Reciprocal Mersenne Subseries

The specific obstacle

What the cited short-interval theorems control. Matomäki and Radziwiłł prove that a real-valued multiplicative function bounded by $1$ has, in almost all windows $[x,x+h]$ with $h\to \mathrm{\infty }$, essentially its long average [11]. Mangerel extends such bounds to a class of divisor-bounded multiplicative functions [12], and Sun describes the almost-all transition profile of the $k$-fold divisor function at its critical polylogarithmic window [13]. These results control short-interval averages. They do not by themselves supply a reset-deviation estimate or its unproved weighted-phase substitute, for two reasons.

First, their conclusions hold for almost all $x$, whereas the proposed application needs every reset row beyond its threshold. An exceptional set of density zero may still contain infinitely many required indices. No density assertion about the reset set, or avoidance of the exceptional set by that orbit, is proved here.

Second, $\tau$ is not bounded by $1$, and the relevant statistic is the distance to $\mathbb{Z}$ of a dyadically weighted sum of $\tau$-values at a precision shrinking exponentially in the window parameter. The first term $(\tau (M+1)-1)/2$ has fixed weight $1/2$, independent of the window length; an unweighted average over that window does not by itself control this term. An application through the phase would need both an exceptional-set argument along the specified indices and control of the weighted sum. It would still have to justify the passage from that sum to the reset deviation, including the skip-set correction.

The required statistic is different. A central limit theorem for prime-factor counts, or an almost-all theorem for unweighted interval averages, is not by itself a bound for the distance to an integer of this dyadically weighted divisor sum. An application must state its precision, its dependence on the growing window length, and its exceptional set. The references cited above do not supply that orbitwise estimate in the form used here. This is a limitation of the proposed application, not a survey claim that every relevant local estimate is unavailable.

The countermodels in the record impose similarly specific limitations. They concern fixed-precision residues (affineBinaryOrbit_mod_twoPow_eq), a decoder without fresh arithmetic input (balancedPulse_no_autonomous_decoder), and fixed-precision signatures (fixedPrecisionTropicalNoGo). They do not partition all possible proof methods. Together with the denominator estimate, they explain why the particular shortcuts examined here are insufficient; they do not prove that a weighted-window method is necessary or impossible.