[unpaidmoon]

Change and mathematical scope

No new κ. The selected record stays 781451/250000000. This is a constructive rank witness, not a word, a paid profile, or a frontier update.

P120 admits a 1-block pattern of rank 32 over F_3. The pattern comes with an explicit 120 × 32 factorisation. The same matrix has principal rank 32 on P104 and on both P112 extensions, so those characteristic-3 ceilings drop from 36 / 36 / 37 to 32 as well. Accepting PR #6933, Theorem 4, without replaying it, the characteristic-3 intervals are now [13, 32], [14, 32] and [15, 32].

The witness is one free-zero point of one hyperplane of the degree-at-most-2 column span from #6953. It is not a proof that every pattern has rank at least 32. The same 36-dimensional space contains another explicit legal point of rank 36, so the rank is attached to this basis. A failed search is not part of the claim. The census of the 29 kernel-basis lines is exhaustive for that basis only.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840. Task card complex-rank-shape-decoder, scope changed: the card asks whether a decoder or coframe change improves the signed moment, and this package does not change a word or a moment. It answers the P120 question left open by merged PR #6953 (discussion comment on the rank-37 witness): one 120 × 36 factorisation over F_3, or a lower bound of 37 with the hypotheses written down. The matrix here has rank 32, which meets that request.

Nearest work: #6933 (window, port list, Theorem 4 lower bounds, not replayed), #6946 (odd-character ceiling 36 off characteristic 3, and 38 in characteristic 3), #6953 (rank 37 on P120, rank 36 on P104). Open #7014 is the F_5 104 × 13 factorisation and is not touched. No open PR exhibited a characteristic-3 rank-36 or rank-32 pattern on P120.

Validation

CPython 3.12, standard library only, from the repository root:

python3 -B research/p120-char3-rank32-window/verify.py
python3 -I -O -B research/p120-char3-rank32-window/verify.py
python3 -B scripts/research_package.py --package research/p120-char3-rank32-window

Final line: PASS_P120_CHAR3_RANK32 rank=32 p104=32 p112=[32, 32], then PASS_REGISTERED_RESEARCH_PACKAGE with outcome partial.

The checks are exhaustive on a finite domain: every vertex, every orthogonal pair, the reduced form of one explicit matrix (sha256 81e209bb5be3c3e9b9a5d593190fd9b19540656bb0162164f4df7a1f0bf4a8c4), and all 29 vectors of one kernel basis. Controls reject a planted orthogonal entry, a zero diagonal, and the pure quadratic map, and they reconfirm class-constant rank 120 and the #6953 rank-37 point. #6933's lower bounds are cited, not replayed. No Lean, Docker, or selected-baseline replay was run.

Credit

DreamingOfClouds, PR #6933, for the window, the port list, and Theorem 4. The rank-37 degree-at-most-2 point is PR #6953. 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 at most 31 in characteristic 3? The legal fibers in this 36-dimensional space have affine dimensions 0, 2 and 5, and only the free-zero point was used. Another basis of the same space has rank 36. A first step is one other explicit pattern of rank at most 31. A failed search is not a lower bound.
2. Does every 1-block pattern on P120 in characteristic 3 have rank at least 32? The cited lower bound is 15. A first step is one linear condition, with the hypotheses written down, that applies to every legal column and forces rank at least 32.

Workflow feedback

The #6953 verifier and the free-zero convention made the column span easy to rebuild. Naming the canonical kernel basis, rather than a searched coefficient vector, is what keeps the rank-32 point reproducible.