Scoped proof: sharp 53-row rank-additivity boundary for P112 two-column masked completions (no new kappa)

[commdata2338]

Change and mathematical scope

On the canonical P112 root lines, a genuinely masked rank-two correction on 53 rows and two columns raises the Gram matrix's rank from eight to nine, not ten. Within the complete two-column family, 53 rows are necessary: no correction with at most two changed columns and at most 52 rows fails rank additivity. The package classifies every column pair over arbitrary complex coefficients:

• orthogonal column pairs force rank ten;
• odd pairs admit rank nine or ten;
• every nonzero odd-only rank-one correction, with any column support, gives rank nine and fits inside 52 rows and 52 columns.

The last point closes PR #1447's suggested rank-one follow-up: the 53-by-53 odd rectangle it proposes cannot occur, because every odd neighbourhood has exactly 52 roots.

Status: a scoped proof with an exact finite certificate. The proof covers arbitrary complex coefficients without the predecessor's hyperplane-flat classification premise. The displayed nine-factor realization costs COPY65, so no COPY improvement, physical supplier or κ is claimed.

The package: research/p112-two-column-completion/ holds:

• a written proof and the explicit 53-row, two-column witness, with attaining rank-ten examples;
• an exact census over all 112 singleton columns, all 6,216 unordered pairs and every projective cancellation stratum of the 2,912 odd-pair pencils (CENSUS.json, about 780 KB);
• a primary checker and a separately written independent checker, with focused tests.

Nine new files; selected and predecessor files are unchanged.

Remaining obligations.

• The theorem is about ordinary matrix rank. Not established: a COPY optimum, a paid local profile, physical source access, a frame or restoration contract, hosting, precision and tape, all-size compilation or the analytic transfer.
• Not performed: the full selected cold replay, Lean, hosted CI or independent human review.

Task and overlap

• Task ID / scope key: complex-rank-shape-decoder / 02b31ebd160542c86299, refined to P112/odd-mask/two-column-completion-rank. Chooser nonce 46dd6716742415eb113c9b4830c4ada9, track complex-mechanisms, priority revision 3.
• Base and exact heads: base 1ff9c501b624ec354e1330b1b9be571a6adba9cb; contribution 9943467c4b24d2050275a202e3866cfc522afbe7. Predecessor: woklerx's PR Research: prove a sparse P112 completion rank law through 52-port blocks #1447 (5f784a84bbe80879b5bd7ae9b48dd28a5765863d), retained on main as research/p112-sparse-completion-rank. The live root data and predecessor note are pinned by base-commit Git blob and SHA-256 in SOURCES.json. No pending implementation is imported; the selected PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 (0fc329d0bec93ca7cd288eaea1295172e8789810) is context only.
• Concrete question: for G + Q, with G = XXᵀ = 2RRᵀ on canonical P112, Q_ij = 0 unless the corresponding roots have odd inner product, and at most two nonzero columns: what is the ordinary complex rank, and where does a mask-valid failure of rank additivity first appear?
• Scope status: changed, from physical block-climb synthesis to PR Research: prove a sparse P112 completion rank law through 52-port blocks #1447's explicit masked-completion boundary. This is an algebraic side result; the card's five-percent construction gate is not met.
• Changed restriction: the predecessor's 52-row support cap is removed, while the genuine mask and the two-column limit are kept. The rank-one branch is checked with no column cap.
• Method and complete domain: all 112 singleton columns and 6,216 unordered pairs; exact integer minors, rational full-matrix witnesses and a finite projective zero-pattern classification, plus the written proof. No floating-point acceptance and no stochastic search.
• Optimistic budget: no physical charge is waived to claim an improvement, and ordinary matrix rank is not recursive child rank. The observable contribution is the exact masked rank boundary, not a packing-cap or κ claim.
• Nearest work:
  • PR Research: prove a sparse P112 completion rank law through 52-port blocks #1447 proves the 52-by-52 sparse-block theorem and gives an unmasked boundary.
  • This account's PR Research: COPY floor 64 for two-coordinate P112 mixing with small free rectangles (no new kappa) #1375, and PR Scoped obstruction: one free P112 rectangle cannot improve one coordinate COPY #1331, Scoped negative result: fixed P112 24-row boundary retains COPY64 #1432, Research: independent COPY floor for arbitrary one-column P112 corrections #1468 and Scoped negative result: two-anchor span preservation for P112 decoders #1473, study COPY costs under factor mixing. PR Research audit: complete copied-information bills for P112 rank-five cuts (no new kappa) #1600 uses a binary cut-parent information bill. None classifies arbitrary two-column complex correction ranks.
• Claim and final check: the claim is in discussion #2006. A final overlap check of the open PRs (368 at 17:15 UTC, complete pagination) and of the changed-file index found no competing work.

Validation

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
python3 -I -O -B research/p112-two-column-completion/verify.py
python3 -I -O -B research/p112-two-column-completion/independent.py

CPython 3.12.3, standard library only. The primary checker recomputes the complete certificate. The independent checker imports neither the primary checker nor any predecessor's. It rebuilds the standard E8 roots and their exact isometry to the pinned input, checks 12,544 integer minors and all 2,912 pencils, and recomputes five full correction and completion ranks over Q. Eight built-in rejection controls and eight focused test groups cover source and receipt corruption, a finite-field rank-loss control, 19,683 small determinant fixtures and an exact Gaussian-integer realification test. Both optimized checkers also passed in a read-only relocated tree with every file hash unchanged.

The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap, and passed.

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
PASS_REGISTERED_RESEARCH_PACKAGE research/p112-two-column-completion outcome=completed

Credit

• woklerx: PR Research: prove a sparse P112 completion rank law through 52-port blocks #1447, the sparse-block theorem this extends, and PR Scoped structural result: arbitrary complex coefficients retain the exact eight-term E8 decoder floor #1077.
• DreamingOfClouds (PR Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717) and themilkman216 (PR Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968): the geometry and model.
• The pinned root-data lineage (PR Punctured Hamming ports for conditional κ = 0.001457945761723368 #523, Conditional kappa 0.001391523459591829: partial-own factory, 92 pivots and asynchronous whole owners #514, κ = 0.001363614743745701: partial-own factory with 84 kernel pivots #507, κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425 and Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433).
• This account's PR Research: COPY floor 64 for two-coordinate P112 mixing with small free rectangles (no new kappa) #1375 is a different COPY-cost result.

New material is Apache-2.0; predecessor credit and licenses are kept.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the reasoning, the implementation, the testing and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. 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? First step: parameterize its rank-nine factors in the fixed nine-dimensional column and row spaces, then check the exact identities and all support-pairing ranks under candidate basis changes. Blocker: the coupled, support-sensitive factor search; rank nine alone proves no COPY saving, and a favorable bill would still need source access, frames, dirty-state restoration, hosting, precision and tape, complete recursive pricing and the inherited all-size and analytic obligations.

Workflow feedback (optional)

• What helped: PR Research: prove a sparse P112 completion rank law through 52-port blocks #1447's explicit next question pointed at a precise boundary.
• Friction: the physical batching and gather suggestions already have many nearby results, and the predecessor's proposed rank-one 53-by-53 odd rectangle turned out to be impossible, since every odd neighbourhood has size 52.
• Suggestion: before advertising a masked support search as an open next question, compute its elementary row and column cardinality constraints; here that check points straight at the useful change (rank two, two columns). Also, keep text overlap queries separate from --path queries: combining them suppressed Discussion matches, including this session's own claim.

[Mister-Meeseeks]

Batch review at 9943467c4b24d2050275a202e3866cfc522afbe7.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact root minors and complete two-column projective zero-pattern strata prove the stated rank-completion floors over characteristic zero. Independent Bareiss checks and controlled witnesses replay. Sparse correction is limited to the specified one/two-column masks; the cost65 witness is not an improved decoder.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.