A Spectral–Entropy Framework for the Mersenne Infinitude Problem (Version v1.2)

## Overview This note develops a **windowed operator–variational framework** for the **Mersenne infinitude problem**, designed to isolate *exactly* where genuinely arithmetic input is required. Let \(M_p := 2^p-1\). Fix \(\alpha\in(0,1)\) and define exponent windows\[I_T := [T,\,T+T^\alpha],\qquad \Omega_T := \mathcal{P}\cap I_T,\]where \(\mathcal{P}\) denotes the primes (candidate exponents).We say \(I_T\) is **Mersenne-empty** if\[\{\,p\in \mathcal{P}\cap I_T:\ M_p \text{ is prime}\,\}=\varnothing.\]Let \(E\) be the set of \(T\) such that \(I_T\) is Mersenne-empty. A standard bookkeeping observation is: - If only finitely many Mersenne primes exist, then \(I_T\) is Mersenne-empty for all large \(T\), hence \(\delta(E)=1\).- Therefore, proving **logarithmic density** \(\delta(E)=0\) forces **infinitely many** prime exponents \(p\) with \(M_p\) prime. The paper’s goal is to provide a **modular pipeline** to establish \(\delta(E)=0\), while keeping the remaining arithmetic bottleneck explicit. ## Modular pipeline (what is proved vs. what is targeted) ### (1) Factor-admissible prime bands and kernel designNaive “small-prime exclusion” kernels are ineffective for Mersenne numbers, since any prime factor \(q\mid (2^p-1)\) satisfies \(q\equiv 1\ (\mathrm{mod}\ 2p)\), hence \(q\ge 2p+1\).We therefore introduce **factor-admissible prime bands** at scale\[Q_T^{\mathrm{sc}} := T^{2+\eta},\qquadQ_T := \{\,q\in\mathcal{P}: Q_T^{\mathrm{sc}}\le q \le 2Q_T^{\mathrm{sc}}\,\},\]and build a **nonnegative symmetric incidence kernel** \(K^{(1)}_{M,T}\) on \(\Omega_T\) from admissible incidences \(q\equiv 1\ (\mathrm{mod}\ 2p)\) (band-limited in the exponent coordinate). ### (2) Markovization and mixing diagnosticsRow-normalization yields a reversible Markov operator \(P_{M,T}\), and a lazy version \(\widetilde P_{M,T}\).A key diagnostic input is a **polylogarithmic mixing bound**, formulated via a spectral-gap / log-Sobolev (LSI) condition outside a log-density-0 exceptional set. - **Conditional bridge (target):** a “sieve-to-mixing” module stating that BV/large-sieve-type control of incidence degrees/overlaps implies a conductance lower bound, hence polylogarithmic gap/LSI diagnostics. ### (3) Multi-scale diagnostic \(S_T^{(\mathrm{ms})}\) (softened survival score)To avoid brittleness of hard indicators, v1.2 uses a softened, multi-band survival score. Fix \(J\ge 2\) and a ladder \(0