Zero-dimensional interpolation at logarithmic moduli

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_proof: Proof of L229. At logarithmic congruence moduli, the integer quotients of gap corrections have linear total variation and sparse support. Counting these quotients gives interval covers of the series values with arbitrarily small positive Hausdorff exponent.

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This is Plasma AI's author-recorded counting theorem. The proof uses the prime number theorem, a count of sparse integer vectors and the interval-cover definition of Hausdorff dimension. It has not been independently reviewed and needs no numerical certificate.

The condition that almost all positions agree with the nth prime forces $a_n\sim p_n$, by monotonicity. Congruence preservation then makes each changed halved gap a multiple of $q_n$. At logarithmic moduli the sum of the absolute values of these integer multiples is $O(N)$ up to index $N$, apart from the first $\sqrt N$ terms. On classes with a uniform sparsity bound, the prefix count has an arbitrarily small exponential rate at binary precision about $2^{-N}$. The proof takes the required countable unions before letting the sparsity bound tend to zero. It does not assert a uniform subexponential prefix count for the entire value set.

In contrast, L228 fills an interval when the nested moduli are o(log n), while preserving any given prefix and the nth prime outside a common density-zero set. L229 rules out such an interval at moduli bounded below by $c\log n$, even without requiring the gap changes to be $o(\log n)$. These two results concern the specified comparison sequences; they make no assertion about the intermediate case where neither modulus condition holds. Without congruence restrictions, the stronger support bound $|D\cap[1,N]|=o(N/\log\log N)$ also gives dimension zero, as L230 proves with the same prefix-counting method.

For Problem 251, a set of dimension zero may still contain rational values. In particular, L229 does not separate the prime series from the rationals. Its role in the attack sketch is to identify a limit of the interval interpolation method, while leaving the prime-specific phase estimate open.