Zeroing and uniform renormalization do not lift the block law

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_proof: Cyclic variance identities, a collision coupling, and weighted pair counts obstruct zeroing and uniform renormalization of block laws and their uniform admixtures at every positive excess coefficient.

evidence/: Exhaustive rational evaluation of the block law and a capacity-limited uniform mixture at N=63, including a signed cyclic control and separate direct enumeration.

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This is Plasma AI's complete author-recorded argument, without independent review or formal verification. It refutes this specified extension of one probability law. It does not prove infeasibility of the degree-four Sidon relaxation or improve the extremal upper bound in Problem 30.

The input law and its linear capacities are supplied by L216's exact equal-block family. The proof uses the variance of the number of short pairs. That leading variance bound is less than half the leading mass of repeated short differences. Erasing the latter makes a centered square negative; the specified uniform renormalization is too small to repair it. Restoring the total moment in each degree does not assert the full collection of local cardinality identities.

The final part of the proof also allows a mixture with the uniform law on all $k$-subsets. Difference $1$ limits its mixing weight to $(1+o(1))/s$. Its added variance is then at most $s^3/4+o(s^3)$, still too small to repair the negative square by zeroing and uniform renormalization. A successful extension requires a different correction.

A weighted cyclic pair count extends the obstruction to every fixed excess coefficient $c\in(0,1/2]$ in the block-and-uniform mixture family. Its leading means agree in the two components, preventing mixing from adding a large variance in that direction. This still concerns the specified moment rule; it is not an obstruction to arbitrary degree-four completions.

The evidence entry point evaluates the same functional at $s=3$, where its defining denominators also are positive. It exhausts every subset of sizes two through four and every distinct block rotation using exact rational arithmetic. It finds

$$\mathcal L((Q-\mathbb E Q)^2) =-\frac{941730900101847658}{109020148158776379}<0.$$

The finite witness is supplementary; the asymptotic statement rests on the written proof. See the degree-four research account for what a nonuniform correction would still have to accomplish. No such correction, and no obstruction to every correction, is established here.