Prove sharp integer inertia for every sunflower downset (no new κ)

research/sunflower-downset-inertia/README.md

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+# Sharp integer inertia certificates for sunflower downsets
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: **scoped mathematical proof with exact finite corroboration**.
+Every downset generated by a finite sunflower of maximal sets has a
+nonsingular, symmetric, nonnegative integer matrix supported on disjoint
+pairs whose number of positive eigenvalues equals its largest star size.
+The construction also gives the exact determinant. The new case has three
+or more maximal sets. It resolves the disjoint-petal first case left in
+Nick Petalas's [#5247](https://github.com/CrocSwap/integer-mult-bounds/pull/5247).
+
+This proves the sunflower subclass of Ellis, Filmus and Friedgut's
+[spectral Conjecture I, section 4](https://arxiv.org/html/2609.28404v1#S4),
+recorded as C070 in David Ganz's
+[atlas #2849](https://github.com/CrocSwap/integer-mult-bounds/pull/2849).
+The general downset conjecture remains open here. Global literature
+priority, independent human review and formal proof-kernel review are
+unclaimed. No new kappa or selected-frontier change is claimed.
+
+## Statement and construction
+
+Let C,A_0,...,A_(r-1) be pairwise disjoint finite sets, with
+
+```text
+r >= 3,  a_0 >= a_1 >= ... >= a_(r-1) >= 1,  a_i = |A_i|,  c = |C|,
+D = union over i of 2^(C union A_i),
+L = 1 + sum_i (2^a_i - 1),  N = 2^c L.
+```
+
+These are precisely the downsets whose three or more maximal sets form a
+sunflower. One or two maximal sets are covered by the cited predecessor.
+Choose injections phi_i:A_i -> A_0 for i>=1, and let phi_0 be the identity.
+The injections define weights only. They do not identify actual ground
+elements from different petals.
+
+First take c=0. List every subset of A_0, then the nonempty subsets of
+each other petal. This lists D once. Define a symmetric base matrix K.
+Within one petal, use ordinary disjointness entries. Between subsets S,T
+from different petals, put k=|phi_i(S) intersect phi_j(T)| and use
+
+```text
+                        k = 0           k >= 1
+one petal is A_0:          1              2^k - 1
+neither is A_0:            1              2^(k+1) - 3.
+```
+
+Every cross-petal pair is disjoint in the actual ground set. All displayed
+weights are positive integers. Within a petal, the only nonzero diagonal
+entry is at the empty set, and it is 1.
+
+For general c, write a set uniquely as R union U, with R subset C and U
+a base label, and set
+
+```text
+M((R,U),(R',V)) = 1[R intersect R' is empty] K(U,V).
+```
+
+Then M is nonnegative, integral, symmetric and zero whenever its two
+indexing sets intersect. Let s denote the largest star size. The formulas
+are
+
+```text
+s = 2^(a_0-1)  if c=0;       s = N/2  if c>=1,
+e = 2^c sum_(i=1)^(r-1) (2^(a_i-1)-1),
+inertia(M) = (s, N-s, 0),
+det(M) = (-1)^(N-s) 3^e.
+```
+
+Thus the optimal number of nonnegative eigenvalues is exactly s, even
+when the competing matrices may have arbitrary real supported entries.
+
+## Proof
+
+The Boolean zeta congruence and the dominant-to-one-petal block below are
+the credited ingredients of #5247. The new step chooses all the other
+cross blocks simultaneously and cancels every off-diagonal Schur block.
+The argument is written in full to specify that new construction; no
+predecessor executable is imported or replayed.
+
+For the cube on a set A, put
+
+```text
+Z_A(S,T) = 1[T subset S],
+J_A(T,T) = (-1)^|T|,
+D_A(S,T) = 1[S intersect T is empty].
+```
+
+In an inclusion-compatible order Z_A is unit lower triangular, and
+`D_A = Z_A J_A Z_A^T`, since the entry is the alternating subset sum
+over S intersect T. For each i>=1, let Z_i' and J_i' be the restrictions
+to nonempty subsets of A_i, and put u_i(T)=(-1)^(|T|+1). Then
+
+```text
+Z_i' u_i = 1,
+D_A_i[nonempty,nonempty] = Z_i' (J_i' + u_i u_i^T) Z_i'^T.
+```
+
+Let e_0 be the empty-subset coordinate of the full A_0 cube. Define E_i
+with columns indexed by nonempty T subset A_i. Its column is the standard
+coordinate vector at phi_i(T) when |T| is even, and zero when |T| is odd.
+Injectivity gives
+
+```text
+J_A_0 E_i = E_i,  e_0^T E_i = 0,
+E_i^T E_i = diag(1[|T| is positive and even]).
+```
+
+Set H_i=e_0 u_i^T+2E_i. Under congruence by
+`Z=diag(Z_A_0,Z_1',...,Z_(r-1)')`, K has the blocks
+
+```text
+B_00 = J_A_0,
+B_0i = H_i,
+B_ii = J_i' + u_i u_i^T,
+B_ij = u_i u_j^T + 4 E_i^T E_j     for distinct i,j >= 1,
+K = Z B Z^T.
+```
+
+To check the explicit weights, Z_A_0 E_i Z_i'^T counts nonempty even
+subsets of S intersect phi_i(T). The analogous cross expression
+`Z_i' E_i^T E_j Z_j'^T` counts nonempty even subsets of
+phi_i(S) intersect phi_j(T). If this intersection has size k, the count
+is zero at k=0 and `2^(k-1)-1` at k>=1. Adding the all-ones term and
+multiplying that count by 2 or 4 gives exactly the two weight formulas.
+
+Eliminate B_00. It is its own inverse. For every i,j>=1,
+
+```text
+H_i^T J_A_0 H_j = u_i u_j^T + 4 E_i^T E_j.
+```
+
+Therefore all off-diagonal Schur blocks vanish, and diagonal block i is
+
+```text
+J_i' - 4 E_i^T E_i
+    = diag(-1 on odd subsets, -3 on nonempty even subsets).
+```
+
+This is negative definite, independently of how many petals there are or
+how their injected images overlap in A_0. Sylvester's law of inertia
+gives exactly `2^(a_0-1)` positive eigenvalues. Every other eigenvalue is
+negative. The zeta and Schur congruence factors have determinant one.
+There are `sum_(i>=1)(2^(a_i-1)-1)` entries -3, which proves the base
+determinant's absolute value and its sign from the negative eigenvalues.
+
+For c>=1, `M=D_C tensor K`. The invertible symmetric D_C has equally
+many positive and negative eigenvalues, by its zeta congruence. Hence
+M has N/2 of each sign. Also |det D_C|=1, so
+`|det M|=|det K|^(2^c)`. This proves both general formulas.
+
+When c=0, an element of A_i belongs to exactly `2^(a_i-1)` sets. When
+c>=1, an element of C belongs to N/2 sets. A petal element belongs to
+`2^c 2^(a_i-1)<=N/2` sets because L>=2^a_i. These counts give s above.
+
+Finally a star of size s indexes a zero principal block in every real
+symmetric disjointness-supported matrix. Its quadratic form vanishes on
+that s-dimensional coordinate subspace. If there were fewer than s
+nonnegative eigenvalues, the negative eigenspace would have dimension
+greater than N-s and intersect this subspace nontrivially. That would
+contradict negative definiteness. This proves the asserted optimality.
+
+## First new example and finite evidence
+
+For three disjoint two-element facets, N=10 and s=2. The construction has
+inertia `(2,8,0)` and determinant 9. Its weights between the full facets
+are 3,3,5. Keeping the two dominant cross weights 3 but changing the last
+weight to 1 gives inertia `(3,7,0)` and determinant -7. Changing that last
+weight to 3 retains sharp inertia but changes the determinant to 5. This
+control distinguishes exact Schur cancellation from mere sharp inertia.
+Weakening the coupling coefficient 2 to 1 gives `(2,6,2)`, with four
+nonnegative eigenvalues, so zero eigenvalues must be counted.
+
+The standalone standard-library verifier enumerates all 50 nonincreasing
+positive petal-size/core tuples with r>=3 and
+`c+sum_i a_i<=7`. Matrices have order at most 64. It builds actual disjoint
+ground-set labels, checks all 23,333 support entries and the largest stars,
+and independently computes exact inertia and determinant by rational
+symmetric elimination with both one- and two-dimensional pivots. It also
+checks the integer zeta congruence and all 785 leaf Schur entries across
+these cases. Thirteen named controls include the new example, changed
+weights, null-eigenvalue counting, zero-diagonal pivots and invalid inputs.
+No two-facet predecessor cases are replayed.
+
+The universal result rests on the written proof. These finite checks
+corroborate the construction and test its failure modes. JSON stdout is
+generated from the actual checks and binds all five package files by
+SHA256. No receipt is handwritten or committed. The verifier writes no
+files and does not fetch or import predecessor sources.
+
+From the repository root, any CPython 3.12 may replace this local path:
+
+```sh
+/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
+```
+
+Expected scientific marker: `PASS_SUNFLOWER_DOWNSET_INERTIA`. The last
+command checks publication bytes and source history after committing.
+
+## Implementation bounds and limitations
+
+Every loop has a finite bound from seven ground coordinates, seven petals,
+matrix order 64 or the fixed five-file inventory. The parameter iterator
+examines at most 1,716 tuples for each of five petal counts; only 50 pass
+the stated domain. Symmetric elimination removes at least one coordinate
+per iteration and uses at most 64 iterations. No recursive calls, dynamic
+code evaluation, subprocesses or external runtime dependencies occur.
+All functions fit within sixty physical lines. Explicit exception checks
+remain active under optimization.
+
+Power-of-Ten rule 3 has a managed-runtime deviation in verify.py: CPython
+allocates lists and exact integers/rationals during execution. Fixed input
+and matrix capacities bound live work; there are no external parameters or
+unbounded streams. A future parameterized checker must enforce size and
+bit-length budgets before lifting these capacities. Rule 9 has six fixed
+local rejection callbacks, used immediately and retaining no state. Their
+only targets are the literal-matrix validator and the label validator.
+Warning-strict normal and optimized execution and AST inspection supply
+available diagnostics; a standalone linter is not installed in this
+interpreter. No shared build configuration is changed.
+
+General overlapping-facet downsets, a weighted-Hoffman certificate,
+physical preparation, recursive callers, precision, stock and supplier
+admission remain outside scope. Lean, Docker and the selected cold replay
+were not run. This is not a complete multiplication theorem.
+
+## Credit and overlap
+
+David Ganz's #2849 supplies atlas C070. Ellis, Filmus and Friedgut supply
+the spectral conjecture. Nick Petalas's #5247 supplies the two-facet
+construction, the Boolean-zeta/one-petal Schur argument and the explicit
+three-facet first question. Classical zeta inversion and Sylvester's law
+retain their credit. The new step is the simultaneous leaf cross-block
+construction, its all-petal cancellation and the resulting determinant.
+Predecessor files and credit lines are unchanged.
+
+Canonical base is `de5ec2e7a6074068452f45b1a191b5c72832df4a`, priority
+revision 8. The selected-baseline binding control passed. TASK.json marks
+the formal-interface card as changed to this algebraic certificate side
+result; no caller-bound formalization is asserted. The
+[advance CLAIM](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19214425)
+records the domain and method.
+
+Live open/account PRs, active Discussions, recent handoffs, local
+reservations, canonical-main README terms and merged topic searches were
+checked. The overlap cache covered 416 open PRs and 5,309 Discussion
+records at 10:01-10:02 UTC on October 27. Sparse local coverage was only
+209 of 7,845 HEAD note paths; separate main README searches supplement it.
+No open PR on this sunflower inertia quantity appeared. These bounded
+searches do not cover unpublished work or establish literature priority.
+
+## Open next questions
+
+1. Can three maximal sets with different nonempty pairwise intersections
+   admit a sharp integer certificate? Start with facets {1,2}, {2,3} and
+   {1,3}. Cross-facet pairs may now intersect, so freely prescribing all
+   Schur-cancelling entries may violate the support restriction. This is
+   the first obstruction to extending the present sunflower proof.
+2. Can a smaller weight alphabet give a uniform sharp certificate for
+   arbitrary sunflowers? The three-two-element-petal control has a sharp
+   alternative determinant 5. Start by classifying its allowed cross
+   weights. Exact cancellation and minimum entry size are different
+   objectives; no general minimum-weight claim is made here.

research/sunflower-downset-inertia/SOURCES.json

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