:reflected[d]+=1 + assert reflected==out + return out,chain + internal=Counter();sources=Counter();pathrows={};sourcepaths={};dirtypaths={} + for s in m.regs: + start=W.gauge[s]['frame']if s in W.gauge else W.w['source_frame'][source_role[s]]if s in source_role else None + H,chain=census_hist(start,physical_path(s)) + if s in source_role:H[1]+=1 + internal.update(H);pathrows['a'+str(s)]=compact(H) + if s in pair_bydirty:dirtypaths[str(s)]=[zero]+chain + for j,r in enumerate(W.g['roots']): + if r['kind']=='center':internal[C.dimf[W.w['root_frame'][j]]]+=1 + for entry in W.k['entries']: + carrier,n=entry['carrier'],entry['passive'];H,chain=census_hist(entry['carrier_chain'][0],[('carrier',j,f)for j,f in enumerate(entry['carrier_chain'][1:])]);sources.update(H) + if n not in m.by_source: + H,chain=census_hist(entry['passive_chain'][0],[('passive',j,f)for j,f in enumerate(entry['passive_chain'][1:])]);sources.update(H);pathrows['s'+str(n)]=compact(H);continue + r=m.by_source[n];s=r['role'];events=physical_path(s) + if s in plainborrow: + r=plainborrow[s];events=[x for x in events if not(x[0]=='op'and x[1]==r['omit_copy'])];at=next(i for i,x in enumerate(events)if x[0]=='op'and x[1]==r['early']);events.insert(at,('mix',n,entry['mix_frame'])) + else: + assert s in gauges;events=[('mix',n,entry['mix_frame']),('gauge',s,W.gauge[s]['frame'])]+events + H,chain=census_hist(W.w['source_frame'][n],events);sources.update(H);pathrows['s'+str(n)]=compact(H) + if s in m.extra_byrole:sourcepaths[str(s)]=chain + assert count==18347 and len(sourcepaths)==56 and len(dirtypaths)==48 and sum(k*v for k,v in sources.items())==1760*23 + gauges=Counter(z['dim']for s,z in W.gauge.items()if s not in W.donor and s not in m.borrow) + assert gauges=={20:2200,18:13,12:18,13:48} + copies=Counter(C.dimf[W.w['root_frame'][j]]for j,r in enumerate(W.g['roots'])if r['kind']=='center');assert copies=={22:24} + withoutcopies=internal.copy();withoutcopies.subtract(copies);assert min(withoutcopies.values())>=0 + helper=internal+sources+target;h=24;v=1760;R=len(m.regs);M=120;stock=4*v+R + assert R==16587 and len(m.borrow)==527 + assert sum(k*n for k,n in helper.items())==h*R+2*v*(h-1)+sum(k*n for k,n in copies.items())-sum(k*n for k,n in gauges.items()) + idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)});five=Counter({r:5*n for r,n in helper.items()});five.update(idle) + for r,n in gauges.items():five[5*r]+=n + mass=sum(k*n for k,n in five.items());assert M*stock-mass==4400 + raw=dict(schema='source527-fresh-raw-ledger/1',source_head=context.get('source_head','source527'),h=h,v=v,physical_R=R,source_aliases=len(m.borrow),terminal_sinks=len(m.terminal),physical_donor_recipient_identifications=len(W.pairs),physical_internal_excluding_center_copies=compact(withoutcopies),paid_center_copy_histogram=compact(copies),physical_source_histogram=compact(sources),physical_target_histogram=compact(target),one_stage_helper_histogram_including_copies=compact(helper),auxiliary_entrance_rank_histogram=compact(gauges),auxiliary_entrance_count=sum(gauges.values()),helper_rank_mass=sum(k*n for k,n in helper.items()),five_stage_profile=dict(m=M,W=stock,histogram=compact(five),idle_histogram=compact(idle),calls=sum(five.values()),rank_mass=mass,deficit=M*stock-mass,maxchild=max(five)),source_aux_paths=count,target_paths=1760,extra_source_paths=len(sourcepaths),paired_dirty_paths=len(dirtypaths),all_reflected_ledgers=True,all_zero_response_relations=True,scope='Fresh complete source527 raw physical source/internal/target and copied-center ledger. Both reflected chains checked; no saved execution receipt consumed.') + assert raw['five_stage_profile']['calls']==494994 and mass==2830840 + return raw + +def rebind_parity(raw,receipt): + """Bind the fresh required-use census to the retained producer source ledger.""" + from copy import deepcopy + def hist(x):return Counter({int(r):n for r,n in x.items()if n}) + assert receipt['status']=='PASS_FRESH_PARITY_FUSION_AND_INDEPENDENT_REQUIRED_FRAME_CENSUS' + assert receipt['both_reflected_ledgers']and receipt['unchanged_all_input_output_frames'] + source=hist(receipt['source_histogram']);target=hist(receipt['target_histogram']);internal=hist(receipt['internal_histogram_including_copies']);copies=hist(receipt['copied_center_histogram']) + assert source==hist(raw['physical_source_histogram']) + assert internal==hist(raw['physical_internal_excluding_center_copies'])+copies + delta=target.copy();delta.subtract(hist(raw['physical_target_histogram']));assert {r:n for r,n in delta.items()if n}=={8:-3,12:-3,20:3} + producer=deepcopy(raw);helper=source+target+internal;five=Counter({r:5*n for r,n in helper.items()});five.update(hist(raw['five_stage_profile']['idle_histogram'])) + for a,n in hist(raw['auxiliary_entrance_rank_histogram']).items():five[5*a]+=n + assert sum(five.values())==494979 and sum(r*n for r,n in five.items())==2830840 + raw['schema']='source527-parity-fused-fresh-raw-ledger/1';raw['producer_ledger']=producer + raw['physical_target_histogram']={str(r):n for r,n in sorted(target.items())};raw['one_stage_helper_histogram_including_copies']={str(r):n for r,n in sorted(helper.items())} + raw['five_stage_profile'].update(histogram={str(r):n for r,n in sorted(five.items())},calls=sum(five.values())) + raw['parity_transform']=receipt;raw['scope']='Fresh retained source527 producer ledger plus independently rebuilt actual surviving-use ledger after F2 payload identity elision and same-stream nested MOVE fusion. Both reflected endpoint inclusions checked. No address arithmetic reduced modulo2.' + return raw + +def rebind_retiming(raw,receipt): + """Recount the complete retimed word from its independently checked gate needs.""" + from copy import deepcopy + hist=lambda x:Counter({int(r):n for r,n in x.items()if n}) + compact=lambda x:{str(r):n for r,n in sorted(x.items())if n} + assert receipt['status']=='PASS_EXACT_COMMON_FRAME_RETIMING_AND_BOTH_REFLECTED_LEDGERS' + assert receipt['both_reflected_ledgers']and receipt['unchanged_all_input_output_frames']and receipt['unchanged_copy_lifetimes'] + assert receipt['selected_gate_count']==364 and receipt['remaining_payload_additions']==786621 + source=hist(receipt['source_histogram']);target=hist(receipt['target_histogram']);internal=hist(receipt['internal_histogram_including_copies']);copies=hist(receipt['copied_center_histogram']) + assert copies==hist(raw['paid_center_copy_histogram'])=={22:24} + helper=source+target+internal;delta=helper.copy();delta.subtract(hist(raw['one_stage_helper_histogram_including_copies'])) + assert {r:n for r,n in delta.items()if n}=={1:-708,2:356,3:-2,4:-3,5:2,7:-2,9:3,10:-1,20:-354,21:707,22:-353} + assert {r:n for r,n in delta.items()if n}==hist(receipt['local_histogram_delta']) + assert sum(source.values())>0 and sum(r*n for r,n in source.items())==1760*23 + assert sum(r*n for r,n in target.items())==1760*23 + assert sum(r*n for r,n in helper.items())==434502 + original=deepcopy(raw);five=Counter({r:5*n for r,n in helper.items()});five.update(hist(raw['five_stage_profile']['idle_histogram'])) + for a,n in hist(raw['auxiliary_entrance_rank_histogram']).items():five[5*a]+=n + assert sum(five.values())==493204 and sum(r*n for r,n in five.items())==2830840 + withoutcopies=internal.copy();withoutcopies.subtract(copies);assert min(withoutcopies.values())>=0 + raw['schema']='source527-parity-retimed-fresh-raw-ledger/1';raw['parity_fused_ledger']=original + raw['physical_source_histogram']=compact(source);raw['physical_target_histogram']=compact(target);raw['physical_internal_excluding_center_copies']=compact(withoutcopies) + raw['one_stage_helper_histogram_including_copies']=compact(helper);raw['five_stage_profile'].update(histogram=compact(five),calls=sum(five.values())) + raw['retiming_transform']=receipt;raw['scope']='Fresh source527 producer, parity-elision census and complete independently rebuilt census of all retimed scalar gate needs; actual frames, both reflections, fixed endpoints and copied-center lifetimes checked. Full physical histogram is used in exact assembly.' + return raw + +def rebind_regauge(raw,receipt): + """Price freshly transported F2 compensations and their actual new entrances.""" + from copy import deepcopy + hist=lambda x:Counter({int(r):n for r,n in x.items()if n}) + compact=lambda x:{str(r):n for r,n in sorted(x.items())if n} + assert receipt['status']=='PASS_SELECTED_GAUGE_TRANSPORT_AND_BOTH_REFLECTED_LEDGERS' + assert receipt['both_reflected_ledgers']and receipt['unchanged_data_input_output_and_dirty_output_frames']and receipt['unchanged_copy_lifetimes'] + assert receipt['selected_gauge_count']==21 and receipt['total_added_entrance_rank']==325 and receipt['remaining_payload_additions']==786623 + transport=receipt['transport'];assert transport['literal_prefix_transfer_matches']and transport['selected_helpers_untouched_before_cut']and transport['noise_before_cut_only_in_targets']and transport['producer_context_unchanged'] + source=hist(receipt['source_histogram']);target=hist(receipt['target_histogram']);internal=hist(receipt['internal_histogram_including_copies']);copies=hist(receipt['copied_center_histogram']);assert copies=={22:24} + helper=source+target+internal;delta=helper.copy();delta.subtract(hist(raw['one_stage_helper_histogram_including_copies'])) + assert {r:n for r,n in delta.items()if n}=={1:12,2:45,3:9,4:8,8:4,9:7,11:-2,12:8,13:-4,14:-6,17:14,18:31,19:-15,20:-41,21:-10} + assert {r:n for r,n in delta.items()if n}==hist(receipt['local_histogram_delta']) + assert source==hist(raw['physical_source_histogram']) + assert sum(r*n for r,n in target.items())==1760*23 + assert sum(r*n for r,n in helper.items())==434177 + original=deepcopy(raw);gauges=hist(raw['auxiliary_entrance_rank_histogram']);gauges.update({9:2,12:5,17:5,18:9});assert gauges=={9:2,12:23,13:48,17:5,18:22,20:2200} + five=Counter({r:5*n for r,n in helper.items()});five.update(hist(raw['five_stage_profile']['idle_histogram'])) + for a,n in gauges.items():five[5*a]+=n + assert sum(five.values())==493525 and sum(r*n for r,n in five.items())==2830840 + withoutcopies=internal.copy();withoutcopies.subtract(copies);assert min(withoutcopies.values())>=0 + raw['schema']='source527-parity-regauged-fresh-raw-ledger/1';raw['retimed_ledger']=original + raw['physical_source_histogram']=compact(source);raw['physical_target_histogram']=compact(target);raw['physical_internal_excluding_center_copies']=compact(withoutcopies) + raw['one_stage_helper_histogram_including_copies']=compact(helper);raw['helper_rank_mass']=434177 + raw['auxiliary_entrance_rank_histogram']=compact(gauges);raw['auxiliary_entrance_count']=2300 + raw['five_stage_profile'].update(histogram=compact(five),calls=sum(five.values())) + raw['regauge_transform']=receipt;raw['scope']='Fresh actual word after transporting twenty-one untouched dirty compensation columns through the exact F2 target prefix. Every scalar column, changed entrance and required frame path is checked; all data endpoints, dirty output endpoints, reflected inclusions and copied-center lifetimes are retained.' + return raw diff --git a/research/multicut-delayed-inverse-checkpoint/regauge-selection.json b/research/multicut-delayed-inverse-checkpoint/regauge-selection.json new file mode 100644 index 0000000000..c9238f2bc5 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/regauge-selection.json @@ -0,0 +1,8771 @@ +{ + "status": "FROZEN_TWENTY_ONE_COMMON_FRAME_GAUGE_SELECTION", + "input_raw_sha256": "eb31ea63d4f9d52226b8caa85401915d92088baee924a6af9f57b06b4c996169", + "input_gzip_sha256": "9b134a4f886265d5c12a52623f0952b96ecf0790f8016445df6d04de25b6b48d", + "input_scalar_sha256": "aeeb17ee9162b8f6469e7205c2cb2abdd1a383e1b4876a68d791d26382160ded", + "cut_core_ordinal": 701779, + "cut_scalar": [ + 1, + 3491, + 3488, + -1, + 25 + ], + "target_prefix_operations": 441, + "old_reads": 84, + "new_reads": 86, 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Twenty-one entrance gauges transport compensation through441actual target-basis operations. Fresh F2 forward and literal inverse, source/dirty restoration, and both reflected local ledgers. Substantial OpenAI Codex assistance. Requires new completed-bank, prime and finite witnesses for global admission.", + "expected_coefficient_counts": { + "3": 1320, + "1": 785303 + }, + "expected_scalar_additions": 786623, + "expected_forward_row_l1": 37631, + "expected_inverse_row_l1": 3295796 +} diff --git a/research/multicut-delayed-inverse-checkpoint/regauge_transform.py b/research/multicut-delayed-inverse-checkpoint/regauge_transform.py new file mode 100644 index 0000000000..af58894749 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/regauge_transform.py @@ -0,0 +1,209 @@ +"""New dirty entrance gauges by exact F2 compensation transport. + +The original scalar producer context is preserved. The transformed execution +context adds only the newly checked entrance frames. Address arithmetic remains +in the retained odd-prime rings. Substantial OpenAI Codex assistance; Apache-2.0. +""" +from array import array +from collections import Counter +from copy import copy +from pathlib import Path +import gzip,hashlib,json,time +if not __debug__:raise SystemExit('assertions required') +HERE=Path(__file__).resolve().parent + +def sha_basis(B): + return hashlib.sha256(json.dumps(B,separators=(',',':')).encode()).hexdigest() + +def transform(producer,selection): + W,C=producer['W'],producer['C'];old=producer['records'];n=20107;v=1760;ZERO=producer['ZERO'];initial=dict(producer['initial_state']);context=producer['context'] + assert hashlib.sha256(old.tobytes()).hexdigest()==selection['input_raw_sha256'] + assert producer['physical']['scalar_projection_sha256']==selection['input_scalar_sha256'] + cats=producer['physical']['category_names'];readcat=cats.index('dirty_read');entries=[];chosen={} + for row in selection['entries']: + r=dict(row);stream=r['stream'];role=r['role'];assert stream not in chosen and context['regs'][stream-2*v]==role + assert role not in W.gauge and role not in W.donor and role not in context['borrow'] and initial[stream]==ZERO + f=W.register(r['basis']);assert C.dimf[f]==r['rank'] and C.nondeg(f) + r['frame']=f;chosen[stream]=r;entries.append(r) + assert len(chosen)==len(selection['entries'])>0 and sum(r['rank']for r in entries)==selection['rank_drop'] + core=[k for k in range(0,len(old),6)if old[k]];cut=core[selection['cut_core_ordinal']] + assert [old[cut+j]for j in(0,1,2,3,5)]==selection['cut_scalar'] + # Derive the transported response directly from the actual prefix, using + # independent selected dirty columns rather than a precomputed adjoint table. + bits={r:1<=0 + out.extend((0,s,before,f,gap,0));state[s]=f + for k in range(0,len(old),6): + op,a,b,c,f,z=old[k:k+6] + if op==1: + if z==readcat and b in chosen and f==ZERO: + assert k<=cut;skipped.append(k//6) + else:move(a,f);move(b,f);out.extend((op,a,b,c,f,z)) + elif op==2: + assert temporary is None;move(a,c);state[b]=f;temporary=(a,b,c);out.extend((op,a,b,c,f,z)) + elif op==3: + assert temporary==(a,b,c) and state[a]==c and state[b]==f + out.extend((op,a,b,c,f,z));del state[b];temporary=None + if k==cut: + assert temporary is None + for row in entries: + r=row['stream'];f=row['frame'];assert state[r]==f + for t in row['targets']: + move(v+t,f);move(r,f);out.extend((1,v+t,r,1,f,readcat));inserted.append((v+t,r)) + assert temporary is None and skipped==deleted and len(inserted)==selection['new_reads'] + for s in sorted(final):move(s,final[s]) + assert state==final + # Only the documented dirty reads are replaced. Every other scalar and + # COPY/ERASE instruction must retain literal chronological order. + def project(a): + result=[] + for k in range(0,len(a),6): + op,x,y,c,f,z=a[k:k+6] + if op:result.append((op,x,y,c,z)) + return result + expected=[] + for k in range(0,len(old),6): + op,a,b,c,f,z=old[k:k+6] + if op and k//6 not in set(deleted):expected.append((op,a,b,c,z)) + if k==cut:expected.extend((1,a,b,1,readcat)for a,b in inserted) + assert project(out)==expected + # Clone only execution metadata. Never expose new gauges to the retained + # source527 producer replay or silently reinterpret its signed decoder. + execution_context=dict(context);execution_W=copy(W);execution_W.gauge=dict(W.gauge);execution_W.w=dict(W.w);execution_W.w['gauges']=list(W.w['gauges']) + for row in entries: + g=dict(role=row['role'],frame=row['frame'],dim=row['rank'],targets=list(row['targets'])) + execution_W.gauge[row['role']]=g;execution_W.w['gauges'].append(g) + execution_context['W']=execution_W + assert all(row['role']not in W.gauge for row in entries) + proof=dict(stable_cut_core_ordinal=selection['cut_core_ordinal'],actual_cut_record=cut//6,target_prefix_operations=targetops,deleted_reads=skipped,inserted_reads=inserted,literal_prefix_transfer_matches=True,selected_helpers_untouched_before_cut=True,noise_before_cut_only_in_targets=True,producer_context_unchanged=True) + return out,initial,final,entries,execution_context,proof + +def scalar_check(records,selection): + expected_digest=selection['expected_scalar_sha256'];expected_count=selection['expected_scalar_additions'];expected_coeff={int(k):v for k,v in selection['expected_coefficient_counts'].items()} + n=20107;v=1760;receipts=[] + for reverse in(False,True): + columns=[1<=2*v) + for t in range(v): + f=final[v+t];assert C.dimf[f]==23 and all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]) + # Census is independently rebuilt only from actual surviving ADD needs, + # frozen COPY source frames, and fixed endpoints, ignoring emitted MOVEs. + needs={i:[] for i in initial};copies=Counter();copy=None + for k in range(0,len(out),6): + op,a,b,c,f,z=out[k:k+6] + if op==1: + assert c%2;needs[a].append(f) + if b!=n:needs[b].append(f) + else:assert copy is not None and f==ZERO + elif op==2: + assert copy is None;copy=(a,b,c);needs[a].append(c);copies[z]+=1 + elif op==3:assert copy==(a,b,c);copy=None + assert copy is None and copies=={22:24} + census=Counter(copies);per_role={};pairs=set() + for s,before in initial.items(): + H=Counter() + for f in needs[s]+[final[s]]: + assert C.sub(before,f);rank=C.dimf[f]-C.dimf[before] + if rank:H[rank]+=1 + pairs.add((before,f));before=f + per_role[s]=H;census.update(H) + endpoint_frames={f for pair in pairs for f in pair};nondeg_bases=set() + for f in endpoint_frames: + assert len(C.B[f])==C.dimf[f] and len(C.A[f])+C.dimf[f]==24 + key=tuple(map(tuple,C.B[f])) + if key not in nondeg_bases:assert C.nondeg(f);nondeg_bases.add(key) + for a,b in pairs: + assert len(C.A[b])<=len(C.A[a]) + assert all(W.module.dot(x,y)==0 for x in C.A[b]for y in C.B[a]) + assert len(C.A[a])-len(C.A[b])==C.dimf[b]-C.dimf[a] + state=dict(initial);copy=None;used=set(initial.values());hist=Counter();cat=Counter();coeff=Counter();centers=[];count=0 + scalar=hashlib.sha256();tagged=hashlib.sha256();category_names=oldmeta['category_names'] + def event(d,row):d.update(json.dumps(row,separators=(',',':')).encode());d.update(b'\n') + for k in range(0,len(out),6): + op,a,b,c,f,z=out[k:k+6] + if op==0: + assert state[a]==b and C.sub(b,c) and C.dimf[c]-C.dimf[b]==f + assert (24-C.dimf[b])-(24-C.dimf[c])==f + state[a]=c;used.update((b,c)) + if f:hist[f]+=1 + elif op==1: + assert c%2 and state[a]==state[b]==f and a!=b + kind=category_names[z];source=copy['source']if b==n else b + event(scalar,[a,source,c]);event(tagged,[a,b,c,f,kind]);count+=1;cat[kind]+=1;coeff[abs(c)]+=1;used.add(f) + elif op==2: + assert copy is None and b==n and state[a]==c and f==ZERO and z==22 + state[b]=f;hist[z]+=1;copy=dict(source=a,frame=c,rank=z,temporary=b,first_event=count,copy_output_frame=f) + else: + assert op==3 and copy is not None and (a,b,c)==(copy['source'],copy['temporary'],copy['frame']) and state[a]==c and state[b]==f==ZERO + copy.update(after_event=count,scatter_reads=count-copy['first_event']);assert copy['scatter_reads']==220 + centers.append(copy);copy=None;del state[b] + assert copy is None and state==final and hist==census + assert count==selection['expected_scalar_additions'] and coeff=={int(k):v for k,v in selection['expected_coefficient_counts'].items()} and len(centers)==24 + assert count==oldmeta['weighted_scalar_events']-selection['old_reads']+selection['new_reads'] + assert scalar.hexdigest()==selection['expected_scalar_sha256'] + delta=Counter(hist);delta.subtract(Counter({int(r):c for r,c in oldmeta['paid_histogram'].items()}));delta={r:c for r,c in delta.items()if c} + assert delta=={int(r):c for r,c in selection['expected_local_histogram_delta'].items()},('combined local delta',delta) + assert sum(r*c for r,c in hist.items())==oldmeta['paid_rank_mass']-selection['rank_drop'] + sourceH=Counter();targetH=Counter();internalH=Counter(copies) + for i,H in per_role.items():(sourceH if i=0 + out.extend((0,q,before,f,rank,0));st[q]=f;used.update((before,f)) + for k in range(0,len(a),6): + op,x,y,c,f,z=a[k:k+6] + if op==0:continue + if op==1: + key=(x,y,c,z) + if key in keys: + assert key not in seen and copy is None + e=keys[key];assert dims[f]==e['old_dimension']and bsha(F[f]['basis'])==e['old_basis_sha256'] + g=e['reuse_frame_id'];assert dims[g]==e['new_dimension']and sub(f,g) + # Exact equality to nominated subspace, rather than trusting frame IDs. + assert all(not sum(b*c for b,c in zip(u,v))for u in e['new_basis']for v in F[g]['annihilator']) + changed.append(dict(input_record=k//6,output_record_before_moves=len(out)//6,scalar=list(key),old_frame=f,new_frame=g));seen.add(key);f=g + if copy is not None:assert x!=copy[0],('COPY source written',x) + move(x,f);move(y,f);assert st[x]==st[y]==f;out.extend((op,x,y,c,f,z)) + elif op==2: + assert copy is None;move(x,c);assert y not in st;st[y]=f;copy=(x,y,c,f) + out.extend((op,x,y,c,f,z));copy_count+=1 + else: + assert op==3 and copy==(x,y,c,f)and st[x]==c and st[y]==f + out.extend((op,x,y,c,f,z));del st[y];copy=None + assert copy is None and seen==set(keys) + for q in sorted(fin):move(q,fin[q]) + assert st==fin and copy_count==24 + before,after=projection(a),projection(out);assert before==after + H=histogram(out);delta=H.copy();delta.subtract(histogram(a));delta={r:n for r,n in delta.items()if n};assert delta==EXPECTED + assert len(out)//6==len(a)//6-1 and set(F)==set(frames if all(isinstance(k,int)for k in frames)else map(int,frames)) + receipt=dict(status='PASS_PHYSICAL_RETIMING_ADAPTER_AND_EXACT_LOCAL_LEDGER',input_raw_sha256=expected_input_sha,raw_sha256=sha(out.tobytes()),selected_gate_count=25,changed_gates=changed,unchanged_scalar_COPY_projection_sha256=sha(after),scalar_COPY_projection_byte_identical=True,unchanged_endpoints=True,copy_windows=copy_count,all_COPY_sources_immutable=True,all_retimed_gates_outside_COPY_windows=True,new_registered_frames=0,exact_new_subspace_pairs=[list(x)for x in sorted(exact_new_pairs)],local_histogram=dict(sorted(H.items())),local_histogram_delta=delta,local_calls=sum(H.values()),local_rank_mass=sum(r*n for r,n in H.items()),normalized_stock=173409,literal_40_replica_stock=867045,scope='Exact emitted frame paths, scalar/COPY identity and cost delta. Inherits source scalar proof; global bank/prime/finite admission is separate.') + return out,receipt + diff --git a/research/multicut-delayed-inverse-checkpoint/retiming_transform.py b/research/multicut-delayed-inverse-checkpoint/retiming_transform.py new file mode 100644 index 0000000000..f9f6e2a6a6 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/retiming_transform.py @@ -0,0 +1,144 @@ +"""Exact common-frame retiming of the parity-filtered scalar ADD word. + +The selected forward-span and backward-cap frames change only address MOVE +partitions. The scalar instructions, dirty/source endpoints and COPY interfaces +are preserved. Prepared with substantial OpenAI Codex assistance; Apache-2.0. +""" +from array import array +from collections import Counter +from pathlib import Path +import gzip,hashlib,json,time +if not __debug__:raise SystemExit('assertions required') +HERE=Path(__file__).resolve().parent + +def sha_basis(B): + return hashlib.sha256(json.dumps(B,separators=(',',':')).encode()).hexdigest() + +def transform(producer,selection): + W,C=producer['W'],producer['C'];old=producer['records'];initial=producer['initial_state'];n=20107 + assert hashlib.sha256(old.tobytes()).hexdigest()==selection['input_raw_sha256'] + assert len(old)//6==selection['source_record_count'] + chosen={};changed=[] + for r in selection['entries']: + i=r['record'];assert i not in chosen + op,a,b,c,f,z=old[6*i:6*i+6] + assert op==1 and [a,b,c,z]==r['scalar'] + assert C.dimf[f]==r['old_dimension'] and sha_basis(C.B[f])==r['old_basis_sha256'] + g=W.register(r['new_basis']) + assert C.dimf[g]==r['new_dimension'] and C.nondeg(g) + assert (C.sub(g,f) if r['direction']=='minimal' else C.sub(f,g)) + chosen[i]=g;changed.append(dict(record=i,old_frame=f,new_frame=g,old_dimension=C.dimf[f],new_dimension=C.dimf[g],basis_sha256=sha_basis(C.B[g]),direction=r['direction'])) + assert len(chosen)==selection['selected_gate_count']==364 + state=dict(initial);final=dict(initial);out=array('i');copy=None + # Read the original physical endpoints before replacing any MOVE. + for k in range(0,len(old),6): + if old[k]==0:final[old[k+1]]=old[k+3] + def move(s,f): + before=state[s] + if before==f:return + assert C.sub(before,f),('retimed frame descent',s,before,f) + gap=C.dimf[f]-C.dimf[before];assert gap>=0 + out.extend((0,s,before,f,gap,0));state[s]=f + for k in range(0,len(old),6): + op,a,b,c,f,z=old[k:k+6] + if op==0:continue + if op==1: + f=chosen.get(k//6,f);move(a,f);move(b,f);out.extend((op,a,b,c,f,z)) + elif op==2: + assert copy is None and b==n;move(a,c);state[b]=f;copy=(a,b,c);out.extend((op,a,b,c,f,z)) + else: + assert op==3 and copy==(a,b,c) and state[a]==c and state[b]==f + out.extend((op,a,b,c,f,z));del state[b];copy=None + assert copy is None + for s in sorted(final):move(s,final[s]) + assert state==final + # The complete scalar projection and COPY events are unchanged byte for byte. + def projection(a): + result=array('i') + for k in range(0,len(a),6): + op,x,y,c,f,z=a[k:k+6] + if op:result.extend((op,x,y,c,z)) + return result.tobytes() + assert projection(old)==projection(out) + return out,final,changed,hashlib.sha256(projection(out)).hexdigest() + +def run(producer,output_dir=None,selection_path=None): + begun=time.monotonic();W,C=producer['W'],producer['C'];v=W.v;n=20107 + path=Path(selection_path)if selection_path else HERE/'retiming-selection.json' + selection=json.loads(path.read_text());old=producer['records'];oldmeta=producer['physical'];initial=producer['initial_state'];ZERO,FULL=producer['ZERO'],producer['FULL'] + out,final,changed,projection=transform(producer,selection) + assert all(final[i]==FULL for i in initial if i=2*v) + for t in range(v): + f=final[v+t];assert C.dimf[f]==23 and all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]) + # Census is independently rebuilt only from actual surviving ADD needs, + # frozen COPY source frames, and fixed endpoints, ignoring emitted MOVEs. + needs={i:[] for i in initial};copies=Counter();copy=None + for k in range(0,len(out),6): + op,a,b,c,f,z=out[k:k+6] + if op==1: + assert c%2;needs[a].append(f) + if b!=n:needs[b].append(f) + else:assert copy is not None and f==ZERO + elif op==2: + assert copy is None;copy=(a,b,c);needs[a].append(c);copies[z]+=1 + elif op==3:assert copy==(a,b,c);copy=None + assert copy is None and copies=={22:24} + census=Counter(copies);per_role={};pairs=set() + for s,before in initial.items(): + H=Counter() + for f in needs[s]+[final[s]]: + assert C.sub(before,f);rank=C.dimf[f]-C.dimf[before] + if rank:H[rank]+=1 + pairs.add((before,f));before=f + per_role[s]=H;census.update(H) + endpoint_frames={f for pair in pairs for f in pair};nondeg_bases=set() + for f in endpoint_frames: + assert len(C.B[f])==C.dimf[f] and len(C.A[f])+C.dimf[f]==24 + key=tuple(map(tuple,C.B[f])) + if key not in nondeg_bases:assert C.nondeg(f);nondeg_bases.add(key) + for a,b in pairs: + assert len(C.A[b])<=len(C.A[a]) + assert all(W.module.dot(x,y)==0 for x in C.A[b]for y in C.B[a]) + assert len(C.A[a])-len(C.A[b])==C.dimf[b]-C.dimf[a] + state=dict(initial);copy=None;used=set(initial.values());hist=Counter();cat=Counter();coeff=Counter();centers=[];count=0 + scalar=hashlib.sha256();tagged=hashlib.sha256();category_names=oldmeta['category_names'] + def event(d,row):d.update(json.dumps(row,separators=(',',':')).encode());d.update(b'\n') + for k in range(0,len(out),6): + op,a,b,c,f,z=out[k:k+6] + if op==0: + assert state[a]==b and C.sub(b,c) and C.dimf[c]-C.dimf[b]==f + assert (24-C.dimf[b])-(24-C.dimf[c])==f + state[a]=c;used.update((b,c)) + if f:hist[f]+=1 + elif op==1: + assert c%2 and state[a]==state[b]==f and a!=b + kind=category_names[z];source=copy['source']if b==n else b + event(scalar,[a,source,c]);event(tagged,[a,b,c,f,kind]);count+=1;cat[kind]+=1;coeff[abs(c)]+=1;used.add(f) + elif op==2: + assert copy is None and b==n and state[a]==c and f==ZERO and z==22 + state[b]=f;hist[z]+=1;copy=dict(source=a,frame=c,rank=z,temporary=b,first_event=count,copy_output_frame=f) + else: + assert op==3 and copy is not None and (a,b,c)==(copy['source'],copy['temporary'],copy['frame']) and state[a]==c and state[b]==f==ZERO + copy.update(after_event=count,scatter_reads=count-copy['first_event']);assert copy['scatter_reads']==220 + centers.append(copy);copy=None;del state[b] + assert copy is None and state==final and hist==census + assert count==786621 and coeff=={1:785301,3:1320} and len(centers)==24 + assert scalar.hexdigest()==oldmeta['scalar_projection_sha256'] + delta=Counter(hist);delta.subtract(Counter({int(r):c for r,c in oldmeta['paid_histogram'].items()}));delta={r:c for r,c in delta.items()if c} + assert delta=={int(r):c for r,c in selection['expected_local_histogram_delta'].items()},('combined local delta',delta) + assert sum(r*c for r,c in hist.items())==434502 + sourceH=Counter();targetH=Counter();internalH=Counter(copies) + for i,H in per_role.items():(sourceH if i2*bound + progress('Checking the signed integer source decoder and dirty restoration') + integer=[] + for direction in(1,-1):integer.append(ns['replay']('Z',direction,bits));gc.collect() + controls={} + for mutation in ['omit_sparse_anchor','wrong_sparse_sign','omit_pair_initialization','omit_pair_restoration','wrong_pair_restore_sign','subtract_paired_alias','omit_transformed_source_mix','omit_transformed_source_read','omit_echelon_setup','omit_echelon_inverse','flip_echelon_sign','omit_rank_setup','omit_rank_inverse','repeat_rank_read','omit_aggregation_setup','omit_aggregation_inverse','repeat_aggregation_read','omit_extra_mix','omit_extra_compensation','omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation','late_phase1_gauge']: + try:ns['replay']('Z'if mutation in('flip_echelon_sign','wrong_pair_restore_sign','wrong_sparse_sign')else'F2',bits=bits,mutation=mutation) + except AssertionError as error:controls[mutation]=dict(rejected=True,reason=str(error)) + else:raise AssertionError('invalid source mutation accepted: '+mutation) + return dict(status='PASS_FRESH527_PRODUCER_SIGNED_DECODER_AND_PARITY_FUSED_F2_INVERSE_AND_CONTROLS',filtered_control=filtered_control,F2=exact,integer=integer,majorant=dict(residual_bound=bound,packing_bits=bits),forward=forward,inverse=inverse,controls=controls,source_sha256=hashlib.sha256(source.encode()).hexdigest(),seconds=time.monotonic()-started,scope='The unchanged source527 producer retains both signed defining-integer decoder proofs and25controls. The emitted parity-filtered word is independently checked on all20107formal F2columns and literalreverse, with its own cancellation-free signed-lift prefix bounds. Its integer endpoint is not asserted equal to the producer decoder; F2payload identity supplies the bit contract. Odd-prime address arithmetic is unchanged.') diff --git a/research/multicut-delayed-inverse-checkpoint/source527/echelon/selection.json b/research/multicut-delayed-inverse-checkpoint/source527/echelon/selection.json new file mode 100644 index 0000000000..291c1760f7 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/source527/echelon/selection.json @@ -0,0 +1,2512 @@ +{ + "groups": [ + { + "targets": [ + 0, + 1, + 2, + 3 + ], + "prefix_last": 131, + "prefix_roles": [ + 14709, + 14714, + 14699 + ], + "frame_basis": [ + [ + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 2 + ], + [ + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + -1 + ], + [ + 0, + 0, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 0, + 0, + 0, + 0, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 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0000000000..bc1cfc52a1 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/source527/pr230/NOTICE @@ -0,0 +1,441 @@ +A sharper exponent for integer multiplication +Copyright 2026 Douglas Colkitt + +This project includes unmodified source files from OpenAI's manuscript +"Integer multiplication below n log n" and its accompanying repository: +https://github.com/openai/math +Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a + +The original manuscript identifies OpenAI as its author. Its Apache-2.0 +license is retained at upstream/LICENSE. The exact source URLs and hashes +are recorded in upstream/manifest.json. Proposed modifications are supplied +as separate files under patches/; upstream source files remain unchanged. + +The original local research note, scripts, certificates, patches, and +documentation were prepared by Douglas Colkitt with assistance from OpenAI +Codex. The community release incorporates work by the contributors identified +in the notices below; it is not solely Douglas Colkitt's work. +This project is not an official OpenAI release or an OpenAI endorsement. + +Related contributions (not imported in this checkpoint): +Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset +motif, finite-alphabet transfer and a different compressed producer: +https://github.com/CrocSwap/integer-mult-bounds/pull/7 +The present checkpoint claims no priority for that motif. Related compressed +complex-network work was submitted by eumemic in PR #3: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +The pending contributions and their stronger bounds require their own review; +this notice does not represent adoption or verification of their claims. + +Additional community work, including parallel and superseded submissions, is +acknowledged in CONTRIBUTORS.md, with pinned submission metadata in +docs/research/contribution-snapshot.json. These acknowledgements do not imply +joint authorship of the checkpoint, incorporation of pending code, or verification +of submitted mathematical claims. Original source-specific notices remain in force. + +Community release (accepted by conditional maintainer audit): +The following original contributor notices apply to the imported PR #39 chain. +The preceding non-import statements describe the preserved 2^-30 checkpoint. + +Ternary five-subset construction and common-four-point resynthesis +Copyright 2026 Zhihao Chen (jacklightChen) + +This is Zhihao Chen's first submission of this specific construction to this +repository. Research, proof development, code and verification were prepared +with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by +the contributor. This is not independent review or an OpenAI endorsement. +Future research building on this contribution should explicitly acknowledge +Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and +retain the preceding contributors' attribution. This notice records research +provenance; the repository's Apache-2.0 license remains unchanged. + +Dependencies retained from eumemic's Claude-assisted PRs #3 and #5: +https://github.com/CrocSwap/integer-mult-bounds/pull/3 +https://github.com/CrocSwap/integer-mult-bounds/pull/5 +PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893 +These include complex_circuit.py, complex_network.py, fast_gaussian.py, +their generators, construction notes and certificates. The Gaussian input +scaling proof is corrected in this derivative; see the modification notice below. +Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank +sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6. +Their results and ideas are credited separately from this new construction. + +Bulk recursion and controlled tensor basis, 8 October 2026 +Copyright 2026 icekylinx + +The new projector batching, controlled simultaneous basis, mixed-width tape +recursions, whole-residual complex recursion, and dependency-path precision +guard in the batched-23 proof package are contributed by icekylinx, developed +with substantial OpenAI GPT-6 Astra assistance. Review corrections and +repository integration were prepared with OpenAI Codex assistance. They build on the retained +PR #7 finite producers and the preceding contributions credited above. +The new package retains the repository's Apache-2.0 license and the original +authors' attribution. It is not an official OpenAI release or endorsement. + +Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian +proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2). +The P=34p precision budget and asymptotic exponent are unchanged. New batching +checks are integrated into make verify. Existing attribution and licenses +are retained. + +Partial-swap extension, 8 October 2026 +Copyright 2026 icekylinx + +The partial-swap framing, binary retained-total triple construction, +dependency-preserving positive labels and carrier reuse, unequal (25,23,57) +common basis, improved complex batching, and exact assembly are contributed +by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance. +This extension directly inherits PR #10, commit +62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters +modify the retained paired-exclusion and shared-point implementations credited +above. The original licenses, attribution, and Gaussian correction are retained. + +Endpoint-gauge extension, 8 October 2026 +Copyright 2026 icekylinx + +The complementary auxiliary endpoint gauges, (32,30,40) controlled basis, +general nondegenerate binary phase residuals, retained-center complex producer, +updated path guard and exact assembly are contributed by icekylinx with +substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits +f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable +adapters retain the paired-exclusion and shared-point contributions credited +above. Original licenses, notices and the Gaussian correction are retained. + +Structured-basis and semantic-bulk extension, 8 October 2026 +Copyright 2026 icekylinx + +The selected (32,30,36) bit construction, A1/A5 block refinements, fixed +middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic +complex centers, complete data-front batching and circuit-specific assembly +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505. + +The semantic/bulk compatibility argument, literal scalar charge, product row +stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23, +commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints +are adopted from the same contributor's PR #21, +commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame +work and the earlier nested-basis contributions. + +Completed-child semantic precision, arbitrary-coordinate routing, phase-cell +Gaussian inversion and bulk tape transfer are credited to the RaD project +(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through +PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof +sources retain their separate CC0 license and original NOTICE under +references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI +Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra +assistance. These are source attributions, not claims about this session's +runtime model. Related contributions by Rohan Arun and Aurel Prosz remain +acknowledged in the imported notices. No exclusive priority or additional +license condition is asserted. + +Copied retained-center extension, 8 October 2026 +Copyright 2026 icekylinx + +The copied retained-center schedule in both orientations, its complex-phase +transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance. This extension inherits PR #32 at +0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work. + +The two-stage topology and paid copied-stream endpoint correction are due +to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab, +through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition, +9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil +Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a. + +The contiguous corner method, incidence factorization and exact partition +certificates are due to Rohan Arun (PR #31), method commit +02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate +specializes Dominik Scholz's parameterization (PR #33), +a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof +snapshots are preserved in references/copied-centers. Historical OpenAI +Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to +the respective imported sources. Existing PR #21/#23 and RaD attribution, +licenses and the Gaussian correction remain in force. + +Copied centers with reversed (23,25) data corners, 2026-10-08 +Rohan Arun with substantial OpenAI Codex assistance supplies the composition, +exact dimension23 controls and cost/assembly integration. James Chang's PR34 +supplies the reversed boundary family; icekylinx's PR36 supplies copied retained +centers and the selected complex graph. The balanced assembly remains attributed +to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan +Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx, +eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses. +This is not a claim of formal verification, new foundational rank identities, +global optimality or measured runtime speedup. + +Fixed middle basis with copied reversed corners (2026-10-08): +Rohan Arun, with substantial OpenAI Codex assistance. Composition and +simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34, +Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures, +eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed. + +Changed-graph composition with copied centers and fixed bases, 8 October 2026 +Prepared with OpenAI Codex assistance. +The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J +projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument, +Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and +Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods, +Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc) +PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40 +reversed integration, modular data tests and both-fixed baseline; +RaD (hipotures) PR41 changed scalar DAGs, alternating point order and +independent role compiler; RaD balanced routing/bulk interfaces, and all +retained predecessors. +Complete attribution and the conditional proof boundary are recorded in +research/copied-fixed/PROOF.md; imported originals and source pins are retained. +Original Apache-2.0 and CC0 notices remain applicable. + +Community follow-up integration (PRs #49, #45 and #26), 8 October 2026: +Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out +orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction +composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and +reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD / +hipotures's alternating pair order and independent physical role compiler (#41). +All predecessor attribution and conditional proof interfaces remain in force. + +Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic +proofs, finite tensor execution, profile precision accounting and mixed-center +reference checks (#45), with substantial OpenAI Codex assistance. See +research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution. +Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic +frame/movement/guard lemmas and independent paired-circuit checks (#26). Original +source notices and historical snapshots are preserved. Neither formal package +claims a full formal proof of the multiplication algorithm or a new kappa. + +Joint frame compilation of skip-prefix strips + +The local compiler and exact finite witnesses were prepared by eumemic with +substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic +Claude assistance supplied the unmodified PR #53 skip-prefix producer. +The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating +Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang, +Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt, +OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors +credited in the retained sources. See references/frame-compiler/pr48/NOTICE +and references/frame-compiler/pr53/NOTICE for their original attribution. +No authorship or review by those predecessors is claimed for this compiler. + +Joint dual-skip/frame-compiler composition (2026-10-08) +--------------------------------------------------- +Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip +scalar graph; original LICENSE and NOTICE are retained under +references/frame-compiler/pr55. eumemic with substantial OpenAI Codex +assistance: PR57 joint reversible frame compiler, serialized physical-word +replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex +assistance: this composition, independent integration and exact certificate. +Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all +predecessor authors retain the notices above and in pinned source files. +No original copyright or Apache-2.0 notice is removed. + +Ranked reclamation after the PR58 joint-dual composition (2026-10-08) +---------------------------------------------------------------- +Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical +slots by descending current frame rank, retaining slot ID as the tie-breaker; +complete replay, exact profiles, comparison and adversarial controls. +The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57 +compiler. Its original shared compiler remains byte-identical. The PR55 +producer and all preceding attribution remain unchanged. The complete PR58 +comparison certificate and its original source/proof provenance are retained +under references/frame-compiler/pr58. + +Skip-prefix strip producer, 8 October 2026 +By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5) +assistance. The skip-prefix leave-one-out strip layout in +research/skip-strips/skip_graph.py is new. Everything else reuses Chafik +Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification +code, exact data recovery and support-order sorting, together with: +- Rohan Arun's PR #44 weighted carrier matching optimizer; +- RaD (hipotures) PR #41 alternating point order; +- every predecessor credited above. +Complete attribution and the conditional proof boundary are recorded in +research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain +applicable. + +Local split-pair skip-prefix research continuation (2026-10-08) +Prepared with substantial OpenAI Codex assistance at the user's request. +New contribution: singleton/pair recursive partition refinement, finite summand +order and moment search, and verified composition with inherited paid clones. +Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips. +Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone +specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51 +whole-chain/source-partition cloning and earlier producer/interfaces. +Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all +stated historical AI assistance. All predecessor notices remain applicable. + +Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones): +Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain +source-partition clone operation is adapted from RaD / hipotures, PR #51, +with original envelopes retained here. The scalar base is Avi Eisenberg's +PR #53 skip-prefix layout, prepared with Anthropic Claude assistance. +All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain +in force. This extension is provided under the repository's Apache-2.0 terms. + +Community review checkpoint #50–#61, 8 October 2026 +Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex. +Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude. +Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance) +and dual-suffix strips (#55, Anthropic Claude assistance). +Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex. +Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked +reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame +composition (#56), with Anthropic Claude and stated Codex validation assistance. +RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones +(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone +composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter +refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with +his disclosed multi-tool and Codex assistance. +Parallel #50/#52 records are reviewed and credited without replacing the earlier +climbed-48 implementation; their exact commit history is retained. The separate +#51 witness is retained as research; its source-graph/negative-basis data proof +has not received the same full replay as #56's consumed compiler machinery. +See docs/research/community-round2-review.md for precise validation boundaries. +All earlier licenses, contributions and notices remain applicable. + +Interval strips and core-aware pair assembly + +The interval leave-one-out strips and the core-aware group-pair assembly in +research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic +Claude assistance. They build on PR #53's skip-prefix pipeline and its +inherited PR #48 sources. The stacked witness in research/pair-assembly/frame +uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier +are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and +verify_skip_frame.py. Original notices are retained. + +Stopped product-ring extension, 9 October 2026 +Copyright 2026 icekylinx + +The opposite-bank generic projector factorization, componentwise atom-ring +adapters, stopped recurrence and outer ordinary-interchange conversion, +aligned pair-star producer with rational centers, exact denominator-21 +grid and selected (23,23)/(24,24) composition are contributed by icekylinx +with substantial OpenAI GPT-6 Astra and Codex assistance. This extension +builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is +integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37. + +The retained ordinary leaf uses the preceding projector, corner and +copied-center proofs. The selected two-stage topology retains Aurel Prosz +(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly +retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching, +original source notices and each imported license remain unchanged. + +Partial-gauge extension, 9 October 2026 +Copyright 2026 icekylinx + +The pair-first cube assembly, compatible backward frame enlargement, +partial source-gauge schedule and stopped whole-projector composition are +contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex +assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834. + +The physical deferred bit word, reflected event accounting, fan-order +clarification and signed two-stage endpoint are adopted from Zhihao Chen +(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with +Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen +witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38, +with the original Claude disclosure retained. Cyclic interval and core-aware +pair primitives build on Avi Eisenberg (ikeboy), PR #62, +ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance. +Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and +precision credits remain in force. Imported source notices and licenses +are preserved under references/partial-gauge/; see SOURCES.json. + +Three-stage cover extension, 9 October 2026 +Copyright 2026 icekylinx + +The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry, +weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring +fallback, uniform batching, internal row borrowing and exact composition +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This +repository integration uses Codex assistance and extends +PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86. + +The selected local complex DAG and support/label replay are adopted from +Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf, +with its original Apache-2.0 terms and Claude assistance disclosure. +The unchanged bit physical word retains Zhihao Chen (jacklightChen), +PR #97, and Swapnil Jain's underlying witness and assistance notices. +The prior opposite-bank, stopped ordinary-leaf, copied-center, topology, +semantic/bulk, precision and analytic credits remain applicable. +Original source notices and license bytes are preserved in +references/three-stage-cover/pr117/ and references/partial-gauge/. + +Paired-cube and shared-core extension, 9 October 2026 +Copyright 2026 icekylinx + +The paired-coordinate triple-cube identity, signed overlap channels, +original-source involution, coordinate-star centers, concrete frame/carrier +compiler, chronological source gauges, PR97 gauge-subset selection, +orthogonal cross-stage routing and exact supplier/assembly integration +are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. +Repository integration uses Codex assistance and extends PR #130 at +6a9970a530119174507904e23592fd59ede19a5d. + +The substantial completed-core workspace-sharing principle is credited to +an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported +OpenAI Codex assistance. No source code, partition or serialized witness +from that PR is imported. The new routing and paired-motif integration +are described separately and claim no exclusive priority for broad sharing, +paired-coordinate, Fourier or carrier-matching ideas. + +Selected coarse queries restrict the retained eumemic PR #117 positive DAG; +its original Apache-2.0 and Claude assistance notices remain unchanged. +The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil +Jain's frozen witness, notices and historical assistance disclosures. +Earlier generalized Clifford, three-stage, uniform weighted recursion, +ordinary-leaf, semantic/bulk, precision and analytic credits remain in force. +Source-specific license bytes are preserved under references/. + +Source-assisted common-frame extension (2026-10-09) +Copyright 2026 icekylinx. Apache-2.0. +Research construction and local certificates: OpenAI GPT-6 Astra assistance. +Repository integration and independent complex scalar replay: Codex assistance. +Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee; +its byte-preserved sources, licenses and original Claude assistance disclosure +are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz. +Additional inherited contributions are recorded in research/source-assisted/CREDITS.md. +The new work concerns source-assisted invertible common-frame lifts, controlled +affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for +broad register reuse, paired cubes, frame theory or ordinary linear elimination. + +Fourth finite bootstrap refinement (2026-10-09) +Prepared by eumemic with substantial OpenAI Codex assistance. +This arithmetic refinement builds on PR207 at +cd14825023b75af4f5919a30e5e6d548b6ade5bc by adding one finite ordinary-leaf +level. The 302 frame overrides, completed banks, scalar words and upstream +assistance disclosures are inherited, not newly attributed to this work. +The Apache-2.0 license and all preceding notices remain applicable. + +PR211 plateau frame refinement (2026-10-09) +Prepared by eumemic with substantial OpenAI Codex assistance. +The additional seven equal-frame component moves build directly on Rohan +Arun's PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347, +prepared with Anthropic Claude assistance. Its original notice is preserved +byte-for-byte in search/PR211-NOTICE. The offline certificate package and +independent stage-private bank admission derive from Dugongue's PR207 at +cd14825023b75af4f5919a30e5e6d548b6ade5bc. Original contributor notices, +licenses, source pins and all upstream AI disclosures remain applicable. + +The final refinement also includes seeded bidirectional cascade search after +the seven initial component moves. Exact canonical subspace comparison +records 274 changed operations beyond PR211. Search, integration and replay +were prepared with substantial OpenAI Codex assistance. + +Shared rank-18 gauges and two donor substitutions (2026-10-09) +Prepared by eumemic with substantial OpenAI Codex assistance. +The new actual gauge/read chronology, gauged-donor handoff, shared target +transitions, rank6 residual banks and donor substitutions extend the same +PR200/PR211 word and PR207/PR197 completed-bank framework. All original +licenses, notices, authorship and upstream AI disclosures remain in force. + +Nested-gauge extension: three rank-19 physical entrances compose with thirteen shared rank-18 entrances; all target transitions and mixed4/5/6/24 banks are explicit. Prepared by eumemic with substantial OpenAI Codex assistance. All retained authorship, licenses and AI disclosures remain applicable. + +Equal-frame plateau refinement on PR210's 471-source word (2026-10-10) +Prepared by sennemmi with substantial OpenAI Codex assistance. The deterministic +frame search, 19 operation retimings, exact price update and repository +integration extend eumemic's PR210 head +df95878d11190518e45ef9717c9ee05011f88ace. They do not claim the inherited +source word, shared gauges, completed banks or upstream proof framework. +All PR200, PR197, PR202/193, PR207, PR211, PR216 and PR210 attributions, +licenses and AI disclosures remain in force; the repository's Apache-2.0 +license is unchanged. diff --git a/research/multicut-delayed-inverse-checkpoint/source527/pr230/PROVENANCE.json b/research/multicut-delayed-inverse-checkpoint/source527/pr230/PROVENANCE.json new file mode 100644 index 0000000000..ff39579387 --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/source527/pr230/PROVENANCE.json @@ -0,0 +1,27 @@ +{ + "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/230", + "author": "sennemmi", + "assistance": "Substantial OpenAI Codex assistance, as disclosed by the original contributor.", + "imported_head": "fee4c05b91d4cdebc0b9d8cbb266a98069928a9a", + "scope": "Nineteen operation-frame changes inherited from the independently verified492-source composition and replayed with493sourcealiases,82equal-response groups,seven integer-response groups andfour signed echelon groups. Original search,result,framewitness andnotice remain byte-preserved.", + "original_files": { + "NOTICE": "7dfbad087ee49b74fd5075d517eaa263c5dd3d4fbd4c69bfba34ee1cb55e837f", + "joint-plateau-result-prdf958.json": "cd5731260af0daa62952c3b82debdb26cfb79e62a5c615688d8ccac6f7a95fcc", + "joint-plateau-search-prdf958.py": "80f3f9e8ba1972b911b701a4de4f811d6d25c9b4c0bc6282d640c2f44caafd0d", + "opframe-bases.json": "387ae1b5e69004426d094f4947b29c532d60fdf31f727f5e8830468278fe0f92" + }, + "final_frame_sha256": "9a2bba5c72aebc9ee4f21e59b10ce810b2da98a1fa1748fbc44c50a080e584ff", + "raw_child_delta": { + "1": -27, + "2": -36, + "3": 51, + "4": -30, + "7": 30, + "8": -15, + "9": -15, + "10": 24, + "11": -24, + "12": 30, + "15": -15 + } +} diff --git a/research/multicut-delayed-inverse-checkpoint/source527/pr230/active-opframe-bases.json b/research/multicut-delayed-inverse-checkpoint/source527/pr230/active-opframe-bases.json new file mode 100644 index 0000000000..1c2c64e3ef --- /dev/null +++ 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i:1)if major else(lambda i:1<=0 + if d:target_hist[d]+=1 + target_frames[t]=f;target_paths[t].append(f) + X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[] + def track(value): + nonlocal top + if major:top=max(top,value) + return value + def value(s): + s=W.phys[s];return x[borrow[s]]if s in borrow else z[s] + def assign(s,val): + s=W.phys[s] + if s in borrow:x[borrow[s]]=track(val) + else:z[s]=track(val) + source_paths={role:[W.w['source_frame'][r['source']],r['mix_frame']]for role,r in extra_byrole.items()} + pair_dirty_paths={r:[zero]for r in pair_bydirty} + def source_move(s,f): + role=W.phys[s] + if role in pair_dirty_paths: + assert C.sub(pair_dirty_paths[role][-1],f),('paired dirty chronology',role,pair_dirty_paths[role][-1],f) + pair_dirty_paths[role].append(f) + if role not in source_paths:return + assert C.sub(source_paths[role][-1],f),('orderedsource chronology',s,role,source_paths[role][-1],f) + source_paths[role].append(f) + responses=W.adjoint() if mutation=='old_adjoint' else adj + def aggregate_restore(k,trigger): + if not aggregation_active[k]:return + group=agg[k];pivot=group['pivot'];assert aggregation_read[k]==set(group['roles']),('early aggregation inverse',k,trigger,aggregation_read[k]) + for t in group['targets']:target_move(t,group['frame']) + if mutation!='omit_aggregation_inverse' or k!=39: + for t in group['targets']: + if t!=pivot:y[t]=track(add(y[t],1,y[pivot])) + aggregation_active[k]=False;aggregation_restores.append(dict(group=k,before_role=trigger,frame=group['frame'])) + def rank_restore(k,trigger): + group=rankgroups[k];assert k in rank_active and rank_seen[k]==group['roles'] + for t in group['targets']:target_move(t,group['frame']) + if mutation!='omit_rank_inverse'or k!=0: + for t,cs in group['dependent'].items(): + for p,c in cs.items():y[t]=track(add(y[t],c,y[p])) + rank_active.remove(k);rank_restores.append(dict(group=k,before_role=trigger,frame=group['frame'])) + def echelon_restore(k,trigger): + group=echelon[k];assert echelon_seen[k]==group['roles'] + for t in group['targets']:target_move(t,group['frame']) + if mutation!='omit_echelon_inverse'or k!=0: + for t,p,c in reversed(group['moves']):y[t]=track(add(y[t],-c,y[p])) + echelon_active.remove(k);echelon_restores.append(dict(group=k,before_role=trigger,frame=group['frame'])) + def read(s): + nonlocal aggregation_pivot_reads,rank_skipped,echelon_reads + if s in pair_byalias: + if mutation=='subtract_paired_alias'and s==next(iter(pair_byalias)): + for t,c in adj[s].items():y[t]=track(add(y[t],-direction*c,value(s))) + return + if s in fresh: + if mutation=='omit_transformed_source_read'and s==next(iter(fresh)):return + for t,c in fresh[s]['transformed_response'].items(): + t=int(t);source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame']);y[t]=track(add(y[t],-direction*c,value(s))) + return + echelon_skip=set() + if s in W.gauge: + for k,group in enumerate(echelon): + if k not in echelon_active:continue + if s not in group['roles']: + if set(responses[s])&set(group['targets']):echelon_restore(k,s) + continue + echelon_seen[k].add(s);echelon_skip.update(group['targets']) + for t,a in group['response'].items(): + c=a.get(s,0) + if not c:continue + if mutation=='flip_echelon_sign'and k==0 and c<0:c=-c + source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame']);y[t]=track(add(y[t],-direction*c,value(s)));echelon_reads+=1 + if s in W.gauge: + for k in sorted({rankowner[t]for t in responses[s]if t in rankowner}): + if k not in rank_active:continue + if s not in rankgroups[k]['roles']:rank_restore(k,s) + else:rank_seen[k].add(s) + if mutation=='omit_gauge_compensation'and s==new_gauge_selection[-1]['role']:return + if mutation=='omit_extra_compensation'and s==extra_selection[0]['role']:return + grouped=role_groups.get(s,[]) + for k in sorted({target_group[t]for t in responses[s]if t in target_group}) if s in W.gauge else []: + if k not in grouped:aggregate_restore(k,s) + skipped=set() + for k in grouped: + group=agg[k];pivot=group['pivot'];assert aggregation_active[k] + assert set(group['targets'])<=set(responses[s])and len({responses[s][t]for t in group['targets']})==1 + c=responses[s][pivot];source_move(s,read_frames[s,k]);target_move(pivot,read_frames[s,k]);y[pivot]=track(add(y[pivot],-direction*c,value(s))) + if mutation=='repeat_aggregation_read'and k==39: + t=next(t for t in group['targets']if t!=pivot);target_move(t,read_frames[s,k]);y[t]=track(add(y[t],-direction*c,value(s))) + skipped.update(group['targets']);aggregation_read[k].add(s);aggregation_pivot_reads+=1 + for t,c in responses[s].items(): + if t in skipped or t in echelon_skip:continue + rk=rankowner.get(t) + if s in W.gauge and rk in rank_active and t in rankgroups[rk]['dependent']: + rank_skipped+=1 + if mutation!='repeat_rank_read'or rk!=0:continue + if s in W.gauge:source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame']) + else:assert target_frames[t]is None,'initial correction leaves target atD0' + y[t]=track(add(y[t],-direction*c,value(s))) + def gate(i,sign): + a,b,_=W.ops[i] + if sign==1: + source_move(a,W.opframe[i]);source_move(b,W.opframe[i]) + assign(a,add(value(a),sign,value(b))) + for s in range(W.R): + if s not in W.gauge and s not in borrow and s not in removed:read(s) + for n,s in W.source.items():assign(s,add(value(s),1,x[n])) + for r in gauge_selection: + if mutation=='omit_transformed_source_mix'and r is next(iter(fresh.values())):continue + if mutation=='omit_extra_mix'and r is extra_selection[0]:continue + if mutation!='omit_gauge_mix' or r is not new_gauge_selection[-1]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']])) + def forward(i): + if i not in omitted:gate(i,1);done.append(i) + for k in schedule['center_prefix_events']: + event=schedule['events'][k];i=event['op'] + if event['kind']=='early_mix': + r=early[i];a,b=r['partner'],r['source'] + if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b])) + else:forward(i) + for r,s in zip(W.g['roots'],W.w['rootroles']): + if r['kind']=='center': + for t in r['targets']: + assert all(f is None for f in target_frames),'copied centers scatter before any nonzero target movement' + center_reads.append((s,t));y[t]=track(add(y[t],direction,value(s))) + for k,group in enumerate(echelon): + assert all(target_frames[t]is None for t in group['targets']) + if mutation!='omit_echelon_setup'or k!=0: + for t,p,c in group['moves']:y[t]=track(add(y[t],c,y[p])) + for k,group in enumerate(rankgroups): + assert all(target_frames[t]is None for t in group['targets']) + if mutation!='omit_rank_setup'or k!=0: + for t,cs in group['dependent'].items(): + for p,c in cs.items():y[t]=track(add(y[t],-c,y[p])) + for k,group in enumerate(agg): + assert all(target_frames[t]is None for t in group['targets']), 'aggregation setup after all D0 center scatters' + if mutation!='omit_aggregation_setup'or k!=39: + for t in group['targets']: + if t!=group['pivot']:y[t]=track(add(y[t],-1,y[group['pivot']])) + for e in terminal: + assert all(target_frames[t]is None for t in e['targets']),'terminal setup afterD0center scatter' + for t in e['targets']: + if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']])) + # Original dirty compensation happened at D0 before these source injections. + for r in pair_selection: + b,f=r['aliased_role'],r['mix_frame'] + for a,c in r['retained_dirty_coefficients'].items(): + source_move(a,f);source_move(b,f) + if mutation=='omit_sparse_anchor'and r is pair_selection[14]and a==next(iter(r['retained_dirty_coefficients'])):continue + if mutation=='wrong_sparse_sign'and r is pair_selection[14]and a==next(iter(r['retained_dirty_coefficients'])):c=-c + if mutation!='omit_pair_initialization'or r is not pair_selection[0]:assign(a,add(value(a),c,value(b))) + for j,i in enumerate(W.rest): + for s in at[j]: + if mutation!='late_phase1_gauge' or s not in new_phase_roles:read(s) + if mutation=='late_phase1_gauge'and j==168: + for s in sorted(new_phase_roles):read(s) + if i in bywrite: + e=bywrite[i];a,b,n=W.ops[i];source_move(b,W.opframe[i]);target_move(e['pivot'],W.opframe[i]);y[e['pivot']]=track(add(y[e['pivot']],direction,value(b))) + else:forward(i) + if i in afterwrite: + e=afterwrite[i] + for t in e['targets']: + target_move(t,e['root_frame']) + if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']])) + for s in at[len(W.rest)]:read(s) + assert not any(aggregation_active),'all quotient inverses precede original side/K reads' + assert not rank_active,'rank quotient inverses precede side/K reads' + assert not echelon_active,'echelon inverses precede side/K reads' + for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])): + if r['kind']=='side' and j not in deletedroots: + for k in sorted({target_group[t]for t in r['targets']if t in target_group}):aggregate_restore(k,('root',j)) + source_move(s,W.w['root_frame'][j]) + for t in r['targets']:target_move(t,W.w['root_frame'][j]);y[t]=track(add(y[t],direction,value(s))) + for e in deliveries[j]: + a,b=e['carrier'],e['passive'] + if b not in by_source:x[a]=track(add(x[a],1,x[b])) + for k in sorted({target_group[t]for t in e['receivers']if t in target_group}):aggregate_restore(k,('K',j)) + for t in e['receivers']:target_move(t,e['deliver_frame']);y[t]=track(add(y[t],direction,x[a])) + assert not any(aggregation_active),'all target inverses completed beforecleanup' + for e in W.k['entries']: + if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']])) + if mutation=='undo_before_restore': + for e in W.k['entries']: + if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']])) + for role in source_paths:source_move(role,W.w['full_frame']) + for role in pair_dirty_paths:source_move(role,W.w['full_frame']) + for i in reversed(done):gate(i,-1) + for r in reversed(pair_selection): + b=r['aliased_role'] + for a,c in reversed(list(r['retained_dirty_coefficients'].items())): + c=-c + if mutation=='wrong_pair_restore_sign'and r is pair_selection[0]:c=-c + if mutation!='omit_pair_restoration'or r is not pair_selection[0]:assign(a,add(value(a),c,value(b))) + if mutation!='undo_before_restore': + for e in W.k['entries']: + if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']])) + for n,s in W.source.items():assign(s,add(value(s),-1,x[n])) + if binary:want=[Y0[t]^X0[t]for t in range(v)] + else: + want=Y0[:];cache={} + for root in W.g['roots']: + n=root['node'] + if n not in cache: + mask=C.sup[n];val=0 + while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1]) + cache[n]=val + for t in root['targets']:want[t]=add(want[t],direction,cache[n]) + for e in W.k['entries']: + val=add(X0[e['carrier']],1,X0[e['passive']]) + for t in e['receivers']:want[t]=add(want[t],direction,val) + # Finish every local target at its codimension-one cap, then reverse the exact + # annihilator stream. All copied-center scatters lie before these movements, + # hence atD0 forward and atD1 after the complementary movements reverse. + reflected=Counter() + for t,chain in enumerate(target_paths): + assert chain + d=W.h-1-C.dimf[chain[-1]];assert d>=0 + if d:target_hist[d]+=1 + previous=[C.cov[t]];dimension=1 + for f in reversed(chain): + assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f]) + basis=C.A[f];step=len(basis)-dimension;assert step>=0 + if step:reflected[step]+=1 + previous=basis;dimension=len(basis) + if W.h>dimension:reflected[W.h-dimension]+=1 + assert reflected==target_hist + assert len(center_reads)==sum(len(r['targets'])for r in W.g['roots']if r['kind']=='center') + assert sum(k*n for k,n in target_hist.items())==W.v*(W.h-1) + source_histogram=Counter();source_reflected=Counter() + for role,path in source_paths.items(): + assert C.dimf[path[0]]==1 and C.dimf[path[-1]]==W.h + for a,b in zip(path,path[1:]): + assert C.sub(a,b)and C.nondeg(a)and C.nondeg(b) + step=C.dimf[b]-C.dimf[a] + if step:source_histogram[step]+=1 + for a,b in zip(reversed(path),list(reversed(path))[1:]): + assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);step=len(C.A[b])-len(C.A[a]);assert step>=0 + if step:source_reflected[step]+=1 + assert source_histogram==source_reflected and sum(k*n for k,n in source_histogram.items())==len(source_paths)*(W.h-1) + paired_hist=Counter();paired_reflected=Counter() + for role,path in pair_dirty_paths.items(): + assert C.dimf[path[0]]==0 and C.dimf[path[-1]]==W.h + for a,b in zip(path,path[1:]): + assert C.sub(a,b)and C.nondeg(a)and C.nondeg(b) + d=C.dimf[b]-C.dimf[a] + if d:paired_hist[d]+=1 + for a,b in zip(reversed(path),list(reversed(path))[1:]): + assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);d=len(C.A[b])-len(C.A[a]) + if d:paired_reflected[d]+=1 + assert paired_hist==paired_reflected and sum(k*n for k,n in paired_hist.items())==len(pair_dirty_paths)*W.h + if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0))) + assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10]) + assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10]) + assert z==Z0,('dirty restoration',mode) + return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True,copied_center_target_reads=len(center_reads),all_forward_centers_at_D0=True,all_reflected_centers_at_D1=True,actual_target_histogram={str(k):n for k,n in sorted(target_hist.items())},actual_reflected_target_histogram={str(k):n for k,n in sorted(reflected.items())},aggregation_groups=len(agg),aggregation_pivot_reads=aggregation_pivot_reads,aggregation_restores=aggregation_restores,actual_extra_source_paths=source_paths,rank_groups=len(rankgroups),rank_targets=len(rankowner),rank_restores=rank_restores,rank_removed_reads=rank_skipped,actual_extra_source_histogram=dict(source_histogram),actual_extra_reflected_source_histogram=dict(source_reflected),echelon_restores=echelon_restores,echelon_reads=echelon_reads,paired_dirty_paths=pair_dirty_paths,paired_dirty_histogram=dict(paired_hist),paired_dirty_reflected_histogram=dict(paired_reflected)) diff --git a/research/multicut-delayed-inverse-checkpoint/verify.py b/research/multicut-delayed-inverse-checkpoint/verify.py new file mode 100644 index 0000000000..28f09028fd --- /dev/null +++ b/research/multicut-delayed-inverse-checkpoint/verify.py @@ -0,0 +1,82 @@ +#!/usr/bin/env python3 +"""Immutable source527 five-stage banks: regenerate every mandatory finite stage. +Prepared with substantial OpenAI Codex assistance; Apache-2.0. +""" +import sys +if not __debug__:raise SystemExit('Assertions required; refusing-O') +sys.dont_write_bytecode=True +from pathlib import Path +from datetime import datetime,timezone +from fractions import Fraction +import argparse,hashlib,importlib.util,json,time,traceback +ROOT=Path(__file__).resolve().parent +REQUIRED={'raw','bit','scalar','primes','banks','complex','math','finite'} + +def progress(message):print(datetime.now(timezone.utc).strftime('%H:%M:%S UTC')+' '+message,flush=True) +def sha(p):return hashlib.sha256(p.read_bytes()).hexdigest() +def serial(x): + if isinstance(x,Fraction):return str(x) + if isinstance(x,dict):return {str(k):serial(v)for k,v in x.items()} + if isinstance(x,(list,tuple)):return list(map(serial,x)) + return x + +def load(name): + s=importlib.util.spec_from_file_location('source527_'+name,ROOT/(name+'.py'));m=importlib.util.module_from_spec(s);sys.modules[s.name]=m;s.loader.exec_module(m);return m + +def integrity(): + p=ROOT/'MANIFEST.json';m=json.loads(p.read_text());actual={} + for q in ROOT.rglob('*'): + assert not q.is_symlink(),'Symlink in package' + if q.is_file()and q.name!='MANIFEST.json':actual[q.relative_to(ROOT).as_posix()]=sha(q) + # Nested source manifests are also ordinary pinned inputs. + for q in ROOT.rglob('MANIFEST.json'): + if q!=p:actual[q.relative_to(ROOT).as_posix()]=sha(q) + assert actual==m['files'],'Missing, changed or unpinned package/source file' + return sha(p),len(actual) + +def validate_required(results): + assert set(results)==REQUIRED,'Missing mandatory proof stage' + assert all(isinstance(results[k],dict)and results[k]for k in REQUIRED) + +def controls(): + out=[];good={k:{'present':True}for k in REQUIRED} + for k in sorted(REQUIRED): + bad=good.copy();bad.pop(k) + try:validate_required(bad) + except AssertionError:out.append('omitted '+k+' stage') + else:raise AssertionError('Missingstageaccepted') + return out + +def main(): + ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--output',type=Path,required=True);args=ap.parse_args() + import sympy + assert sys.version_info>=(3,11)and sympy.__version__=='1.14.0' + digest,count=integrity();output=args.output.resolve();assert not output.is_relative_to(ROOT)and not output.exists(),'Output must be new and outside package';output.mkdir(parents=True) + results={};timings={};start=time.monotonic() + def save(name,value):(output/name).write_text(json.dumps(serial(value),sort_keys=True,indent=2)+'\n') + def stage(name,call): + t=time.monotonic();value=call();results[name]=value;timings[name]=time.monotonic()-t;save(name+'.json',value);progress(name+' PASS '+format(timings[name],'.1f')+'s');return value + try: + progress('Regenerating source527 from pinned source recipes') + context=load('prepare').prepare() + raw=stage('raw',lambda:load('raw_ledger').run(context)) + portable=load('portable_bit');bit=portable.run(prepared_context=context,raw=raw,output_dir=output,run_geometry=True,package_root=ROOT);raw=bit['raw'];results['raw']=raw;save('raw.json',raw);results['bit']=portable.summary(bit);timings['bit']=bit['seconds'];save('bit.json',results['bit']);progress('bit physical/global/geometry PASS') + scalar=stage('scalar',lambda:load('scalar_check').run(context,bit['records'],progress)) + primes=stage('primes',lambda:load('prime_check').run(bit['context'],bit['physical'],progress)) + bankresult=stage('banks',lambda:load('bank_check').run(bit['context'],bit['global_result'],bit['lower'])) + banks=bankresult['banks'];banked=bankresult['banked_result'] + complex_result=stage('complex',lambda:load('portable_complex').run(package_root=ROOT,progress=progress)) + assert complex_result['precision_guard']['retained_row_coefficient']==20161 + math=stage('math',lambda:load('math_check').run(raw,complex_result['five_stage_histogram'],banks,bit['global_result'])) + finite=stage('finite',lambda:load('finite_check').run(raw,bit['physical'],scalar,primes,math,banks,banked,bit['global_result'])) + assert math['mathematics']['finite_bridge']['rows']['coefficient']==complex_result['precision_guard']['retained_row_coefficient']==finite['row_reserve']['external_complex_coefficient'] + validate_required(results);assert len(scalar['controls'])==25 and len(primes['controls'])==6 + assert Fraction(math['mathematics']['kappa'])==Fraction(1777613869061,2500000000000000) + after,_=integrity();assert after==digest,'Package changed during replay' + certificate=dict(schema='source527-parity-regauged-five-stage-banks/1',**math['mathematics'],scope='Conditionalfiniteconstruction; inheritedallsizecompiler,weightedselector,commonancestorchart,restoredrow,routing,prime,recovery,complexsymbolicandanalyticinterfacesremainassumptions.') + save('certificate.json',certificate) + report=dict(status='PASS_IMMUTABLE_PARITY_REGAUGED_SOURCE527_FIVE_STAGE_BANKED_CONSTRUCTION',manifest_sha256=digest,files=count,kappa=math['mathematics']['kappa'],kappa_scientific=math['mathematics']['kappa_scientific'],fresh_stages=sorted(REQUIRED),inputs_unchanged=True,timings=timings,seconds=time.monotonic()-start,negative_controls=controls()) + save('verification.json',report);print(json.dumps(report),flush=True) + except BaseException as exc: + save('failure.json',dict(status='FAIL',error=str(exc),completed_stages=sorted(results),timings=timings,traceback=traceback.format_exc()));raise +if __name__=='__main__':main()

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