Reciprocal Mersenne Subseries

Definition 9.1 (257-reset). For every upper or middle reset row $r\ge 10$, require ${\mathrm{\Delta }}_{r+1}^2>2^{r+5}$, where ${\mathrm{\Delta }}_{r+1}=\mathrm{rem}(r+1)-2^{r+1}$. The finite range $10\le r<31$ must be covered as well as the tail. Section 12.5 gives the exact conditional implication from this hypothesis. A proposed bound on right-run lengths is a different statement until the pulse corrections and endpoint conventions are matched. Finite observations establish neither failure of every constant run bound nor logarithmic asymptotic growth.

Definition 9.2 (257-cofinal-rows). For every $N$ there is $n\ge N$ with an exact row at depth $n$ (Definition 6.63); that is, there are finite sets at unboundedly many depths whose quotient sums land exactly on the half-value target. This condition implies $1/2\in \mathcal{A}$ unconditionally (half_mem_mersenneAchievementSet_of_cofinalExactLocalRows). No coherence between the witnessing rows is demanded and none should be assumed: the rows may be mutually incompatible; only their depths need tend to infinity. Requiring compatibility would add a condition on the witnesses. Whether that condition can be obtained from this existence statement needs a separate extraction argument; it is not a strict logical comparison merely because the witnesses are described differently.

Definition 9.3 (257-universal). The original question is the base-$2$ statement: $\sum _{n\in A}(2^n-1{)}^{-1}$ is irrational for every infinite $A\subseteq {\mathbb{N}}_{\ge 1}$. Asking the same for every integer base $b\ge 2$ is a stronger all-base extension, not the definition of Problem #257 itself.