[petalas]

Summary

Scoped mathematical proof with finite exact corroboration: every downset with at most two maximal members and a nonempty member has a nonsingular, symmetric, nonnegative integer disjointness-supported matrix with exactly the largest-star number of nonnegative eigenvalues. This settles the two-facet case of Ellis–Filmus–Friedgut's Conjecture I, recorded as C070 in David Ganz's PR2849. Credit those predecessors and the classical Boolean zeta/Sylvester tools; the construction and proof here were produced by Nick Petalas.

D = 2^(C union A) union 2^(C union B), a=|A| >= b=|B|
within either exclusive cube: ordinary disjointness entries
cross weight: h(0)=1; h(k)=2^k-1 for k>=1
zeta congruence: J_a plus a negative definite Schur complement
inertia = (largest star, remaining sets, 0)

The package also gives the exact determinant. The general conjecture remains open; no global literature priority or new kappa is claimed. The primary paper, section 4 supplies the conjecture, not an assumed construction.

Evidence

• Before: for two disjoint two-element facets, the ordinary disjointness matrix has inertia (3,4,0), above the largest star size 2.
  After: changing the full-facet cross weight from 1 to 3 gives inertia (2,5,0) and determinant -3. Weight 2 has a zero eigenvalue and still three nonnegative eigenvalues.
• The all-size proof uses an explicit Boolean zeta congruence and a diagonal negative Schur complement. The standalone checker independently computes exact rational inertia and determinant for all 49 parameter triples with 1<=a+b+c<=6 and a>=b, including single-cube presentations, up to matrix order 64. It checks every support entry, every star and the integer congruence, with eight enumerated controls.
• CPython 3.12.15, standard library. Both registered commands passed locally in 0.49 s and 0.48 s; package replay passed in 1.19 s. The committed-byte/source-history preflight passed for head 834e574 and base ff3cc58. Normal and optimized verifier JSON agree exactly, including source hashes.

Exact reproduction from the repository root:

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

Any CPython 3.12 may replace the local interpreter path. Expected markers: PASS_TWO_FACET_DOWNSET_INERTIA, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. The verifier writes no files; its JSON receipt is generated from package hashes.

Limitations

This is a scoped written proof, not a proof-kernel formalization or a multiplication theorem. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, selected cold replay, physical supplier, precision/storage implementation or supplier admission was run. The selected frontier is unchanged. Atlas source bytes are pinned as the originating question; the proof and direct verifier have no mathematical/runtime dependency on atlas code or the external paper.

Open next questions

Can three maximal sets admit the same sharp integer certificate? First write the zeta cross blocks for three disjoint two-element facets. The missing step is simultaneous control of three Schur interactions; the present two-facet negative complement does not supply it.

Related open work

No matching open PR on the two-facet downset inertia quantity appeared in the targeted live/cached searches. The overlap cache covered 311 open PRs and 4,483 Discussion records; sparse notes and unpublished work remain coverage limits.

The completion refresh read the titles and bodies of all eight other open account PRs created since the CLAIM: #5245 studies Hermitian qubit repair, #5246 F4 Hermitian hyperplane intersections, #5248 symplectic partial-spread completion, #5249 P112 diagonal-completion rank, #5250 matrix-pencil determinant images, #5251 a uniform projective design, #5252 finite etale tensor determinant images and #5253 characteristic-two odd-type fitting rank. None studies the two-facet downset inertia quantity or supplies a competing value for this certificate. Several share atlas provenance, with distinct objects and objectives. No additional computations were run to validate or compare those results.

CI notes

Both hosted checks passed at exact head 834e5749d0797796af92bdd3031108e5e337fce2 in Registered research packages run 39966036094: planning in 3m49s and the focused registered replay in 1m16s. Run metadata confirms completed status, success and the matching head SHA. The registered replay executes both warning-strict normal and optimized VERIFY.json commands.

All local registered commands and the committed-byte/source-history checks listed above also passed. Hosted replay certifies execution of the bounded checks and controls. Independent human and formal proof-kernel review of the written theorem remain unclaimed; the listed unrun obligations are unchanged.

Merge Danger

Door: two-way.

Blast Radius: isolated research package. Five new files; no shared implementation, predecessor credit, pinned evidence or selected-frontier changes.