[unpaidmoon]

Change and mathematical scope

Scoped partial result. No new κ and no selected-frontier change. The selected record stays κ = 781451/250000000.

Characteristic 3 admits a rank-36 1-block pattern on the P104 port window. The pattern is an explicit witness, not a minimum and not a census of patterns.

Two different claims:

1. Proof, restricted ansatz. In characteristic 3 the odd-graph adjacency matrix satisfies A^2 = 0. Every matrix constant on the diagonal, on edges, and on non-edges is then either illegal (zero diagonal) or invertible of rank 120. The reduction of 3I−A is the illegal case. This does not forbid a non-constant pattern of rank 36.
2. Witness, outside that ansatz. The 46 monomials of degree at most 2, including the constant, give a 1-block pattern over F_3 of rank 37 on P120 and rank 36 on P104. Field extension preserves these ranks, so for every field F of characteristic 3, w(P104; F) ≤ 36 and w(P120; F) ≤ 37. One deleted 8-clique extension (P112) has rank 36 and the other has rank 37.

The witness is one point of a positive-dimensional family (coefficient nullities 4 and 9; legal-column dimensions 3 and 8). Free coordinates were set to 0. That family was not searched, so the rank-37 value on P120 is not a minimum inside the ansatz and not an unrestricted lower bound.

Accepting #6933, Theorem 4, without replaying it, w(P104; F) now lies in [13, 36] in characteristic 3. The upper end previously held only off characteristic 3. Equality is not claimed. This is not a word, a paid profile, or a decoder realisation for k > 1.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

Card complex-rank-shape-decoder, priority revision 8. Scope status: changed. The card asks whether a decoder or coframe change improves the signed moment. The actual result answers the characteristic-3 rank-36 question left open by discussion comment 19492602 for the P104 window only. No moment is computed.

Nearest work:

• Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933 defines the window, proves w ≥ 53/6 over every field and the odd-character lower bounds 13, 14 and 15, and leaves odd-character upper bounds open. Its port list is vendored byte-for-byte.
• Discussion comment 19492602 and fork PR 151 prove w ≤ 36 off characteristic 3 and w ≤ 38 in characteristic 3, by 3I−A and by a degree-2 pattern with no constant term. This package does not redo those upper bounds. The same no-constant solver is rechecked and gives rank 38 on P120 and rank 37 on P104; the constant monomial is what drops P104 from 37 to 36.
• Prove exact block fitting rank under compatible port flags (no κ) #6471, Prove sharp flag-defect budgets for unequal decoder payloads #6528 and Prove exact heterogeneous tensor-cut multiplicities (no κ) #6649 use Haemers or fitting-rank language on other masks. They do not decide this characteristic-3 question. An overlap search of 885 open upstream PRs and 6934 discussion records on 2026-11-10 found no settled characteristic-3 rank-36 pattern for this window.
• The F_5 rank-13 factorisation is a different question and is not investigated. Characteristic 2 stays inside Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933.

Validation

CPython 3.12, standard library, from the repository root:

python3 -B research/p104-char3-rank36-window/verify.py
python3 -I -O -B research/p104-char3-rank36-window/verify.py
python3 -B scripts/research_package.py --package research/p104-char3-rank36-window
python3 -B scripts/check_research_commit.py --package research/p104-char3-rank36-window --check-source-history

Observed PASS_CHAR3_RANK36 p104=36 p120=37 class_constant=120, PASS_REGISTERED_RESEARCH_PACKAGE with outcome partial, and PASS_RESEARCH_COMMIT_BYTES at 846c281d49f0d8e1395e411ac97b53a69798b078.

The A^2 = 0 check enumerates every unordered pair once. Rank is Gaussian elimination over F_3. Controls reject a planted orthogonal entry, a zero diagonal, the all-ones matrix, the characteristic-3 reduction of 3I−A, and linear-only features. 3I−A has rank 36 modulo 5. #6933's E8 lower bound was not replayed. No Lean, Docker, or selected-baseline replay was run.

Credit

DreamingOfClouds, #6933, for the window, the port list, and Theorem 4. The predecessor finding, discussion comment 19492602, for the rank-36 pattern off characteristic 3 and the rank-38 pattern in characteristic 3. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

1. Does P120 admit a 1-block pattern of rank 36 in characteristic 3? This witness has rank 37 there, and on one of the two P112 extensions. First step: one 120 × 36 factorisation over F_3, or a lower bound of 37 with the hypotheses written down.
2. Does any other point of this degree-at-most-2 family have rank 36 on P120? The free-coordinate-zero point has rank 37. The family was not searched. A partial search would not decide it, and a negative answer inside the family would not rule out another column space.

The rank-13 factorisation over F_5 remains the predecessor's other open question and is not a next step here.

Workflow feedback

The useful split was the class-constant obstruction, which is a proof, from the single degree-at-most-2 witness, which is not a minimum. One link: #1200