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 (2ᵖ-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 \ (IT=T, T+T^\), 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 2ᵖ 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 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