A Spectral–Entropy Framework for the Mersenne Infinitude Problem: Multi-scale Density–0 Rigidity and Transduction Diagnostics

## Overview This preprint develops a **windowed operator–variational framework** for the Mersenne infinitude problem(Mersenne primes \(M_p = 2^p-1\)). The paper is deliberately structured to **isolate exactly where genuinelyMersenne-specific arithmetic input is required**, while keeping the remaining pipeline rigorous and modular. The guiding idea is to treat “Mersenne-empty exponent windows” as an **exceptional set** controlled by anentropy–rigidity functional, once (i) a **mixing diagnostic** is available and (ii) a uniform **energy-gap**lower bound can be enforced through a multi-scale exceptional mask. --- ## What is new in v0.9 - **Data recency tightened via an explicit GIMPS freeze-date.** The catalogue check and the “true-positive” stress test are synchronized with the public GIMPS list (access date 2026-01-04; 52 known Mersenne-prime exponents; most recent exponent \(p=136279841\), discovered 2024-10-12 and announced 2024-10-21).- **Graphical abstract included.** A one-page diagram visualizes the multi-scale windowing, entropy dissipation, and the density–0 closure pipeline, emphasizing the location of the transduction barrier.- **Barrier/necessity statements polished.** The paper explicitly records why band-limited factor diagnostics cannot, by themselves, certify Mersenne primality, and why a transduction-type input is internally required for unconditional closure within this framework.- **Exploratory empirical appendix strengthened.** Pilot ROC/AUC and calibration (reliability) plots are provided on a bounded range (e.g. \(p\le 10^7\)), together with robustness/negative-control language, to support the plausibility of the empirical transduction threshold \(\theta_{\mathrm{emp}}\) as a *screening* criterion. --- ## Core program (high-level) ### 1) Windowed exponent modelWe work on prime exponents \(p\) in variable windows\[I_T = [T,\;T+T^\alpha],\qquad \alpha\in(0,1),\]with a finite probability space \((\Omega_T,m_T)\) induced by the window. ### 2) Factor-admissible prime bands and incidence kernelWe introduce factor-admissible prime bands and construct a symmetric, band-limited incidence kernel\(K^{(1)}_{M,T}\) on \(\Omega_T\), designed to avoid degeneracy and to support conductance estimates afterMarkovization. ### 3) Sieve-to-mixing bridge (conditional)A conditional bridge is formulated: BV / large-sieve type control on incidence degrees and overlaps yieldsconductance bounds, and hence **polylogarithmic** spectral-gap / logarithmic-Sobolev (LSI) diagnostics for thelazy Markov chain \(\widetilde P_{M,T}\). ### 4) Entropy–rigidity functional and density–0 closure (rigorous module)For an explicit exceptional mask \(E_T\subset \Omega_T\), the paper studies\[\widetilde{\mathcal{G}}_{M,T}[\rho]=\mathcal{H}(\rho|m_T)+\lambda_1\|\widetilde P_{M,T}\rho-\rho\|_2^2+\lambda_2\int_{\Omega_T}\rho^2\mathbf{1}_{E_T}\,dm_T.\]Assuming (i) a mixing diagnostic and (ii) an energy-gap premise tied to a **multi-scale Mask A**, the paperproves a **logarithmic density–0 closure** result: the set of Mersenne-empty windows has logarithmic density \(0\). ### 5) Transduction (open target)The only genuinely Mersenne-specific obstacle is isolated as a **transduction hypothesis**:a quantitative implication asserting that sufficiently strong diagnostics force the existence of a Mersenneprime in the window. v0.9 emphasizes this step as the remaining “last mile” for unconditional infinitudewithin the present program. --- ## Numerical appendix (exploratory, not a proof substitute) Numerics are used strictly to validate diagnostics (gap estimators, band stability, robustness of rescaledquantities such as \(\widehat{\gamma}_T\log^2 T\), and pilot threshold calibration). They are **not** used toreplace the transduction or energy-gap steps. --- ## Scope and non-claims - This paper does **not** claim an unconditional proof of infinitely many Mersenne primes from classical sieve bounds alone.- The barrier results are **methodological**: they do not claim formal independence from ZFC/PA.- Mellin–trace identities, when mentioned, are used only as **bookkeeping** for windowed diagonal observables, not as an analytic-continuation route to infinitude. --- ## Suggested citation Lee, Byoungwoo. *A Windowed Operator–Variational Framework for the Mersenne Infinitude Problem:Multi-scale Density–0 Rigidity, Transduction Diagnostics, and Methodological Barriers* (v0.9). Zenodo, 2026. --- ## Keywords Mersenne primes; analytic number theory; sieve methods; large sieve; Bombieri–Vinogradov;Markov chains; conductance; spectral gap; logarithmic Sobolev inequality; entropy dissipation;rigidity functional; multi-scale windows; exceptional sets; logarithmic density; ROC/AUC; calibration;GIMPS; reproducibility.