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:** \(Ω_T=P∩[T,T+T^α]\), 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 \( PM,T\) and fix \(m_T\) as its reversible stationary measure.4. **Mixing diagnostics:** conductance \(Φ_T\) 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 \(Diag⁽ᵐˢ⁾(T)≤ θ_<1\) on intended prime-free windows, supporting Gate–T by contrapositive once \(θ_≤ θempty<θ\). - **BDH-type second-moment implication (conditional):** A BDH/mean-square style band-occupancy dispersion input can populate the COV certificate module, turning the vacancy route into a standard “mean-square ⇒ variance/covariance ⇒ vacancy” chain. A surrogate progression model is also formalized as a closed (model-level) theorem. --- # What is new in v1.7r4 (journal-facing) - Add a one-page **BVinc-to-kernel input map** (standard BV form ↔ degree/overlap/cond).- Add a one-page **Claims–Inputs ledger** (Closed / Conditional / Target).- Provide theorem-shaped sufficient-condition templates for Gate–T (dispersion/decoupling ⇒ a concrete \(θ<1\)).- Re-expose the **COV-certificate vacancy route** as a setup-level corollary and an explicit appendix proposition.- Add **BDH-type mean-square dispersion ⇒ COV certificate** implication (referee-friendly: mean-square input → auditable fields).- Add a **surrogate BDH ⇒ band-occupancy dispersion** proposition (closed in the progression model) + a clear remark delimiting the genuine order-hit upgrade.- Provide a **COV certificate PASS template** (JSON + audit script) to make the vacancy route mechanically auditable.- Keep rigor fixes: the “energy-gap from mask mass” proof keeps \(v:=m_T(E)\) throughout; duplicate statements removed; stationary-measure equations numbered. --- # Related Zenodo records / cited inputs - **Mersenne Infinity v1.6** (previous public release). DOI: 10.5281/zenodo.18754063 - **Mersenne Infinity v1.4.1** (earlier public release). DOI: 10.5281/zenodo.18751855 - **Large sieve bounds for prime-index powers modulo primes (v1.0)**. DOI: 10.5281/zenodo.18744040 - **Multi-scale Band Occupancy and Certificate-Controlled Vacancy Bounds for Mersenne Divisors (v1.0)**. DOI: 10.5281/zenodo.18752642