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Claim. For a finite nonempty set $P$ of primes, put $h_P(a)=\prod_{p\in P}p^{v_p(a)}$. Cook claims that, for every integer $b\ge2$ and infinite set $A$ of positive integers,

$$\sum_{a\in A}\frac{h_P(a)}{a(b^{h_P(a)}-1)}<\infty \quad\Longrightarrow\quad \sum_{a\in A}\frac1{b^a-1}\notin\mathbb Q.$$

The selected declaration divisibilityWeightedClaim also states that if a host $H$ satisfies this condition at base two for some such $P$, every infinite subset $A\subseteq H$ gives an irrational sum at every integer base $b\ge2$. The registration includes four further declarations about positive displacements tending to zero and divisor-cover extensions, with their own summability and majorization hypotheses.

Covers. At $b=2$ this proves instances of Problem 257 with the displayed weighted hypothesis. It contains the reciprocal-summable supports of Cook's earlier note: for $h\ge1$ and $b\ge2$, $h/(b^h-1)\le1$, so the weighted sum is bounded by $\sum_{a\in A}1/a$. The registration explicitly excludes arbitrary infinite supports from its conclusion; failure of its criterion asserts nothing about rationality. It does not decide whether $1/2$ or $1/21$ is a reciprocal Mersenne subseries sum.

Standing. Claimed. The registry entry and the five declarations in the pinned Challenge.lean were read at statement depth on 2026-10-16. The registry reports successful mechanical verification of the matching solutions and a neutral editorial review, and makes no claim of independent human peer review. The deliberate sorry bodies in the Challenge file specify the statements to be compared; they are not the submitted solution proofs. This corpus has not replayed those proofs or checked the manuscript's proofs. Registration alone does not make this an accepted project result or settle the full problem.

Depends on. Nothing in this wiki. The displayed summability condition is an explicit hypothesis, not a known property of every infinite support.

Submission note. Palomar registered the selected result on 2026-09-25. The record does not state the underlying submission date, so submitted remains null. The links identify the registry's selected statements and the manuscript version it cites; no claim of complete manuscript coverage is made.