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 \ (Mₚ = 2ᵖ-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: ** \ (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: ** conductance \ (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) _<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. 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 \ (<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: =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