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"7a8d17b04da1e6c9d0ec217fb5bda72faa09d9a21695128722dd1e2827870fcf", + "notices/research/gauge44-slots-t16/UPSTREAM-NOTICE.md": "a75967ebc34e3bf2ff2c25c08e6c7f1a61b0c0d171cfd1b62196ae2caaf2c610", + "notices/research/gauge44-slots-t16/construction/NOTICE.md": "defe7e44af610e25a6c50c1c3b2efd0972d0690574995c0c872fc50eebf4c816", + "notices/research/gauge44-slots-t16/t16/NOTICE.md": "902370e6bfcd39dbea2ece474f15fccbcce3b8e966b2b94066bd03e5174591b5", + "replay.py": "9fb77efa21b9ebfb7b6325899cdc3274ca75ca4d1a45a0675b68ea06d475dac6" + }, + "schema": 1, + "scope": "Immutable package inventory; generated output is external" +} diff --git a/research/finite-tagged-centre-reset/NOTICE.md b/research/finite-tagged-centre-reset/NOTICE.md new file mode 100644 index 0000000000..371aec48dc --- /dev/null +++ b/research/finite-tagged-centre-reset/NOTICE.md @@ -0,0 +1,46 @@ +# Attribution and license + +New research, code, packaging, and verification used substantial OpenAI +Codex assistance under the user's direction. Separate AI agents supplied +the construction, exact policy enumeration, and independent reviews. +No independent human review is claimed. New material uses Apache-2.0. +The [license](notices/LICENSE) is retained without changes. + +The scalar word and native frame data retain the historical gauge44/t16 +lineage at commit `bce54fb3f895cbe5cd4c0eb477c3afeed100190f`. Its original +notices are copied byte for byte: + +- [Gauge44 notice](notices/research/gauge44-slots-t16/NOTICE.md). +- [Construction notice](notices/research/gauge44-slots-t16/construction/NOTICE.md). +- [T16 integration notice](notices/research/gauge44-slots-t16/t16/NOTICE.md). +- [Complete upstream notice](notices/research/gauge44-slots-t16/UPSTREAM-NOTICE.md). + +Those notices credit icekylinx for the source gauge and supplier pipeline; +Hao Yan (@FanPUYen) for t16 construction and integration; Jeremy Kothe for +the retained [PR #799](https://github.com/CrocSwap/integer-mult-bounds/pull/799) +word; Chafik Boukhalfa for the +[PR #741](https://github.com/CrocSwap/integer-mult-bounds/pull/741) frames +and [PR #685](https://github.com/CrocSwap/integer-mult-bounds/pull/685) +owner construction; and Rohan Arun for the exact-complex native recurrence +correction. The retained notices give the full earlier credit and license +record. These contributions are not new results of this package. + +Within the centre-encoding campaign, history-encoding supplied the +full-mode word, source rows, and exact cut response. Circuit-interface +supplied the matrix. Schur-selection supplied the finite tagged word, +its corrected version 2 proof, and the fixed nonzero carrier witness. +Temporal-reuse supplied the native response work, separate record formats, +and the complete pure-reset policy theorem. Interface-proof reviewed the +native typing and the conditional record-format proof. Paired-package-review +supplied the corrected physical phase check, found the parent return-grid +error, independently checked the local word and all policy cases, and +proved the sharp surplus 17. Portable-source-codes supplied the separate +historical complete-profile comparison; that comparison is not a claim +about the changed selected construction. + +The replay uses both independent checker files without changes. The +frozen directory keeps their relative source layout. Producer scripts +are retained for source comparison. They are not imported by the replay. +[SOURCES.json](SOURCES.json) binds every copied file and records the +historical native and precision source pins. This package changes no +frozen evidence and makes no complete supplier or kappa claim. diff --git a/research/finite-tagged-centre-reset/PROOF.md b/research/finite-tagged-centre-reset/PROOF.md new file mode 100644 index 0000000000..739e4f72aa --- /dev/null +++ b/research/finite-tagged-centre-reset/PROOF.md @@ -0,0 +1,259 @@ +# Proof map and exact claim scope + +This document states the two retained results in one place. The full +proofs and the later independent reviews are copied without changes. +All event indices and mode numbers start at zero. + +## Fixed local word + +Use the raw backward word in +[FULL-MODE-PROGRAMS.json.gz](frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/FULL-MODE-PROGRAMS.json.gz), +with decompressed SHA-256 +`f271842a741584389e166e700e262f161b0dd570ce8c7a540f2717e90ead5205`. +Its 704 inputs contain 224 data fields and 480 arbitrary helper fields. +The actual matrix `R` is in +[INTERFACE.json](frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json), +SHA-256 `5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`. +Put `K=R^T/4`. + +Stop before event 12995. The centres are +`[92,57,81,224,11,240,9,256]` in frames 415 through 422. +Their initial helper roles are `[225,226,229,232,241,242,257,258]`. +The [exact cut proof](frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CENTRE-RESET-INTERFACE.md) +gives centre rows `u_k+R_kY/4`. Replace their logical values by `v_k`, +omit the eight signs at events 12996 through 13003, and resume at 13004. +The unchanged suffix gives + +```text +Y_out = Y, +X_out = X-Y+R^T(v-u), +u_out = v. +``` + +The other 472 helper contents are restored. Thus `v=u` gives the unit +backward data map and restores all 480 helpers. For centre frame `B_k`, +the physical replacement must be `B_k u_k`. The construction below acts +on physical coefficients and keeps each binary access in one actual +frame. It does not commute a scalar map through unequal frames. + +## Finite local reset + +The governing proof is +[GRID-RESET-PROOF-V2.md](frozen/work/centre-encoding-campaign-20261014/lanes/schur-selection/finite-grid/GRID-RESET-PROOF-V2.md), +SHA-256 `40fa4d2cbd2a6cfa3a7ec6facd81e04af727b18d77bed11643b13f19e38428e7`. +Its [independent review](frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/tagged-grid-review/REVIEW.md) +accepts this corrected local theorem. + +Use the maximum absolute real or imaginary coefficient as the norm. +For `h,q` on `Delta Z[i]`, suppose + +```text +0 < delta <= 1, +|delta q| <= Delta/16, +|e| <= delta Delta/16. +``` + +For `z=h+(1+delta)q+e`, the operations + +```text +b = Round_Delta(z), +q_hat = Round_Delta((z-b)/delta), +h_hat = b-q_hat +``` + +give `b=h+q`, `q_hat=q`, and `h_hat=h`. The first error is at most +`Delta/8`, and the second is at most `Delta/16`. Both are strictly +inside the nearest-grid rounding cell. A coarse bound on `e` alone +does not suffice for the second operation. + +Tag both data banks by `(1+delta)` with the owned field `C0`: +copy the data to zero `C0`, scale `C0` by `delta`, add it to the data, +and clear `C0`. At the cut, round 696 non-centre fields to the ordinary +untagged prefix state. Decode the eight centres with the formula above. +Then run the suffix and round all 704 outputs to their known grid. +The centre decoder uses the same owned field and clears it after use. +All scale factors are dyadic units. No arbitrary helper is cleared. + +The literal [word](frozen/work/centre-encoding-campaign-20261014/lanes/schur-selection/finite-grid/REDUNDANT-GRID-WORD.json.gz) +adds 232 COPY, 240 SCALE, 240 ADD, 233 CLEAR, and 1,416 ROUND passes: +2,361 passes in total. The native word retains 3,486 climbs, with rank +mass 5,522, and adds no child. These passes have a new finite bit cost; +the old scalar coefficient does not include them. + +### Physical grid and error premise + +Let every initial physical coefficient lie on `2^-g Z` with magnitude +at most one. For tensor width `lambda <= e`, define + +```text +H = 5522 lambda + 16339, +G = 5522 lambda + 49012, +L = 2^G, +Delta = 2^(-g-H), +delta = 2^(-g-H-G-4), +Gamma = 2^(-g-2H). +``` + +The exact scalar denominator and growth sums are 16,339 and 49,012. +The inherited raw native recipes give denominator exponent at most +`r lambda` and component row norm at most `2^(r lambda)` for a rank-r +child. Their monomial wrappers preserve the grid and component norm. +These facts imply that both cut contributions lie on `Delta Z[i]`, +their data part has norm at most `L`, and the suffix output lies on +`Gamma Z[i]`. The native phase premise is retained explicitly in +[FULL-MODE-PORT-REVIEW.json](frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/FULL-MODE-PORT-REVIEW.json). +The separate [physical correction](frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/centre-reset-review/PHYSICAL-GRID-V2.md) +checks actual canonical phase recipes and rejects the plain-product +interpretation of their listed basis vectors. + +Assume `g <= t+B e`, with fixed nonnegative integer `B`, and set + +```text +P = 2B e + 2H + 4G + 32. +``` + +Give each child relative computation and normalized-input error at +most `2^(-a_j t-P)`, where `a_j` is one or two. Separately bound each +parent scalar or return conversion by absolute error `2^(-2t-P)`. +The separate return allowance is necessary for a tiny child input. +A half-parent-grid native output can incur relative error at least +one half on conversion even when the child computation is exact. + +There are at most `25457+3486+2361=31304 < 2^16` insertion sites. +Exact suffix propagation gives coarse cut, fine centre, and final +output error bounds + +```text +4 * 2^16 * L^2 * 2^(-t-P), +4 * 2^16 * L^2 * 2^(-2t-P), +4 * 2^16 * L^3 * 2^(-t-P). +``` + +Relative to `Delta/16`, `delta Delta/16`, and `Gamma/16`, the exponent +margins are at least `B e+H+2G+10`, `G+6`, and `B e+G+10`. +The strict margins make all cut and final snaps exact. The stated +format has `2t+O(e)` fractional bits and `O(e)` magnitude bits. +Tag arithmetic fits because `2g+H+G+4 <= 2t+P`. Scaling the residual +does not form a large scaled baseline. + +Set `a_j=2` for every child with a nonzero response at an observed +centre and `a_j=1` for the others. The exact response set has 124 +rank-one and eight rank-four children, of mass 156. Every such input +has both data and helper terms. Treat each child error as an additive +output vector on its actual input; the exact response calculation +then applies even when that error depends on earlier errors. +This is a sufficient assignment under arbitrary independent relative +errors. It does not bound a correlated or lattice-restricted contract. + +The theorem starts on the declared input grid. Coarse conversion of +finer caller records is outside this local fine-error allowance. +The reviewed [record-format proof](frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.md) +is conditional on a complete node endpoint and finite error proof. +It does not supply either obligation for this local word. + +### Local resolution scope + +The actual minor on columns `[0,7,20,27,36,46,57,111]` obeys +`R_A R_A^T=4I`. Assume `g>=3` and choose `s_k` in +`2^-g Z intersect [-1/8,1/8]`. Set `v_k=s_k e0`, `Y_A=R_A^T v`, and +`u_k=e0/8-v_k`. These inputs lie on the declared input grid. +The physical tagged carrier is `(1/8+delta s_k) B_k e0`. +For adjacent values spaced by `2^-g` and `delta <= 1/2`, its relative +separation is between `(16/3) delta 2^-g` and `16 delta 2^-g`. +The nonzero common vector cancels in the ratio. For positive scalar +carriers `a= 17` for every nonempty +`S` and every permitted policy. Equality occurs for `S={5}` or `S={7}` +with no resets, with 21 rank-one and one rank-four job. For +`0 <= sigma <= 1`, put `x=2^sigma`. Then `1 <= x <= 2` and +`8^sigma-4^sigma=x^2(x-1) <= 4`. Therefore + +```text +sum_(j in C(S)) r_j^sigma - |S|*8^sigma + = n1(S)-|S|*(8^sigma-4^sigma) + >= 17. +``` + +The two one-query witnesses attain equality at `sigma=1`. Thus 17 is +sharp over this family and interval, without numerical sampling. + +For nonnegative query allocations `gamma_k`, charge each child the +maximum allocation over queries in its support. Integrate the subset +inequality over `S(t)={k:gamma_k>t}` to obtain + +```text +sum_j r_j^sigma max_(k in support(j)) gamma_k + >= 8^sigma sum_k gamma_k + 17 max_k gamma_k. +``` + +The empty-support maximum is zero. The same two witnesses show +sharpness. The result also holds for a policy that changes by bit layer +when costs add by layer; it does not emit or admit such a native word. +The separate mass bound `mass(C(S)) >= (65/4)|S|` gives the stated +non-improvement threshold `beta >= 97/65` for uniform width multiplier +`beta`, as proved in the source. + +These cost conclusions assume the declared independent relative-error +width model. The family excludes new access edges, new bases or codes, +joint resets, other reset times, changed native interfaces, and special +child error contracts. It grants the original width to retained private +centre jobs without claiming a complete partial-centre controller. +No complete supplier, universal lower bound, or kappa improvement follows. diff --git a/research/finite-tagged-centre-reset/README.md b/research/finite-tagged-centre-reset/README.md new file mode 100644 index 0000000000..ae6103b1fc --- /dev/null +++ b/research/finite-tagged-centre-reset/README.md @@ -0,0 +1,103 @@ +# Finite tagged centre reset and exact pure-reset cost + +This package contains a local nonlinear backward reset on a declared +physical dyadic input grid, and an exact cost theorem for every policy +in a fixed family of pure resets. It gives no improvement to the complete +kappa calculation. + +The local word returns `(X-Y,Y)`, restores all 480 arbitrary helper +contents in their inherited output frames, and clears one existing owned +private field `C0`. It adds 2,361 field passes and no native child. Its +sufficient fine-accuracy profile contains 132 retained children: +124 of rank one and eight of rank four, with rank mass 156. The eight +removed private centre jobs have rank mass 64. This local choice has an +adverse precision cost under the declared width rule. + +The complete policy theorem covers all `2^36` effective reset policies +through 12,288 exact group cases. The minimum all-query profile is +121 rank-one and eight rank-four jobs, with mass 153. For each nonempty +query subset, its added rank moment exceeds the removed centre moment +by at least 17 for `0 <= sigma <= 1`. The constant 17 is sharp. This is +a conditional cost theorem for the stated independent relative-error +width model. It is not a universal precision lower bound. + +Read [PROOF.md](PROOF.md) for the exact hypotheses and the proof map. +[SOURCES.json](SOURCES.json) gives the source hashes. +[NOTICE.md](NOTICE.md) gives the source credit and license. + +## Reproduce + +Use CPython 3.10 or later with its standard library. No network, Git +database, compiled package, or external campaign path is required. +Keep at least 3 GiB of disk space free. From this package directory, run: + +```sh +python3 -I -O -B replay.py --out /tmp/finite-tagged-centre-reset-replay +``` + +The output directory must not exist. Choose a new path for a later run. +The replay runs the two unchanged independent checkers in sequence and +compares every field of each new receipt with the frozen review receipt. +It writes logs, two mathematical receipts, and `REPLAY.json` outside the +package. Expected final status: + +```text +PASS_FINITE_TAGGED_CENTRE_RESET_PACKAGE +``` + +The first check covers all 704 input columns, all 480 helper contents, +132 direct native error impulses, the complete added pass count, 11,025 +exact decoder cases, 96 margin cases, and 27 carrier cases. Its controls +reject a coarse-only decoder allowance and a relative-only parent return +allowance. The second check reconstructs all 860 eligible reset sites, +all 12,288 group policies, all 255 nonempty query subsets, and 528 direct +native error impulses. It checks the sharp surplus witnesses. + +The original producer scripts are retained as source evidence. The +supported portable command is `replay.py`; it uses the independent +checkers and does not rerun the producer scripts. + +## Review state and limits + +Use the corrected [version 2 local proof](frozen/work/centre-encoding-campaign-20261014/lanes/schur-selection/finite-grid/GRID-RESET-PROOF-V2.md) +and its [independent review](frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/tagged-grid-review/REVIEW.md). +Parent return-grid rounding has a separate absolute error budget. It +cannot be included in the relative child error at a fixed parent width. +The fine-centre margin is `G+6`. The older proof is retained as history. + +The [complete family proof](frozen/work/centre-encoding-campaign-20261014/lanes/temporal-reuse/EXACT-TAGGED-RESETS.md) +was accepted by the later [full-family review](frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/exact-reset-policy-review/REVIEW.md). +That review proves the sharp surplus 17. The proof's older sentence +that review was pending is historical. The frozen files are unchanged. +The reviews used separate AI agents; they are not human reviews. + +Actual native frames, phase recipes, and their tensor extension remain +explicit inherited premises. The retained [physical phase correction](frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/centre-reset-review/PHYSICAL-GRID-V2.md) +uses the actual quadratic conjugation. A listed frame basis is not a +plain product of transforms on its vectors. The portable replay checks +scalar accesses and frame identities; it does not repeat the complete +phase audit or the historical row and tape precision audit. + +The local theorem starts after every role input is on its declared grid. +Conversion from finer caller records, the complete node endpoint, owner +admission, and all new finite bit and tape costs remain open. Exact local +helper restoration refers to those grid inputs. It does not prove exact +restoration of finer records before entry conversion. + +The publication base is `b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`. +Its [selected record](context/selected-result-20261022.json) uses the changed +conditional construction in [PR #4470](https://github.com/CrocSwap/integer-mult-bounds/pull/4470). +The historical source checks here do not establish admission to that +construction. No frontier update or complete supplier is claimed. + +## Open next questions + +- Prove a complete node endpoint and its entry-conversion error before + importing the local grid word into a caller. The first check must + include every row used as borrowed space. That endpoint is not proved. +- Evaluate a different native error contract or an access change only + with a new input and response proof. Such a change can leave the policy + theorem's class; the current theorem gives no cost outside that class. +- Price all new scans, rounding, record conversion, and tape movement + if a complete candidate first meets its precision moment. 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Native endpoint mixing and a changed controller are not admitted.", + "status": "PASS_CENTRE_FREE_SCALAR_INTERFACE", + "target_endpoint": "T_q = F C_q^-1; distinct q have distinct frames", + "word_sha256": "4ddb29fb9bc6f27e0c0fbfcf19a2a93a3496ad900d0f6d89059d0305c9576c0c" +} diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CENTRE-RESET-INTERFACE.md b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CENTRE-RESET-INTERFACE.md new file mode 100644 index 0000000000..11677dc871 --- /dev/null +++ b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CENTRE-RESET-INTERFACE.md @@ -0,0 +1,112 @@ +# Exact interface for a coarse centre reset + +An exact coarse-component reset at the backward centre cut gives the +original unit shear and restores all 480 arbitrary helper inputs. The +same cut has no effect on either forward data output. This is an exact +operator statement with an oracle reset. It is not a rounding circuit +or a precision certificate. + +## Literal cut and carrier values + +Use the pinned raw `FULL-MODE-PROGRAMS.json.gz`, decompressed SHA-256 +`f271842a741584389e166e700e262f161b0dd570ce8c7a540f2717e90ead5205`. +Execute events 0 through 12994. Event 12995 is the centre phase marker. +Events 12996 through 13003 negate the eight centre registers. Omit those +eight signs for the reset experiment. Resume the suffix at event 13004, +the phase B marker. + +The centre registers are 92, 57, 81, 224, 11, 240, 9, and 256. Their +native frames are 415 through 422, each of rank eight. In the mode input +basis, their complete input rows are + +``` +forward: w = u + R x; +backward: w = u + R y/4. +``` + +Each row has exactly 56 source coefficients and one mode coefficient. +It has no initial target coefficient and no other helper coefficient. +The corresponding mode input roles are 225, 226, 229, 232, 241, 242, +257, and 258. All statements include the inherited native frame views. + +Both data banks can therefore be scaled by a common delta without +changing the coarse part of a centre row. In physical centre frame H_k, +the backward carrier is H_k(u_k+delta R_k y/4). An exact coarse reset +would replace it by H_k u_k. A grid and error proof must show that a +permitted rounding operation makes this replacement. + +## The two suffix maps + +Let w_old be the incoming centre value and replace it by an arbitrary +w_new before the retained suffix. No centre sign is applied. The exact +maps are + +``` +F: Xout=x; + Yout=y+x-K w_old; + uout=w_new-Rx. + +B: Yout=y; + Xout=x-y+R^T(w_new-u); + uout=w_new. +``` + +Here K=R^T/4. All 472 other helpers return to their input contents. +These formulas follow from the complete suffix impulse responses. A +forward centre increment d changes only the mode output by d. A backward +increment d changes the mode output by d and the X output by R^T d. +The same identities were proved in `CROSS-CUT-PROOF.md` and +`CROSS-CUT.json`; the new checker replays them directly. + +For B choose w_new=u. Then the output is (x-y,y,u), with all 480 helper +contents restored. The retained native rank is 5522. There is no new +native child in this oracle interface. The eight removed complex signs +reduce the retained scalar bound from 204418 to 204402. This figure +excludes all reset, data-scaling, precision, and address costs. + +For F the data output is independent of w_new. This remains true when +w_new is a nonlinear function of the cut values. A coarse reset, a fine +reflection, or a negative coarse value cannot remove the term -KRx. +An input correction Y+=Ku cancels only the old helper term. An output +mode correction cannot change Y. A forward improvement needs an edit +at another carrier or an additional target access. + +## Exact checks and precision boundary + +`check_centre_reset_interface.py` checks all 704 initial columns, the +full eight-column suffix response, all 480 backward helper outputs, +and the exact eight sign locations. A control that erases the arbitrary +coarse input fails. Forward reset and reflection have equal data outputs; +neither gives the required unit shear. + +The result is `CENTRE-RESET-INTERFACE.json.gz`, with decompressed SHA-256 +`fa0b2c75f8fdcbe75aa2f35f77559a06477d99adcf0d69040cba075743108868`. + +``` +python3 -I -O -B work/centre-encoding-campaign-20261014/lanes/history-encoding/check_centre_reset_interface.py +``` + +Expected status: + +``` +PASS_EXACT_CENTRE_RESET_ORACLE_INTERFACE +``` + +The separate frozen source packet is `INTERNAL-MODE-SOURCE-ROWS.json.gz`, +decompressed SHA-256 +`ff1ae64f2c81d3192ef1f2f0cf0c5d984eed9215b200bbe9d4d54bf630925d38`. +Its exact reverse scalar cones contain 132 native events of total rank +156 for the backward centres, and 616 events of total rank 784 for the +forward centres. Every listed input has an initial-helper component, +even if both data banks are scaled. The packet lists endpoint views +separately. These counts describe the centre read cones. They are not +the full final-data error cones after a reset. + +The digit study must prove the coarse lattice, the separation margin, +the effect of each approximate native job, the output decoding error, +and the complete finite cost. In particular, errors on paths outside +the reset carriers can remain in the final data. This note makes no +precision, supplier, or kappa claim. + +OpenAI Codex performed this exact interface check under the user's +direction. A separate backward error-cone review is in progress. diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CROSS-CUT-PROOF.md b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CROSS-CUT-PROOF.md new file mode 100644 index 0000000000..90e815fde1 --- /dev/null +++ b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/CROSS-CUT-PROOF.md @@ -0,0 +1,87 @@ +# Transfer through a common centre cut + +This result applies to two separate local words in the same native chart. +The first word is forward. The second word is backward. It does not prove +that the selected global owner schedule can stop both words at this cut. + +The base is `FULL-MODE-PROGRAMS.json.gz`, with decompressed SHA-256 +`f271842a741584389e166e700e262f161b0dd570ce8c7a540f2717e90ead5205`. +The checker pins the canonical word as well. All values can be arbitrary. +In particular, neither of the two eight-mode inputs must be zero. + +Write `K=R^T/4`. At the cut just before `retained_centre_scale`, the eight +centre registers contain + + W_F = R x_F + u_F, + W_B = R y_B/4 + u_B. + +The corresponding centre registers have the same native frame in the +two words. Each frame has rank eight. Remove the eight scales by -1 +from each word and apply a two by two matrix C to the two values at +each centre. Then execute both retained suffixes. The result is + + u_F_new = C00 W_F + C01 W_B - R x_F, + u_B_new = C10 W_F + C11 W_B, + y_F_new = y_F + x_F - K W_F, + x_B_new = x_B - y_B + R^T (u_B_new-u_B). + +The other two data banks stay fixed. All 944 ordinary helpers are +restored. A dyadic invertible C gives a reversible local interface. + +To prove these formulas, use the exact suffix responses. A change d +in the forward cut changes only its mode output, by d. A change d in +the backward cut changes its mode output by d and its X output by +`R^T d`. These responses include all 704 outputs of each word. Linearity +then gives the formulas, including the complete arbitrary helper claim. + +The checker independently derives the two cut queries and the two +suffix responses from the pinned programs. It also emits three literal +instances and checks all 1,408 input columns, all scalar frame accesses, +and all ordinary helper outputs. + +| C | Scalar gates at the cut | Complete real component bound | +| --- | ---: | ---: | +| `[[0,-1],[-1,0]]` | 32 | 409012 | +| `[[1,1],[0,1]]` | 8 | 408868 | +| `[[1,1],[0,-1]]` | 16 | 408884 | + +These bounds include the two retained local scalar words. They exclude +native child cost, address cost, and any global owner repair. The native +mass is twice 5522. No new child or array is added by the cut coupling. + +## Physical maps at the cut + +The scalar theorem also gives the exact order for an operator transfer. +Let `B_k` be the physical native frame of centre k. For a physical +pointwise monomial M between the matching centre registers, the logical +coefficient is + + C_k = B_k^-1 M B_k. + +For the Walsh retarget, replace `B_k` by its exact retargeted frame. +The common-frame relation remains true. A physical sign, translation, +unit phase, or address permutation can therefore be inserted without a +new recursive child. Its logical value must use the displayed ordered +conjugation. A logical monomial need not remain a physical monomial under +that conjugation. This result does not identify a port-dependent map on +112 data families with one of these eight centre maps. + +The same suffix proof applies to operator entries of C. No entry is +commuted through R or K. A reversible block monomial can be lowered by +its explicit address and phase rules. Such rules, their cost, and their +native chart are required before an operator instance is admitted. + +## Reproduction and scope + +Run `check_cross_cut.py` with the pinned canonical word, the pinned full +mode program, and `verify_echo.py`. The output program is +`CROSS-CUT-PROGRAMS.json.gz`; the receipt is `CROSS-CUT.json`. The status is +`PASS_EXACT_COMMON_CUT_TRANSFER_AND_MIXING`. + +The local role sets in this check are separate. A simultaneous cut in the +selected supplier needs a new owner audit. The sequential endpoint-pair +owner patch does not supply that audit. No complete supplier, final +decoder, or kappa improvement is claimed here. + +Research and verification: OpenAI Codex under the user's direction, with +substantial AI assistance. No independent human review is claimed. diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/FULL-MODE-PROGRAMS.json.gz b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/FULL-MODE-PROGRAMS.json.gz new file mode 100644 index 0000000000..d53ce4b3cf Binary files /dev/null and b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/history-encoding/FULL-MODE-PROGRAMS.json.gz differ diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/FULL-MODE-PORT-REVIEW.json b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/FULL-MODE-PORT-REVIEW.json new file mode 100644 index 0000000000..23276ce660 --- /dev/null +++ b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/FULL-MODE-PORT-REVIEW.json @@ -0,0 +1,51 @@ +{ + "checks": { + "backward": { + "all_persistent_columns": 704, + "native_wrapper_placement": "none", + "new_monomial_maps_after_lowering": 0, + "new_rank_one_children": 0, + "ordinary_helper_columns_restored": 472, + "paid_rank_mass": 5522, + "real_component_scalar_primitive_bound": 204418, + "retained_native_events": 5056 + }, + "forward": { + "all_persistent_columns": 704, + "native_wrapper_placement": "none", + "new_monomial_maps_after_lowering": 0, + "new_rank_one_children": 0, + "ordinary_helper_columns_restored": 472, + "paid_rank_mass": 5522, + "real_component_scalar_primitive_bound": 204418, + "retained_native_events": 5056 + } + }, + "complete_native_operator_equalities": 0, + "global_phase_controls": 0, + "inverse_transpose_scope": "Logical scalar coordinates after the stated native entry and exit frames are stripped", + "limits": [ + "The mode output is the full frame; its phase type changes under repeated routed calls.", + "Eight previous cold helper roles are now productive state roles.", + "No common-coordinate global controller or owner schedule is certified." + ], + "mode_labels": [ + 11, + 11, + 11, + 11, + 11, + 11, + 11, + 11 + ], + "mode_type": "full", + "native_scope": "All tensor powers by exact operator tensoring and the common-frame scalar lemma; arbitrary data and ordinary borrowed inputs", + "normal_recipe_source_sha256": "6c856f4d321bd63fc0b4869dcd8385b5e9851f38f9a1b8fb19a1dbd9c73e0eb9", + "program_payload_sha256": "f271842a741584389e166e700e262f161b0dd570ce8c7a540f2717e90ead5205", + "source_compressed_sha256": { + "CANONICAL-WORD.json.gz": "16bc0ebb04b4be6c0ade74badc6a0c0cdba4215eda86e4ff9f13abf8ecebd219", + "LOCAL-PROGRAMS.json.gz": "241fde27d4e43f5a9cb6b128f9f76c5131a53d09f0371676163f914d0b0fef3a" + }, + "status": "PASS_INDEPENDENT_EIGHT_MODE_NATIVE_PAIR" +} diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.json b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.json new file mode 100644 index 0000000000..0e99974eb6 --- /dev/null +++ b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.json @@ -0,0 +1,52 @@ +{ + "archive_members": { + "candidate/strict/global/audit/OWNERS-AND-PROFILE.json": "30bafff060afcfb3644e951214d66cfaffe58242881c652e28003958b42127cc", + "candidate/strict/global/hosted/GLOBAL-TEMPLATE.json.gz": "2acf02b418078e37090495c601fe550448751d4f35912757b13a1f778efc7760" + }, + "archive_sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6", + "arithmetic": { + "controls": [ + "missing normalization rejected", + "external unscaled helper rejected", + "single-accuracy parent return grid rejected" + ], + "direct_kernel_relative_cases": 300, + "exact_public_rounding_cases": 100, + "normalization_exponent_range": [ + -82, + 17 + ], + "return_grid_cases": 36, + "zero_cases": 5 + }, + "checker_sha256": "3147a25d6c799df463a03c82397f1511ea538ea79197dae5f7d5134fd0c406f1", + "potential": { + "controls": [ + "root-width cap rejected", + "insufficient additive potential rejected" + ], + "format_potential_cases": 864, + "maximal_rank_doubling_depth": 38, + "stack_factor_bound": "7168/7167" + }, + "review_inputs": { + "HETEROGENEOUS-PRECISION-CONTRACT.md": "4c47389d756d61fea7a672e03494e5219d83bbd7d1b2f7526a62cf411edc6607", + "PRECISION-FORMAT-CHECKS.json": "daf2f3040578884ce2a5d5c1c0c01dc5c43e6f25deea5e5f1dd20c195943e719", + "check_precision_formats.py": "ecbdd2d717510ded0ffb696e13d5544956075d4de94cec4a1f9719dd4b454356" + }, + "scope": "Exact finite controls and pinned source checks. The all-size conditional proof review is separate. No changed circuit census, error-transfer premise, supplier, or finite invoice is certified.", + "selected_constants": { + "W": 14336, + "m": 73, + "maximum_child_rank": 48 + }, + "source_revision": "3ae9a6f3ac74ded022ab3b4872a7357a621af6fb", + "source_sha256": { + "notes/batched-complex-rows.tex": "64c70926a8705874f1c9b42a2f703e42cad4e90f927f544458cfdf729540236c", + "notes/compact-control-layout.tex": "2bc28e50f3eb092195294cafa81213e520f6d42e7935e9905ad8a6199e2a7c87", + "upstream/build/sections/02-streams.tex": "606c80db61cad13aa0c3b6060dc4b88dbbe7ac60cdd831aa93f28d908fd09dab", + "upstream/build/sections/04-swap.tex": "412b170ccaab38dba96e95e3cbfae09f4876a9818f8d278776a746b0f8044906", + "upstream/build/sections/05-layers.tex": "20cfc8d12099d4e4ed9e3e7eeb6162edd9b6e0e076f24693364cd24377252594" + }, + "status": "PASS_INDEPENDENT_PRECISION_INTERFACE_CONTROLS" +} diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.md b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.md new file mode 100644 index 0000000000..0da17c1bf9 --- /dev/null +++ b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/REVIEW.md @@ -0,0 +1,197 @@ +# Independent review of separate child precision + +Result: PASS for the conditional arithmetic and tape interface. This +review found one scope error in the first proof version. The author fixed +it. No material defect remains in the revised statement. + +The reviewed proof is +`temporal-reuse/HETEROGENEOUS-PRECISION-CONTRACT.md`, SHA-256 +`4c47389d756d61fea7a672e03494e5219d83bbd7d1b2f7526a62cf411edc6607`. +Its first version had SHA-256 +`73713ee5390f879cc7306978a1560a56f39b6738e8fcb04e4fbebfd716fcc2aa`. +The first version stated a `2^O(e)` bound for every finite prefix and +suffix. An explicit `delta^-1` border has a factor `2^Theta(t+e)`. +The revised paragraph applies the bound to the unscaled core and fixed +graphs only. The error-transfer proof and record format must pay for the +border scale separately. This is the required distinction. + +## Physical child and helper contract + +The source revision is +`3ae9a6f3ac74ded022ab3b4872a7357a621af6fb`. I read Section 05, the grouped +complex recurrence, the complete compact-field layout, the fixed-tape +schedule in Section 04, and the signed stream arithmetic in Section 02. +The checker binds all five source +files by SHA-256. It also binds the original selected archive and its +template and owner-profile members. These give `m=73`, `W=14336`, and +maximum child rank 48. + +Section 05 splits the entire parent row by `u=Wg+w`. One child receives +one complete role stream. Its next recursive row split takes place +inside that stream. Thus the values that become its internal borrowed +helpers are part of its input, even when they are correlated or large. +They must be included in the normalization maximum. The caller's other +roles remain on the LIFO tape and are not available as unscaled helpers. + +This distinction is necessary. An approximate identity on `(x,h)` may +return `(x+epsilon*h,h)` and have error at most +`epsilon*max(abs(x),abs(h))`. If only `x=delta` is scaled and an external +`h=1` is borrowed, its principal error is not at most `epsilon*abs(x)`. +The reviewed source layout does not require that invalid call. + +The width change occurs during child-input and return copies. Each +scalar or packed address operation sees one common, fixed record width +at its node. The packed operation remains an exact permutation of those +encodings. No coefficient rounding takes place while it holds temporary +XOR combinations. No new owner field or address axis is required for +the numerical normalization. Its copies use a fixed number of work +tapes, as do the retained row scheduler and record arithmetic. + +## Numerical proof + +For a nonzero complete input with maximum component `M`, choose `b` with +`2^(b-1) <= M < 2^b`. An absolute error at most `2^(-t-1)` after +normalization gives an error at most `2^-t*M` after multiplication by +`2^b`. This holds for negative `b` and for arbitrary helper contents. +The zero branch is valid because the required principal endpoint is +linear and no external helper is touched. + +The real block row sum of the raw tensor is at most `2^e`. Thus input +truncation with `t+e+O(1)` fractional bits has an allocated error of +order `2^-t`. A separate output conversion allowance is required. +This is not a claim that the internal approximate word is homogeneous. +Its error guarantee must hold on the actual rounded input. + +The parent return grid is also a separate allowance. For the stated +single source restoration, let +`delta^-1 <= 2^(t+D*e+D0)` and bound an unscaled suffix by +`2^(S*e+S0)`. If there are at most `N` conversion errors, a grid with +`2t+(D+S)*e+O(log N+D0+S0)` fractional bits bounds their restored sum +by a fixed share of `2^-t`. The finite constants can make this share +strictly smaller than the remaining budget. A different border with +more powers of `delta^-1` must change the coefficient of `t` in this +grid. Any fixed such coefficient still fits `O(t+e)` bits; it does not +follow from the displayed two-copy grid without a new error bound. + +The normalization exponent can be computed by a highest-bit scan on +absolute signed components. A second scan writes the shifted input. +Both scans, the sign conversion, and all required reversals have linear +bit cost. A large negative exponent does not require repeated shifts. +On return, the stream is clipped to the parent's format. It is not +necessary to retain every unscaled low bit. The first input scan must +read the old format in full; its cost belongs to the caller's input-copy +allowance and is not charged only at the smaller child width. + +The magnitude bound for the new word must follow from its finite +perturbation estimate. The old exact-child dependency guard cannot be +used for this purpose. The reviewed note makes this restriction explicit. +Likewise, the finite error-transfer schedule must be proved in the norm +used here, with the complete child input and error-output spaces. The +core sensitivity census does not prove that schedule for a changed +controller or its graphs. + +## All-size time and simultaneous space + +Put `theta=r_max/m < 1`. Choose an integer +`c >= A/(1-theta)` and define `w(e,t)=t+c*e`. For +`e_j=r_j*floor(e/m)` and `t_j=a_j*t+A*e`, where `a_j` is 1 or 2, + + w(e_j,t_j) <= a_j*w(e,t). + +This is an all-size inequality; integer remainders only decrease its +left side. Fixed additive precision constants can be included in `A*e` +at internal nodes. The finite number of remainder axes is part of the +node work. + +Choose one integer format multiplier `L`, common to all nodes, large +enough for both integer and fractional fields. A real component can use +exactly `L*w(e,t)` bits. This removes any ambiguity from independent +rounding of record lengths. Each child has one `W`th of the coefficient +count. Its physical coefficient volume is therefore at most +`a_j/W` times that of the caller. Fixed headers and work areas change +the common space constant only. + +In particular, one active descent contracts the potential volume by at +most `2/W`. The total potential of simultaneously parked ancestors and +the active node is at most + + V_root * sum_(k>=0) (2/W)^k + = V_root * W/(W-2). + +For the selected `W`, this factor is `7168/7167`. The scheduler uses a +fixed number of copies at each active level. Hence total simultaneous +data space is `O(V_root)`, even though a deep child's component width +can exceed the root component width. No ancestor record is widened. +Section 04 also gives a stack schedule for the address subroutine. Its +child volumes contract by `1/W_bit`, its padding factors are fixed, +and its work areas are cleared at each return. The same geometric sum +bounds that temporary data while it is used inside one active numerical +node. Descriptor stacks fit the retained polynomial-record allowance. +This is a space result for the physical scheduler, not a statement that +the maximum precision itself stays at the root value. + +Let `v_j` be a node's potential volume divided by the root potential +volume. Then its children satisfy + + sum_child v_child * e_child^sigma + <= v_j * e_j^sigma * Psi_prec(sigma), + +where `Psi_prec(sigma)=(1/W)*sum a_j*(r_j/m)^sigma`. If this moment is +strictly below 1, the grouped source proof applies with these weights. +At a stopping leaf, `e= g+e+B+3` put the error strictly below half a grid unit. Nearest +grid rounding then recovers the exact raw output on every principal +row. This includes every row that served as an internal helper. It +does not repair a wrong encoded endpoint or an unproved owned-mode +restoration. Internal approximate children need not use this exact +public rounding rule. + +The independent checker imports no producer. It uses the direct tensor +kernel `C[y,x]=alpha^(e-h)*beta^h`, with `h=popcount(x xor y)`, rather +than the producer's butterfly replay. Its exact rational checks include +300 relative cases with a separate internal error, 100 exact public +rounding cases, 36 parent-return cases, five zero cases, and 864 +potential cases. It checks a 38-edge maximal-rank descent with precision +doubling on every edge. Five controls reject missing normalization, +external unscaled helpers, a coarse return grid, a root-width cap, and +an insufficient additive potential. + +Run from the workspace root: + +```text +python3 -I -O -B work/centre-encoding-campaign-20261014/lanes/interface-proof/heterogeneous-precision-review/check_review.py --workspace /Users/aliromman/Projects/IntegerMultiplicationBounds --out /tmp/heterogeneous-precision-review.json +``` + +Expected first line: + +```text +PASS_INDEPENDENT_PRECISION_INTERFACE_CONTROLS +``` + +The finite checks support the proof; they do not prove it by enumeration. +No approximate native supplier, changed error census, complete numerical +invoice, or new kappa value was verified. This review is independent of +the failed direct terminal-border construction. It makes no claim about +the cause of that failure. + +OpenAI Codex assisted this local research. No PR was prepared. 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+ }, + "plain_product_phase_failures": [ + { + "actual_phase": 2, + "frame": 415, + "input": 0, + "output": 34, + "plain_product_phase": 3, + "subset": 8 + }, + { + "actual_phase": 2, + "frame": 416, + "input": 0, + "output": 5, + "plain_product_phase": 3, + "subset": 64 + }, + { + "actual_phase": 2, + "frame": 417, + "input": 0, + "output": 17, + "plain_product_phase": 3, + "subset": 16 + }, + { + "actual_phase": 2, + "frame": 418, + "input": 0, + "output": 36, + "plain_product_phase": 3, + "subset": 8 + }, + { + "actual_phase": 2, + "frame": 419, + "input": 0, + "output": 66, + "plain_product_phase": 3, + "subset": 4 + }, + { + "actual_phase": 2, + "frame": 420, + "input": 0, + "output": 65, + "plain_product_phase": 3, + "subset": 4 + }, + { + "actual_phase": 2, + "frame": 421, + "input": 0, + "output": 65, + "plain_product_phase": 3, + "subset": 4 + }, + { + "actual_phase": 2, + "frame": 422, + "input": 0, + "output": 68, + "plain_product_phase": 3, + "subset": 4 + } + ], + "scope": "Actual canonical phases and sufficient physical lattice bounds. No cancellation of listed basis vectors, native accuracy choice, finite tagged word, or supplier invoice.", + "status": "PASS_ACTUAL_CANONICAL_PHASES_AND_WHOLE_CUT_GRID", + "whole_cut_fields": 704 +} diff --git a/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/centre-reset-review/PHYSICAL-GRID-V2.md b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/centre-reset-review/PHYSICAL-GRID-V2.md new file mode 100644 index 0000000000..72a460b37a --- /dev/null +++ b/research/finite-tagged-centre-reset/frozen/work/centre-encoding-campaign-20261014/lanes/paired-package-review/centre-reset-review/PHYSICAL-GRID-V2.md @@ -0,0 +1,151 @@ +# Actual canonical phases and the whole backward cut grid + +This note corrects the physical-kernel argument in the first reset +review, receipt `f86a6122d2f883730206b37f8d21273a8b7f20d59738cf48f277423ef7fe17b5`. +That review expanded a product of intrinsic C_v gates for each listed +frame basis. The actual canonical frame includes a quadratic phase +conjugation. Its listed basis does not specify that plain product. +The earlier phase claim is withdrawn. The centre lattice and norm +conclusions remain valid, for the corrected reason proved here. +The exact cut response, rounding formula, and error-cone census in +the first review are unchanged. Its files are preserved. + +## Source and actual centre operators + +The selected source archive is +`f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6`. +Its exact native recipes are pinned at +`d0768f2164d5f64a2fdbec422f1cfde4e400818792f1904ca541958377243731`. +The canonical operator definition is read from `clifford_exact.py`, +SHA-256 `2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962`. +The checker imports none of that code. + +Let G be the exact canonical basis extension. Put + + Q=I+(G^-1)^T G^-1 over F2, + K=D_Q P_G. + +The rank-r canonical frame is `K F_r K*`. The phase of D_Q is + + q(x)=sum_i Q_ii x_i + 2 sum_(i