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"e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e", + "payload_sha256": "44290efbd61ee8d4d7a2698cf083892eff0c1f601605f3d4d28b284ff42c0b68" + }, + { + "file": "inputs/pr1319-extra.json.gz", + "commit": "603defe58aac018c6a8457840481390a15846fe4", + "source_path": "research/global-mixed-t74-exact-fill/vendor/pr1297/extra-word.json.gz", + "git_blob": "4d464f55a2955e5c832b5ce789d5d0e0a39fb4e1", + "sha256": "7b0e9fbda32749aabc7e156d9fe09bceb0624a178628fe69e5dfea783462f905", + "payload_sha256": "60f07abaafd6ede2bb42033dacab5b785b36bfa4fd6b153480608c9ca07af4f0" + } + ], + "orientation_program": { + "file": "inputs/selected-programs.json.gz", + "commit": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f", + "source_path": "research/community-frontier1042-audit/selected-replay.tar.gz", + "git_blob": "6f4dc6fe7631d8aacf8081fcaecf32197181685b", + "archive_member": "t16/global/base/providers/gauge44/LOCAL-PROGRAMS.json.gz", + "sha256": "241fde27d4e43f5a9cb6b128f9f76c5131a53d09f0371676163f914d0b0fef3a", + "payload_sha256": "9f41d53659e9be84e20c661560c5687abbf5cd0660cdecbe01ad5eb29e6b8ee3" + } +} diff --git a/research/complex-block-rank-optimum/PROOF.md b/research/complex-block-rank-optimum/PROOF.md new file mode 100644 index 0000000000..31c6c2bdf2 --- /dev/null +++ b/research/complex-block-rank-optimum/PROOF.md @@ -0,0 +1,323 @@ +# Complete moment optimum for twelve fixed block interfaces + +This note proves an all-order and all-length result. It does not use a bound +on the number of scalar gates or a list of coordinate changes. The finite +checker verifies the interface premises. The argument below proves the result +for every program in the stated class. + +## 1. Interface and program class + +A replacement uses the retained single-word provider interface. One canonical +forward word defines both orientations. The backward scalar program is the +same-order inverse-transpose dual of that word. Each ADD exchanges its source +and destination and negates its coefficient; each SCALE takes its reciprocal. +Both orientations use the same array-frame itinerary. Independent redesign of +the backward supplier with a different boundary contract is outside this class. + +Each block has eight helper arrays of the same volume. Four helpers start at +one-dimensional frames contained in a common frame U. Four helpers start at U. +Write u = dim(U), where u is 4 or 5. Their eight initial scalar values are +independent and arbitrary. Let x be the four values that start at U, and let y +be the other four values. + +There are eight read-only external input arrays. Their independent values are +z_i. Input i has a fixed frame F_i. Every F_i contains U. The F_i are pairwise +incomparable. Their dimensions f_i are 5 or 6. If u=5, every f_i is 6. + +There are four additive output ports. Port j has its retained late entry frame +G_j, of dimension 7 or 8, and its distinct final hyperplane H_j, of dimension 8. +The source word gives G_j immediately before the first late read from this +block into that target. For every i,j, dim(F_i+G_j) is at least 8. Earlier target +reads and all exterior operations remain fixed. This is the late block +interface: output retiming to an earlier exterior frame is not part of it. + +The required increment at these ports is R y + Q z. There is no x term. +Every column of Q is nonzero at exactly two distinct output ports. All eight +helpers must retain their independent dirty information and end at the full +nine-dimensional frame. The retained helper endpoint map can be imposed in +addition; the proof grants any invertible helper endpoint map for its lower +bound. + +A program may use any finite number and order of invertible scalar additions +and nonzero scalar multiplications on the eight helpers. Its coefficients are +fixed, not functions of the input values. It may read an +external input repeatedly into helpers, and add helpers to the four output +ports repeatedly. Each addition uses one common frame on its two arrays. +Frames only grow by inclusion. We allow every such frame in the lower bound, +including one that might fail a separate native phase condition. The exact +native transport for a rank-r growth +is charged as one rank-r recursive child. Splitting a growth is allowed and +every part is charged. The class does not allow an extra array, a frame descent, +a target used as scratch, a write to an external input, a changed centre path, +or communication between different blocks. Coefficients may be arbitrary +rational numbers; the argument also works over any characteristic-zero field. + +The sources contain twelve blocks with helper bases +592, 600, 608, 624, 632, 640, 648, 656, 672, 680, 688, and 696. +For a base b, its helpers are b,...,b+7. The scalar correction can move past +intervening exterior events to the input segment: it touches only these eight +helpers, and no intervening exterior event uses them. No new operation or +frame is needed for this move. The different times at which the external ports +are ready remain part of the exterior interface; granting all fixed input +ports for the whole block only enlarges the program class used in the lower +bound. + +### Binding the interface to the emitted words + +Move only the helper-only correction to immediately before its eight input +additions. The checker verifies that these additions are contiguous, use eight +distinct external arrays, and have no intervening exterior operation. It also +verifies that every external array has its specified F_i at this common +boundary. The low helpers still have their old one-dimensional frames before +the deferred correction. The high helpers still have frame U. Thus this is +one common boundary with sixteen distinct physical main arrays. + +These sixteen values can vary independently. All main arrays in the retained +native contract have arbitrary initial data. Each addition, nonzero scaling, +and frame transport is invertible. The copied-centre cut adds into target +arrays from copies of non-target arrays. No cut source is a cut target. The +cut is therefore an invertible shear on the main arrays, even though the +copied centre arrays are not independent new inputs. The prefix after the +unchanged preconditioner is invertible. Removing the deferred invertible +helper correction preserves this fact. Any sixteen distinct coordinates of +these main arrays are independent at the boundary. + +The certificate gives an additional exact scalar witness for each block. It +pulls the sixteen boundary rows back through the literal A, cut, and B prefix, +with only that correction removed. It selects sixteen original scalar columns +and gives a nonzero 16-by-16 minor. A second calculation replays those columns +forward through the prefix and verifies the same minor. Every minor uses only +eight original source columns and the same eight dirty helper columns. The +checker verifies that the unchanged preconditioner touches only targets. +Thus these witnesses need no independently variable target value. They do not +assume clean helper values or introduce extra physical arrays. + +The checker derives R and Q from all subsequent emitted gates that touch the +eight helpers. It records each input index, each late output index, the target +frame immediately before its first late output, and the final target frame. +It verifies that no late target is subsequently scaled or used as a source. +Other additions into those targets commute. Thus the four accumulated port +increments are the actual required increments in the emitted word, and not a +separate handwritten correction matrix. + +Universal replacement of this block must therefore satisfy the displayed +matrix identity on all sixteen inputs. Earlier asynchronous availability does +not create a relation between these values. Giving the replacement all eight +external inputs from the start only enlarges the class in the lower bound. + +## 2. Exact finite premises + +The checker reconstructs Q and all frames from each frozen word. It checks: + +1. Q has rank 4. For every set S of at most four coordinates, projection of + ker(Q) onto S has rank at least ceil(3|S|/4). It computes this rank as + |S|-4+rank(Q restricted to the complementary columns). +2. No nonzero vector in ker(Q) is supported on a set of input frames whose + union has dimension at most 6. +3. If a kernel support has frame union F of dimension 7, every input frame + contained in F has dimension 5, and F is contained in at most one H_j. + +These statements concern all subsets of eight columns, not a sampled set. +They also apply to an arbitrary frame F of dimension at most 7. If F contains +a kernel support, premise 2 forces that support's frame union to have dimension +7. Hence that union equals F, and premise 3 applies. + +## 3. Input barrier, with repeated reads allowed + +Each external input must be read, because its required output column is +nonzero. A helper cannot read two different inputs: it would have to visit +both of their incomparable fixed frames. There are exactly eight helpers and +eight inputs. Thus input-to-helper assignment is a bijection. + +This does not assume one read. A helper can read its assigned input repeatedly +at that input's frame. It cannot mix with an as-yet-unread helper after its own +first read: their common frame would contain its F_i and would have to lie in +the other helper's incomparable F_j. When two already-read helpers mix, both +advance strictly beyond their individual input frames. Neither can read its +input again. A target read also precludes any later input read, because target entry rank +is at least 7, above every input rank. In particular, a helper cannot emit +an output before its first assigned input: that would leave one of the eight +required inputs without an available helper. + +It follows that all pre-input mixing can be moved before all input reads, and +all post-input mixing after them. Operations moved past each other use disjoint +helpers. Repeated reads and scalar multiplications combine to a nonzero +coefficient d_i for input i. A zero coefficient would erase that input from +all outputs. The helper values at this barrier are therefore + + h = P (y,x) + D z, + +up to an input permutation, where P is invertible and D is invertible diagonal. +No helper is assumed clean. An invertible scalar program on eight independent +dirty values cannot create an identically zero helper. A helper may evaluate +to zero on a particular input. That fact cannot change the required linear +identity on arbitrary inputs, or the common-frame requirement of an addition. +There is no read-once assumption or value-based frame reset in this argument. + +Let P_x be the four columns of P for x. The output map from h is Q D^-1. +Since x has zero output response, Q D^-1 P_x=0. The matrix P_x has rank 4, +so its scaled column space is exactly ker(Q). Thus the projection bounds in +premise 1 apply to every choice of helper coordinates in the program. + +## 4. Cost before the input barrier + +For 04->6, its possible saving is at most + + A(s)=3^s+2^s-4^s-1. + +The x information in these k rows must come through helpers that entered the +rank-5 region carrying x information from rank 4. Each such carrier supplies +at most one new independent x direction when it first enters that region. +It cannot return to rank 4. Later mixing within rank 5 adds no new direction. +A low helper that first acquires x at rank 5 is already counted in S, so it is +not a new source of x information. Thus, if c is the number of carriers, + + c >= rank(P_x restricted to S) >= ceil(3k/4). + +Every carrier must be assigned an input of rank 6: a helper assigned a +different rank-5 input cannot meet another helper at that frame. Each carrier +therefore pays a rank-4 stop, a rank-5 stop, and its rank-6 input stop. It is +not in S. Relative to a direct 4->6 growth, its extra cost is at least + + B(s)=2-2^s. + +The elementary inequality 4A(s)<=3B(s), proved below, gives kA(s)<=cB(s). +All possible savings are paid by distinct carrier costs. If n_5 and n_6 are +the counts of input ranks 5 and 6, the complete pre-input cost is at least + + 4*3^s + n_5 + n_6*2^s. + +For the power inequality, put t=2^s in (1,2) and p=log_2(3). We need + + f(t)=4t^2-7t-4t^p+10 >= 0. + +The integer inequalities 9>8 and 243<256 give 3/20 +on [1,2], while f'(2)=9-6p<0. Thus f is decreasing on [1,2], and f(t)>=f(2)=0. +The inequality is strict for 0=8. The output caps are the distinct +rank-8 hyperplanes H_j. + +Each helper must mix with another helper before it first reaches rank 8. +Until that first mixing, its injected variable occurs in no other helper. +If it reached rank 8 first, every later path for that variable would contain +that rank-8 frame and could reach at most one of the distinct H_j. Its required +column has two nonzero entries. This is a contradiction. The first mixing has +rank at least f_i+1, because it includes another incomparable input frame. + +It remains to account for a helper that has already helped deliver outputs +and then jumps from rank at most 7 directly to rank 9. This case is permitted; +the proof does not impose a rank-8 stop on every helper. + +Just before such a jump, put a nonzero perturbation on that helper and zero +on all other current helpers and outputs. Reverse the full executed prefix. +At each reversed addition, the perturbation can reach only a register whose +frame is contained in the retiring frame F. Thus it cannot reach an earlier +target read, all of which have rank at least 8. All outputs already written +have zero response to this perturbation. At the input barrier it is a nonzero +vector v supported only on input frames contained in F. It is nonzero because +the helper transformation is invertible. + +After the jump to rank 9, no path from that helper can reach a rank-8 target +cap. Hence all later outputs also have zero response. Exactness implies Qv=0. +This argument includes arbitrary information splitting and all earlier output +writes, not only a terminal kernel condition. + +Premises 2 and 3 now imply dim(F)=7, f_i=5, and F is eligible for at most one +output port. This helper must have mixed earlier at rank 6. If its first +post-input mixing occurred at rank 7, that frame would be F by monotonicity; +the still unique injected variable could then reach at most one output, a +contradiction. Its rank path therefore includes 5->6->7->9 and costs at least + + 2+2^s. + +If a rank-5 input helper does visit rank 8, its first mixing is at rank 6 or 7. +The cheapest path through such a stop, rank 8 and final rank 9 has the same +cost 2+2^s. A rank-6 input helper cannot retire below rank 8 by premise 3. +Its path includes 6->7->8->9 and costs at least 3. The total post-input cost is +therefore at least + + n_5*(2+2^s) + 3*n_6. + +## 6. Attainment and complete supplier moment + +Combining the two parts gives the following optimum for one block: + + u=4: 24 + 8*2^s + 4*3^s; + u=5: 32 + 4*4^s. + +The exact native source word attains these expressions. The checker recounts +all eight helper paths, including their final growth to rank 9. This is an +optimum over arbitrary finite scalar factorizations in the declared class, +not a finite search optimum. + +The orientation rule matters for the controller conclusion. The pinned +`hpe_emit_local.py` computes the frame climbs from the canonical gate before +it applies its `dual` scalar rule. Swapping an ADD's two arrays leaves its +common-frame requirement unchanged. A reciprocal SCALE uses the same array. +Thus every scalar replacement emits the same ranked helper climbs in both +orientations. Algebraically, same-order inverse-transpose replaces a complete +matrix M by M^-T. It does not require reversing the frame itinerary. The centre +copy path reverses direction but retains each absolute rank change and the +same complete charge. Centres are fixed in this theorem. + +The selected emitted LOCAL-PROGRAMS file is included as exact evidence. The +checker verifies equality of all 3486 climb events, inverse-transpose duality +of every non-centre scalar, and equality of the complete paid profile. The +same retained emitter is the native provider path for the two pinned PR1319 +words. This is a source-bound orientation contract, not a claim that every +possible independently synthesized backward circuit has equal cost. + +Let L(a) be the sum of block costs at s=1-a. For any replacement of any subset +of the twelve blocks, L_new(a)>=L_old(a) for every 0= [H_rest(a) + sum_calls L_new(a)] / (W*73^(1-a)) + >= Psi_old(a). + +Any additional paid source, phase, or restoration work explains the first +inequality. The retained word attains equality with its original block costs. +The comparison holds pointwise, not just at one claimed endpoint. The complex +admissible exponent cannot increase under this replacement class. The bit +constraint and all other outer-recipe conditions are unchanged. Every admissible tuple +of exponents and outer parameters for a replacement is therefore also +admissible for the retained supplier. This feasible-set inclusion proves that +the optimum complete outer kappa cannot improve through this class. It does +not identify the bit primitive exponent c with the outer kappa, or turn the +complex-only moment root into an outer exponent. + +The selected PR1094 word has seven u=4 and five u=5 blocks here. Their total is +328H1+56H2+28H3+20H4. The PR1319 base provider has ten u=4 and two u=5 blocks, +with total 304H1+80H2+40H3+8H4. Its extra provider has eleven u=4 and one +u=5 block, with total 296H1+88H2+44H3+4H4. The conclusion applies separately to each +provider and hence to the stronger pending mixed construction as well. + +This closes the fixed eight-array late-interface coalescing approach. It does +not exclude changed external availability or output frames, target scratch, +extra arrays, cross-block coding, frame descents, an independently designed +backward interface, or another native operation alphabet. Such changes need their own complete cost and interface proof. diff --git a/research/complex-block-rank-optimum/README.md b/research/complex-block-rank-optimum/README.md new file mode 100644 index 0000000000..77bd353fda --- /dev/null +++ b/research/complex-block-rank-optimum/README.md @@ -0,0 +1,173 @@ +# Fixed eight-array block decoders have no recursive moment headroom + +**Definitive dead end for the declared block class. No new kappa is claimed.** + +The twelve late helper blocks in the selected complex word already attain the +minimum recursive moment over every finite rational scalar program on their +eight arrays with the retained input frames, late output frames, and dirty +helper endpoint contract. The result permits arbitrary gate order and length, +repeated source reads, information splitting, and changed helper coordinates. +It includes fusion with all later work inside each block. The retained provider +uses one canonical word for both orientations; the backward program is its +same-order inverse-transpose dual with the same frame climbs. + +For a block with four low helpers and common frame dimension u, the exact +minimum at every 0, with substantial +OpenAI Codex assistance. New work is under Apache-2.0. + +The fixed local supplier is Jeremy Kothe's seven-slot k8 word in PR1074, +commit `052afc128954f7b6a2b3c409eb4d72248e4020f2`. The seven-slot step comes +from PR1018 and credits Rohan Arun's PR987 search. The earlier local word and +its scalar, frame, phase, and source-access construction retain all notices +in `vendor/pr1074/NOTICE.md` and `vendor/pr1074/UPSTREAM-NOTICE.md`. +This package does not claim authorship of those local changes. + +The signed bin prices come from Chafik Boukhalfa's PR795. The proof method +builds on the PR890/PR942 owner bounds and the prior all-replica T37 theorem. +The new right-hand side, exact attaining flow, and complete R480 owner map +are specific to the changed seven-slot inventory. The old theorem did not +cover this word. + +The T16 source-event routines are by Hao Yan (@FanPUYen). Their source notice +is kept in `vendor/pr1074/t16/NOTICE.md` and their source headers. The derived +independent owner checker comes from `research/complex-t21-owners` at the +exact repository base. `INDEPENDENT-DERIVATION.patch` records its changes. + +The global admission code comes from the exact PR1074 package. Its sources +and notices are retained. `CODE-DERIVATION.patch` records all changes needed +for 37 replicas and the 1024-bit complex prefix allowance. The arithmetic +engines and full PR589 bit supplier are unchanged and source-pinned. + +PR1034 proved a mixed bound for 71 older catalogue entries and excluded +R480. PR1069 independently checked the rank-mass identity for recent claims. +These contributions informed the scope and cost audit. Neither admits the +new physical owner map by itself. + +The concurrent mutable-leaver sprint worker rebuilt the unchanged local k8 +and seven-slot chains, including their scalar and phase checks. This package +retains that source-bound evidence. It does not use the worker's new local +word experiments. No predecessor notice, license, or AI disclosure is replaced +by this notice. + +Josie Elliston's PR1091 independently checked fifteen T16 owner maps and +made the disjoint-pending-axis obligation explicit. The independent T37 +audit includes that check and a targeted negative control. Rohan Arun's +PR1094 gives a later same-word T16 repack. Jeremy Kothe's PR1117 changes +the local word; it is a concurrent comparison outside the proved inventory. diff --git a/research/complex-block-rank-optimum/notices/k8-coordinated-owners-NOTICE.md b/research/complex-block-rank-optimum/notices/k8-coordinated-owners-NOTICE.md new file mode 100644 index 0000000000..281e6404ca --- /dev/null +++ b/research/complex-block-rank-optimum/notices/k8-coordinated-owners-NOTICE.md @@ -0,0 +1,41 @@ +# Attribution and scope + +Prepared for Thomas DiFiore with OpenAI Codex assistance, October 13, 2026. +Licensed under the repository's Apache-2.0 license. + +This contribution changes the whole-helper ownership map for the circuit from +Jeremy Kothe's PR1018 (`eac7547cb191f875d5f5ea5f9bbb1d2b8c51c4ef`). The +eight no-helper mutable-source cells, R480 circuit, frame optimization and +construction controls come from Jeremy Kothe and the credited contributors to +PR1005. The seven additional triangle slots come from PR1018 using Rohan Arun's +PR987 search and NewBArch's PR869 builder. None of those circuit improvements +are claimed as new here. + +The starting ownership map is G44Pack3's K8c8, descended from chafreaky's PR685 +and PR741 maps through G44Pack's relabeling and width changes. This package adds +coordinated complete-call reassignment by grouped integer packing, with +explicitly serialized owners and an unchanged independent admission verifier. + +The separate all-replica upper bound adapts jon314159's PR942 arithmetic +implementation and proof, using chafreaky's PR795/890 signed-dual prices. Its +new contribution is the application to PR1018's changed local profile and +helper inventory. The two rational numerical engines retain their upstream +authorship. `independent/UPSTREAM.json` records the source file and hash. + +All construction, controller, spectator, native phase, finite invoice, exact +moment and outer-transfer implementations retain their upstream authorship +and notices. The runtime copies PR1018's verifier files byte for byte. It +changes the owner nomination, its derived PLAN fields, and expected-value SPEC. +The original packages remain unchanged. + +The bit supplier is the unchanged PR589/PR555 punctured-star construction; +this run reprices its pinned profile and reruns its admission controls. It does +not rebuild the bit circuit. Earlier complex ancestry is inherited through the +pinned PR741 initial word and PR622/PR558 reviewed parent artifacts. The run +freshly rebuilds the k8 leaver circuit, all later layers, and the seven slots. + +This is a conditional finite construction, not an unconditional integer +multiplication theorem. All-size compilation, common-label and mutable-source +realization, whole-call hosting, precision/tape, analytic transfer and bit H1–H5 +assumptions remain. No independent human peer review is claimed. Optimizer +outputs and floating relaxations are discovery evidence, not admission proofs. diff --git a/research/complex-block-rank-optimum/notices/k8-eight-ends-NOTICE.md b/research/complex-block-rank-optimum/notices/k8-eight-ends-NOTICE.md new file mode 100644 index 0000000000..a756f9da58 --- /dev/null +++ b/research/complex-block-rank-optimum/notices/k8-eight-ends-NOTICE.md @@ -0,0 +1,9 @@ +Apache-2.0. Prepared for Thomas DiFiore with OpenAI Codex assistance. + +This package composes earlier community constructions and retains their mathematical assumptions. It claims the displayed finite composition and its checks, not original authorship of E8, the mutable-source k8 circuit, the endpoint technique, or the admission machinery. + +* CrocSwap/integer-mult-bounds PR1018 (`research/gauge44-slots`) supplies the R480 k8 + seven-slot circuit, all native scalar/phase checkers, and strict global admission. +* PR1030, commit d71857cd0ae7824e935aac17327e4ec27f66322b, supplies the rank-seven endpoint-pair lead. Its U8 helpers 588/589 correspond to k8 helpers 574/575. The new composition also uses 572/573, 580/581, and 582/583, with all four reductions refitted together. +* `discovery/frame_model.py` adapts the complete frame model from PR864 (navinor / Claude), retained in the earlier local research run. The adaptation permits a new initial frame assignment after changing the scalar graph or endpoint constraints. It is nomination code; unchanged independent native checks decide admission. +* `research/k8-coordinated-owners` supplies the initial whole-owner map and coordinated complete-call search. Earlier owner machinery and dual prices originate with chafreaky, G44Pack, and jon314159; see that package's NOTICE.md. +* All analytic, compiler, boundary, common-label, hosting, and bit-supplier assumptions remain those of the upstream source chain. diff --git a/research/complex-block-rank-optimum/notices/k8-eight-ends-t16-NOTICE.md b/research/complex-block-rank-optimum/notices/k8-eight-ends-t16-NOTICE.md new file mode 100644 index 0000000000..d0359f2e08 --- /dev/null +++ b/research/complex-block-rank-optimum/notices/k8-eight-ends-t16-NOTICE.md @@ -0,0 +1,9 @@ +Apache-2.0. Prepared for Thomas DiFiore with OpenAI Codex assistance. + +This package composes community work. It claims the new finite owner assignment, the documented pattern-pricing search, and the inventory-specific ceiling certificate, not original authorship of E8, the k8 word, the sixteen-replica controller, or the admission machinery. + +* `../k8-eight-ends` supplies the unchanged endpoint word, local verification, and initial T8 owners. Its NOTICE.md credits PR1018's R480 k8 + seven-slot supplier, PR1030's rank-seven endpoint lead, PR864's frame model, and earlier owner research. +* CrocSwap/integer-mult-bounds PR1074, commit `052afc128954f7b6a2b3c409eb4d72248e4020f2`, supplies the consumed controller and strict admission sources. Every file under `vendor/pr1074` is byte-identical to that upstream commit and pinned in `UPSTREAM.json`. The original `UPSTREAM-NOTICE.md` is retained there. PR1074's owner research was contributed by jeremykothe2 with Claude Opus 5.5. +* The T16 controller, direct-sum parent, actual-window reconstruction, and independent checker originate in Hao Yan / FanPUYen's PR858, with subsequent PR870 and PR1074 adaptations. This package does not modify their checks. +* `independent/all_t_bound.py` adapts `../k8-coordinated-owners/independent/all_t_bound.py`, itself adapted from jon314159's PR942 certificate and chafreaky's PR795/890 signed dual prices. The two rational arithmetic engines are retained upstream code. +* The bit supplier remains the cited PR589 supplier. All analytic, compiler, boundary, common-label, hosting, and bit-supplier assumptions remain in force. diff --git a/research/complex-block-rank-optimum/notices/k8-three-demotions-NOTICE.md b/research/complex-block-rank-optimum/notices/k8-three-demotions-NOTICE.md new file mode 100644 index 0000000000..9f759cfade --- /dev/null +++ b/research/complex-block-rank-optimum/notices/k8-three-demotions-NOTICE.md @@ -0,0 +1,7 @@ +Apache-2.0. Prepared for Thomas DiFiore with OpenAI Codex assistance. + +* `../k8-eight-ends` and `../k8-eight-ends-t16` supply the unchanged scalar word, endpoint construction, initial owner map, and pinned T16 controller. Their notices retain the PR1018, PR1030, PR864, PR858, PR870, PR1074, and earlier E8, k8, frame, owner, and bit-supplier credits. +* PR1111 by seslly / shipstuff, commit `381831ea554ea983e16224dabed8921b7d0ffdbc`, supplied the entrance-demotion lead: reversing a promotion can profitably trade wider helpers for a different recursive profile. Its README/PR claim was consulted; no PR1111 code or owner payload is consumed. This package instead screens all codimension-one subspaces of the six current dimension-five groups, composes three groups on the eight-endpoint word, and constructs its own complete T16 owner assignment. The older promotion mechanism is credited to PR664 and its descendants. +* The frame refit uses the preceding adaptation of PR864's complete frame model (navinor / Claude). Solver output is nomination data; independent native replay and unchanged global engines decide admission. +* The coordinated pattern-pricing search is retained from `../k8-eight-ends-t16`. Earlier owner algorithms and signed duals are credited in its notice. The bound adapts jon314159's PR942 certificate and chafreaky's PR795/890 prices, using the existing two rational arithmetic engines. +* The fresh native and global checks retain every inherited analytic, compiler, boundary, common-label, hosting, precision/tape, and bit H1–H5 assumption. This package does not claim original authorship of those constructions or an unconditional theorem. diff --git a/research/complex-block-rank-optimum/verify.py b/research/complex-block-rank-optimum/verify.py new file mode 100644 index 0000000000..5b08fcad86 --- /dev/null +++ b/research/complex-block-rank-optimum/verify.py @@ -0,0 +1,512 @@ +"""Check the source-bound premises of the block moment theorem.""" + +import argparse +from collections import Counter +from copy import deepcopy +from fractions import Fraction +import gzip +import hashlib +import itertools +import json +import math +from pathlib import Path + + +HERE = Path(__file__).resolve().parent +BASES = (592, 600, 608, 624, 632, 640, 648, 656, 672, 680, 688, 696) + + +def f2_basis(values): + basis = {} + for value in values: + while value: + pivot = value.bit_length() + if pivot in basis: + value ^= basis[pivot] + else: + basis[pivot] = value + break + return tuple(basis[p] for p in sorted(basis, reverse=True)) + + +def rank(values): + return len(f2_basis(values)) + + +def contains(frame, sub): + return rank(frame + sub) == rank(frame) + + +def qrank(rows): + rows = [list(map(Fraction, row)) for row in rows] + if not rows: + return 0 + pivot = 0 + for column in range(len(rows[0])): + index = next((i for i in range(pivot, len(rows)) if rows[i][column]), None) + if index is None: + continue + rows[index], rows[pivot] = rows[pivot], rows[index] + factor = rows[pivot][column] + rows[pivot] = [x / factor for x in rows[pivot]] + for i in range(pivot + 1, len(rows)): + factor = rows[i][column] + rows[i] = [x - factor * y for x, y in zip(rows[i], rows[pivot])] + pivot += 1 + if pivot == len(rows): + break + return pivot + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def scalar(gate, rows): + if gate[0] == "add": + _, dst, src, numerator, denominator, _ = gate + coefficient = Fraction(numerator, denominator) + rows[dst] = [a + coefficient * b for a, b in zip(rows[dst], rows[src])] + elif gate[0] == "scale": + _, dst, numerator, denominator, _ = gate + rows[dst] = [Fraction(numerator, denominator) * a for a in rows[dst]] + else: + raise ValueError(f"Unknown scalar gate: {gate[0]}") + + +def incident(gate): + return gate[1:3] if gate[0] == "add" else gate[1:2] + + +def canonical(word): + if "frames" in word: + return word + frames = [] + keys = {} + + def frame(values): + key = tuple(values) + if key not in keys: + keys[key] = len(frames) + frames.append(list(values)) + return keys[key] + + count = 2 * len(word["ports"]) + word["report"]["R"] + start = [frame([p]) for p in word["ports"]] + [frame([])] * (count - len(word["ports"])) + for r, f in word["start_extra"].items(): + start[int(r)] = frame(f) + full = frame([1 << i for i in range(9)]) + final = [full] * count + for j, p in enumerate(word["ports"]): + final[len(word["ports"]) + j] = frame(f2_basis(x for x in range(512) if (x & p).bit_count() % 2 == 0)) + for r, f in word["helper_final"].items(): + final[int(r)] = frame(f) + result = {s: [g[:-1] + [frame(g[-1])] for g in word[s]] for s in ("A", "B")} + result.update(start=start, final=final, frames=frames, ports=word["ports"], + cut=word["cut"], precondition=word["precondition"]) + return result + + +def frame_states(word): + state = word["start"].copy() + frames = word["frames"] + for gate in word["A"]: + for register in incident(gate): + require(contains(frames[gate[-1]], frames[state[register]]), "A frame descends") + state[register] = gate[-1] + states = [] + for gate in word["B"]: + states.append(state.copy()) + for register in incident(gate): + require(contains(frames[gate[-1]], frames[state[register]]), "B frame descends") + state[register] = gate[-1] + states.append(state.copy()) + return states + + +def determinant(matrix): + """Use the integer Bareiss identity after row denominator clearing.""" + if not matrix: + return Fraction(1) + integers = [] + scale = 1 + for row in matrix: + denominator = math.lcm(*(Fraction(x).denominator for x in row)) + scale *= denominator + integers.append([int(Fraction(x) * denominator) for x in row]) + previous = sign = 1 + for k in range(len(integers) - 1): + pivot = next((i for i in range(k, len(integers)) if integers[i][k]), None) + if pivot is None: + return Fraction(0) + if pivot != k: + integers[k], integers[pivot] = integers[pivot], integers[k] + sign = -sign + value = integers[k][k] + for i in range(k + 1, len(integers)): + for j in range(k + 1, len(integers)): + numerator = integers[i][j] * value - integers[i][k] * integers[k][j] + require(numerator % previous == 0, "Bareiss division is not exact") + integers[i][j] = numerator // previous + integers[i][k] = 0 + previous = value + return Fraction(sign * integers[-1][-1], scale) + + +def gram_rank(rows): + """A rational Gram matrix has the same rank as its row matrix.""" + gram = [[sum(a * b for a, b in zip(x, y)) for y in rows] for x in rows] + for size in range(len(rows), 0, -1): + for indices in itertools.combinations(range(len(rows)), size): + if determinant([[gram[i][j] for j in indices] for i in indices]): + return size + return 0 + + +def span_set(values): + """Enumerate the binary span without a pivot basis.""" + result = {0} + for value in values: + result |= {x ^ value for x in result} + return result + + +def prefix_witness(word, registers, inputs, first_input, correction): + """Bind all sixteen boundary values to the actual scalar prefix.""" + count = len(word["start"]) + ports = len(word["ports"]) + targets = set(range(ports, 2 * ports)) + cut = word["cut"] + require(all(set(incident(g)) <= targets for g in word["precondition"]), + "The preconditioner changes an original source or helper") + require(all(g[0] == "add" and g[1] in targets and g[2] not in targets for g in cut), + "The copied-centre cut is not a shear on the main arrays") + removed = set(correction) + prefix = word["A"] + cut + [g for i, g in enumerate(word["B"][:first_input]) if i not in removed] + for gate in prefix: + require(gate[0] in ("add", "scale"), "A prefix gate is outside the scalar alphabet") + require(gate[2 if gate[0] == "scale" else 3] != 0, "A prefix scalar is zero") + if gate[0] == "add": + require(gate[1] != gate[2], "A prefix addition has equal source and target") + variables = registers + inputs + pulled = [{r: Fraction(1)} for r in variables] + for gate in reversed(prefix): + dst = gate[1] + if gate[0] == "add": + src, coefficient = gate[2], Fraction(gate[3], gate[4]) + for row in pulled: + if dst in row: + row[src] = row.get(src, Fraction(0)) + coefficient * row[dst] + if not row[src]: + del row[src] + else: + coefficient = Fraction(gate[2], gate[3]) + for row in pulled: + if dst in row: + row[dst] *= coefficient + dense = [[row.get(i, Fraction(0)) for i in range(count)] for row in pulled] + reduced = [row.copy() for row in dense] + columns = [] + for column in range(count): + pivot = len(columns) + index = next((i for i in range(pivot, 16) if reduced[i][column]), None) + if index is None: + continue + reduced[pivot], reduced[index] = reduced[index], reduced[pivot] + value = reduced[pivot][column] + reduced[pivot] = [x / value for x in reduced[pivot]] + for i in range(pivot + 1, 16): + factor = reduced[i][column] + if factor: + reduced[i] = [x - factor * y for x, y in zip(reduced[i], reduced[pivot])] + columns.append(column) + if len(columns) == 16: + break + require(len(columns) == 16, "The sixteen interface values are not independent") + require(all(c < ports or c in registers for c in columns), + "The independence witness uses a target or an exterior dirty helper") + require(sum(c < ports for c in columns) == 8, + "The witness does not use eight original source columns") + minor = [[row[j] for j in columns] for row in dense] + det = determinant(minor) + require(det != 0, "The interface minor is singular") + forward = [[Fraction(int(r == c)) for c in columns] for r in range(count)] + for gate in prefix: + scalar(gate, forward) + require([forward[r] for r in variables] == minor, "The forward prefix and pullback minor differ") + return {"boundary": "before-inputs-with-helper-correction-deferred", + "registers": variables, "initial_scalar_columns": columns, + "minor": [[str(x) for x in row] for row in minor], "determinant": str(det), + "removed_correction_indices": sorted(removed), "scalar_prefix_length": len(prefix)} + + +def block(word, states, base): + registers = list(range(base, base + 8)) + regset = set(registers) + frames = word["frames"] + gates = word["B"] + inner = [(i, g) for i, g in enumerate(gates) if g[0] == "add" and g[1] in regset and g[2] in regset] + all_incoming = [(i, g) for i, g in enumerate(gates) if g[0] == "add" and g[1] in regset and g[2] not in regset] + incoming = [(i, g) for i, g in all_incoming if rank(frames[g[-1]]) > 1] + outgoing = [(i, g) for i, g in enumerate(gates) if g[0] == "add" and g[1] not in regset and g[2] in regset] + require(len(incoming) == 8, "There must be eight input gates") + require(len({g[1] for _, g in incoming}) == 8, "There must be eight input recipients") + require(len({g[2] for _, g in incoming}) == 8, "There must be eight distinct input arrays") + first_input = incoming[0][0] + last_input = incoming[-1][0] + require(last_input - first_input == 7, "The input gates must form one segment") + correction = [(i, g) for i, g in inner if i < first_input] + require(correction, "The block must have a correction") + start = correction[0][0] + require(all(i < start for i, g in all_incoming if (i, g) not in incoming), "An additional input occurs after the module starts") + early = [(i, g) for i, g in outgoing if i < start] + late = [(i, g) for i, g in outgoing if i > last_input] + require(len(early) + len(late) == len(outgoing), "An output lies inside the correction or input segment") + require(len(late) == 8, "There must be eight late output gates") + targets = sorted({g[1] for _, g in late}) + require(len(targets) == 4, "There must be four output targets") + require(all(not (g[0] == "scale" and g[1] in targets) and + not (g[0] == "add" and g[2] in targets) + for g in gates[late[0][0]:]), "A late target is scaled or read by the exterior") + initial = {r: frames[states[start][r]] for r in registers} + low = [r for r in registers if len(initial[r]) == 1] + high = [r for r in registers if len(initial[r]) > 1] + require(len(low) == len(high) == 4, "There must be four low and four high helpers") + common = initial[high[0]] + u = rank(common) + require(u in (4, 5), "The common frame must have rank four or five") + require(all(contains(common, initial[r]) and contains(initial[r], common) for r in high), "The high frames differ") + require(all(contains(common, initial[r]) for r in low), "A low frame is outside the common frame") + require(all(contains(common, frames[g[-1]]) for _, g in correction), "A correction gate is above the common frame") + require(all(contains(frames[g[-1]], common) for _, g in correction), "A correction gate is below the common frame") + for i in range(start, first_input): + if regset.intersection(incident(gates[i])): + require(set(incident(gates[i])) <= regset, "The correction cannot move past an exterior event") + + input_frames = [frames[g[-1]] for _, g in incoming] + for (index, gate), frame in zip(incoming, input_frames): + require(rank(frame) in (5, 6) and contains(frame, common), "An input frame has the wrong rank or common subspace") + require(contains(frames[states[first_input][gate[2]]], frame), "An external input is not ready at its declared frame") + require(contains(frame, frames[states[index][gate[2]]]), "An external input has advanced beyond its declared frame") + require(all(not contains(a, b) and not contains(b, a) for a, b in itertools.combinations(input_frames, 2)), "Two input frames are comparable") + entry_indices = [next(i for i, g in late if g[1] == t) for t in targets] + target_entry = [frames[states[i][t]] for i, t in zip(entry_indices, targets)] + target_caps = [frames[word["final"][t]] for t in targets] + require(all(rank(f) in (7, 8) for f in target_entry), "A target entry frame does not have rank seven or eight") + require(all(rank(f) == 8 for f in target_caps), "A target cap does not have rank eight") + require(all(rank(a + b) == 9 for a, b in itertools.combinations(target_caps, 2)), "Two target caps coincide") + require(all(rank(f + g) >= 8 for f in input_frames for g in target_entry), "An input can reach a target below rank eight") + + variables = registers + [g[2] for _, g in incoming] + require(len(set(variables)) == 16, "The sixteen inputs are not distinct arrays") + rows = {r: [Fraction(int(i == j)) for i in range(16)] for j, r in enumerate(variables)} + rows.update({t: [Fraction(0)] * 16 for t in targets}) + for i, gate in enumerate(gates): + if i >= start and regset.intersection(incident(gate)): + require(set(incident(gate)) <= set(rows), "An undeclared array occurs in the block") + scalar(gate, rows) + output = [rows[t] for t in targets] + high_indices = [registers.index(r) for r in high] + require(all(row[j] == 0 for row in output for j in high_indices), "A high dirty input reaches a target") + q = [row[8:] for row in output] + require(qrank(q) == 4, "The injection output matrix must have rank four") + require(all(sum(row[j] != 0 for row in q) == 2 for j in range(8)), "An input does not need two distinct output targets") + require(all(qrank([[v for i, v in enumerate(row) if i != j] for row in q]) == 4 for j in range(8)), "The kernel has a zero coordinate") + projection_ranks = Counter() + for size in range(1, 5): + for selected in itertools.combinations(range(8), size): + complement = [j for j in range(8) if j not in selected] + projection_rank = size - 4 + qrank([[row[j] for j in complement] for row in q]) + require(projection_rank >= (3 * size + 3) // 4, "A kernel coordinate projection is too small") + projection_ranks[(size, projection_rank)] += 1 + + spectrum = Counter() + kernel_frames = {} + for mask in range(1, 256): + indices = [j for j in range(8) if mask >> j & 1] + frame = [v for j in indices for v in input_frames[j]] + frame_rank = rank(frame) + restricted = [[row[j] for j in indices] for row in q] + matrix_rank = qrank(restricted) + require(gram_rank(restricted) == matrix_rank, "The rational rank methods disagree") + require(len(span_set(frame)) == 1 << frame_rank, "The binary span methods disagree") + spectrum[(frame_rank, len(indices), matrix_rank)] += 1 + if frame_rank <= 6: + require(matrix_rank == len(indices), f"A kernel is present below rank seven: {base}, {mask}") + if frame_rank == 7 and matrix_rank < len(indices): + eligible = [j for j, cap in enumerate(target_caps) if contains(cap, frame)] + require(eligible == [j for j, cap in enumerate(target_caps) if span_set(frame) <= span_set(cap)], + "The cap containment methods disagree") + require(len(eligible) <= 1, "A rank-seven kernel can reach two target caps") + closure = tuple(j for j, f in enumerate(input_frames) if contains(frame, f)) + require(all(rank(input_frames[j]) == 5 for j in closure), "A rank-six input lies in a rank-seven kernel frame") + kernel_frames[closure] = {"indices": list(closure), "frame": list(f2_basis(frame)), + "eligible_targets": [targets[j] for j in eligible]} + + profile = Counter() + current = initial.copy() + for i, gate in enumerate(gates): + if i >= start: + for r in regset.intersection(incident(gate)): + delta = rank(frames[gate[-1]]) - rank(current[r]) + require(delta >= 0 and contains(frames[gate[-1]], current[r]), "A helper frame descends") + if delta: + profile[delta] += 1 + current[r] = frames[gate[-1]] + for r in registers: + require(rank(frames[word["final"][r]]) == 9, "A helper does not end at the full frame") + delta = 9 - rank(current[r]) + if delta: + profile[delta] += 1 + expected = Counter({1: 24, 2: 8, 3: 4}) if u == 4 else Counter({1: 32, 4: 4}) + require(profile == expected, f"The selected profile does not attain the proposed floor: {base}, {profile}") + return {"base": base, "helpers": registers, "low": low, "high": high, + "common_frame": common, "common_rank": u, + "first_correction": start, "first_input": first_input, + "input_gate_indices": [i for i, _ in incoming], + "output_gate_indices": [i for i, _ in late], + "output_entry_indices": entry_indices, + "R": [[str(row[registers.index(r)]) for r in low] for row in output], + "independence_witness": prefix_witness(word, registers, [g[2] for _, g in incoming], + first_input, [i for i, _ in correction]), + "input_arrays": [g[2] for _, g in incoming], "input_frames": input_frames, + "target_arrays": targets, "target_entry_frames": target_entry, + "target_caps": target_caps, "Q": [[str(x) for x in row] for row in q], + "profile": dict(sorted(profile.items())), + "kernel_projection_ranks": [{"size": k[0], "rank": k[1], "count": v} + for k, v in sorted(projection_ranks.items())], + "rank_seven_kernel_frames": [v for _, v in sorted(kernel_frames.items())], + "subset_spectrum": [{"frame_rank": k[0], "size": k[1], "matrix_rank": k[2], "count": v} + for k, v in sorted(spectrum.items())]} + + +def load_payload(entry, data=None): + if data is None: + data = (HERE / entry["file"]).read_bytes() + require(hashlib.sha256(data).hexdigest() == entry["sha256"], "A frozen input hash differs") + payload = gzip.decompress(data) + require(hashlib.sha256(payload).hexdigest() == entry["payload_sha256"], "An input payload hash differs") + return json.loads(payload) + + +def load_source(entry, data=None): + return canonical(load_payload(entry, data)) + + +def failure_controls(word, states, first_block, entry): + controls = [] + + def reject(name, operation, message): + try: + operation() + except ValueError as error: + require(str(error) == message, f"The {name} control failed for another reason: {error}") + controls.append({"name": name, "rejection": message}) + else: + raise ValueError(f"The {name} control was accepted") + + changed = deepcopy(word) + i = first_block["first_correction"] + changed["B"][i][3] += changed["B"][i][4] + reject("wrong-helper-correction", lambda: block(changed, states, BASES[0]), + "A high dirty input reaches a target") + changed = deepcopy(word) + i = first_block["first_input"] + changed["B"][i + 1][2] = changed["B"][i][2] + reject("duplicate-external-input", lambda: block(changed, states, BASES[0]), + "There must be eight distinct input arrays") + changed = deepcopy(word) + changed["final"][BASES[0]] = changed["final"][first_block["target_arrays"][0]] + reject("missing-full-helper-end", lambda: block(changed, states, BASES[0]), + "A helper does not end at the full frame") + data = bytearray((HERE / entry["file"]).read_bytes()) + data[-1] ^= 1 + reject("changed-frozen-input", lambda: load_source(entry, data), "A frozen input hash differs") + return controls + + +def orientation_contract(entry): + programs = load_payload(entry) + forward, backward = programs["forward"], programs["backward"] + climbs = [[e for e in program if e[0] == "climb"] for program in (forward, backward)] + require(climbs[0] == climbs[1], "The provider orientations have different frame climbs") + profiles = [] + scalars = [] + for program in (forward, backward): + profile = Counter() + operations = [] + phase = None + for event in program: + if event[0] == "phase": + phase = event[1] + elif event[0] == "climb": + profile[event[5]] += 1 + elif event[0] == "center": + profile[event[4]] += 1 + elif event[0] == "scalar" and phase != "centers": + operations.append(event) + profiles.append(profile) + scalars.append(operations) + require(profiles[0] == profiles[1] == Counter({1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}), + "The two native orientation profiles differ") + require(len(scalars[0]) == len(scalars[1]), "The two orientation scalar counts differ") + for fwd, bwd in zip(*scalars): + gate = deepcopy(fwd[1]) + if gate[0] == "add": + gate[1], gate[2] = gate[2], gate[1] + gate[3] = -gate[3] + elif gate[0] == "scale": + gate[2], gate[3] = gate[3], gate[2] + else: + raise ValueError("An orientation scalar is outside the dual alphabet") + require(bwd == ["scalar", gate, fwd[2]], "The backward scalar is not the retained inverse-transpose dual") + return {"file": entry["file"], "source_sha256": entry["payload_sha256"], + "contract": "same-order-inverse-transpose-from-one-canonical-word", + "equal_climb_events": len(climbs[0]), "dual_noncentre_scalars": len(scalars[0]), + "equal_complete_profile": dict(sorted(profiles[0].items()))} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-certificate", action="store_true", + help="Write the exact interface certificate after all checks pass.") + args = parser.parse_args() + manifest = json.loads((HERE / "INPUTS.json").read_text()) + receipts = [] + expected_totals = ({1: 328, 2: 56, 3: 28, 4: 20}, + {1: 304, 2: 80, 3: 40, 4: 8}, + {1: 296, 2: 88, 3: 44, 4: 4}) + require([e["file"] for e in manifest["inputs"]] == + ["inputs/selected.json.gz", "inputs/pr1319-base.json.gz", "inputs/pr1319-extra.json.gz"], + "The source set differs") + for entry, expected in zip(manifest["inputs"], expected_totals): + word = load_source(entry) + states = frame_states(word) + blocks = [block(word, states, base) for base in BASES] + profile = Counter() + for item in blocks: + profile.update(item["profile"]) + require(dict(profile) == expected, "The total attained moment differs") + require(any(b["rank_seven_kernel_frames"] for b in blocks), + "The rank-seven retirement case is no longer exercised") + receipts.append({"file": entry["file"], "source_sha256": entry["payload_sha256"], + "blocks": blocks, "total_profile": dict(sorted(profile.items())), + "subsets": 255 * len(blocks), + "controls": failure_controls(word, states, blocks[0], entry)}) + certificate = {"schema_version": 1, "status": "PASS_FIXED_BLOCK_MOMENT_PREMISES", + "sources": receipts, "source_block_pairs": 36, "subset_checks": 9180, + "independence_minors": 36, "failure_controls": 12, + "orientation_contract": orientation_contract(manifest["orientation_program"])} + output = json.dumps(certificate, indent=2, sort_keys=True) + "\n" + target = HERE / "CERTIFICATE.json" + if args.write_certificate: + target.write_text(output) + else: + require(json.loads(target.read_text()) == json.loads(output), "The interface certificate differs") + print("PASS_FIXED_BLOCK_MOMENT_PREMISES blocks=36 subsets=9180 minors=36 controls=12") + print("The finite checks support PROOF.md; they are not a formal proof of the all-length theorem.") + + +if __name__ == "__main__": + main()