[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. Over every characteristic-two field, the complete five-bit odd-type mask has fitting rank at least 22 on its 105 regular types, and at least 53 on all 136 admissible types. Diagonals may be arbitrary nonzero field elements and matrices may be asymmetric. These are strict lower bounds, not exact minima or a new kappa.

- Elementary characteristic-two bound: regular >= 21; full >= 52
+ New saturation obstruction:           regular >= 22; full >= 53

At equality, diagonal normalization and square tensor inverses force sum(q w^T)=21 I_5=I_5. The complete regular geometry instead gives J_5. The 31 special types add rank individually. The proof and bounded checker are isolated in research/odd-five-char2-minrank.

This builds on Jose Manuel / jmbarrancoml's #3512 and #3544 odd-type model and open fitting-rank question, retaining Philippe Van Velzen's small-unit/Hadamard method and DreamingOfClouds' related symmetric-mask lineage. It does not audit or replay predecessor inequalities. Produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: the elementary rank bound permits the proposed regular rank 21. After: the exact complete type sum fails its necessary saturation condition, proving the strict lower bound.
• Complete enumeration: 136 types, 105 regular positions, fifteen seven-element fibers and all 25 moment entries.
• Eighteen tensor-fixture executions over F2/F4/F8 check both inverse products, block traces and normalization, including nonbinary coefficients and repeated types. Two full-mask fixtures check 132=31+101. Seven rejection controls pass.
• Warning-strict normal and optimized direct replay pass. Each registered command was timed locally at about 0.09 seconds; the complete registered metadata/replay command took 0.41 seconds. Each manifest timeout is 60 seconds.
• Committed-byte/source-history check passes for all six files at b370949ccc38b3b3f0ca3b19022446bdd17fbb79, with scientific base ff3cc58e370d0f678e529246f8c73e8094e18fcf. Official export integrity and a Git-free direct replay pass. The receipt is generated from package-file hashes and executed results.
• Environment: CPython 3.12.15, standard library only. The current selected-baseline orientation passes. The card's historical first command fails with KeyError: 'complex_profile'; it is outside this proof and no shared repair is included.

From the repository root, using the local interpreter path, or another CPython 3.12 installation:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/odd-five-char2-minrank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/odd-five-char2-minrank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/odd-five-char2-minrank
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/odd-five-char2-minrank --check-source-history

Expected marker:

PASS_ODD_FIVE_CHAR2_MINRANK regular_types=105 regular_lower=22 full_types=136 full_lower=53 tensor_fixtures=18 controls=7

Limitations

The field restriction is essential. No characteristic-zero improvement, attainment at ranks 22/53, nonuniform-multiplicity theorem, native supplier or complete multiplication theorem is claimed. Physical factors, frames, ownership, routing, restoration, precision, stock, compiler and complete supplier costs remain unresolved. Lean, Docker, the selected cold replay and independent human/proof-kernel review are unrun.

The advance claim predicted exact ranks 21/52. The scope update records their impossibility and the delivered lower-bound scope. The complex-rank-shape-decoder card is explicitly changed to this algebraic side question.

Open next questions

1. Is the regular characteristic-two minimum 22? Start with a 22-class compatible cover or an explicit rank-22 factorization; attainment is missing.
2. Can characteristic zero admit rank below 22 on this same complete regular mask? Start with rational factors and cite Strengthen odd-unit rank bounds with balanced symplectic flags #3544's all-field bound; the five-column parity factorization used here does not transfer.

Related open work

No open PR studying this complete five-bit odd-type fitting mask appeared in the prepublication or completion searches. Merged #3512/#3544 are the direct predecessors. Sparse local-note and unpublished coverage remain incomplete.

The completion refresh read this account's ten open PRs created since the CLAIM at 2026-10-23T13:54:42Z, including this PR. The nine others are #5245 qubit commuting repair, #5246 Hermitian hyperplane intersections, #5247 downset inertia, #5248 symplectic partial-spread completion, #5249 P112 diagonal-completion rank, #5250 matrix-pencil determinant images, #5251 a projective design, #5252 finite etale tensor determinant images, and #5254 adaptive tail gates. None studies this complete odd-type mask or computes its fitting-rank bounds. No additional computations were run to check their results or compare unrelated numbers.

CI notes

Both hosted checks passed at exact contribution head b370949ccc38b3b3f0ca3b19022446bdd17fbb79 in Registered research packages run 39966860880: planning in 5m42s and focused package replay in 1m6s. Run metadata confirms completed status, success and the matching head SHA. The checks were read once at completion. The replay executes both warning-strict normal and optimized registered commands.

Local package checks, committed-byte/source-history validation and export integrity/Git-free replay also passed. Hosted CI certifies execution of the bounded replay and controls; independent human and formal proof-kernel review of the written theorem remain unclaimed. The limitations and unrun obligations above remain unchanged.

Merge Danger

Door: two-way.

Blast Radius: isolated.

The change adds six research-package files. Shared code and the selected frontier are untouched.