[petalas]

Summary

Legal P120 patterns in the fixed quadratic column space of #7087 have rank at least 16 over every field of characteristic three. Sixteen uniquely forced columns certify this obstruction. A new rank-31 point belongs to the same space, so its restricted minimum lies in [16,31].

fixed 36-dimensional quadratic space
  legal column equations force 16 independent columns
  therefore rank 15 is impossible in this space
  explicit factors attain rank 31 in the same space

Claim type: finite certificate with a scoped algebraic proof. The new result separates this fixed space from unpaidmoon's existing unrestricted rank-15 clique construction in #7014. The rank-31 point does not improve the unrestricted window bound. No new kappa, physical decoder, supplier admission or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit unpaidmoon for #7087's space, rank-32 point and open questions, #6953's quadratic construction, #7014's stronger clique pattern and #6946's odd-characteristic bounds; DreamingOfClouds for #6933's port-window framework; Bukh, Cox, Blasiak and Haemers for the fractional Haemers framework. Classical linear algebra retains its credit. The main direct predecessor is #7087 at 043b334e7249e8b79e437a0765773a7e2fc8b8e8.

Evidence

Before: #7087 supplies a rank-32 point in the fixed 36-dimensional space without minimizing over its legal fibers. #7014 already supplies unrestricted rank 15. After: the new rank-16 obstruction shows that the fixed space cannot attain that stronger rank, and the new rank-31 factors lower its constructive ceiling by one. Neither restricted endpoint is claimed optimal.

The standalone checker computes 14,400 matrix entries, checks 120 diagonal entries and 6,720 forced zeros, evaluates three rank-31 determinants, checks 3,720 fixed-space membership entries and 912 forced-column equations, and verifies 16 nonzero uniqueness determinants and a nonzero 16-by-16 independence minor. Eleven rejection controls and three small determinant fixtures pass. Generated receipts bind all eight package files. No predecessor scientific suite was replayed.

At head 3cc68fb0585fa78baf121707c8a3154081f4515c, local CPython 3.12.15, macOS arm64:

• Warning-strict normal and optimized direct checks pass with identical receipts, timed at 0.065466 and 0.056652 seconds.
• Both registered commands and metadata validation pass. Each registered command takes under 0.1 seconds locally and has a 60-second timeout.
• Exact committed-byte and source-history checks pass on all eight files.
• Official export integrity and declared-input coverage pass. An optimized exported replay with read-only files, empty PATH and no Git access passes.
• Syntax compilation and whitespace checks pass. The worktree is clean.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev research/p120-char3-rank31-window/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev -O research/p120-char3-rank31-window/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p120-char3-rank31-window
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p120-char3-rank31-window --check-source-history

Another CPython 3.12 path may replace the local path. Expected scientific marker: PASS_P120_CHAR3_RANK31 no_new_kappa. There are no third-party requirements or live external inputs. The finite checker needs no Git or network access; the publication preflight needs full Git history. Catalogue and selected-record provenance are pinned at canonical main e54523517d0f03e58352e647c403a5af3e525a57.

Limits and scope correction

The initial CLAIM sought an unrestricted rank-31 improvement. The broader live-PR check exposed #7014's stronger prior construction. A public scope correction preceded publication. This PR delivers the fixed-space obstruction and the new point inside that space, rather than a global upper-bound improvement.

The theorem is uniform over characteristic-three fields. Exhaustive computations concern only the displayed finite certificate. No characterization of every rigid column, exact restricted optimum, characteristic-zero realization, physical/native decoder, complete paid invoice, caller, stock, precision or movement guarantee is supplied. The five selected assumptions remain unchanged. The selected scientific replay, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the exact minimum rank in this fixed space. Start with a hyperplane meeting every legal affine fiber in the delivered rank-31 span; a failed search would not settle the minimum.
• Identify an enlargement or change of feature space admitting a rank-15 point. Start by expressing the cited clique columns in a larger space. No physical cost follows from such membership alone.
• Find a separate characteristic-zero factorization and complete decoder invoice if this finite-field geometry is to help the complex supplier.

Related open work

• Witness a characteristic-3 rank-32 pattern on P120. #7087 is the direct quadratic-space predecessor. Its reported rank-32 point is cited, not reverified; this adds a rank-16 restriction and a different rank-31 point in that same space.
• Exhibit a 104 by 13 factorisation of the P104 window over F_5. #7014 already supplies the stronger unrestricted rank-15 clique construction over every field. This PR proves that no rank-15 point has all its columns in the fixed space considered here; it does not audit that clique construction.
• P104 port window is exactly 9 in characteristic 2 (no κ) #7114 studies all block patterns and the characteristic-two window, with related general field statements. This PR fixes a particular characteristic-three column space; it makes no competing unrestricted window claim.
• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 studies real/complex E8 decoder COPY floors and reductions. The present finite-characteristic matrix rank gives no COPY comparison.
• One-sided padding lemma: P112 to P120 at cost exactly +8 (L120 suffices for Delta >= n_nf) #3074 transfers one-sided P112 decoders to P120 at an explicit COPY cost. This PR supplies no one-sided decoder or COPY tariff.
• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 records bounded masked-E8 completion and basis searches with physical COPY objectives. The fixed characteristic-three feature-space rank here is a different objective.
• Prove the E8 ports-per-dimension law N_e = 2^(e-2) + 4 in closed form (no κ) #7127 counts P120 ports in binary subspaces. This contribution computes no competing ports-per-dimension number.
• Exclude the proposed 96-line E8 complement at inner dimension 8k (no κ) #7174 excludes one 96-line E8 complement at inner dimension 8k. This PR concerns the full P120 mask in a fixed quadratic space and computes no competing complement value.
• Bound scalar COPY savings at one 93-line E8 puncture (no κ) #7170 bounds complex scalar COPY savings at a 93-line puncture. This PR provides no puncture or scalar COPY comparison.
• Prove the exact E8 Gram-class minimum through 24 deletions (no κ) #7168 gives Gram-class COPY minima through 24 deleted E8 lines. This PR computes neither a Gram-class nor a deleted-line minimum.
• Prove the star-pattern floor n+1 on the wheel W4 and K33 at minimum degree 3 (no κ) #7129 proves block-fitting floors for star-configuration graphs. This PR concerns a fixed column space on the full P120 mask and computes no competing star-graph value.

Completion refresh inspected the nine other open PRs from this account created since the CLAIM: #7261, #7262, #7263, #7264, #7265, #7267, #7268, #7269 and #7270. None studies this P120 fixed quadratic column space or computes its restricted rank. No additional same-object comparison or scientific computation was run.

CI notes

CI did not certify this package. At unchanged head 3cc68fb0585fa78baf121707c8a3154081f4515c, run 43029993808, plan job 130554266928, failed in Onboarding helper tests. I read the completed checks once and inspected the job log. Exactly one of 760 tests failed, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found in the CI verification document

At canonical base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still opens with Current selected-family replay: #6030 and documents make frontier6030-verify. Both files and the failing test are unchanged by this PR. The package replay job was skipped because plan failed. The evidence for this package remains the local checks listed above.

I also inspected the actual logs of 42905094314, 42903313790, 43029125610, 43029249083 and 43029427643. Each has the same sole failing test, the same missing-command AssertionError, 760 tests and 24 skips. No fix, retrigger commit, push or job rerun was made. The repository documentation repair remains with maintainers.

Merge danger

Door: two-way. Blast radius: package. Eight additive files in one opt-in research directory. Shared code, frozen predecessor packages and the selected frontier are unchanged.