Reciprocal Mersenne Subseries

Theorem 6.84 (A square-root carry bound implies half-membership). Write $G$ for the greedy support and $C_G(N)$ for its centred carry. If

then the greedy selected support $G$ is infinite and $X_G(2)=1/2$. Nonnegativity of $C_G$ is unconditional (see Theorem 6.11 and the greedy case above); only the upper $2\sqrt{N}+4$ bound remains open. This is a different hypothesis from the dyadic-band condition and the largest-skip condition; logical independence is not asserted.

Theorem 6.85 (Terminal bounds at unbounded depths). Suppose that for every $N\ge 0$ there are $M\ge max\{N,1\}$ and $D\subseteq \{2,\dots ,M\}$ such that

Then some infinite $A\subseteq {\mathbb{N}}_{>0}$ satisfies $X_A(2)=1/2$. No compatibility between different finite supports or bound on their earlier carries is required.

These are not the exact-row conditions of Definition 6.66. For such a finite support,

so an exact row has carry $1$, whereas the displayed bound allows several positive and negative values. At $M=6$, for example, $D=\{2,3\}$ has $Q(D,6)=30$ and carry $2$: it satisfies the terminal bound but is not an exact row. Both cofinal existence statements nevertheless imply, and are implied by, half-membership, as explained in Proposition 2.7. The difference between their finite data must not be confused with a strict logical weakening of the membership problem.

Theorem 6.86 (Cofinal returns of the greedy carry). Suppose that for every $N\ge 0$ there is $M\ge N$ such that

Then $G$ is infinite and $X_G(2)=1/2$. Unlike Theorem 6.84, this hypothesis bounds the actual greedy carry only at unboundedly many indices. It still concerns the same fixed support $G$, not independently chosen finite supports. Comparing this carry with the carry obtained by fixing a finite prefix introduces a contribution from later omitted ranks. No equivalence with the finite-support hypothesis of Theorem 6.75 follows without controlling that additional contribution. The hypothesis is not established.