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

# Overview This record releases **Mersenne Infinity v1. 7**, a windowed spectral–entropy / Markov-mixing frameworkfor organizing *logarithmic density–0 rarity* statements around Mersenne numbers \ (Mₚ = 2ᵖ-1\). The design goal is to keep the proof interface modular and to isolate the uniquely Mersenne-specificarithmetic obstacle as an explicit **transduction barrier (Gate–T) **. **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 material is explicitly **supplementary** and **not used in proofs**. --- # What the paper does (pipeline) 1. **Exponent windows: ** \ (T = P, T+T^\), 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 \ (P₌, ₓ\) and fix \ (mT\) as its reversible stationary measure. 4. **Mixing diagnostics: ** use conductance \ (T\) and the spectral gap as primary diagnostics (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) _<1\) on intended prime-free windows, supporting Gate–T by contrapositive once \ (_ ₄₌₏ₓₘ<\). - **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. --- # What is new in v1. 7 (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 \ (<1\) ). - Re-expose the **COV-certificate vacancy route** as a short setup-level corollary. - Add a **BDH-type second-moment dispersion ⇒ COV certificate** implication (referee-friendly mean-square input → auditable certificate fields). - Tighten numerics tone: all ROC/calibration material is supplementary and not used in proofs. - Keep rigor fixes: the “energy-gap from mask mass” proof keeps \ (v: =mT (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