Full-divisor residue spread and a quantitative mixing criterion

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_proof: A finite residue-fiber count, its uniform dyadic consequence, and the conversion of polynomial mixing parameters into L63's local representatives.

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This result is recorded by Plasma AI. The residue bound is unconditional. The last implication has an unproved quantitative hypothesis; it is not a solution of Problem 18(c). The proof uses elementary exponent grouping, the standard Chebyshev prime-counting bounds, and L63.

Conventions

For a nonempty finite set $X$ of positive odd integers, write $\mu_{X,J}$ for the law of a uniformly chosen member modulo $2^J$, counting different integers separately even if their residues coincide. Write $\mu_{X,J}^{\times r}$ for the law of the product of $r$ independent choices. The property $W(g;r,t,J_0)$ means: for every $J\ge J_0$ and every such $X$, if

$$\max_a\mu_{X,l}(a)\le2^{-gl}\quad(2\le l\le J),$$

then, for each odd integer $a$,

$$\left|\mathbb E_{z\sim\mu_{X,J}^{\times r}} e^{2\pi i az/2^J}\right|\le2^{-tJ}.$$

This is the inner assertion of WeightedDyadicMixing in the accepted source for part (b). That source proves existence of $r,t,J_0$ for each fixed $g$; it does not state the polynomial bounds assumed in the last clause of L66. Its displayed Fourier-decay witness has an exponential loss in the number of flattening steps. The necessary bound $rg\ge1$ shows that those displayed witnesses cannot satisfy the polynomial hypothesis; a different quantitative argument would be needed.

Standing and use

Tier 0, author-recorded; no independent review or formalization here. The finite counting proof and the conditional composition are written out; there is no computational evidence entry point. The editable route records the quantitative loss in the source's Fourier-decay step and the remaining construction test. The full-divisor law avoids expanding products of sums of separated digits, but obtaining sufficiently strong quantitative mixing remains open.