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 \ (Mₚ = 2ᵖ-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 \ (T=P, T+T^\). 2. Markovize to a **lazy reversible chain** \ (P₌, ₓ\) with stationary measure \ (mT\). 3. Use **conductance \ (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) _<1\) on intended prime-free windows, which supports Gate–T by contrapositive once \ (_ ₄₌₏ₓₘ<\). --- # 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 \ (<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: =mT (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. )