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

## Overview This preprint proposes a **spectral–entropy / density–0 rigidity pipeline** for the *Mersenne infinitude problem*, i.e. the existence of infinitely many Mersenne primes \(M_p=2^p-1\). The goal of the paper is **not** to claim an unconditional proof at v0.7, but to present a **self-contained closure mechanism** in which the remaining arithmetic input is isolated into an explicit “transduction” premise and an empirically testable diagnostic threshold. The central message is: - If “Mersenne-empty windows” satisfy a quantitative diagnostic deficit (transduction),- then an **entropy–rigidity functional** forces the set of empty windows to have **logarithmic density \(0\)**,- and this density–0 closure implies **infinitude**, conditional on the stated transduction hypothesis. --- ## What is new in v0.7 - **Multi-scale Mask A** is adopted as the main technical device, tying exceptional penalties to *multiple disjoint factor bands* (rather than a single cutoff), making the energy-gap target substantially more plausible.- The **energy-gap premise is closed inside the paper** via two program-target propositions: 1) *MS-Deficit*: an empty window forces nontrivial mask mass, and 2) *Energy-gap from mask mass*: a self-contained entropy + \(L^2\) penalty lower bound.- Appendix C is strengthened with a **reproducible pilot ROC/AUC and calibration illustration** for \(p\le 10^7\), using a conservative “trial-factor proxy” score \(\widehat{S}\) to motivate an empirical threshold \(\theta_{\mathrm{emp}}\).- A **GIMPS catalogue contrast (52 known exponents)** is included to show that known Mersenne-prime exponents are not assigned low diagnostic scores under the stated proxy/budget (illustrative validation, not a proof). --- ## Core objects and claims (high-level) ### Variable windows on prime exponentsWe work on prime exponents \(p\) in variable windows\[I_T=[T,\,T+T^\alpha],\qquad \alpha\in(0,1),\]and consider the induced probability space \((\mathcal{O}_T,m_T)\) of exponents in the window. ### Lazy Markov operator and LSI diagnosticA reversible/lazy Markov operator \(\widetilde{P}_{M,T}\) is constructed on \(\mathcal{O}_T\). Assuming a window-uniform logarithmic Sobolev inequality (LSI) for \(\widetilde{P}_{M,T}\), one obtains quantitative entropy dissipation for the chain. ### Entropy–rigidity functionalFor a measurable mask \(E_T\subset\mathcal{O}_T\),\[\widetilde{\mathcal{G}}_{M,T}[\rho]:= \mathcal{H}(\rho\,|\,m_T)+ \lambda_1 \|\widetilde P_{M,T}\rho-\rho\|_2^2+ \lambda_2 \int_{\mathcal{O}_T} \rho^2\,\mathbf{1}_{E_T}\,dm_T.\]The multi-scale Mask A, \(E_T^{(A,\mathrm{ms})}\), is defined through a multi-band diagnostic score \(S_T^{(\mathrm{ms})}(p)\). ### Density–0 closureUnder (i) the LSI diagnostic and (ii) the closed energy-gap premise for multi-scale Mask A, the set \(\mathcal{E}\) of **Mersenne-empty windows** satisfies\[\logdens(\mathcal{E})=0.\]This yields infinitude **conditional on the stated transduction hypothesis** (and its threshold). --- ## Empirical / reproducibility capsule (Appendix C) Appendix C provides a small-range, fully reproducible illustration: - ROC curve and an illustrative calibration diagram for \(p\le 10^7\),- using a conservative “trial-factor proxy” diagnostic score \(\widehat{S}(p)\),- plus a GIMPS catalogue contrast (52 known exponents). These experiments are included **only to support plausibility of the transduction threshold** and to document a reproducible validation loop; they are not presented as a proof. --- ## Scope, limitations, and reading guide - v0.7 should be read as a **program paper**: it isolates the analytic–operator pipeline and reduces the remaining arithmetic content to an explicit transduction hypothesis with an empirically testable threshold.- The paper intentionally avoids over-claiming: the main theorem is a **conditional density–0 closure** plus a clear list of remaining targets.- A natural next step is to strengthen the transduction premise from “empirical threshold justification” to a theorem-level arithmetic statement (or to replace it with a known sieve/dispersion input). --- ## Citation If you use or build on this work, please cite as: Lee, Byoungwoo. *A Spectral–Entropy Framework for the Mersenne Infinitude Problem: Multi-scale Density–0 Rigidity and Transduction Diagnostics*, Zenodo (v0.7), 2026. --- ## Keywords Mersenne primes; sieve methods; Markov chains; logarithmic Sobolev inequality; entropy dissipation; rigidity functional; density–0 closure; variable windows; ROC/AUC; reproducible diagnostics.