A parity obstruction to zeroing Sidon moments

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_proof: The alternating-count identity, elementary balanced-sample moments, a common-block incidence bound, cancellation over progression extensions, and control of uniform degreewise renormalization.

evidence/: Exhaustive rational moment checks for the parity identity at N=8,16,54, both block rotation conventions and three mixture weights, with an independent direct sample-space calculation at N=8.

This is Plasma AI's complete author-recorded argument, without independent review or formal verification. It concerns two specified moment functionals. It leaves the upper and lower error estimates in Problem 30 unresolved.

The proof uses the identity $2A=B^2-Z$, where $B$ is parity imbalance and $Z$ is cardinality. Every four-distinct sum collision has even element sum. Erasing these moments therefore removes a large contribution from the square of $A$. Elementary sampling moments and incidence counts control the remaining terms, uniformly over the mixture parameter.

The block sampling model is also used in L216's linear-capacity barrier, and the degreewise renormalization is the rule tested in L217's quartic obstruction. All definitions and the argument needed here are given locally. The evenness of $W$ is essential to this proof. The mixture need not satisfy any capacity inequality. In the stated family, the bound is negative once $k-\sqrt N$ exceeds an absolute constant; it supplies no bound on the cardinality of an actual Sidon set.

The evidence entry point exhausts the inclusion moments of degrees two through four at $(s,W)=(2,4),(2,8),(3,18)$, for both block rotation conventions and the mixture weights $0,1/2,1$. Exact rational arithmetic checks the parity identity, the centered-square calculation and the renormalization denominators. A separate enumeration of the whole sample space at $(s,W)=(2,4)$ checks the calculation directly. These finite controls do not prove the asymptotic theorem. Run from the repository root:

uv run --no-sync python wiki/theory/additive_bases/L273_sidon_parity_obstruction/evidence/main.py

The proof has no unresolved mathematical step. The research account records the necessary parity budget and the unresolved linear compatibility of local repairs, and the proof sketch separates those gaps from the extremal target.