A Spectral--Entropy Framework for the Mersenne Infinitude Problem (v1.3)

# Overview This record releases **v1.3** of the preprint **“A Spectral--Entropy Framework for the Mersenne Infinitude Problem”** Author: **Lee Byoungwoo** The paper develops a **windowed operator--variational framework** for the Mersenne infinitude problem,with the explicit goal of isolating which parts of the argument are structural/analytic and which partsrequire genuinely arithmetic input. The framework combines: - factor-admissible prime bands for Mersenne divisors \(q (2^p-1)\),- a band-limited incidence kernel and lazy Markovization,- a sieve-to-mixing bridge (dispersion / overlap control \(⇒\) conductance / spectral diagnostics),- and a logarithmic density--\(0\) closure module driven by an explicit exceptional-window mask. The main contribution is **not** an unconditional proof of infinitely many Mersenne primes.Instead, the paper gives a rigorous modular pipeline in which the remaining bottleneck is isolated as aquantitative **transduction (Gate--T)** target. # Closed results - **Factor-admissible windowed kernel construction.** A non-degenerate, band-limited symmetric incidence kernel is constructed on exponent windows \(I_T=[T,T+T^α]\), tailored to the arithmetic shape of Mersenne divisors. - **Sieve-to-mixing bridge (conditional module).** The paper formulates a precise bridge from dyadic-band dispersion/overlap estimates to Markov-chain diagnostics (conductance, spectral gap, and logarithmic-Sobolev type controls). - **Density--\(0\) closure module (self-contained once inputs are supplied).** Assuming (i) a mixing diagnostic and (ii) a uniform energy-gap premise tied to a masked exceptional set, the framework yields a logarithmic density--\(0\) closure statement. - **Gate--T isolation (program bottleneck).** The only genuinely Mersenne-specific obstruction is isolated as a transduction condition, making the remaining target explicit and externally testable. # What is new in v1.3 - Added an explicit **Gate--T external input remark** linking the transduction bottleneck to a dyadic prime-moduli exponential large sieve input for the prime-index sequence \(p ↦ 2^p q\).- Added a direct citation to the companion analytic note: **Lee Byoungwoo, _Large sieve bounds for prime-index powers modulo primes (v1.0)_, Zenodo (2026)**, DOI: **10.5281/zenodo.18744040**.- Restored a missing `remark` environment boundary in the TeX source (compilation-consistency fix).- Source and bibliography alignment improved for clean reproducible builds. # Scope & non-toy status - This is a **program paper** with a rigorous closure architecture: it cleanly separates structural modules (already formalized) from the remaining arithmetic input target (Gate--T).- The paper does not present a full unconditional Mersenne infinitude proof.- Its value is in providing a mathematically explicit, modular, and auditable route from sieve-style inputs to density-level consequences, with the bottleneck localized. # Program closure and targets A concrete next target is a **dyadic prime-moduli dispersion/large-sieve statement** strong enough to certify Gate--T.The paper now explicitly points to a companion large-sieve package for prime-index powers as a candidate external input: - **Large sieve bounds for prime-index powers modulo primes (v1.0)** Zenodo DOI: **10.5281/zenodo.18744040** Future progress can proceed by strengthening this analytic input (or replacing it with a sharper one)and feeding it into the Gate--T transduction certificate route. # Keywords / tags Mersenne primes; spectral methods; entropy methods; logarithmic Sobolev inequality; conductance;Markov chains; exponential sums; large sieve; analytic number theory; density-zero methods # 2020 MSC (suggested) Primary: 11A41 Secondary: 11K38, 11L07, 37A30