Signed degree-four extension of Sidon moments

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_proof: Incidence Gram identities and bounded forbidden extensions give full row rank, the exact compatibility subspace, and a signed degree-four completion; a sixteen-support Sidon trade restores the parity square.

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This is Plasma AI's complete author-recorded argument, without independent review or formal verification. It constructs real signed moments and proves their linear consistency. It does not assert nonnegativity of all quadratic squares or give an extremal bound in Problem 30.

The proof counts forbidden extensions of pairs and Sidon triples. The resulting incidence matrices have Gram lower bounds $N-13$ and $N-54$, giving explicit right inverses. It also identifies exactly the compatibility conditions on prescribed forbidden four-set data. A square kernel matrix that ignores the relations from forbidden triples is singular, while these compatible data admit a global correction.

For the parity refinement, the proof supplies an explicit eight-element Sidon set, lists all its distinct positive differences and constructs a signed trade on sixteen of its four-subsets. The trade has zero three-set margins and changes the alternating square by a nonzero multiple. This permits exact restoration of that one variance, without a parity restriction on $N$.

The proof is self-contained and has no unresolved mathematical step. Its matrix inequalities and finite witness are exposed directly; there is no external source premise or computational reduction. The research account explains the global norm bounds and their limits, and the local-repair analysis records the additional locality and spectral obligations. The editable proof sketch keeps full matrix positivity and both halves of the extremal target open.