From 9943467c4b24d2050275a202e3866cfc522afbe7 Mon Sep 17 00:00:00 2001 From: commdata2338 <199849969+commdata2338@users.noreply.github.com> Date: Wed, 14 Oct 2026 12:18:24 -0500 Subject: [PATCH] Research: prove sharp P112 two-column masked completion boundary On the canonical P112 root lines, an odd-only rank-two correction on exactly 53 rows and two columns gives completion rank nine instead of ten, and no correction with at most two changed columns and at most 52 rows fails rank additivity. Orthogonal column pairs force rank ten; odd pairs admit nine or ten. Every nonzero odd-only rank-one correction gives rank nine within 52 rows and columns, so PR #1447's suggested 53-by-53 rank-one follow-up cannot occur. Exact certificate over all 112 columns and 6,216 pairs with an independent checker. No COPY saving or kappa. Adds research/p112-two-column-completion/ only. Submitted by @commdata2338 with substantial AI assistance (OpenAI Codex research and implementation; packaged with Claude Code). --- .../p112-two-column-completion/CENSUS.json | 9249 +++++++++++++++++ research/p112-two-column-completion/README.md | 210 + .../p112-two-column-completion/RESULT.json | 922 ++ .../p112-two-column-completion/SOURCES.json | 30 + research/p112-two-column-completion/TASK.json | 120 + .../p112-two-column-completion/VERIFY.json | 42 + .../p112-two-column-completion/independent.py | 241 + .../p112-two-column-completion/test_verify.py | 118 + research/p112-two-column-completion/verify.py | 348 + 9 files changed, 11280 insertions(+) create mode 100644 research/p112-two-column-completion/CENSUS.json create mode 100644 research/p112-two-column-completion/README.md create mode 100644 research/p112-two-column-completion/RESULT.json create mode 100644 research/p112-two-column-completion/SOURCES.json create mode 100644 research/p112-two-column-completion/TASK.json create mode 100644 research/p112-two-column-completion/VERIFY.json create mode 100644 research/p112-two-column-completion/independent.py create mode 100644 research/p112-two-column-completion/test_verify.py create mode 100644 research/p112-two-column-completion/verify.py diff --git a/research/p112-two-column-completion/CENSUS.json b/research/p112-two-column-completion/CENSUS.json new file mode 100644 index 0000000000..4226746926 --- /dev/null +++ b/research/p112-two-column-completion/CENSUS.json @@ -0,0 +1,9249 @@ +{ +"schema_version": 1, +"singletons": [ + [0,60,[10,3,5,58,0,26,13,27],[0,1,2,3,4,5,6,7]], + [1,60,[10,2,4,58,1,26,13,27],[0,1,2,3,4,5,6,7]], + [2,60,[8,2,4,58,1,24,15,14],[0,1,2,3,4,5,6,7]], + [3,60,[8,3,5,58,0,24,15,14],[0,1,2,3,4,5,6,7]], + [4,60,[8,2,4,56,1,26,15,14],[0,1,2,3,4,5,6,7]], + [5,60,[8,3,5,56,0,26,15,14],[0,1,2,3,4,5,6,7]], + [6,60,[10,3,5,56,0,24,13,6],[0,1,2,3,4,5,6,7]], + [7,60,[10,2,4,56,1,24,13,7],[0,1,2,3,4,5,6,7]], + 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This is a sharp failure of +rank additivity in the two-column family. Every correction in that family +supported on at most 52 rows is additive. The new proof uses explicit finite +P112 geometry and does not need the E8 hyperplane-flat classification premise. + +The sparse-block theorem in [PR1447](../p112-sparse-completion-rank/README.md) +remains valid: its row and column caps are both 52. Its displayed 53-row +counterexample is unmasked. The construction here supplies a masked boundary +example, and settles that package's proposed rank-one follow-up: a nonzero +odd-only rank-one correction has at most 52 rows **and** 52 columns, so a +53-by-53 rank-one odd rectangle cannot exist. + +## Exact model and result + +Let `X` be `root_lines[8:]` in the pinned +[PROTOTYPE.json](../direct-delivery504-global/inputs/hamming/PROTOTYPE.json). +Its 112 integer rows have squared norm four. Put `R=X/sqrt(2)` and +`G=XX^T=2RR^T`. All transposes are ordinary transposes. Off-diagonal entries +of G are `0,+2,-2`. The binary pairing is `B_ij=(G_ij/2+1) mod 2`. +An **odd-only correction** satisfies `Q_ij=0` unless `G_ij=+2` or `-2`, +equivalently unless `B_ij=0`. Diagonal edits are forbidden. + +All ranks below are ordinary matrix ranks over **C**. For a correction with +at most two nonzero columns, the complete classification is: + +| Correction | Possible rank of G+Q | +| --- | --- | +| Q=0 | 8 | +| rank(Q)=1 | 9 | +| rank(Q)=2; the two changed columns are orthogonal roots | 10 | +| rank(Q)=2; the two changed columns are an odd root pair | 9 or 10, both attained for every such pair | + +There is an exact criterion, not a coefficient search: + +``` +rank(G+Q) = 8 + rank((56 I - G) Q) (at most two changed columns). +``` + +In addition, **every nonzero odd-only rank-one correction**, with no restriction +on its number of columns, gives `rank(G+Q)=9`. + +A rank-nine completion requires at least nine rank-one factors, including +cost-zero factors. This is not a lower bound on binary COPY cost or on recursive +child rank. In fact the explicit nine-factor construction below has separate +COPY ranks `(8,8,8,8,8,8,8,8,1)`, totaling **65**; it supplies no COPY improvement. +No optimization over its other factorizations is claimed. + +## The boundary construction + +For **any** odd pair of columns j,k, let e_j be the j-th coordinate vector and +write `g_j=G e_j`. Set + +``` +Q[:,j] = -g_j + 4 e_j, +Q[:,k] = e_j, +Q[:,other] = 0. +``` + +The first column has exactly 52 nonzero entries, all in the odd neighborhood +of j. The second changes only `(j,k)`, another allowed entry. Their supports +are disjoint, so Q has rank two, exactly 53 nonzero rows, and 53 nonzero entries. + +The completed column j is `4 e_j`. Every column outside `{j,k}` is unchanged +and those 110 columns span the eight-dimensional root code `C=col(X)`. +Also `e_j` is outside C. All completed columns lie in `C + span(e_j)`. +Therefore the completion has rank exactly nine. + +For explicit factors, let A equal `X^T` with column j set to zero and put +`b=4 e_j^T+e_k^T`. Then `G+Q = X A + e_j b`. The last term's diagonal entry +is part of this factorization; the **net correction Q** still has zero diagonal. +The verifier checks every entry of this identity and all nine binary COPY ranks. +The stored witness uses zero-based columns `(0,1)`. + +Rank ten is also attainable for every pair: choose two distinct row indices +u,v with `G_uj` and `G_vk` odd, and take `Q=e_u e_j^T+e_v e_k^T`. +Each odd neighborhood has 52 choices. Deleting any two rows leaves X of rank +eight, so `span(e_u,e_v)` meets C trivially. This proves rank ten. The stored +examples cover an odd pair `(0,1)` and an orthogonal pair `(0,3)`. + +## Proof of the classification and the 53-row minimum + +The certificate establishes four finite facts, by explicit nonzero integer +minors or exact integer identities: + +1. `X^T X=56 I_8`. Every root has 52 odd neighbors. For every j, the roots on + `S_j={i:G_ij is not +2 or -2}` span dimension eight. +2. Deleting any two rows of X leaves rank eight. +3. For all 6,216 unordered column pairs, the roots on `S_j intersect S_k` + span dimension eight when j,k are orthogonal, and dimension six when + they are an odd pair. In the latter case their span is precisely the + perpendicular complement of the two independent roots `X_j,X_k`. +4. For every odd pair, every nonzero complex vector in the pencil + `span(G[:,j],G[:,k])` has support at least 53, and 53 is attained. + +The first three facts give the rank classification. Unchanged columns outside +the two-column support J span C, so the columns of `G+Q` span exactly +`C+col(Q)`. The map `56I-G` has kernel C, giving the displayed rank criterion. +No nonzero allowed column of Q lies in C: such a column would be `Xa`, vanish +on S_j, and force a=0 by fact 1. Thus a nonzero correction adds at least one +quotient dimension. A rank-two correction can add only one or two. +For orthogonal j,k, any vector in `col(Q) intersect C` vanishes on +`S_j intersect S_k`; fact 3 forces it to be zero, so both dimensions are added. + +For the 53-row minimum, a failure of additivity needs a nonzero `Xa` in +`col(Q) intersect C`. Only an odd column pair can permit this. By fact 3, +`a` lies in `span(X_j,X_k)`, so `Xa` belongs to the pencil in fact 4. +Consequently its support has at least 53 rows. Its support is contained in +the row support of Q. This proves the minimum, without appealing to a global +root-code distance theorem. The construction attains it. One changed column +has rank at most one and cannot fail additivity, so two columns are minimal too. + +Fact 4 covers all complex coefficients by an elementary finite arrangement +argument. A row of a pencil has value `a G_ij+b G_ik`. If at least one of +those two integers is nonzero, it vanishes at precisely one projective +direction `(a:b)=(G_ik:-G_ij)`. Enumerate these primitive integer directions; +all other complex directions have the same support as the generic `(1:3)`. +The certificate records every exceptional direction and its exact support for +all 2,912 odd pairs. Each pencil has minimum 53. This is an exhaustive +classification of zero patterns, not numerical sampling of C. + +Finally, for an arbitrary nonzero rank-one correction `Q=u v^T`, the mask +forces `supp(u) x supp(v)` to be an odd rectangle. Each support is contained +in one 52-element odd neighborhood. Fact 1 shows both u and v are outside C. +Both `[X,u]` and `[X,v]` have full column rank nine, so their product +`XX^T+u v^T` has rank nine over C. This proves the unrestricted rank-one claim. + +## Certificate and reproduction + +Python 3.12, standard library only, from the repository root: + +```sh +python3 -B scripts/research_package.py --package research/p112-two-column-completion --check-only +python3 -B scripts/research_package.py --package research/p112-two-column-completion +``` + +Expected replay markers: + +``` +PASS_P112_TWO_COLUMN_COMPLETION pairs=6216 odd=2912 orthogonal=3304 boundary_rows=53 correction_rank=2 completion_rank=9 controls=8 +PASS_P112_TWO_COLUMN_INDEPENDENT integer_minors=12544 pencils=2912 witnesses=5 root_isometry=120 +``` + +The primary checker reconstructs every census record and compares both complete +JSON artifacts. It uses modular elimination only for nonzero-minor lower bounds; +upper bounds come from dimension, orthogonality or exact factorizations. +The separately written checker imports neither the primary nor predecessor +code. It enumerates the standard E8 roots used by PR1447, applies the exact +block transform `(x,y) -> ((x+y)/2,(x-y)/2)`, and checks equality with all 120 +pinned root lines up to sign. Thus the two P112 coordinate conventions are +explicitly related; the transformed removed frame is the first eight axes. +It checks **12,544 integer minors** with Bareiss determinants, all **2,912 +projective pencils**, and five full correction/completion matrices with +integer elimination over Q. It also recomputes the separate COPY65 invoice. + +[CENSUS.json](CENSUS.json) contains all singleton and pair minor witnesses, +pair-deletion witnesses, and pencil strata. [RESULT.json](RESULT.json) contains +summary counts, the canonical census hash, five sparse exact matrices and eight +failure controls. There are 2,912 odd pairs and 3,304 orthogonal pairs. These +implementations are by the same assisted contributor, not independent human +review or proof-assistant formalization. + +Both direct checkers support `-I -O -B` and ordinary replay is read-only. +`verify.py --write-to NEW_DIRECTORY` regenerates both certificates without +overwriting existing files; `--write` creates initial certificates in the package. Source bytes and historical provenance are pinned in +[SOURCES.json](SOURCES.json); the registered hook declares both live inputs. +Focused tests exercise exact arithmetic, the mathematical boundaries, malformed +certificates, source changes and a Gaussian-integer completion. + +## Scope and attribution + +The claim is in [discussion #2006](https://github.com/CrocSwap/integer-mult-bounds/discussions/2006). +The chooser selected `complex-rank-shape-decoder`, priority revision 3, nonce +`46dd6716742415eb113c9b4830c4ada9`, at main +`1ff9c501b624ec354e1330b1b9be571a6adba9cb`. The scope is explicitly **changed** +from physical block-climb synthesis to an algebraic masked-completion boundary. +The original five-percent construction gate is not met. Rank additivity does +not provide a paid local profile, so applying the packing-ceiling calculator to +these ordinary matrix ranks would be invalid. The selected word, its κ and all +predecessor files remain unchanged. + +Credit woklerx / PR1447, head `5f784a84bbe80879b5bd7ae9b48dd28a5765863d`, +for the sparse-block theorem and its explicit open masked-boundary question; +DreamingOfClouds / PR717 for the masked decoder model; themilkman216 / PR968 +and woklerx / PR1077 for the geometry and complex ordinary-transpose conventions; +and the PR523/514/507/425/433 lineage for the pinned Hamming/root data. +This account's PR1375, and PR1331/1432/1468/1473, concern separate COPY costs +under restricted factor mixing. This result concerns completion matrix rank +and does not repeat or extend those COPY lower bounds. No pending PR code is +imported. Source licenses and notices are retained; new material is Apache-2.0. + +## Open next questions + +Does another exact factorization of the displayed +53-by-2 completion improve on its direct COPY65 bill, perhaps approaching or +beating the original COPY64 decoder? A feasible first step is to parameterize +its rank-nine factors in the fixed nine-dimensional column and row spaces, +then check exact identities and all support-pairing ranks under candidate +basis changes. The blocker is the coupled support-sensitive factor search; +rank nine alone proves no COPY saving. Even a favorable COPY bill would still +need source access, frames, dirty-state restoration, hosting, precision/tape, +complete recursive pricing and the inherited all-size/analytic obligations. +Those obligations, a full selected cold replay, Lean, hosted CI and independent +human review have not been supplied. Structured next questions and overlap +context are recorded in [TASK.json](TASK.json). + +Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance. diff --git a/research/p112-two-column-completion/RESULT.json b/research/p112-two-column-completion/RESULT.json new file mode 100644 index 0000000000..d6723f4d54 --- /dev/null +++ b/research/p112-two-column-completion/RESULT.json @@ -0,0 +1,922 @@ +{ + "base_commit": "1ff9c501b624ec354e1330b1b9be571a6adba9cb", + "boundary": "Rank-two odd-only corrections on 53 rows and two columns attain completion rank 9; rank-one odd rectangles have at most 52 rows and 52 columns.", + "census_sha256": "588436fb2db1a46c601c610685407117d32ed0fde8e2608478715e0cfc35c121", + "controls": [ + "nonmasked_diagonal", + "rank1_correction_mislabel", + "false_additivity_rank10", + "false_rank8_completion", + "omitted_second_column", + "orthogonal_pair_low_witness", + "wide_rectangle_as_two_columns", + "wrong_projection_normalization" + ], + "frame_constant": 56, + "gram_rank": 8, + "normalization": "X has squared row norm 4; R=X/sqrt(2); G=XX^T=2RR^T", + "odd_degree": 52, + "odd_pairs": 2912, + "odd_pencils_checked": 2912, + "orthogonal_pairs": 3304, + "pair_class_histogram": { + "odd|32|6": 224, + "odd|34|6": 2688, + "orthogonal|28|8": 168, + "orthogonal|30|8": 2688, + "orthogonal|32|8": 448 + }, + "result_kind": "scoped-proof-with-finite-certificate", + "schema_version": 1, + "scope": "Ordinary complex matrix rank, not binary COPY cost, recursive child rank, a new supplier or kappa.", + "singletons_checked": 112, + "source_sha256": "327d2a868043e846381aa58a33053c24425e8cad522ff401b10ec8af1d649a3f", + "theorem": "For <=2 changed columns: rank(G+Q)=8+rank((56I-G)Q). 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"research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json", + "usage": "input", + "git_blob": "9efe30ee107b9c67f15f1c726e1a1cb5cef03929", + "sha256": "327d2a868043e846381aa58a33053c24425e8cad522ff401b10ec8af1d649a3f" + }, + { + "path": "research/p112-sparse-completion-rank/README.md", + "usage": "input", + "git_blob": "1118f0a09c80c4ab54bb048d21a8d3d2cb59d283", + "sha256": "771893ee9c728d8fe84b69ec9883133f9284d5f0319c890ab37d3949f152777a" + } + ] +} diff --git a/research/p112-two-column-completion/TASK.json b/research/p112-two-column-completion/TASK.json new file mode 100644 index 0000000000..31c3caa1cb --- /dev/null +++ b/research/p112-two-column-completion/TASK.json @@ -0,0 +1,120 @@ +{ + "schema_version": 1, + "task_id": "complex-rank-shape-decoder", + "priority_revision": 3, + "base_commit": "1ff9c501b624ec354e1330b1b9be571a6adba9cb", + "original_card": { + "id": "complex-rank-shape-decoder", + "question": "Can a decoder deliver the local paid mass in fewer, larger-rank children (block frame climbs instead of rank-1 to rank-3 steps)?", + "starting_points": [ + "research/complex-local-word-ceiling/README.md", + "research/complex-decoder-field-extension-seed44/README.md", + "research/complex-frame-pair-decoders/README.md" + ], + "methods": [ + "Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.", + "Bound the best effective child rank achievable for one fixed support alphabet, including copies, frames and restoration." + ], + "acceptance_gate": "A per-call local profile, with all copied information, frames and restoration counted, whose ceiling exceeds the selected word's by at least 5%; masked or overcomplete supports must change an assumption of the eight-term all-pairs bound.", + "first_step": { + "command": [ + "python3", + "-B", + "research/complex-local-word-ceiling/verify.py" + ], + "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING", + "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.", + "scope": "Positive control and planning budget only; this is not a new construction or full cold replay." + }, + "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin. Ideal budget: rank-16 delivery is 2.31x (levers.py).", + "bottleneck_link": "Rank-1 children are about half of recursive cost; the selected word's effective child rank is about 3.9. Larger-rank delivery lowers recursive cost per unit of paid mass.", + "allowed_variations": [ + "One masked, overcomplete or block decoder family with exact labels and supports.", + "A scoped lower bound on effective rank for one alphabet.", + "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope." + ], + "scope_change_triggers": [ + "Repacking idle remainders of an unchanged word (covered by the packing ceiling).", + "Repeating the eight-term all-pairs search inside its proved scope." + ], + "overlap_terms": [ + "rank shape", + "block climb", + "masked decoder", + "overcomplete", + "effective rank" + ] + }, + "scope_status": "changed", + "actual_scope": "Classify ordinary complex matrix ranks of canonical P112 odd-only completions with at most two changed columns; prove a sharp 53-row rank-two additivity failure and the unrestricted rank-one correction result.", + "scope_change_reason": "The physical block-climb card is changed to the explicit masked-completion boundary left by PR1447. This is an algebraic side result, not a paid local profile or a claim to meet the five-percent construction gate.", + "bottleneck_link": "A masked completion fixes the scalar matrix that a decoder must factor. Its ordinary rank bounds the number of scalar rank-one factors, but is distinct from binary COPY charges and recursive child ranks. The displayed nine-factor witness costs COPY65, so it is not an improvement over COPY64.", + "budget_minutes": 60, + "stop_rule": "Stop after all singleton/two-column finite geometry and the coefficient-universal rank proof; no physical synthesis, factor-optimization expansion or selected-frontier change.", + "outcome": "completed", + "evidence": "README.md proof; CENSUS.json complete 112 singleton / 6216 pair / 2912 projective-pencil records; RESULT.json five witnesses and eight controls; independent integer-minor and rational-rank replay; focused tests.", + "author": "commdata2338", + "ai_assistance": "Substantial OpenAI Codex assistance in scope choice, mathematics, implementations, testing and writing. No subagents or independent human review.", + "open_questions": [ + { + "question": "Can another exact nine-term factorization of the displayed 53-by-2 completion reduce its direct COPY65 bill?", + "why_it_matters": "Ordinary rank nine alone does not lower the COPY64 incumbent or its recursive cost. The explicit masked boundary gives a concrete matrix for a support-sensitive search.", + "first_step": "Parameterize rank-nine factors in the fixed column and row spaces, check the complete matrix identity, and price every separate binary support pairing under exact basis changes.", + "blocked_by": "Unrestricted coupled support search and, beyond a favorable COPY bill, all physical source/frame/restoration/hosting/precision and recursive pricing obligations." + } + ], + "assignment_nonce": "46dd6716742415eb113c9b4830c4ada9", + "track": "complex-mechanisms", + "scope_key": "02b31ebd160542c86299/P112/odd-mask/two-column-completion-rank", + "coordination": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2006", + "predecessors": [ + { + "pr": 1447, + "head": "5f784a84bbe80879b5bd7ae9b48dd28a5765863d", + "author": "woklerx", + "scope": "Sparse-block rank additivity through 52 rows and columns; unmasked 53-row boundary and open masked follow-up." + } + ], + "overlap": { + "checked_at": "2026-10-14T17:05:07.140958+00:00", + "nearest": [ + "PR1447: predecessor; new masked boundary and full two-column classification", + "PR1331/1375/1432/1468/1473: binary COPY costs under factor mixing; different object", + "PR1600: binary parent cut information; different field and objective", + "PR1814/1865/1960: chronological gather/fanout/diagonal screens; different object" + ], + "coverage_limit": "Anonymous cached open-PR bodies and Discussions with complete reported pagination, local package notes and ledger; no absence-of-overlap proof.", + "coverage": { + "open_prs": { + "fetched_at": "2026-10-14T17:03:27.953145+00:00", + "records": 381, + "complete_pagination": true + }, + "discussions": { + "fetched_at": "2026-10-14T17:05:07.140958+00:00", + "records": 646, + "complete_pagination": true + }, + "changed_files": { + "fetched_at": "2026-10-14T16:41:39.242517+00:00", + "records": 404, + "complete_pagination": true + } + }, + "final_findings": "Text queries return this claim in discussion 2006 and distinct COPY/deletion-screen packages; path-filter query has no open-PR match. No code-diff-complete or novelty claim." + }, + "unchecked_obligations": [ + "No COPY optimum or paid local profile", + "No physical supplier or native/compiler admission", + "No dirty-state restoration, hosting, precision/tape or all-size/analytic transfer", + "No selected cold replay, Lean proof, hosted CI or independent human review" + ], + "verification": { + "python": "CPython 3.12.3, GCC 13.3.0, standard library only", + "registered_replay": "passed; primary, independent, and 8 focused test groups", + "metadata_and_source_pins": "passed", + "optimized_read_only_relocation": "both standalone checkers passed with -I -O -B; all copied file hashes unchanged", + "arithmetic_oracle": "all 19683 integer 3x3 matrices over {-1,0,1} for Bareiss versus permutation determinants; all 81 2x2 matrices for rational rank; modular rank loss control", + "complex_fixture": "exact rational realification of G+(1+i)Q has rank 18, hence complex rank 9" + } +} diff --git a/research/p112-two-column-completion/VERIFY.json b/research/p112-two-column-completion/VERIFY.json new file mode 100644 index 0000000000..282d41ac16 --- /dev/null +++ b/research/p112-two-column-completion/VERIFY.json @@ -0,0 +1,42 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json", + "research/p112-sparse-completion-rank/README.md" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Complete exact geometry, projective pencils, rank witnesses, source pins, COPY65 invoice and eight rejection controls." + }, + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/independent.py" + ], + "timeout_seconds": 120, + "checks": "Independent E8 root isometry, 12544 integer minors, 2912 pencils and five rational rank witnesses." + }, + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/test_verify.py" + ], + "timeout_seconds": 120, + "checks": "Arithmetic oracles, exact complex realification, source/receipt rejection and mask boundary controls." + } + ] +} diff --git a/research/p112-two-column-completion/independent.py b/research/p112-two-column-completion/independent.py new file mode 100644 index 0000000000..f0118eece0 --- /dev/null +++ b/research/p112-two-column-completion/independent.py @@ -0,0 +1,241 @@ +#!/usr/bin/env python3 +"""Independent integer-minor audit; imports no primary/predecessor checker. +Apache-2.0. Prepared for commdata2338 with substantial OpenAI Codex assistance. +""" +from collections import Counter +from hashlib import sha256 +from fractions import Fraction +from itertools import combinations, product +from math import gcd +from pathlib import Path +import json + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SOURCE = 'research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json' +SOURCE_SHA = '327d2a868043e846381aa58a33053c24425e8cad522ff401b10ec8af1d649a3f' + + +def check(ok, message): + if not ok: + raise ValueError(message) + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) + + +def determinant(rows): + """Bareiss determinant over Z, including row swaps and divisibility checks.""" + a = [list(r) for r in rows] + n = len(a) + check(all(len(r) == n for r in a), 'square minor') + previous, sign = 1, 1 + for col in range(n - 1): + pivot = next((i for i in range(col, n) if a[i][col]), None) + if pivot is None: + return 0 + if pivot != col: + a[col], a[pivot] = a[pivot], a[col] + sign = -sign + value = a[col][col] + for i in range(col + 1, n): + for j in range(col + 1, n): + numerator = a[i][j] * value - a[i][col] * a[col][j] + check(numerator % previous == 0, 'Bareiss exact division') + a[i][j] = numerator // previous + a[i][col] = 0 + previous = value + return sign * (a[-1][-1] if n else 1) + + +def rational_rank(rows): + """Exact row rank by integral elimination with primitive-row reduction.""" + if not rows: + return 0 + a = [list(r) for r in rows] + width, height = len(a[0]), len(a) + check(all(len(r) == width for r in a), 'rectangular integer matrix') + r = 0 + for col in range(width): + pivot = next((i for i in range(r, height) if a[i][col]), None) + if pivot is None: + continue + a[r], a[pivot] = a[pivot], a[r] + p = a[r][col] + for i in range(r + 1, height): + q = a[i][col] + if not q: + continue + tail = [p * a[i][j] - q * a[r][j] for j in range(col, width)] + divisor = 0 + for value in tail: + divisor = gcd(divisor, value) + if divisor: + tail = [v // divisor for v in tail] + a[i][col:] = tail + r += 1 + if r == min(width, height): + break + return r + + +def independent_roots(source): + # The standard E8 model used by PR1447, at squared norm eight. + standard = set() + for i, j in combinations(range(8), 2): + for a, b in product((-2, 2), repeat=2): + x = [0] * 8; x[i], x[j] = a, b + standard.add(max(tuple(x), tuple(-v for v in x))) + for x in product((-1, 1), repeat=8): + if sum(v < 0 for v in x) % 2 == 0: + standard.add(max(x, tuple(-v for v in x))) + # Block matrix H/2 takes (2,+/-2) frame roots to the coordinate axes. + rotated = set() + for x in standard: + y = tuple(v for i in range(0, 8, 2) + for v in ((x[i] + x[i + 1]) // 2, (x[i] - x[i + 1]) // 2)) + check(all((x[i] + x[i + 1]) % 2 == 0 for i in range(0, 8, 2)), 'integral rotation') + rotated.add(max(y, tuple(-v for v in y))) + exported = [tuple(x) for x in source['root_lines']] + check(len(standard) == len(rotated) == 120, 'complete independent E8 enumeration') + check(len(exported) == 120 and set(exported) == rotated, 'source root-set isometry') + check(exported[:8] == [tuple(2 * int(i == j) for j in range(8)) for i in range(8)], + 'same punctured frame') + return exported[8:] + + +def check_minor(roots, rows, columns, allowed, size): + check(len(rows) == len(set(rows)) == size, 'minor row multiplicity') + check(len(columns) == len(set(columns)) == size, 'minor column multiplicity') + check(all(type(i) is int and i in allowed for i in rows), 'minor outside required roots') + check(all(type(j) is int and 0 <= j < 8 for j in columns), 'minor coordinate range') + value = determinant([[roots[i][j] for j in columns] for i in rows]) + check(value != 0, 'singular claimed minor') + return abs(value) + + +def audit_census(roots, gram, cert): + singles, pairs = cert['singletons'], cert['pairs'] + check(len(singles) == 112 and len(pairs) == 6216, 'complete census cardinality') + determinants = Counter() + histogram = Counter() + for j, record in enumerate(singles): + check(len(record) == 4 and record[0] == j, 'singleton coverage') + allowed = {i for i in range(112) if gram[i][j] not in (-2, 2)} + check(record[1] == len(allowed) == 60, 'singleton constrained row count') + determinants[check_minor(roots, record[2], record[3], allowed, 8)] += 1 + for (j, k), record in zip(combinations(range(112), 2), pairs): + check(len(record) == 8 and record[:3] == [j, k, gram[j][k]], 'pair coverage and parity') + allowed = {i for i in range(112) if gram[i][j] not in (-2, 2) and gram[i][k] not in (-2, 2)} + odd = gram[j][k] in (-2, 2) + size = 6 if odd else 8 + check(record[3] == len(allowed), 'common constrained row count') + if odd: + check(all(dot(roots[i], roots[j]) == dot(roots[i], roots[k]) == 0 for i in allowed), + 'complex-rank-six upper-bound premise') + check(16 - gram[j][k] ** 2 != 0, 'independent normals') + determinants[check_minor(roots, record[4], record[5], allowed, size)] += 1 + determinants[check_minor(roots, record[6], record[7], set(range(112)) - {j, k}, 8)] += 1 + histogram[('odd' if odd else 'orthogonal') + '|' + str(len(allowed)) + '|' + str(size)] += 1 + expected_pairs = [(j, k) for j, k in combinations(range(112), 2) if gram[j][k] in (-2, 2)] + check(len(cert['odd_pencils']) == 2912, 'all odd pencils') + for (j, k), record in zip(expected_pairs, cert['odd_pencils']): + check(len(record) == 4 and record[:2] == [j, k], 'pencil coverage') + active = sum(bool(row[j] or row[k]) for row in gram) + check(record[2] == active, 'pencil generic support') + # Independent classification by finite slopes b/a, with None for infinity. + slopes = {(-Fraction(row[j], row[k]) if row[k] else None) + for row in gram if row[j] or row[k]} + seen = set() + supports = [] + for a, b, size in record[3]: + check(type(a) is int and type(b) is int and (a or b), 'nonzero pencil direction') + check(gcd(a, b) == 1 and (a > 0 or (a == 0 and b > 0)), 'canonical direction') + slope = Fraction(b, a) if a else None + check(slope not in seen, 'duplicate pencil stratum') + seen.add(slope) + actual = sum(a * row[j] + b * row[k] != 0 for row in gram) + check(size == actual and size >= 53, 'exact pencil support floor') + supports.append(size) + check(seen == slopes and min(supports) == 53, 'complete projective exceptional slopes') + check(Fraction(3) not in slopes and sum(row[j] + 3 * row[k] != 0 for row in gram) == active, + 'generic complex-direction support') + return dict(sorted(histogram.items())), dict(sorted(determinants.items())) + + +def audit_result(roots, gram, result, cert): + canonical = json.dumps(cert, sort_keys=True, separators=(',', ':')).encode() + check(sha256(canonical).hexdigest() == result['census_sha256'], 'census commitment') + check(result['source_sha256'] == SOURCE_SHA, 'result source binding') + check(result['singletons_checked'] == 112 and result['unordered_pairs_checked'] == 6216, + 'reported census size') + check(result['odd_pairs'] == 2912 and result['orthogonal_pairs'] == 3304, 'pair class counts') + check(result['odd_pencils_checked'] == 2912 and result['two_column_failure_row_floor'] == 53, 'pencil summary') + check(result['gram_rank'] == rational_rank(gram) == 8, 'Gram rank') + check(result['odd_degree'] == 52 and all(sum(v in (-2, 2) for v in row) == 52 for row in gram), + 'rank-one rectangle cardinality bound') + check(result['frame_constant'] == 56, 'projection factor') + check([[sum(r[a] * r[b] for r in roots) for b in range(8)] for a in range(8)] == + [[56 * int(a == b) for b in range(8)] for a in range(8)], 'projection kernel') + expected_names = ['odd_pair_rank9', 'odd_pair_rank10', 'orthogonal_pair_rank10', + 'maximal_one_column_rank9', 'wide_rank1_rectangle'] + check([w['name'] for w in result['witnesses']] == expected_names, 'witness inventory') + for w in result['witnesses']: + q = [[0] * 112 for _ in range(112)] + seen = set() + for i, j, value in w['nonzero_entries']: + check(type(value) is int and value != 0, 'nonzero integer witness entry') + check(0 <= i < 112 and 0 <= j < 112 and (i, j) not in seen, 'unique witness entry') + check(gram[i][j] in (-2, 2), 'witness violates odd mask') + seen.add((i, j)); q[i][j] = value + cols = sorted({j for i, j in seen}) + check(cols == w['columns'] and len({i for i, j in seen}) == w['row_support'], 'witness support') + y = [[row[j] for j in cols] for row in q] + total = [[gram[i][j] + q[i][j] for j in range(112)] for i in range(112)] + check(rational_rank(y) == w['correction_rank'], 'independent correction rank') + check(rational_rank(total) == w['completion_rank'], 'independent completion rank') + residual = [[56 * y[i][a] - sum(gram[i][t] * y[t][a] for t in range(112)) + for a in range(len(cols))] for i in range(112)] + check(rational_rank(residual) == w['projection_rank'], 'independent projection rank') + sharp = result['witnesses'][0] + j, k = sharp['columns'] + supports = [([i for i in range(112) if roots[i][a]], + [i for i in range(112) if roots[i][a] and i != j]) for a in range(8)] + supports.append(([j], [j, k])) + copy_ranks = [] + for left, right in supports: + # Set symmetric differences, with smallest pivots, instead of bit elimination. + basis = {} + for i in left: + row = {col for col in right if (gram[i][col] // 2 + 1) % 2} + while row and min(row) in basis: + row ^= basis[min(row)] + if row: + basis[min(row)] = row + copy_ranks.append(len(basis)) + check(copy_ranks == sharp['displayed_factorization_copy_ranks'] == [8] * 8 + [1] and + sharp['displayed_factorization_copy_total'] == sum(copy_ranks) == 65, 'separate COPY invoice') + check(sharp['row_support'] == 53 and len(sharp['columns']) == 2 and + sharp['correction_rank'] == 2 and sharp['completion_rank'] == 9, 'sharp masked boundary') + + +def run(): + raw = (ROOT / SOURCE).read_bytes() + check(sha256(raw).hexdigest() == SOURCE_SHA, 'input bytes') + source = json.loads(raw) + roots = independent_roots(source) + gram = [[dot(x, y) for y in roots] for x in roots] + cert = json.loads((HERE / 'CENSUS.json').read_text()) + result = json.loads((HERE / 'RESULT.json').read_text()) + hist, determinants = audit_census(roots, gram, cert) + check(hist == result['pair_class_histogram'], 'reported class histogram') + check(sum(determinants.values()) == 12544, 'all exact minors') + audit_result(roots, gram, result, cert) + return {'integer_minors': 12544, 'witnesses': len(result['witnesses']), + 'absolute_determinant_histogram': determinants} + + +if __name__ == '__main__': + out = run() + print('PASS_P112_TWO_COLUMN_INDEPENDENT integer_minors=12544 pencils=2912 witnesses=5 root_isometry=120') diff --git a/research/p112-two-column-completion/test_verify.py b/research/p112-two-column-completion/test_verify.py new file mode 100644 index 0000000000..86a34cacc8 --- /dev/null +++ b/research/p112-two-column-completion/test_verify.py @@ -0,0 +1,118 @@ +"""Mathematical and fail-closed controls for the P112 completion certificate.""" +from copy import deepcopy +from importlib.util import module_from_spec, spec_from_file_location +from itertools import permutations, product +from pathlib import Path +from tempfile import TemporaryDirectory +from unittest.mock import patch +import json +import unittest + +HERE = Path(__file__).resolve().parent + + +def load(name, filename): + spec = spec_from_file_location(name, HERE / filename) + module = module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +V = load('p112_primary', 'verify.py') +I = load('p112_independent', 'independent.py') + + +class ArithmeticTests(unittest.TestCase): + def test_all_small_determinants(self): + perms = list(permutations(range(3))) + signs = [(-1) ** sum(p[i] > p[j] for i in range(3) for j in range(i + 1, 3)) + for p in perms] + for values in product((-1, 0, 1), repeat=9): + matrix = [values[:3], values[3:6], values[6:]] + expected = sum(sign * matrix[0][p[0]] * matrix[1][p[1]] * matrix[2][p[2]] + for sign, p in zip(signs, perms)) + self.assertEqual(I.determinant(matrix), expected) + for values in product((-1, 0, 1), repeat=4): + matrix = [values[:2], values[2:]] + expected = 2 if values[0] * values[3] != values[1] * values[2] else int(any(values)) + self.assertEqual(I.rational_rank(matrix), expected) + + def test_modular_rank_is_only_a_lower_bound(self): + matrix = [[V.PRIME, 0], [0, 1]] + self.assertEqual(V.rank(matrix), 1) + self.assertEqual(I.rational_rank(matrix), 2) + self.assertEqual(I.determinant(matrix), V.PRIME) + + +class CertificateTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.source = V.bind_sources() + cls.roots, cls.gram = V.geometry(cls.source) + cls.cert = json.loads((HERE / 'CENSUS.json').read_text()) + cls.result = json.loads((HERE / 'RESULT.json').read_text()) + + def test_missing_pair_and_false_minor_reject(self): + bad = deepcopy(self.cert); bad['pairs'].pop() + with self.assertRaisesRegex(ValueError, 'cardinality'): + I.audit_census(self.roots, self.gram, bad) + bad = deepcopy(self.cert); bad['pairs'][0][4][1] = bad['pairs'][0][4][0] + with self.assertRaisesRegex(ValueError, 'multiplicity'): + I.audit_census(self.roots, self.gram, bad) + + def test_missing_projective_direction_rejects(self): + bad = deepcopy(self.cert); bad['odd_pencils'][0][3].pop() + with self.assertRaisesRegex(ValueError, 'projective exceptional'): + I.audit_census(self.roots, self.gram, bad) + + def test_false_row_threshold_rejects(self): + bad = deepcopy(self.result); bad['two_column_failure_row_floor'] = 54 + with self.assertRaisesRegex(ValueError, 'pencil summary'): + I.audit_result(self.roots, self.gram, bad, self.cert) + bad = deepcopy(self.cert); bad['odd_pencils'][0][3][0][2] += 1 + with self.assertRaisesRegex(ValueError, 'exact pencil support'): + I.audit_census(self.roots, self.gram, bad) + + def test_source_change_and_missing_source_reject(self): + with TemporaryDirectory(prefix='source-control-', dir=HERE) as tmp: + root = Path(tmp) + package = root / 'research' / 'p112-two-column-completion' + package.mkdir(parents=True) + manifest = json.loads((HERE / 'SOURCES.json').read_text()) + for entry in manifest['files']: + if entry['usage'] == 'input': + target = root / entry['path']; target.parent.mkdir(parents=True, exist_ok=True) + target.write_bytes((V.ROOT / entry['path']).read_bytes()) + (package / 'SOURCES.json').write_text(json.dumps(manifest)) + with patch.object(V, 'ROOT', root), patch.object(V, 'HERE', package): + V.bind_sources() + source = root / V.SOURCE + source.write_bytes(source.read_bytes() + b' ') + with self.assertRaisesRegex(ValueError, 'source bytes changed'): + V.bind_sources() + manifest['files'] = [e for e in manifest['files'] if e['path'] != V.SOURCE] + (package / 'SOURCES.json').write_text(json.dumps(manifest)) + with self.assertRaisesRegex(ValueError, 'input inventory'): + V.bind_sources() + + def test_complex_completion_by_exact_realification(self): + # G+(1+i)Q. This is an exact Gaussian-integer fixture; no floating point. + q = V.low_witness(self.gram, 0, 1) + a = [[self.gram[i][j] + q[i][j] for j in range(112)] for i in range(112)] + realification = [a[i] + [-x for x in q[i]] for i in range(112)] + realification += [q[i] + a[i] for i in range(112)] + self.assertEqual(I.rational_rank(realification), 18) + + def test_mask_is_necessary_for_the_rank_one_claim(self): + # An unmasked rank-one correction can decrease rank to seven. + # For a root normal x_j, G-(G e_j)(G e_j)^T/4 removes one root-code direction. + g = [row[0] for row in self.gram] + scaled = [[4 * self.gram[i][j] - g[i] * g[j] for j in range(112)] for i in range(112)] + self.assertEqual(I.rational_rank(scaled), 7) + correction = [[-g[i] * g[j] for j in range(112)] for i in range(112)] + with self.assertRaisesRegex(ValueError, 'not odd-only'): + V.validate_mask(correction, self.gram) + + +if __name__ == '__main__': + unittest.main() diff --git a/research/p112-two-column-completion/verify.py b/research/p112-two-column-completion/verify.py new file mode 100644 index 0000000000..4828be3f03 --- /dev/null +++ b/research/p112-two-column-completion/verify.py @@ -0,0 +1,348 @@ +#!/usr/bin/env python3 +"""Exact P112 two-column masked-completion certificate. Apache-2.0. +Prepared for commdata2338 with substantial OpenAI Codex assistance. +""" +from collections import Counter +from hashlib import sha256 +from itertools import combinations +from math import gcd +from pathlib import Path +import argparse +import json + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SOURCE = 'research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json' +PRIME = 65521 +BASE = '1ff9c501b624ec354e1330b1b9be571a6adba9cb' + + +def need(condition, message): + if not condition: + raise ValueError(message) + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(',', ':')) + + +def dot(x, y): + return sum(a * b for a, b in zip(x, y)) + + +def rank_certificate(matrix): + """A modular lower bound and a specific nonzero square minor. + + This NEVER supplies a characteristic-zero upper bound. Those are proved + separately by ambient dimension, orthogonality, or exact factorizations. + """ + if not matrix: + return 0, [], [] + width = len(matrix[0]) + need(all(len(row) == width for row in matrix), 'rectangular matrix') + a = [[v % PRIME for v in row] for row in matrix] + ids = list(range(len(a))) + chosen_columns = [] + r = 0 + for col in range(width): + pivot = next((k for k in range(r, len(a)) if a[k][col]), None) + if pivot is None: + continue + a[r], a[pivot] = a[pivot], a[r] + ids[r], ids[pivot] = ids[pivot], ids[r] + scale = pow(a[r][col], -1, PRIME) + a[r] = [(v * scale) % PRIME for v in a[r]] + for k in range(r + 1, len(a)): + scale = a[k][col] + if scale: + a[k] = [(x - scale * y) % PRIME for x, y in zip(a[k], a[r])] + chosen_columns.append(col) + r += 1 + if r == min(len(a), width): + break + return r, ids[:r], chosen_columns + + +def rank(matrix): + return rank_certificate(matrix)[0] + + +def bind_sources(): + manifest = json.loads((HERE / 'SOURCES.json').read_text()) + need(manifest['base_commit'] == BASE, 'source base changed') + expected_inputs = {SOURCE, 'research/p112-sparse-completion-rank/README.md'} + inputs = [entry for entry in manifest['files'] if entry['usage'] == 'input'] + need({entry['path'] for entry in inputs} == expected_inputs and len(inputs) == 2, + 'complete live input inventory') + for entry in inputs: + need(sha256((ROOT / entry['path']).read_bytes()).hexdigest() == entry['sha256'], + 'source bytes changed: ' + entry['path']) + return json.loads((ROOT / SOURCE).read_text()) + + +def geometry(source): + all_roots = source['root_lines'] + roots = all_roots[8:] + need(len(all_roots) == 120 and len(roots) == 112, 'P112 inventory') + need(all(type(x) is int for r in roots for x in r), 'integer root coordinates') + need(len({tuple(r) for r in roots}) == 112 and all(len(r) == 8 for r in roots), + 'distinct eight-coordinate roots') + need(all_roots[:8] == [[2 * int(i == j) for j in range(8)] for i in range(8)], + 'removed coordinate frame') + gram = [[dot(r, s) for s in roots] for r in roots] + need(all(gram[i][i] == 4 for i in range(112)), 'root normalization') + need(all(gram[i][j] in (-2, 0, 2) for i, j in combinations(range(112), 2)), + 'off-diagonal Gram alphabet') + labels = source['binary_ports'][8:] + need(len(labels) == 112 and len(set(labels)) == 112, 'label inventory') + need(all(((labels[i] & labels[j]).bit_count() % 2) == + ((gram[i][j] // 2 + 1) % 2) for i in range(112) for j in range(112)), + 'all binary pairing / odd-mask bindings') + frame_gram = [[sum(r[i] * r[j] for r in roots) for j in range(8)] for i in range(8)] + need(frame_gram == [[56 * int(i == j) for j in range(8)] for i in range(8)], + 'X transpose X = 56 I') + need(all(sum(abs(v) == 2 for v in row) == 52 for row in gram), 'odd degree 52') + return roots, gram + + +def minor_on_rows(roots, indices, expected): + value, row_ids, cols = rank_certificate([roots[i] for i in indices]) + need(value == expected, 'wrong finite root-span lower bound') + return [[indices[i] for i in row_ids], cols] + + +def pencil_profile(gram, j, k): + """All projective zero patterns in span(G[:,j], G[:,k]), over C.""" + directions = set() + active = 0 + for row in gram: + x, y = row[j], row[k] + if x or y: + active += 1 + divisor = gcd(x, y) + a, b = y // divisor, -x // divisor + if a < 0 or (a == 0 and b < 0): + a, b = -a, -b + directions.add((a, b)) + strata = [[a, b, sum(a * row[j] + b * row[k] != 0 for row in gram)] + for a, b in sorted(directions)] + need((1, 3) not in directions and + sum(row[j] + 3 * row[k] != 0 for row in gram) == active, + 'generic pencil direction') + need(min(row[2] for row in strata) == 53, 'two-root pencil minimum support') + return [j, k, active, strata] + + +def census(roots, gram): + singles, pairs, pencils = [], [], [] + counts = Counter() + for j in range(112): + constrained = [i for i in range(112) if abs(gram[i][j]) != 2] + minor = minor_on_rows(roots, constrained, 8) + singles.append([j, len(constrained), *minor]) + for j, k in combinations(range(112), 2): + odd = abs(gram[j][k]) == 2 + constrained = [i for i in range(112) if abs(gram[i][j]) != 2 and abs(gram[i][k]) != 2] + expected = 6 if odd else 8 + # For odd pairs, every constrained row is perpendicular to both roots. + # The two independent roots supply the matching characteristic-zero upper bound. + if odd: + pencils.append(pencil_profile(gram, j, k)) + need(all(gram[i][j] == gram[i][k] == 0 for i in constrained), + 'rank-six orthogonality upper bound') + need(gram[j][j] * gram[k][k] - gram[j][k] ** 2 != 0, + 'two independent root normals') + common_minor = minor_on_rows(roots, constrained, expected) + kept = [i for i in range(112) if i not in (j, k)] + deletion_minor = minor_on_rows(roots, kept, 8) + pairs.append([j, k, gram[j][k], len(constrained), *common_minor, *deletion_minor]) + counts[(('odd' if odd else 'orthogonal'), len(constrained), expected)] += 1 + return {'schema_version': 1, 'singletons': singles, 'pairs': pairs, 'odd_pencils': pencils}, { + '|'.join(map(str, key)): count for key, count in sorted(counts.items())} + + +def zeros(): + return [[0] * 112 for _ in range(112)] + + +def validate_mask(q, gram, maximum_columns=None): + need(len(q) == 112 and all(len(row) == 112 for row in q), 'correction dimensions') + active = sorted({j for row in q for j, value in enumerate(row) if value}) + need(maximum_columns is None or len(active) <= maximum_columns, 'column support outside scope') + need(all(not q[i][j] or abs(gram[i][j]) == 2 for i in range(112) for j in active), + 'correction is not odd-only') + return active + + +def low_witness(gram, j, k): + need(j != k and abs(gram[j][k]) == 2, 'low witness needs odd columns') + q = zeros() + for i in range(112): + if i != j: + q[i][j] = -gram[i][j] + q[j][k] = 1 + return q + + +def project_columns(roots, y): + # P=56I-XX^T; no floating point, and no Hermitian substitution. + width = len(y[0]) + xty = [[sum(roots[i][a] * y[i][b] for i in range(112)) + for b in range(width)] for a in range(8)] + return [[56 * y[i][b] - sum(roots[i][a] * xty[a][b] for a in range(8)) + for b in range(width)] for i in range(112)] + + +def inspect_witness(name, q, roots, gram, q_rank, completion_rank, projection_rank): + columns = validate_mask(q, gram) + y = [[row[j] for j in columns] for row in q] + m = [[gram[i][j] + q[i][j] for j in range(112)] for i in range(112)] + need(rank(y) == q_rank, 'correction rank claim fails') + need(rank(m) == completion_rank, 'completion rank lower-bound claim fails') + need(rank(project_columns(roots, y)) == projection_rank, 'quotient rank claim fails') + # All entries are stored, sparsely, for an independent rational rank check. + entries = [[i, j, q[i][j]] for i in range(112) for j in columns if q[i][j]] + return {'name': name, 'columns': columns, 'row_support': len({i for i, j, v in entries}), + 'nonzero_entries': entries, 'correction_rank': q_rank, + 'completion_rank': completion_rank, 'projection_rank': projection_rank} + + +def binary_copy_rank(gram, left, right): + pivots = {} + for i in left: + row = sum((((gram[i][j] // 2 + 1) % 2) << bit) for bit, j in enumerate(right)) + while row: + pivot = row.bit_length() - 1 + if pivot in pivots: + row ^= pivots[pivot] + else: + pivots[pivot] = row + break + return len(pivots) + + +def witnesses_and_controls(roots, gram): + odd = next((j, k) for j, k in combinations(range(112), 2) if abs(gram[j][k]) == 2) + even = next((j, k) for j, k in combinations(range(112), 2) if gram[j][k] == 0) + j, k = odd + low = low_witness(gram, j, k) + witnesses = [inspect_witness('odd_pair_rank9', low, roots, gram, 2, 9, 1)] + need(witnesses[0]['row_support'] == 53, 'sharp 53-row boundary') + copy_ranks = [binary_copy_rank(gram, [i for i in range(112) if roots[i][a]], + [i for i in range(112) if roots[i][a] and i != j]) + for a in range(8)] + [binary_copy_rank(gram, [j], [j, k])] + need(copy_ranks == [8] * 8 + [1], 'explicit factorization COPY invoice') + witnesses[0]['displayed_factorization_copy_ranks'] = copy_ranks + witnesses[0]['displayed_factorization_copy_total'] = sum(copy_ranks) + # Check the exact rank-nine factorization X A + e_j b on every entry. + for row in range(112): + for col in range(112): + factored = (0 if col == j else gram[row][col]) + if row == j: + factored += 4 * int(col == j) + int(col == k) + need(gram[row][col] + low[row][col] == factored, 'rank-nine factorization') + for label, pair in [('odd_pair_rank10', odd), ('orthogonal_pair_rank10', even)]: + u, v = pair + a = next(i for i in range(112) if abs(gram[i][u]) == 2) + b = next(i for i in range(112) if i != a and abs(gram[i][v]) == 2) + q = zeros(); q[a][u] = q[b][v] = 1 + witnesses.append(inspect_witness(label, q, roots, gram, 2, 10, 2)) + q = zeros() + for i in range(112): + if abs(gram[i][j]) == 2: + q[i][j] = 1 + witnesses.append(inspect_witness('maximal_one_column_rank9', q, roots, gram, 1, 9, 1)) + a, b = 0, 1 + cols = [c for c in range(112) if abs(gram[a][c]) == abs(gram[b][c]) == 2] + q = zeros() + for i in (a, b): + for c in cols: + q[i][c] = 1 + need(len(cols) > 2, 'rank-one example has more than two columns') + witnesses.append(inspect_witness('wide_rank1_rectangle', q, roots, gram, 1, 9, 1)) + + controls = [] + def rejects(name, fn): + try: + fn() + except ValueError: + controls.append(name) + else: + raise ValueError('control did not reject: ' + name) + bad = [row[:] for row in low]; bad[j][j] = 1 + rejects('nonmasked_diagonal', lambda: validate_mask(bad, gram, 2)) + rejects('rank1_correction_mislabel', lambda: inspect_witness('bad', low, roots, gram, 1, 9, 1)) + rejects('false_additivity_rank10', lambda: inspect_witness('bad', low, roots, gram, 2, 10, 1)) + rejects('false_rank8_completion', lambda: inspect_witness('bad', low, roots, gram, 2, 8, 1)) + omitted = [row[:] for row in low]; omitted[j][k] = 0 + rejects('omitted_second_column', lambda: inspect_witness('bad', omitted, roots, gram, 2, 9, 1)) + rejects('orthogonal_pair_low_witness', lambda: low_witness(gram, *even)) + rejects('wide_rectangle_as_two_columns', lambda: validate_mask(q, gram, 2)) + wrong = [[55 * roots[i][a] - sum(gram[i][t] * roots[t][a] for t in range(112)) + for a in range(8)] for i in range(112)] + rejects('wrong_projection_normalization', lambda: need(all(not x for row in wrong for x in row), + 'projection must kill the root code')) + return witnesses, controls + + +def compute(): + need(all(PRIME % d for d in range(2, 256)), 'prime modulus') + source = bind_sources() + roots, gram = geometry(source) + cert, histogram = census(roots, gram) + witnesses, controls = witnesses_and_controls(roots, gram) + result = { + 'schema_version': 1, + 'result_kind': 'scoped-proof-with-finite-certificate', + 'base_commit': BASE, + 'source_sha256': sha256((ROOT / SOURCE).read_bytes()).hexdigest(), + 'normalization': 'X has squared row norm 4; R=X/sqrt(2); G=XX^T=2RR^T', + 'gram_rank': 8, 'frame_constant': 56, 'odd_degree': 52, + 'singletons_checked': len(cert['singletons']), + 'unordered_pairs_checked': len(cert['pairs']), + 'odd_pairs': sum(abs(row[2]) == 2 for row in cert['pairs']), + 'orthogonal_pairs': sum(row[2] == 0 for row in cert['pairs']), + 'odd_pencils_checked': len(cert['odd_pencils']), + 'two_column_failure_row_floor': 53, + 'pair_class_histogram': histogram, + 'census_sha256': sha256(canonical(cert).encode()).hexdigest(), + 'witnesses': witnesses, 'controls': controls, + 'theorem': 'For <=2 changed columns: rank(G+Q)=8+rank((56I-G)Q). Nonzero rank-one Q gives 9. Rank-two Q gives 9 or 10; orthogonal changed columns force 10. Every odd pair attains both 9 and 10. Every nonzero odd-only rank-one correction, of any column support, gives 9.', + 'boundary': 'Rank-two odd-only corrections on 53 rows and two columns attain completion rank 9; rank-one odd rectangles have at most 52 rows and 52 columns.', + 'scope': 'Ordinary complex matrix rank, not binary COPY cost, recursive child rank, a new supplier or kappa.' + } + return cert, result + + +def main(): + parser = argparse.ArgumentParser() + group = parser.add_mutually_exclusive_group() + group.add_argument('--write', action='store_true', help='write initial certificates; refuse existing paths') + group.add_argument('--write-to', type=Path, help='write certificates in a new output directory') + args = parser.parse_args() + output = args.write_to if args.write_to is not None else HERE + writing = args.write or args.write_to is not None + if writing: + need(all(not (output / name).exists() for name in ('CENSUS.json', 'RESULT.json')), + 'refuse existing certificate output') + output.mkdir(parents=True, exist_ok=True) + cert, result = compute() + for name, value in [('CENSUS.json', cert), ('RESULT.json', result)]: + path = output / name + if writing: + with path.open('x') as out: + if name == 'CENSUS.json': + blocks = ['"schema_version": 1'] + for key in ('singletons', 'pairs', 'odd_pencils'): + blocks.append('"' + key + '": [\n' + + ',\n'.join(' ' + canonical(row) for row in value[key]) + '\n]') + out.write('{\n' + ',\n'.join(blocks) + '\n}\n') + else: + out.write(json.dumps(value, indent=2, sort_keys=True) + '\n') + else: + need(canonical(json.loads(path.read_text())) == canonical(value), name + ' differs from exact replay') + print('PASS_P112_TWO_COLUMN_COMPLETION pairs=6216 odd=2912 orthogonal=3304 boundary_rows=53 correction_rank=2 completion_rank=9 controls=' + str(len(result['controls']))) + + +if __name__ == '__main__': + main()