Mersenne Infinity v1.6 — A Spectral–Entropy Framework for the Mersenne Infinitude Problem

# Overview This record releases **Mersenne Infinity v1.6**, a windowed spectral–entropy / Markov-mixing frameworkfor organizing *density–0 rarity* statements around Mersenne numbers \(M_p = 2^p-1\).The paper is explicitly modular: it isolates the unique Mersenne-specific arithmetic bottleneckas a **transduction barrier (Gate–T)**, while keeping the remaining steps operator-theoretic and variational. **Author:** Lee Byoungwoo (leeclinic@protonmail.com)**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/calibration material is **supplementary** and **not used in proofs**. --- # What the paper does Pipeline:1. Build **factor-admissible incidence kernels** on exponent windows \(Ω_T=P∩[T,T+T^α]\).2. Markovize to a **lazy reversible chain** \( PM,T\) with stationary measure \(m_T\).3. Use **conductance \(Φ_T\)** and the **spectral gap** as primary mixing diagnostics (LSI is optional).4. Prove a closed “Mask A \(⇒\) energy gap” mechanism once transduction is certified.5. Derive a **conditional density–0 closure module** for Mersenne-empty windows. --- # Main editor-facing conditional statements - **Main conditional corollary (BV-type incidence control):** BV-type incidence/dispersion control + a mixing diagnostic + the 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, which supports Gate–T by contrapositive once \(θ_≤ θempty<θ\). --- # What is new in v1.6 (journal-facing) - Add a one-page **BVinc-to-kernel input map** (standard BV form \(↔\) degree/overlap/cond) to make the bridge referee-friendly.- Add a one-page **Claims–Inputs ledger** (Closed / Conditional / Target) near the front.- Provide a theorem-shaped **sufficient-condition template for Gate–T** (dispersion/decoupling \(⇒\) a concrete threshold \(θ<1\)).- Re-expose the **COV-certificate vacancy route** (mean-hit floors + COV \(⇒\) strict diagonal bound) as a short setup-level corollary.- Keep rigor fixes: the energy-gap proof keeps \(v:=m_T(E)\) throughout; duplicate statements removed; stationary-measure equations are numbered.- Tighten numerics tone: ROC/calibration is explicitly supplementary (not used in proofs). --- # Related Zenodo records / inputs cited - 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- (Previous public release) Mersenne Infinity v1.4.1. DOI: 10.5281/zenodo.18751855 --- # Suggested citation Lee Byoungwoo,*Mersenne Infinity v1.6 — A Spectral–Entropy Framework for the Mersenne Infinitude Problem*,Zenodo (2026).(After upload, insert the Zenodo DOI assigned to this record.)