[unpaidmoon]

Change and mathematical scope

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

The odd graph on the 120 weight-3-or-7 vectors in F_2^9 is connected and strongly regular with parameters (120, 63, 30, 36). The matrix H = 3I − A has rank 36 over Q, nonzero diagonal in every characteristic other than 3, and zeros on non-edges. Every nonempty subset, including the P104 port set and both 112-port extensions, therefore has window at most 36 outside characteristic 3.

In characteristic 3 that matrix is illegal. A degree-at-most-2 factorisation gives a 1-block pattern of rank 38, so the window is at most 38.

Accepting the lower bounds in #6933, Theorem 4, without replaying them:

field	P104	P112	P120
char ≠ 2, 3	[13, 36]	[14, 36]	[15, 36]
char 3	[13, 38]	[14, 38]	[15, 38]

Characteristic 2 is unchanged: #6933 already has w(P104; F_2) ∈ [53/6, 9] and w = 9 on P112 and P120. 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 open question 3 of #6933: what w(P104; F) is in odd characteristic, above the lower bound 13. 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, proves the odd-character lower bounds 13, 14 and 15, and leaves the odd-character upper bound open. Its port list is vendored byte-for-byte.
• Prove exact block fitting rank under compatible port flags (no κ) #6471 is a flag-compatible fitting-rank theorem. It cites the fractional Haemers framework and does not bound this graph.
• The characteristic-2 Gram rank 9 is Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933's and is not improved here.

Validation

CPython 3.12, standard library, from the repository root:

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

Observed PASS_ODD_CHAR_WINDOW rank36=36 char3_rank=38, PASS_REGISTERED_RESEARCH_PACKAGE with outcome partial, and PASS_RESEARCH_COMMIT_BYTES.

The census covers every unordered pair, twice. Rank 36 is checked modulo 5, 7, 11 and 10007. Controls reject a planted orthogonal entry, the characteristic-3 reduction of 3I−A, the all-ones matrix, and linear-only features on eight columns. #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, the characteristic-2 rank, and Theorem 4. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

1. Is w(P104; F) equal to 13 for some odd characteristic? The pattern proved here has rank 36, or 38 in characteristic 3, so equality needs weights outside the Bose-Mesner algebra. First step: one 104 × 13 factorisation over F_5.
2. Does characteristic 3 admit rank 36? 3I−A is illegal there, and the degree-2 matrix has rank 38.

Workflow feedback

The useful split was the strongly regular identity, which gives rank 36 in every characteristic except 3, plus one separate factorisation for that prime. No further session report.