Mersenne Infinity v1.7r4 — A Spectral–Entropy Framework for the Mersenne Infinitude Problem

# Overview This record releases **Mersenne Infinity v1.7r4**, a windowed spectral–entropy / Markov-mixing frameworkfor organizing *logarithmic density–0 rarity* statements around Mersenne numbers \(M_p = 2^p-1\).The paper is modular by design: it isolates the uniquely Mersenne-specific arithmetic obstacle as an explicit**transduction barrier (Gate–T)**, while keeping the remaining steps operator-theoretic, mixing-based, and variational. **Author:** Lee Byoungwoo **Date:** 2026-02-24 Scope (critical):- This is a *methods/program* paper.- **No unconditional infinitude claim** is made from classical sieve bounds alone.- All ROC/AUC and calibration figures are **supplementary diagnostics** and **not used in proofs**. --- # Pipeline (what the paper does) 1. **Exponent windows:** \(\Omega_T=\mathbb{P}\cap[T,T+T^\alpha]\), with logarithmic-density bookkeeping for “empty windows.”2. **Factor-admissible kernels:** build symmetric incidence kernels from dyadic prime-moduli bands.3. **Lazy reversible Markovization:** construct \(\widetilde P_{M,T}\) and fix \(m_T\) as its reversible stationary measure.4. **Mixing diagnostics:** conductance \(\Phi_T\) and the spectral gap are primary (LSI is optional).5. **Transduction barrier (Gate–T):** keep the only Mersenne-specific upgrade explicit.6. **Variational closure module:** once a uniform energy-gap premise is certified on empty windows, the entropy–rigidity functional yields a logarithmic density–0 elimination of empty windows. --- # Main editor-facing conditional statements - **Main conditional corollary (BV-type incidence control):** BV-type incidence/dispersion control + a mixing diagnostic + a transduction hypothesis imply that the set of Mersenne-empty windows has logarithmic density \(0\), hence infinitely many Mersenne primes exist (conditional on the stated hypotheses). - **Certified Gate–T route via vacancy certificates (conditional):** A “mean-hit floors + covariance-control (COV) certificate” route yields a strict diagonal bound \(\mathrm{Diag}^{(\mathrm{ms})}(T)\le \theta_\star