From ea2f6bf0940580dfaeb393ec0786abc8f7bbb7b4 Mon Sep 17 00:00:00 2001 From: Hanson Sun <182361798+bbMisk@users.noreply.github.com> Date: Tue, 13 Oct 2026 21:19:32 -0400 Subject: [PATCH] Research: certify two-anchor P112 span preservation and COPY64 --- .../CERTIFICATE.json | 22844 ++++++++++++++++ .../INDEPENDENT.json | 84 + .../OVERLAP.json | 212 + .../README.md | 306 + .../SOURCES.json | 30 + .../TASK.json | 132 + .../VALIDATION.json | 144 + .../VALIDATION.md | 37 + .../VERIFY.json | 32 + .../independent_check.py | 230 + .../verify.py | 413 + 11 files changed, 24464 insertions(+) create mode 100644 research/p112-multiline-free-rectangles-seed42/CERTIFICATE.json create mode 100644 research/p112-multiline-free-rectangles-seed42/INDEPENDENT.json create mode 100644 research/p112-multiline-free-rectangles-seed42/OVERLAP.json create mode 100644 research/p112-multiline-free-rectangles-seed42/README.md create mode 100644 research/p112-multiline-free-rectangles-seed42/SOURCES.json create 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Recheck relevant heads before submission.", + "terms": [ + "free rectangle", + "two-coordinate", + "two-column", + "two common odd neighbors" + ], + "nearest_work": [ + { + "pr": 1375, + "head": "4a533aff2a7a87c9b36e077a31774b332a7f64bd", + "author": "commdata2338", + "contribution": "Two-coordinate small-rectangle floor and first free-compatible erasure boundary." + }, + { + "pr": 1432, + "head": "4bcc49e3f4041018b0607da46a87d8928a6e0fd0", + "author": "Hanson Sun / bbMisk", + "contribution": "Fixed 24-by-one boundary and coupled-factor proof." + }, + { + "pr": 1331, + "head": "f08b698cc2bd74e2425486728a008e531fc35f0e", + "author": "Arham Tahir", + "contribution": "One-coordinate arbitrary rectangles; two-coordinate handoff." + }, + { + "pr": 1373, + "head": "3eaef3e4427068b33847dc1147c8a2e5709a970e", + "author": "Cinna Davis", + "contribution": "Arbitrary rectangles with three active projected products at pair (4,6); leaves exactly-two branch open." + }, + { + "discussion": 1434, + "author": "Gareth Davidson / bitplane", + "contribution": "Earlier independent unrestricted singleton-column investigation; released our overlapping scope." + }, + { + "pr": 1460, + "head": "8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6", + "author": "Hanson Sun / bbMisk", + "contribution": "Concurrent earlier arbitrary-rectangle COPY64 at fixed pair (4,6), via compensation-size argument. Our distinct evidence is the stronger per-active-term span lemma, explicit all-pair symmetry and maximal-erasure check." + }, + { + "pr": 1468, + "head": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a", + "author": "Gareth Davidson / bitplane", + "contribution": "Independent all-pair one-column theorem, complementary to our at-least-two-indices-per-side scope." + } + ], + "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1316#discussioncomment-18912476", + "initial_overlap": { + "open_prs": { + "fetched_at": "2026-10-14T00:57:55.369472+00:00", + "complete_pagination": true, + "records": 820 + }, + "discussions": { + "fetched_at": "2026-10-14T00:58:57.154931+00:00", + "complete_pagination": true, + "records": 154 + } + }, + "note": "Initial paginated scan discovered Discussion1434 and triggered the single pivot. Final scan found PR1460 and PR1468; both are credited and the independent evidence distinction is explicit.", + "presubmit_refresh_utc": "2026-10-14T01:18:37.644858+00:00", + "presubmit_latest_prs": [ + { + "number": 1471, + "title": "Scoped audit: PR1417 P112 tensor rank is field-specific (no new kappa)", + "head": "3695ad16e7bc4c0f6248a9e5a64fc069fa096733" + }, + { + "number": 1469, + "title": "Research: rank-8 masked decoders are gauge-rigid on every P109 unit \u2014 19 new orbit certificates (no new \u03ba)", + "head": "d9d101b3123b6f1d17afe80e5e70a71fb3b29e39" + }, + { + "number": 1468, + "title": "Research: independent COPY floor for arbitrary one-column P112 corrections", + "head": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a" + }, + { + "number": 1466, + "title": "Research audit: exact truncation transport boundary (no new kappa)", + "head": "8230e01b60d321ca10f2789e27cde037c40e6d6c" + }, + { + "number": 1465, + "title": "Prove a complete-cost obstruction for eight two-accumulator rewrites", + "head": "bb7fc427c79c26669a9e8de06e55f31cadaa992d" + }, + { + "number": 1464, + "title": "Conditional \u03ba 0.0016768552290713218862064567722745926195124487395104752628033651837667029477836: verified fine64 endpoint refinement, unchanged construction", + "head": "2ffb640826106f01184c818ecd0cccb683f2aabf" + }, + { + "number": 1461, + "title": "Tooling: exact annihilator containment for historical frame audits", + "head": "1a092583b989051a089cdba68b52cedf36a8d78e" + }, + { + "number": 1460, + "title": "Scoped negative result: arbitrary free rectangles retain COPY64 for P112 pair (4,6)", + "head": "8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6" + }, + { + "number": 1459, + "title": "Tooling: accelerate exact historical frame containment", + "head": "69db3d7f10706ae897b6a7ae030db5f0b4892be8" + }, + { + "number": 1458, + "title": "Conditional \u03ba = 0.001676855229071321: finer 1e-60 certified endpoints and a deeper outer witness on #1424's construction (unchanged)", + "head": "fec30ff23f972fdc59aeec24bc5bd64b8638d8aa" + }, + { + "number": 1457, + "title": "Independent verification: PR1407 complex profile arithmetic (no new kappa)", + "head": "43a4920cfb08398fdc949fc50dcdab794d2f37f8" + }, + { + "number": 1456, + "title": "Research: extend the one-frame-relation bound to arbitrary register counts", + "head": "83b6ad2ace819a96128131a3600ee914a66dd0dd" + }, + { + "number": 1455, + "title": "Research: sharpen the fixed P104 first-drop boundary to 54 copy cost", + "head": "0ebd49c31300e67698fd812fc00f2a3b2b805a67" + }, + { + "number": 1454, + "title": "Scoped negative result: one frame relation cannot save paid rank at any register count", + "head": "2971637b4a291c355353742f3f187ad1b1e4c58d" + }, + { + "number": 1453, + "title": "Prove the source-independent boundary obstruction for fixed R480 cleanup", + "head": "3bceac90599a02a96e9b744736b3feeb00171504" + }, + { + "number": 1452, + "title": "Performance: stream research provenance through one Git process", + "head": "f785c7a847a396d1947bb9fdf3a936af487a70ab" + }, + { + "number": 1451, + "title": "Research: certify when precision refinement preserves inverse increments", + "head": "450b0f228807bbd88c6a37606070db54adf1dcfa" + }, + { + "number": 1450, + "title": "Docs: bind the selected k8 provider to the historical catalogue scope", + "head": "b8f150781b0a7a1edc5d686e8ee9c7ee653ea171" + }, + { + "number": 1449, + "title": "Finite bit construction: two neutral R439 moves jointly save two ADDs", + "head": "52a487a0c85e83d875f30a74f795588399bac093" + }, + { + "number": 1447, + "title": "Research: prove a sparse P112 completion rank law through 52-port blocks", + "head": "5f784a84bbe80879b5bd7ae9b48dd28a5765863d" + }, + { + "number": 1446, + "title": "Research: exclude all neutral two- and three-ADD omissions in PR967 D", + "head": "83bab0ac63d7297309e769641d176ca3f12b0124" + }, + { + "number": 1445, + "title": "Research: joint rank-five star cuts beat one-sided support capacity", + "head": "7af6f56bf9ed58eacb3164d7b87b1c92716d8f07" + }, + { + "number": 1444, + "title": "Conditional \u03ba = 0.001676855229071321: deeper outer-witness refinement of #1424's construction (unchanged)", + "head": "e8281be5d0123527bd467e92c4f7005d2ab33b3f" + }, + { + "number": 1443, + "title": "Research: prove the eight-call floor for two dihedral commutator targets", + "head": "965ff75aabb16cdd0978f4bf9222bcdedf514021" + }, + { + "number": 1442, + "title": "Research: prove the connector optimum under output delivery cuts", + "head": "7c573bc769798ce4e336fd5be630060bd9d69a71" + } + ], + "presubmit_neighbor_heads": { + "1460": "8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6", + "1468": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a" + } +} diff --git a/research/p112-multiline-free-rectangles-seed42/README.md b/research/p112-multiline-free-rectangles-seed42/README.md new file mode 100644 index 0000000000..909971c944 --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/README.md @@ -0,0 +1,306 @@ +# Free rectangles with multiple rows and columns retain COPY64 + +**Scoped negative result, with a written proof and exact finite premises.** +On the fixed P112 coordinate decoder, add any nonzero free rank-one correction +whose support contains at least two rows and two columns. Mix that correction +with any two original coordinate terms by an arbitrary complex invertible +3-by-3 factor change. Keep the other six coordinate terms unchanged. +At least two replacement terms still have binary pairing rank eight or more. +The sharp whole-decoder separate COPY cost is therefore **64**, and its COPY +power bill cannot improve for `0 < q <= 1`. + +The contribution is the **two-anchor span-preservation lemma** below and its +explicit extension to all 28 coordinate pairs. Concurrent PR1460 already gives +a COPY64 theorem for arbitrary nonzero rectangles at pair (4,6), using a +compensation-size argument. This package supplies a different, stronger +per-active-term rank-eight statement when both supports have at least two +indices, together with a complete independent two-anchor census. PR1375's +small rectangles and PR1373's three-active-product case are also credited. +Singleton-sided rectangles and zero factors are excluded; PR1468 independently +addresses the one-column class. None of those pending theorems is assumed. + +No physical supplier, complete child histogram, improved kappa, full +multiplication theorem, or proof-assistant verification is claimed. + +## Reproduce + +From the repository root, Python 3.12 and the standard library: + +```sh +python3.12 -B scripts/research_package.py --package research/p112-multiline-free-rectangles-seed42 +``` + +Expected final marker: + +```text +PASS_REGISTERED_RESEARCH_PACKAGE research/p112-multiline-free-rectangles-seed42 outcome=negative-result +``` + +The explicit replay runs both checkers after validating metadata and source +pins. Their default modes only read files. `--write PATH` regenerates the +respective receipt, and `--check PATH` checks an alternative receipt. + +- `verify.py`: all 512 binary characters and five support strata, producing + 2,553 nonempty cuts; 28 explicit coordinate symmetries; seven exact + Gaussian-rational decoder fixtures, 87,808 matrix entries; 12 controls. +- `independent_check.py`: constructs Euclidean E8 roots without importing the + primary checker or predecessor code. Set-based elimination checks all 6,216 + two-anchor erasures and 31,080 span identities. It independently validates + the 28 proposed symmetries, including 351,232 pairing entries, and a literal + integer compensation identity on 12,544 entries; six controls. + +Both implementations are by the same AI-assisted contributor. Separate code +and exact finite checks are not independent human review or formalization of +the quantified theorem. Execution details are in `VALIDATION.md`. + +## Model and statement + +Use the PR845/PR717 P112 geometry pinned in `SOURCES.json`. Enumerate doubled +E8 roots, choose the lexicographically larger sign of each root line, remove +the eight frame roots `2e_(2j) +/- 2e_(2j+1)`, and sort the remaining 112 lines. +The frame order is increasing j, plus before minus. Define + +``` +G_ij = (r_i dot r_j) / 4 +X_ik = (r_i dot f_k) / 4 +B_ij = (1 + G_ij) mod 2. +``` + +Then `XX^T = 2G`, `X_ik in {0,1,-1}`, and the symmetric binary matrix B has +rank nine. For index sets A,C, let `rho(A,C) = rank_F2 B[A,C]`. The inherited +separate-total model charges a scalar rank-one term d t^T by +`rho(supp(d),supp(t))`. Complex coefficients are not reduced modulo two. +Entries with `B_ij=1` are constrained; a correction on a rectangle U x V is +free precisely when `B[U,V]=0`. + +Choose any distinct coordinate modes p,r, and put x=X[:,p], y=X[:,r]. Let +u,v be arbitrary complex vectors with actual supports U,V satisfying + +``` +|U| >= 2, |V| >= 2, B[U,V] = 0. +``` + +For every `M in GL(3,C)`, set + +``` +D = [x,y,u] M, T = [x,y,v] M^(-T). +``` + +Ordinary transpose gives `DT^T = xx^T + yy^T + uv^T`. Thus these three terms, +together with the other six unchanged coordinate terms, form a masked decoder. +Zeros within the chosen support containers are allowed as long as the actual +supports still satisfy the displayed conditions. + +**Theorem.** Write `r_j=rho(supp(D[:,j]),supp(T[:,j]))`. At least two r_j are +at least eight. Consequently + +``` +6*8 + sum_j r_j >= 64, +6*8^q + sum_j r_j^q >= 8*8^q (0 < q <= 1). +``` + +Identity mixing M=I attains equality for every allowed u,v: the replacement +ranks are `(8,8,0)`. This is a sharp statement for this family and objective. + +## Finite lemma: two common odd neighbors prevent span loss + +For the seed-selected representative coordinate pair `(2,4)`, every nonzero +complex combination of x,y has one of five supports. Representatives are +`x, y, x+y, x-y, x+2y`; call the supports S_0,...,S_4. This classification is +exhaustive over C: when both coordinates at a row are nonzero, cancellation +can occur only at ratio +1 or -1. The generic ratio 2 represents all other +nonzero ratios. Their support sizes are `56,56,76,76,88` and their binary row +span dimensions are `8,8,9,9,9`. + +The unmodified pairing-rank matrix is + +``` +8 8 8 8 8 +8 8 8 8 8 +8 8 9 9 9 +8 8 9 9 9 +8 8 9 9 9. +``` + +The primary certificate enumerates the entire 512-element row space of B. +For every character c and every S_i, form E=S_i intersect supp(c). Whenever +E is nonempty, it has **at most one common odd neighbor**: at most one index b +satisfies `B[e,b]=0` for every e in E. The receipt lists all 2,553 nonempty +cuts, their exact masks and their complete common-neighbor sets; seven cuts +are empty. No random coefficient search enters this claim. + +Why these characters suffice: a linear functional on the row space of B +extends to a dot product with a vector in F2^112. Its evaluations on the rows +of B lie in the column space of B, which equals its row space by symmetry. +Thus all possible binary separating functionals occur in the census. + +Suppose a set U has at least two common odd neighbors. If deleting U from +S_i decreased the span of the corresponding full rows of B, some binary +functional would vanish on all surviving rows but not on all original rows. +Its nonempty cut E would be contained in U, and would inherit those two +common odd neighbors. This contradicts the finite certificate. Hence + +``` +span_F2 B[S_i minus U, :] = span_F2 B[S_i, :]. (1) +``` + +The second checker verifies (1) by a different finite reduction. Any such U +is contained in `N(beta) intersect N(gamma)` for two distinct common odd +neighbors. It checks the largest possible deletion for **every** unordered +pair beta,gamma, then uses rank monotonicity for every smaller U. The maximal +deletion sizes and counts are + +| Deleted rows | Anchor pairs | +|---:|---:| +| 20 | 168 | +| 22 | 2,688 | +| 24 | 672 | +| 26 | 2,688 | + +All five span dimensions remain unchanged in all 6,216 cases. In particular, +this includes free-compatible deletions beyond the earlier 23-row cap. + +The certificate also gives 28 explicit permutations of the eight coordinates, +with signed bijections of the 112 root lines, carrying `(2,4)` to every +unordered coordinate pair. Both checkers verify their complete binary pairing +matrices and root-coordinate equations. Row signs do not alter supports or +binary pairings. Reversing the two chosen coordinates merely exchanges the +first two factor coordinates. Thus (1) and the five-stratum pairing table hold +for every pair, without assuming an unverified symmetry group classification. + +## Proof of the COPY floor over all complex factors + +Write the j-th columns of M and N=M^(-T) as `(a_j,c_j)` and `(b_j,d_j)`, with +`a_j,b_j in C^2`. The top-left block of `MN^T=I_3` gives + +``` +sum_j a_j b_j^T = I_2. +``` + +A sum of fewer than two nonzero rank-one matrices has rank at most one. +Therefore at least two terms have both a_j and b_j nonzero. For each such +term, its left support contains `S_i minus U` for some support stratum S_i, +and its right support contains `S_l minus V` for some S_l. Corrections cannot +cancel entries outside their actual supports. + +Since V has at least two elements and `B[U,V]=0`, equation (1) applies to U. +Similarly it applies to V, using two elements of U and symmetry of B. +Deleting U preserves the full row span of S_i; deleting V preserves the full +column span of S_l. Restricting a spanning set of rows to any fixed columns +still spans their restrictions, and the analogous statement holds for +columns. Consequently + +``` +rho(S_i minus U, S_l minus V) = rho(S_i,S_l) >= 8. +``` + +Enlarging supports cannot lower matrix rank. Each of the two active terms +therefore costs at least eight. This proves the two displayed bounds and +covers exactly two or three active projected products alike. Pure correction +terms remain in the bill; their rank is nonnegative. If all three products +are active, the whole separate COPY rank is at least 72. + +The arbitrary-complex quantifiers are supplied by this argument. The finite +Gaussian-rational fixtures check implementation and conventions, not a sample +from which those quantifiers are inferred. + +## Controls, budget, and boundary + +The identity fixture has a nonzero 24-by-2 correction and attains `(8,8,0)`. +A literal compensated cancellation fixture has ranks `(8,8,6)`, hence whole +cost 70. Omitting its third term changes constrained entry `(8,8)` by -1 in +the independent integer fixture. Other fixtures include nonreal coefficients, +a 4-by-8 rectangle, a maximal 26-by-2 rectangle, transposition, and dense and +triangular factor changes. +Wrong inverse-transpose, Hermitian substitution, singular mixing and nonfree +support mutations are rejected. + +A singleton-sided control shows why the two-anchor premise matters. For +coordinate x=X[:,2] and singleton column 42, deleting +`{i : x_i != 0 and B_i42=0}` lowers the row span from eight to seven. This is a +counterexample to extending **the span lemma** to one anchor, not to a COPY64 +theorem for singleton corrections. Those cases require a different argument. +Missing source pins, altered source digests, incomplete character/orbit +inventories, altered geometry and corrupted receipts also fail closed. + +The optimistic COPY screen retains the width, normalization, multiplicity and +separate-call convention of the original decoder. Its complete replacement +bill includes all three terms and the other six unchanged terms. Grant zero +additional preparation, source-access, frame-change, helper, restoration, +hosting and recursive-management expenses. Even this relaxation cannot reduce +this COPY bill. A physical implementation still must account for all those +operations; none is constructed here. No complete supplier moment or kappa is +inferred, and savings elsewhere in a supplier or a changed sharing convention +are outside the obstruction. + +## Provenance and overlap + +Exact base: `410026dc5513b87db92ae98571d1d5984c69a983`. The selected certificate +at that base is PR1094's conditional +`kappa=836330098538459/500000000000000000=0.001672660197076918`. +Its construction/source manifest and maintainer review were consulted; the +baseline source-binding check passed. This is not its full cold replay. +Selected, frozen, and upstream files are unchanged. + +OS assignment seed 42 selected structural decoder research. `TASK.json` +records three candidates, the stopping rule and exact predecessor heads. +The initial one-column scope was released when the complete overlap scan +found Gareth Davidson / bitplane's earlier +[Discussion1434](https://github.com/CrocSwap/integer-mult-bounds/discussions/1434). +The one allowed pivot tested two-column erasures. The same finite lemma then +proved the larger multiple-row, multiple-column statement above; this was a +logical consequence, not another search or pivot. The +[claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1316#discussioncomment-18912476) +records that change. `OVERLAP.json` records the final search coverage and scope +comparison. No absence of search hits is presented as proof of novelty. + +Credit and distinctions: + +- commdata2338 / PR1375: small-rectangle floor and the free-compatible erasure + boundary. The new result removes the size cap when both supports have at + least two elements. +- Cinna Davis / PR1373: arbitrary rectangles with three active products at + pair (4,6), and identification of the exactly-two-product branch. The present + proof handles that branch and all pairs under its explicit support premise. +- Arham Tahir / PR1331: one-coordinate arbitrary-rectangle predecessor and + the two-coordinate handoff; Discussion1360 considers all nine factors of a + single-entry completion and is distinct. +- Hanson Sun / PR1432: fixed 24-by-one boundary and coupled-factor accounting. +- Hanson Sun / concurrent PR1460, head + `8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6`: earlier arbitrary nonzero + rectangle COPY64 proof at (4,6), including the two-product branch. This + package adds the stronger span-preservation lemma, explicit all-pair + transport and an independent maximal-erasure method, not first priority for + that COPY64 subcase. +- Gareth Davidson / bitplane / PR1468, head + `5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a`: independent all-pair one-column + theorem after Discussion1434. Its singleton-sided class is outside this + proof's two-anchor premise; the coverage is complementary. +- Mihir Srivastava / PR845: pinned geometry implementation, imported only by + the primary checker. DreamingOfClouds / PR717 and PR446/595: masked decoder + and separate-copy framework. themilkman216 / PR968: pairing-cost model. + Joel Pulikkan / GamingPuzzled / PR620: coordinate COPY64 construction. + woklerx / PR1077: complex-coefficient orientation check. + +Pending predecessor theorems are contextual credit, not assumed proof steps. +The finite geometry and every premise used here are recomputed. All-size +compilation, common-label/mutable-source realization, legal frames/owners, +physical gathering, restoration/hosting/precision, analytic transfer and bit +H1-H5 obligations remain outside this result. + +Prepared for **Hanson Sun (@bbMisk)** with substantial **OpenAI Codex** +assistance in mathematics, both implementations, checking and writing. +Apache-2.0. + +## Open next questions + +- **Three original modes.** Fix (2,4,6), classify their exact complex support + strata, and test character cuts with two common odd neighbors before trying + a GL(4,C) synthesis. The five-stratum argument here does not extend without + a new classification and four-term cost proof. +- **Shared COPY calls.** An explicit shared endpoint contract could change + the separate-term objective. First produce a small legal two-term sharing + construction and price gathering, frames and restoration. No such contract + or complete physical invoice is supplied here. + +`TASK.json` records why these questions matter and their blockers. diff --git a/research/p112-multiline-free-rectangles-seed42/SOURCES.json b/research/p112-multiline-free-rectangles-seed42/SOURCES.json new file mode 100644 index 0000000000..16ed95b4aa --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/SOURCES.json @@ -0,0 +1,30 @@ +{ + "schema_version": 1, + "base_commit": "410026dc5513b87db92ae98571d1d5984c69a983", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb", + "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce" + }, + { + "path": "research/complex-frame-pair-decoders/PROOF.md", + "usage": "input", + "git_blob": "889f3a653ed1bdbb7df61ebe3af4f2103602a1f4", + "sha256": "ba53c1d8a9bc2d935d546bde46ca7341bd65f8515013e07b073296b4db2a9a74" + }, + { + "path": "research/p112-single-lift/geometry.py", + "usage": "input", + "git_blob": "34c1abf059d7be5fe5e5044cf01a09b843071c68", + "sha256": "3857c11ec731a602b8059d85d58fbb3b641bab687bb33016a0e54f7567752c8a" + } + ] +} diff --git a/research/p112-multiline-free-rectangles-seed42/TASK.json b/research/p112-multiline-free-rectangles-seed42/TASK.json new file mode 100644 index 0000000000..beefda98d4 --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/TASK.json @@ -0,0 +1,132 @@ +{ + "schema_version": 1, + "task_id": "complex-masked-decoder", + "priority_revision": 2, + "base_commit": "410026dc5513b87db92ae98571d1d5984c69a983", + "original_card": { + "id": "complex-masked-decoder", + "question": "Can one specified masked or overcomplete decoder improve the complete moment budget?", + "starting_points": [ + "research/complex-decoder-field-extension-seed44/README.md", + "research/complex-frame-pair-decoders/README.md" + ], + "methods": [ + "Synthesize a decoder with exact coefficients and fully priced helper endpoints.", + "Bound the best possible budget for a fixed support alphabet before synthesis." + ], + "acceptance_gate": "Change an assumption of the eight-term all-pairs bound and count all copied information, frames and restoration.", + "first_step": { + "command": [ + "python3", + "-B", + "research/complex-decoder-field-extension-seed44/verify.py" + ], + "expected": "PASS_COMPLEX_DECODER_FIELD_EXTENSION_S44", + "inspect": "Locate the excluded classes: masked identities, at least nine terms, and deeper punctures.", + "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay." + }, + "first_experiment": "Choose one finite support alphabet and test exact decoding constraints before building a whole network.", + "bottleneck_link": "A cheaper fully realized decoder could increase deficit or lower recursive cost on the binding complex side.", + "allowed_variations": [ + "Select a specific masked or overcomplete family with exact labels and supports." + ], + "scope_change_triggers": [ + "Repeating the exact eight-term all-pairs search inside its proved scope." + ], + "overlap_terms": [ + "masked decoder", + "overcomplete", + "eight-term" + ] + }, + "scope_status": "refined", + "actual_scope": "Every nonzero free P112 rectangle with at least two rows and two columns, coherently mixed with any two coordinate terms by GL(3,C); the other six coordinate terms are unchanged.", + "scope_change_reason": "The initial unrestricted singleton-column scope overlapped Discussion1434. A single pivot to two-column corrections gave a finite two-anchor span-preservation lemma, which also covers every rectangle with at least two indices on each side. No second mechanism or coefficient search was introduced.", + "bottleneck_link": "Tests whether masked rank-one corrections can reduce the binding complex supplier separate COPY power bill. The bound grants all additional physical costs zero and rules out this specified COPY-saving route.", + "budget_minutes": 45, + "stop_rule": "At most nine minutes orientation; one overlap-driven pivot. Stop synthesis if the optimistic complete COPY bill cannot improve. Reserve final quarter for verification and handoff.", + "outcome": "negative-result", + "evidence": "README.md contains the quantified proof; CERTIFICATE.json and INDEPENDENT.json carry exact finite-premise checks; verify.py and independent_check.py replay with semantic controls.", + "author": "Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, two implementations, verification and writing. Both implementations have the same AI-assisted contributor; no independent human review.", + "open_questions": [ + { + "question": "Can mixing three original coordinates with one free rectangle reduce the COPY bill?", + "why_it_matters": "The two-coordinate five-stratum classification no longer applies to GL(4,C).", + "first_step": "Fix coordinate modes (2,4,6); classify their exact complex support strata, then test binary character slices with at least two common odd neighbors.", + "blocked_by": "A new support classification and four-term coupled-factor proof are required." + }, + { + "question": "Can shared-copy billing exploit this rectangle family despite the separate-term floor?", + "why_it_matters": "The present proof lower-bounds individual COPY ranks, not an implementation with legal sharing between terms.", + "first_step": "Exhibit an explicit two-term shared endpoint contract and charge gathering, frames and restoration before considering a full supplier.", + "blocked_by": "No shared endpoint construction or full physical invoice is provided here." + } + ], + "assignment_seed": 42, + "assignment_seed_source": "Python secrets.randbelow(100), OS randomness; generated once at 2026-10-14T00:54:48Z", + "assigned_track": "structural decoder research", + "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1316#discussioncomment-18912476", + "candidate_tasks": [ + "Unrestricted singleton-column correction with two original coordinate modes (released: Discussion1434 already claimed it).", + "Three original coordinate modes on the fixed PR1432 24-row boundary.", + "Sparse-dependency nine-normal all-pairs family adjacent to PR1224/1325." + ], + "representative_coordinate_pair": [ + 2, + 4 + ], + "seed_diversification": "42 mod 3 selects initial candidate; 42 mod 28 selects pair (2,4); seed 42 selects fixture amplitudes and sites.", + "protected_invariant": "Every replacement term is charged by the exact binary label-pairing rank of both actual complex supports; no compensation term is omitted. Finite premises and written all-complex proof are distinct.", + "risk_and_authorization": "Only new isolated research package files are mutated; contribution and focused PR are authorized by the donation request. Selected frontier, upstream and frozen packages remain unchanged.", + "failure_modes": [ + "Confuse support cardinality with rank.", + "Drop the pure correction term or decouple inverse factors.", + "Treat a finite coefficient sample as a proof over C.", + "Forget singleton or zero-factor exclusions.", + "Infer a kappa from a COPY-only necessary-condition screen." + ], + "predecessors": [ + { + "pr": 1375, + "head": "4a533aff2a7a87c9b36e077a31774b332a7f64bd", + "author": "commdata2338", + "contribution": "Two-coordinate small-rectangle floor and first free-compatible erasure boundary." + }, + { + "pr": 1432, + "head": "4bcc49e3f4041018b0607da46a87d8928a6e0fd0", + "author": "Hanson Sun / bbMisk", + "contribution": "Fixed 24-by-one boundary and coupled-factor proof." + }, + { + "pr": 1331, + "head": "f08b698cc2bd74e2425486728a008e531fc35f0e", + "author": "Arham Tahir", + "contribution": "One-coordinate arbitrary rectangles; two-coordinate handoff." + }, + { + "pr": 1373, + "head": "3eaef3e4427068b33847dc1147c8a2e5709a970e", + "author": "Cinna Davis", + "contribution": "Arbitrary rectangles with three active projected products at pair (4,6); leaves exactly-two branch open." + }, + { + "discussion": 1434, + "author": "Gareth Davidson / bitplane", + "contribution": "Earlier independent unrestricted singleton-column investigation; released our overlapping scope." + }, + { + "pr": 1460, + "head": "8d53667d9bb09d9b1ffb0f865b5c8dc230eb02f6", + "author": "Hanson Sun / bbMisk", + "contribution": "Concurrent earlier arbitrary-rectangle COPY64 at fixed pair (4,6), via compensation-size argument. Our distinct evidence is the stronger per-active-term span lemma, explicit all-pair symmetry and maximal-erasure check." + }, + { + "pr": 1468, + "head": "5e2d8ec8ebff17c85e813ef1bd6f06312c1fe16a", + "author": "Gareth Davidson / bitplane", + "contribution": "Independent all-pair one-column theorem, complementary to our at-least-two-indices-per-side scope." + } + ] +} diff --git a/research/p112-multiline-free-rectangles-seed42/VALIDATION.json b/research/p112-multiline-free-rectangles-seed42/VALIDATION.json new file mode 100644 index 0000000000..9cb884d40a --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/VALIDATION.json @@ -0,0 +1,144 @@ +{ + "status": "PASS_SCOPED_LOCAL_VALIDATION", + "base_commit": "410026dc5513b87db92ae98571d1d5984c69a983", + "refreshed_upstream": "4040a0f46d67b41c36a472a7cd4af791fd4c36cc", + "environment": { + "os": "macOS-27.0.1-arm64-arm-64bit-Mach-O", + "registered_python": "3.12.2", + "optimized_python": "3.14.6", + "dependencies": "standard library only" + }, + "implementation_and_proof_sha256": { + "verify.py": "bab689f9b7805cee54accdb7b1ff73524374b2babd4dd76ad860deda6f845d98", + "independent_check.py": "a76f4fe9ad3bf7600a5bb474c5f675af3f16b622cc449d0e41109165d4f9f82a", + "README.md": "dd01a5f293c9f5f3ede914feb00827063a3d6ff5b2036b5204c7581d1eb386e0", + "CERTIFICATE.json": "80e885d5f864db8572ae15a45b95b055cfb42750a6d50309104ea88e89bc4cec", + "INDEPENDENT.json": "a6f12b12aad3e9d4d0af3bc2bff21dfc196348ca07eb3f0e1de67f01ccd75ea2", + "SOURCES.json": "d1571e772c5e22e96673514abca050a9f1eb660582f1e99a5d28764bd37fb6b4" + }, + "checks": [ + { + "command": [], + "returncode": 0, + "expected_fragment": null, + "elapsed_seconds": null, + "registered_python": "Python 3.12.2" + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-B", + "verify.py", + "--check", + "omitted-cut.json" + ], + "returncode": 1, + "expected_fragment": "complete certificate mismatch", + "elapsed_seconds": 1.378, + "registered_python": null + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-B", + "verify.py", + "--check", + "false-bound.json" + ], + "returncode": 1, + "expected_fragment": "complete certificate mismatch", + "elapsed_seconds": 1.369, + "registered_python": null + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-B", + "independent_check.py", + "--check", + "false-independent.json" + ], + "returncode": 1, + "expected_fragment": "complete independent receipt mismatch", + "elapsed_seconds": 13.999, + "registered_python": null + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-B", + "verify.py" + ], + "returncode": 1, + "expected_fragment": "complete input inventory", + "elapsed_seconds": 0.172, + "registered_python": null + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-B", + "verify.py" + ], + "returncode": 1, + "expected_fragment": "source digest", + "elapsed_seconds": 0.174, + "registered_python": null + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-S", + "-B", + "verify.py" + ], + "returncode": 0, + "expected_fragment": "PASS_P112_MULTILINE_FREE_RECTANGLES_S42", + "elapsed_seconds": 1.376, + "registered_python": null + }, + { + "command": [ + "python3.14", + "-I", + "-O", + "-S", + "-B", + "independent_check.py" + ], + "returncode": 0, + "expected_fragment": "PASS_P112_MULTILINE_INDEPENDENT_S42", + "elapsed_seconds": 14.099, + "registered_python": null + } + ], + "source_binding_check": "scripts/verify_selected_result.py passed; not a cold replay", + "orientation": "research_baseline.py and complex-decoder-field-extension-seed44/verify.py passed", + "read_only_export": { + "files_checked": 10, + "byte_identical_before_after": true, + "isolated": true, + "optimized": true, + "site_disabled": true + }, + "not_run": [ + "full selected-supplier cold replay", + "physical or native supplier admission", + "proof-assistant formalization", + "hosted CI at local validation time", + "Ruff: not installed" + ], + "limits": "Finite premises and exact fixtures support the written proof. No new kappa, independent human review or maintainer acceptance." +} diff --git a/research/p112-multiline-free-rectangles-seed42/VALIDATION.md b/research/p112-multiline-free-rectangles-seed42/VALIDATION.md new file mode 100644 index 0000000000..7729ae20ce --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/VALIDATION.md @@ -0,0 +1,37 @@ +# Executed local validation + +The exact implementation, proof and certificate digests are recorded in +`VALIDATION.json`. No external numerical dependencies are used. + +- Registered Python 3.12.2 replay passed both commands and the package's + metadata, live input and Git provenance checks. +- The primary certificate checks 2,553 cuts, all 28 coordinate symmetries, + seven Gaussian-rational fixtures / 87,808 entries and 12 internal controls. + The separate checker checks 6,216 anchor pairs / 31,080 spans, all proposed + symmetries / 351,232 pairing entries, a 12,544-entry integer identity and + six internal controls. +- Five external corrupted-input runs failed as intended: one omitted cut, a + falsified baseline rank, a missing independent span-check count, a removed + source pin and altered source bytes. +- Both commands passed under Python 3.14.6 with `-I -O -S -B` from a minimal + relocated, read-only source export. All ten exported file hashes were + identical before and after. Documentation added afterward is outside that + runtime-input export; the checked source and certificate bytes are pinned. +- The repository's 20 `test_research_*.py` tests passed under Python 3.12.2. + An earlier 3.14 run skipped its one explicitly 3.12-only replay test; the + 3.12 run covers it. A mistyped discovery pattern initially selected zero + tests; that empty run is not counted as evidence. +- `scripts/verify_selected_result.py` passed. The selected-record baseline + and the task card's complex-coefficient positive control also passed. + These are the stated source/arithmetic checks, not a selected cold replay. +- The additive package was checked against refreshed upstream + `4040a0f46d67b41c36a472a7cd4af791fd4c36cc`; its two live inputs, selected + record, contribution instructions and registered replay runner are unchanged + from the research base. Patch application and whitespace are checked before + submission. + +The scope excludes physical realization, all-size compilation, shared-copy +billing, full supplier recurrence, native admission and a multiplication +proof. No proof assistant or independent human review was used. Hosted CI and +maintainer review are separate from these local checks. Ruff was unavailable +in the local interpreter and is not claimed as a passed check. diff --git a/research/p112-multiline-free-rectangles-seed42/VERIFY.json b/research/p112-multiline-free-rectangles-seed42/VERIFY.json new file mode 100644 index 0000000000..ac5c3ab0f6 --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/VERIFY.json @@ -0,0 +1,32 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/complex-frame-pair-decoders/PROOF.md", + "research/p112-single-lift/geometry.py" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Complete binary-character census, coordinate automorphisms, exact complex fixtures, source and semantic rejection controls." + }, + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/independent_check.py" + ], + "timeout_seconds": 120, + "checks": "Direct Euclidean geometry, independent set elimination on all 6216 two-anchor erasures, coordinate symmetry and integer decoder identities." + } + ] +} diff --git a/research/p112-multiline-free-rectangles-seed42/independent_check.py b/research/p112-multiline-free-rectangles-seed42/independent_check.py new file mode 100644 index 0000000000..0a91a7c4d7 --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/independent_check.py @@ -0,0 +1,230 @@ +#!/usr/bin/env python3 +"""Independent finite-premise method: all maximal two-anchor erasures. + +No primary or predecessor imports. Euclidean roots and set elimination replace +the primary's lattice coordinates, packed-bit character census and cut proof. +Same AI-assisted author; this is not independent human review. Apache-2.0. +""" +import argparse +from collections import Counter +from copy import deepcopy +import hashlib +from itertools import combinations, product +import json +from pathlib import Path + +HERE = Path(__file__).resolve().parent +UNIVERSE = frozenset(range(112)) + + +def check(ok, reason): + if not ok: + raise ValueError(reason) + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) + + +def canonical(r): + return max(tuple(r), tuple(-x for x in r)) + + +def dimension(rows): + # Sparse Gaussian elimination with smallest-column pivots and symmetric + # differences; no integer bit-packing or calls to the primary rank engine. + echelon = {} + for raw in rows: + row = set(raw) + while row: + leading = min(row) + if leading in echelon: + row.symmetric_difference_update(echelon[leading]) + else: + echelon[leading] = row + break + return len(echelon) + + +def geometry(): + roots = {canonical(signs) for signs in product((-1, 1), repeat=8) + if product_sign(signs) == 1} + for a, b in combinations(range(8), 2): + for sign in (-1, 1): + r = [0] * 8 + r[a], r[b] = 2, 2 * sign + roots.add(canonical(r)) + check(len(roots) == 120, 'complete 120 E8 root lines') + frame = [] + for i in range(0, 8, 2): + for sign in (1, -1): + r = [0] * 8 + r[i], r[i + 1] = 2, 2 * sign + frame.append(tuple(r)) + roots = sorted(roots - set(frame)) + check(len(roots) == 112, 'P112 puncture') + x = [[dot(r, f) // 4 for f in frame] for r in roots] + matrix = [{j for j, s in enumerate(roots) if (1 + dot(r, s) // 4) % 2} + for r in roots] + for i in range(112): + for j in range(112): + check(dot(roots[i], roots[j]) % 4 == 0, 'integral root Gram') + check(dot(x[i], x[j]) == dot(roots[i], roots[j]) // 2, + 'coordinate/root identity') + check((j in matrix[i]) == (i in matrix[j]), 'pairing symmetry') + check(dimension(matrix) == 9, 'ambient binary rank') + return roots, x, matrix + + +def product_sign(values): + result = 1 + for value in values: + result *= value + return result + + +def supports(x): + return [{i for i, row in enumerate(x) if a * row[2] + b * row[4]} + for a, b in ((1, 0), (0, 1), (1, 1), (1, -1), (1, 2))] + + +def direct_erasures(matrix, ss): + original = [dimension(matrix[i] for i in s) for s in ss] + check(original == [8, 8, 9, 9, 9], 'five original spans') + stream, histogram = bytearray(), Counter() + for beta, gamma in combinations(range(112), 2): + deleted = UNIVERSE - matrix[beta] - matrix[gamma] + histogram[len(deleted)] += 1 + for wanted, s in zip(original, ss): + actual = dimension(matrix[i] for i in sorted(s - deleted)) + check(actual == wanted, 'two-anchor span preservation') + stream.append(actual) + check(sum(histogram.values()) == 6216 and len(stream) == 31080, + 'complete unordered-anchor domain') + return {'anchor_pairs': 6216, 'span_checks': len(stream), + 'original_spans': original, + 'maximal_rectangle_row_sizes': {str(k): v for k, v in sorted(histogram.items())}, + 'rank_stream_sha256': hashlib.sha256(stream).hexdigest()} + + +def verify_orbit(x, matrix, maps): + check(len(maps) == 28 and {tuple(m['pair']) for m in maps} + == set(combinations(range(8), 2)), 'all 28 coordinate pairs') + lookup = {canonical(r): i for i, r in enumerate(x)} + for m in maps: + p = m['permutation'] + check(sorted(p) == list(range(8)), 'coordinate permutation') + check(sorted([p[2], p[4]]) == m['pair'], 'selected pair image') + rows = [] + for i, r in enumerate(x): + moved = [0] * 8 + for k, coordinate in enumerate(p): + moved[coordinate] = r[k] + j = lookup.get(canonical(moved)) + check(j is not None, 'preserved live root set') + sign = 1 if moved == x[j] else -1 + check(j == m['row_map'][i] and sign == m['signs'][i], 'recorded signed root map') + rows.append(j) + check(len(set(rows)) == 112, 'root bijection') + for i in range(112): + check({rows[j] for j in matrix[i]} == matrix[rows[i]], 'transported pairing row') + return 28 * 112 * 112 + + +def term_cost(matrix, a, b): + right = {j for j, value in enumerate(b) if value} + return dimension(matrix[i] & right for i, value in enumerate(a) if value) + + +def integer_fixture(x, matrix): + v = [int(j in (42, 43)) for j in range(112)] + u_set = UNIVERSE - matrix[42] - matrix[43] + u = [x[i][2] if i in u_set else 0 for i in range(112)] + a, b = [row[2] for row in x], [row[4] for row in x] + # Literal algebra, without a matrix inverse or the primary fixture builder. + left = [[a[i] - u[i], b[i], u[i]] for i in range(112)] + right = [[a[i], b[i], a[i] + v[i]] for i in range(112)] + entries = 0 + for i in range(112): + for j in range(112): + check(dot(left[i], right[j]) == a[i] * a[j] + b[i] * b[j] + u[i] * v[j], + 'literal compensated identity') + entries += 1 + costs = [term_cost(matrix, [r[k] for r in left], [r[k] for r in right]) + for k in range(3)] + omission = next((i, j) for i in range(112) for j in matrix[i] + if u[i] * (a[j] + v[j]) != 0) + i, j = omission + error = -u[i] * (a[j] + v[j]) + check(error != 0, 'omission changes constrained entry') + return {'identity_entries': entries, 'replacement_ranks': costs, + 'whole_copy_cost': 48 + sum(costs), + 'omission_entry': [i, j], 'omission_error': error} + + +def rejects(fn, reason): + try: + fn() + except ValueError as exc: + check(reason in str(exc), 'wrong rejection: ' + str(exc)) + return reason + raise ValueError('negative control accepted: ' + reason) + + +def run(): + roots, x, matrix = geometry() + primary = json.loads((HERE / 'CERTIFICATE.json').read_text()) + packed = [sum(1 << j for j in row) for row in matrix] + obj = {'roots': roots, 'X': x, 'B': packed} + actual_hash = hashlib.sha256(json.dumps(obj, sort_keys=True, + separators=(',', ':')).encode()).hexdigest() + check(actual_hash == primary['geometry_sha256'], 'geometry agreement') + ss = supports(x) + baseline = [[dimension(matrix[i] & t for i in s) for t in ss] for s in ss] + check(baseline == primary['character_census']['baseline_pairings'], 'baseline agreement') + check(min(map(min, baseline)) == 8, 'baseline cost eight') + erasures = direct_erasures(matrix, ss) + orbit_entries = verify_orbit(x, matrix, primary['coordinate_orbit']) + fixture = integer_fixture(x, matrix) + controls = [] + controls.append(rejects(lambda: verify_orbit(x, matrix, primary['coordinate_orbit'][:-1]), + '28 coordinate')) + bad = deepcopy(primary['coordinate_orbit']) + bad[0]['row_map'][0], bad[0]['row_map'][1] = bad[0]['row_map'][1], bad[0]['row_map'][0] + controls.append(rejects(lambda: verify_orbit(x, matrix, bad), 'signed root map')) + deleted = ss[0] - matrix[42] + after = dimension(matrix[i] for i in ss[0] - deleted) + check(after == 7, 'single-anchor rank drop') + controls.append(rejects(lambda: check(after == 8, 'single-anchor span claim'), 'single-anchor')) + controls.append(rejects(lambda: check(fixture['omission_error'] == 0, 'omitted compensation'), + 'omitted compensation')) + controls.append(rejects(lambda: check(dimension([matrix[0] & {0}]) == 0, + 'diagonal rectangle is free'), 'diagonal rectangle')) + bad_baseline = deepcopy(baseline) + bad_baseline[0][0] -= 1 + controls.append(rejects(lambda: check(bad_baseline == baseline, 'changed rank receipt'), + 'changed rank receipt')) + return {'status': 'PASS_P112_MULTILINE_INDEPENDENT_S42', + 'geometry_sha256': actual_hash, 'geometry_entries': 12544, + 'baseline_pairings': baseline, 'direct_erasures': erasures, + 'orbit_pairing_entries': orbit_entries, 'integer_fixture': fixture, + 'negative_controls': controls, + 'scope': 'Independent two-anchor finite-premise check; quantified theorem is in README.'} + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--write', type=Path) + ap.add_argument('--check', type=Path, default=HERE / 'INDEPENDENT.json') + args = ap.parse_args() + result = run() + if args.write: + args.write.write_text(json.dumps(result, indent=2, sort_keys=True) + '\n') + else: + check(result == json.loads(args.check.read_text()), 'complete independent receipt mismatch') + print(result['status'], 'anchor_pairs=6216 span_checks=31080', + 'orbit_pairs=28 integer_entries=12544 controls=6') + + +if __name__ == '__main__': + main() diff --git a/research/p112-multiline-free-rectangles-seed42/verify.py b/research/p112-multiline-free-rectangles-seed42/verify.py new file mode 100644 index 0000000000..66f8eb291e --- /dev/null +++ b/research/p112-multiline-free-rectangles-seed42/verify.py @@ -0,0 +1,413 @@ +#!/usr/bin/env python3 +"""Exact finite premises for README's quantified proof, not a theorem prover. + +Hanson Sun / bbMisk, substantial OpenAI Codex assistance. Apache-2.0. +Geometry: Mihir Srivastava / PR845, following DreamingOfClouds / PR717. +""" +import argparse +from collections import Counter +from copy import deepcopy +from fractions import Fraction as F +import hashlib +import importlib.util +from itertools import combinations, permutations +import json +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +INPUTS = {'research/p112-single-lift/geometry.py', + 'research/complex-frame-pair-decoders/PROOF.md'} +PAIR = (2, 4) +STRATA = ((1, 0), (0, 1), (1, 1), (1, -1), (1, 2)) +ALL = (1 << 112) - 1 + + +def need(ok, why): + if not ok: + raise ValueError(why) + + +def digest(obj): + data = json.dumps(obj, sort_keys=True, separators=(',', ':')).encode() + return hashlib.sha256(data).hexdigest() + + +def check_sources(sources): + rows = [r for r in sources['files'] if r['usage'] == 'input'] + need(len(rows) == len(INPUTS) and {r['path'] for r in rows} == INPUTS, + 'complete input inventory') + for row in rows: + need(hashlib.sha256((ROOT / row['path']).read_bytes()).hexdigest() + == row['sha256'], 'source digest') + + +def basis(rows): + pivots = {} + for value in rows: + while value: + pivot = value.bit_length() - 1 + if pivot in pivots: + value ^= pivots[pivot] + else: + pivots[pivot] = value + break + return list(pivots.values()) + + +def rank(rows): + return len(basis(rows)) + + +def indices(mask): + return [i for i in range(112) if mask >> i & 1] + + +def support(values): + return sum(bool(v) << i for i, v in enumerate(values)) + + +def cost(pairing, left, right): + return rank(pairing[i] & right for i in indices(left)) + + +def canonical(row): + return max(tuple(row), tuple(-v for v in row)) + + +def model(): + spec = importlib.util.spec_from_file_location( + 'source_geometry', ROOT / 'research/p112-single-lift/geometry.py') + geo = importlib.util.module_from_spec(spec) + spec.loader.exec_module(geo) + data = geo.build() + geo.check_geometry(data) + order = sorted(range(112), key=lambda i: canonical(data['roots'][i])) + roots, x, pairing = [], [], [] + for i in order: + r = data['roots'][i] + sign = 1 if tuple(r) == canonical(r) else -1 + roots.append(list(canonical(r))) + x.append([sign * v for v in data['X'][i]]) + pairing.append(sum(((data['duals'][i] & data['labels'][j]).bit_count() % 2) + << k for k, j in enumerate(order))) + check_geometry(roots, x, pairing) + return roots, x, pairing + + +def check_geometry(roots, x, pairing): + need(len(roots) == len(x) == len(pairing) == 112, 'geometry size') + need(rank(pairing) == 9, 'ambient pairing rank nine') + for i in range(112): + need(set(x[i]) <= {-1, 0, 1}, 'coordinate alphabet') + for j in range(112): + gram4 = sum(a * b for a, b in zip(roots[i], roots[j])) + need(gram4 % 4 == 0, 'integral Gram') + need(sum(a * b for a, b in zip(x[i], x[j])) == gram4 // 2, + 'coordinate Gram identity') + need((pairing[i] >> j & 1) == (1 + gram4 // 4) % 2, + 'pairing bound to Euclidean Gram') + + +def common_odd(pairing, subset): + covered = 0 + for i in indices(subset): + covered |= pairing[i] + return ALL ^ covered + + +def strata(x, pair=PAIR): + return [support(a * row[pair[0]] + b * row[pair[1]] for row in x) + for a, b in STRATA] + + +def characters(pairing): + values = [0] + for b in basis(pairing): + values += [v ^ b for v in values] + return sorted(values) + + +def check_character_space(pairing, values): + need(len(values) == len(set(values)) == 512 and 0 in values, + 'complete 512-character space') + need(rank(values) == 9 and all(v in values for v in pairing), + 'character space equals full pairing row space') + + +def census(pairing, supports, values): + check_character_space(pairing, values) + records, hist = [], Counter() + for i, s in enumerate(supports): + for char in values: + cut = s & char + if not cut: + continue + neighbors = common_odd(pairing, cut) + need(neighbors.bit_count() <= 1, 'cut has at most one common odd neighbor') + records.append([i, format(char, 'x'), format(cut, 'x'), + format(neighbors, 'x')]) + hist[(i, cut.bit_count(), neighbors.bit_count())] += 1 + need(len(records) == 2553, 'complete nonempty cut inventory') + base = [[cost(pairing, s, t) for t in supports] for s in supports] + need(min(map(min, base)) == 8, 'baseline pairing floor eight') + return {'records': records, + 'histogram': [list(k) + [v] for k, v in sorted(hist.items())], + 'nonempty_cuts': len(records), 'empty_cuts': 5 * 512 - len(records), + 'support_sizes': [s.bit_count() for s in supports], + 'support_spans': [rank(pairing[i] for i in indices(s)) for s in supports], + 'baseline_pairings': base} + + +def orbit_maps(x): + lookup = {canonical(row): i for i, row in enumerate(x)} + need(len(lookup) == 112, 'distinct coordinate root lines') + maps = {} + for perm in permutations(range(8)): + target = tuple(sorted((perm[PAIR[0]], perm[PAIR[1]]))) + if target in maps: + continue + row_map, signs = [], [] + for row in x: + moved = [0] * 8 + for k in range(8): + moved[perm[k]] = row[k] + j = lookup.get(canonical(moved)) + if j is None: + break + row_map.append(j) + signs.append(1 if moved == x[j] else -1) + if len(row_map) == 112: + maps[target] = {'pair': list(target), 'permutation': list(perm), + 'row_map': row_map, 'signs': signs} + if len(maps) == 28: + break + return [maps[p] for p in sorted(maps)] + + +def check_maps(x, pairing, maps): + need(len(maps) == 28 and {tuple(m['pair']) for m in maps} + == set(combinations(range(8), 2)), 'complete 28-pair orbit') + for m in maps: + perm, rows, signs = m['permutation'], m['row_map'], m['signs'] + need(sorted(perm) == list(range(8)) and sorted(rows) == list(range(112)), + 'coordinate and root permutations') + need(sorted([perm[PAIR[0]], perm[PAIR[1]]]) == m['pair'], 'pair image') + for i in range(112): + need(signs[i] in (-1, 1) and all( + x[i][k] == signs[i] * x[rows[i]][perm[k]] for k in range(8)), + 'signed coordinate image') + for j in range(112): + need((pairing[i] >> j & 1) == (pairing[rows[i]] >> rows[j] & 1), + 'binary pairing preserved by orbit map') + + +# Gaussian rationals. Complex coefficients are never reduced modulo two. +def q(a=0, b=0): + return F(a), F(b) + + +def add(a, b): + return a[0] + b[0], a[1] + b[1] + + +def neg(a): + return -a[0], -a[1] + + +def mul(a, b): + return a[0] * b[0] - a[1] * b[1], a[0] * b[1] + a[1] * b[0] + + +def inv(a): + norm = a[0] ** 2 + a[1] ** 2 + need(norm != 0, 'invertible scalar') + return a[0] / norm, -a[1] / norm + + +def total(values): + out = q() + for value in values: + out = add(out, value) + return out + + +def inverse(m): + a = [row[:] + [q(i == j) for j in range(3)] for i, row in enumerate(m)] + for j in range(3): + pivot = next((i for i in range(j, 3) if a[i][j] != q()), None) + need(pivot is not None, 'invertible mixing matrix') + a[pivot], a[j] = a[j], a[pivot] + scale = inv(a[j][j]) + a[j] = [mul(v, scale) for v in a[j]] + for i in range(3): + if i != j: + scale = a[i][j] + a[i] = [add(v, neg(mul(scale, w))) for v, w in zip(a[i], a[j])] + return [row[3:] for row in a] + + +def check_rectangle(pairing, u, v): + su = support(z != q() for z in u) + sv = support(z != q() for z in v) + need(su.bit_count() >= 2 and sv.bit_count() >= 2, 'two indices on each side') + need(cost(pairing, su, sv) == 0, 'free rectangle') + return su, sv + + +def factors(x, u, v, m, wrong_transpose=False): + mi = inverse(m) + left = [[q(row[PAIR[0]]), q(row[PAIR[1]]), u[i]] for i, row in enumerate(x)] + right = [[q(row[PAIR[0]]), q(row[PAIR[1]]), v[i]] for i, row in enumerate(x)] + d = [[total(mul(row[k], m[k][j]) for k in range(3)) for j in range(3)] + for row in left] + t = [[total(mul(row[k], mi[k][j] if wrong_transpose else mi[j][k]) + for k in range(3)) for j in range(3)] for row in right] + return d, t + + +def validate_identity(x, u, v, d, t, conjugate=False, constrained=False, + pairing=None): + for i in range(112): + for j in range(112): + if constrained and not (pairing[i] >> j & 1): + continue + actual = total(mul(d[i][k], (t[j][k][0], -t[j][k][1]) + if conjugate else t[j][k]) for k in range(3)) + expected = q(sum(x[i][k] * x[j][k] for k in PAIR)) + if not constrained: + expected = add(expected, mul(u[i], v[j])) + need(actual == expected, 'exact ordinary-transpose decoder identity') + + +def rejection(fn, fragment): + try: + fn() + except ValueError as exc: + need(fragment in str(exc), 'unexpected rejection: ' + str(exc)) + return fragment + raise ValueError('negative control accepted: ' + fragment) + + +def fixtures(x, pairing): + zero, one, imag = q(), q(1), q(0, 1) + ident = [[q(i == j) for j in range(3)] for i in range(3)] + shear = [[one, zero, zero], [zero, one, zero], [q(-1), zero, one]] + dense = [[one, one, one], [zero, one, imag], [one, zero, one]] + triangle = [[one, q(1, 1), zero], [zero, one, imag], [zero, zero, one]] + v_mask = (1 << 42) | (1 << 43) + u_mask = common_odd(pairing, v_mask) + u = [q(x[i][PAIR[0]]) if u_mask >> i & 1 else zero for i in range(112)] + varied = [q(1 + (i + 42) % 3, (-1, 1)[i % 2]) if u_mask >> i & 1 else zero + for i in range(112)] + v = [imag if i == 42 else one if i == 43 else zero for i in range(112)] + wide_v_mask = common_odd(pairing, sum(1 << i for i in indices(u_mask)[:4])) + narrow_u = [q(1, i % 2) if i in indices(u_mask)[:4] else zero for i in range(112)] + wide_v = [q(1, -(i % 2)) if wide_v_mask >> i & 1 else zero for i in range(112)] + maximal_pair = next((a, b) for a, b in combinations(range(112), 2) + if common_odd(pairing, (1 << a) | (1 << b)).bit_count() == 26) + maximal_rows = common_odd(pairing, sum(1 << j for j in maximal_pair)) + maximal_u = [q(1, i % 2) if maximal_rows >> i & 1 else zero for i in range(112)] + maximal_v = [one if i == maximal_pair[0] else imag if i == maximal_pair[1] else zero + for i in range(112)] + cases = [('sharp identity with nonzero correction', varied, v, ident), + ('coupled cancellation and compensation', u, v, shear), + ('dense nonreal factor change', varied, v, dense), + ('triangular complex factor change', varied, v, triangle), + ('four-row wide rectangle', narrow_u, wide_v, dense), + ('transposed rectangle', v, varied, dense), + ('maximal 26-row free rectangle', maximal_u, maximal_v, dense)] + result, controls = [], [] + for name, uu, vv, m in cases: + su, sv = check_rectangle(pairing, uu, vv) + d, t = factors(x, uu, vv, m) + validate_identity(x, uu, vv, d, t) + costs = [cost(pairing, support(row[k] != zero for row in d), + support(row[k] != zero for row in t)) for k in range(3)] + need(sum(r >= 8 for r in costs) >= 2, 'two charged rank-eight terms') + result.append({'name': name, 'rows': indices(su), 'columns': indices(sv), + 'replacement_ranks': costs, 'whole_copy_cost': 48 + sum(costs), + 'identity_entries': 112 * 112}) + need(result[0]['replacement_ranks'] == [8, 8, 0], 'sharp COPY64 fixture') + d, t = factors(x, varied, v, dense) + controls.append(rejection(lambda: validate_identity(x, varied, v, d, t, True), + 'ordinary-transpose')) + wd, wt = factors(x, varied, v, dense, True) + controls.append(rejection(lambda: validate_identity(x, varied, v, wd, wt), + 'ordinary-transpose')) + d, t = factors(x, u, v, shear) + for row in d: + row[2] = zero + controls.append(rejection(lambda: validate_identity(x, u, v, d, t, + constrained=True, pairing=pairing), 'ordinary-transpose')) + controls.append(rejection(lambda: inverse([[one] * 3 for _ in range(3)]), + 'invertible mixing matrix')) + bad_v = v[:] + witness = next(i for i, z in enumerate(varied) if z != zero) + bad_v[witness] = one # B_ii=1, so this is a constrained entry. + controls.append(rejection(lambda: check_rectangle(pairing, varied, bad_v), + 'free rectangle')) + controls.append(rejection(lambda: check_rectangle(pairing, varied, + [one if i == 42 else zero for i in range(112)]), + 'two indices on each side')) + return result, controls + + +def run(): + sources = json.loads((HERE / 'SOURCES.json').read_text()) + check_sources(sources) + roots, x, pairing = model() + supports, values = strata(x), characters(pairing) + finite = census(pairing, supports, values) + maps = orbit_maps(x) + check_maps(x, pairing, maps) + examples, controls = fixtures(x, pairing) + changed = deepcopy(sources) + changed['files'] = [r for r in changed['files'] if r['path'] != sorted(INPUTS)[0]] + controls.append(rejection(lambda: check_sources(changed), 'input inventory')) + changed = deepcopy(sources) + next(r for r in changed['files'] if r['usage'] == 'input')['sha256'] = '0' * 64 + controls.append(rejection(lambda: check_sources(changed), 'source digest')) + controls.append(rejection(lambda: check_character_space(pairing, values[:-1]), + '512-character')) + controls.append(rejection(lambda: check_maps(x, pairing, maps[:-1]), '28-pair')) + bad_pairing = pairing[:] + bad_pairing[0], bad_pairing[1] = bad_pairing[1], bad_pairing[0] + controls.append(rejection(lambda: check_geometry(roots, x, bad_pairing), + 'Euclidean Gram')) + # Removing the second anchor breaks the span lemma, not necessarily the + # COPY theorem: singleton cases need their own coupled-factor argument. + boundary = supports[0] & (ALL ^ pairing[42]) + before = rank(pairing[i] for i in indices(supports[0])) + after = rank(pairing[i] for i in indices(supports[0] & ~boundary)) + need((before, after) == (8, 7), 'singleton boundary really loses span') + controls.append(rejection(lambda: need(after == before, 'singleton erasure preserves span'), + 'singleton erasure')) + return {'status': 'PASS_P112_MULTILINE_FREE_RECTANGLES_S42', + 'base_commit': sources['base_commit'], 'assignment_seed': 42, + 'representative_pair': list(PAIR), 'strata': [list(v) for v in STRATA], + 'geometry_sha256': digest({'roots': roots, 'X': x, 'B': pairing}), + 'character_space_sha256': digest([format(v, 'x') for v in values]), + 'character_census': finite, 'coordinate_orbit': maps, + 'fixtures': examples, 'negative_controls': controls, + 'singleton_scope_control': {'column': 42, 'deleted': indices(boundary), + 'before': before, 'after': after}, + 'scope': 'Nonzero free rectangles with >=2 indices on each side; two original modes mixed by arbitrary GL(3,C); separate COPY objective only.'} + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--write', type=Path) + ap.add_argument('--check', type=Path, default=HERE / 'CERTIFICATE.json') + args = ap.parse_args() + result = run() + if args.write: + args.write.write_text(json.dumps(result, indent=2, sort_keys=True) + '\n') + else: + need(result == json.loads(args.check.read_text()), 'complete certificate mismatch') + print(result['status'], 'cuts=2553 pairs=28 fixtures=7 entries=87808', + 'controls=' + str(len(result['negative_controls'])), 'copy_floor=64') + + +if __name__ == '__main__': + main()