[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. Every downset generated by three or more sunflower facets has an explicit nonsingular, symmetric, nonnegative integer disjointness-supported matrix whose number of positive eigenvalues equals its largest star size. The construction also gives the exact determinant. This resolves the disjoint-petal first case left by merged #5247, extending its two-facet result to any finite number of petals and any common core.

The new step chooses the leaf cross blocks simultaneously:

B_ij = u_i u_j^T + 4 E_i^T E_j = H_i^T J_0 H_j
all off-diagonal leaf Schur blocks = 0
leaf diagonal blocks = diagonal(-1 on odd subsets, -3 on nonempty even subsets)

Credit Ellis, Filmus and Friedgut for spectral Conjecture I, David Ganz's #2849 for atlas C070, and Nick Petalas's #5247 for the two-facet zeta/Schur construction and explicit open question. Classical zeta inversion and Sylvester's law retain their credit. This contribution was produced by Nick Petalas. Predecessor files and credit lines are unchanged.

Evidence

• Before: Prove sharp integer inertia for two-facet downsets (no new κ) #5247 leaves three disjoint facets open. For three two-element petals, keeping the dominant cross weights 3,3 and a leaf cross weight 1 gives inertia (3,7,0) and determinant -7.
• After: the new cross weights 3,3,5 give inertia (2,8,0) and determinant 9, attaining largest-star size 2. The written proof covers every sunflower size and petal count. The determinant is (-1)^(N-s) 3^e, where e=2^c sum_(i>=1)(2^(a_i-1)-1).
• The standalone checker enumerates all 50 nonincreasing positive petal/core tuples with at least three petals and at most seven ground elements. It checks 23,333 actual matrix entries, literal largest stars, exact rational inertia and determinants, integer zeta congruences, all 785 leaf Schur entries and 13 named controls. Largest matrix order is 64. No predecessor verifier or two-facet case is replayed.
• Local CPython 3.12.15 warning-strict normal and optimized checks pass in 0.182 s and 0.184 s. The registered runner passes in 0.656 s. Both direct receipts agree exactly. Each registered command has a 120-second cap. These are verification durations.
• At head b59f9c29bedfb3e2e227f63afb5f630e73a3cd6a, committed-byte/source-history validation passes for all five files and canonical base de5ec2e7a6074068452f45b1a191b5c72832df4a. Official export integrity and both read-only Git-free empty-PATH replays pass, with receipts identical to local execution. AST inspection and staged whitespace checks pass; the longest function is 44 lines.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/sunflower-downset-inertia/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/sunflower-downset-inertia/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sunflower-downset-inertia
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sunflower-downset-inertia --check-source-history

Any CPython 3.12 executable may replace that local path. Expected markers are PASS_SUNFLOWER_DOWNSET_INERTIA, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Generated stdout receipts bind the five package files by SHA256; no receipt is handwritten or committed.

The advance CLAIM states the delivered scope. Live open/account PRs, active Discussions, local reservations, main README searches and merged topic searches were refreshed before committing. The cached search covered 416 open PRs and 5,309 Discussion records; sparse local-note coverage is incomplete. No literature-priority guarantee is claimed.

Limitations

The universal theorem rests on the written informal proof; finite execution corroborates it. Independent human and formal proof-kernel review are unclaimed. General overlapping-facet downsets and weighted-Hoffman certificates remain open here. No physical supplier, caller, precision, stock, full multiplication theorem, new kappa or selected-frontier change is claimed. Lean, Docker and the selected cold replay were not run. No local registered check was left solely to CI. The verifier's fixed finite domains and managed-runtime/callback deviations are documented in the README.

Open next questions

1. Extend the certificate to three facets with distinct nonempty pairwise intersections, starting with {1,2}, {2,3}, {1,3}. Cross-block entries must respect intersecting-pair zeros, so the present free cross-block prescription does not apply.
2. Determine whether smaller integer weights give sharp certificates for arbitrary sunflowers. The three-petal pairwise-weight control retains sharp inertia with determinant 5; no general minimum-weight assertion is made.

Related open work

No matching open PR studying sunflower-downset inertia or this determinant appeared in the refreshed topic/body searches. #5247 is the merged mathematical predecessor. Shared atlas provenance alone does not identify the same object or quantity. Unpublished work remains outside this overlap check.

The completion refresh read the titles and bodies of this account's nine other open PRs created since the CLAIM at 2026-10-27T10:02:18Z: #5865, #5866, #5867, #5868, #5869, #5871, #5872, #5873 and #5874. None studies the sunflower-downset inertia or determinant proved here. #5867 mentions this PR as adjacent matrix work but counts finite-field connector repair ranks, a different quantity. No extra computations were run to validate those contributions or compare unrelated numbers.

CI notes

At exact head b59f9c29bedfb3e2e227f63afb5f630e73a3cd6a, run 40486899975 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. Its ten failure/error entries are all in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22), exit code 1.

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card transformation counts, missing-reference rejection and sparse omission lists. All ten failure/error headers and the final counts match the failed log of PR #5862, run 40479932917. This is the reported shared onboarding failure on main de5ec2e7a; no shared test or implementation was changed to bypass it. Maintainer repair remains outstanding.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks listed above have run. Independent human and formal proof-kernel review remain unclaimed. The contribution commit and package are unchanged.

Merge Danger

Two-way door. Blast radius: one isolated research package. Five new files; no shared implementation, frozen evidence, predecessor credit or selected record is changed.