Reciprocal Mersenne Subseries

Theorem 6.89 (The next floor quotient). For $d\ge 2$ and $M\ge 0$, the floor quotient satisfies

Indeed, $q(M,d)=\sum _{j=1}^{\lfloor M/d\rfloor }{2}^{M-jd}$. Increasing $M$ doubles each existing term and adds $1$ exactly when $M+1$ is divisible by $d$. Thus a quotient doubles or doubles plus one; it does not remain unchanged except when both values are zero. Summing over a fixed finite support gives the corresponding update with added term $c_D(M+1)$.

Theorem 6.90 (The next quotient sum). For a fixed finite set $D\subseteq \{2,3,\dots \}$ and $M\ge 0$,

This is the sum of Theorem 6.89 over $d\in D$. Keeping $D$ fixed is essential: changing the support between steps adds a separate difference of quotient sums. The exclusion of $d=1$ is also essential for the displayed correction term, since $q(M,1)=2^M$ doubles without an added unit.

Theorem 6.91 (The signed endpoint recurrence). The signed next-step expression is $H(D,k,n)=2S(D,k,n-1)+1-c_D(n)$. The last term counts the selected exponents dividing the new endpoint. This is the same recurrence shape as the generic tempered-orbit recurrence $u(N+1)=2u(N)-v\cdot c(N+1)$ from the tempered-orbit criterion (Theorem 6.105 below), specialised to Mersenne local repair. A matching recurrence alone does not imply rationality: Theorem 6.105 also requires one fixed coefficient sequence, integer states and a vanishing scaled limit. Those additional conditions must be checked before that criterion can be applied to these finite-row quantities.