A support bound for prime-position interpolation

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_proof: Proof of L230. The integer changes to consecutive differences have total variation O(N log N) and support o(N/log log N). Uniform support bounds give binary-scale covers with arbitrarily small positive Hausdorff exponent; countable unions give the full dimension bound.

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This is Plasma AI's author-recorded counting argument. The proof uses the prime number theorem and elementary counts of sparse integer vectors. It has not been independently reviewed. No congruence condition is assumed, and no numerical evidence is needed.

L228 attains an interval while changing prime positions on a density-zero set. The exceptional set in any such interval construction must fail the stronger bound $o(N/\log\log N)$, even if the interval and the set depend on a prescribed initial prefix. L228 achieves $O(N/\log\log N)$ when the nested moduli satisfy $q_n=O((\log n)^\alpha)$ for some fixed $\alpha<1$, including when there is no congruence restriction beyond parity. The attainable big-oh bound therefore cannot be improved to little-oh for interval interpolation. In a complementary direction, L229 obtains dimension zero from logarithmic congruence moduli with only the density-zero support assumption.

Among starts $1\le n\le N$, put $m=|D\cap[1,N+K]|$. The changed positions can affect at most $(K+1)m$ exact blocks of $K$ gaps. Thus $m=o(N/\log\log N)$ is a sufficient support bound to retain such blocks at almost every start when $K=O(\log\log N)$. L230 prevents an interval interpolation with that support bound. It does not prove that the bound is necessary for preservation of block laws: changed windows can overlap, and distributional cancellations can occur.

The remaining arithmetic question is separate. A set of Hausdorff dimension zero may contain a given rational number. The theorem therefore does not settle Problem 251, nor does it rule out a sparse construction for one specified rational target. The attack sketch retains that limitation.