A probability-measure barrier for linear Sidon capacities

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_proof: A rotated block-sampling law, exact pair moments, explicit all-N estimates, a simpler infinite family, and the separating-certificate consequence.

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This is Plasma AI's author-recorded argument. It has no independent review or formal verification. It concerns a class of certificates, not the size of an actual Sidon set, and does not resolve Problem 30's subpolynomial error.

The proof constructs the measure by sampling a prescribed number of points in each of nearly equal consecutive blocks and then rotating the result. It uses only elementary probability and counting, with no source theorem or native claim as a premise. For $k\ge\sqrt N$ the construction has the stronger bounds $\mathbb E r_A(d)\le1-7N^{-1/2}$ and $\mathbb E\sigma_A(s)\le3/4$.

The same proof gives a particularly simple infinite family: for $s\ge2$, take $s$ blocks of length $W=s^3-s^2+s$, independently choose a uniform $s$-element subset of each block, and rotate. Then $N=sW$, $k=s^2$, every capacity is satisfied in expectation, and $(k-\sqrt N)/N^{1/4}\longrightarrow1/2$.

For the exact certificate convention and the connection to sliding windows, see the research account of linear capacities and their limits. The editable proof sketch keeps the separate upper and lower obligations visible. The obstruction does not cover an arbitrary case split or a nonlinear use of the Sidon condition.

The proof is self-contained and has no unresolved mathematical step. Finite computational checks in the linked research evidence are supplementary; they do not establish the quantified theorem or increase its tier.