A Spectral–Entropy Framework for the Mersenne Infinitude Problem (Version v1.1)

## Overview This preprint develops a **windowed spectral–entropy / operator–variational framework** for the **Mersenne infinitude problem**. The design goal is modularity: to make it explicit **which steps are fully rigorous** (given standard mixing diagnostics) and **where genuinely arithmetic input is still required**. We work on exponent windows\[I_T := [T,\, T+T^\alpha],\qquad \alpha\in(0,1),\]with prime-exponent state space\[\Omega_T := \mathbb{P}\cap I_T,\]and study the set \(E\) of **Mersenne-empty windows**:\[I_T\ \text{is empty if}\ \{p\in\Omega_T:\ 2^p-1\ \text{is prime}\}=\varnothing.\]A basic fact is that if **only finitely many** Mersenne primes exist, then all sufficiently large windows are empty, hence the **logarithmic density** would be \(\delta(E)=1\). Therefore,\[\delta(E)=0 \ \Longrightarrow\ \text{infinitely many Mersenne primes}.\] --- ## Core architecture (pipeline) The paper builds and connects the following modules: 1) **Factor-admissible incidence kernels**- For a prime exponent \(p\), any prime divisor \(q\mid(2^p-1)\) satisfies \(q\equiv 1\ (\mathrm{mod}\ 2p)\), so \(q\ge 2p+1\). This motivates “small-prime” avoidance and instead uses **factor-admissible prime bands**.- For each \(T\), define prime bands \(Q_{T,j}\subset [T^{2+\eta_j},\,2T^{2+\eta_j}]\) and an incidence indicator \[ a_T(p,q):=\mathbf{1}_{\{q\in Q_{T,j}\}}\mathbf{1}_{\{q\equiv 1\ (\mathrm{mod}\ 2p)\}}. \]- A symmetric, band-limited kernel \(K^{(1)}_{M,T}\) is constructed from overlaps of these incidences, producing a non-degenerate similarity structure on \(\Omega_T\). 2) **Markovization + mixing diagnostics**- The kernel is row-normalized to a reversible Markov operator \(P_{M,T}\), and then “lazified” as \[ \widetilde P_{M,T}:=\tfrac12(\mathrm{Id}+P_{M,T}). \]- Standard implications are recorded: conductance \(\Phi_T\) ⇒ spectral gap ⇒ (log-)Sobolev/entropy dissipation diagnostics. 3) **A sieve-to-mixing bridge (conditional)**- A clean “bridge statement” is formulated: **BV / large-sieve type** control on incidence degrees and overlaps yields a **polylogarithmic conductance lower bound**, hence a gap/LSI diagnostic for \(\widetilde P_{M,T}\).- This isolates the arithmetic difficulty: proving the required degree/overlap controls at the relevant scales. 4) **Multi-scale diagnostics + explicit Mask A**- A multi-scale survival score is defined from band hit-counts: \[ H_{T,j}(p):=\sum_{q\in Q_{T,j}} \mathbf{1}_{\{q\equiv 1\ (\mathrm{mod}\ 2p)\}} \mathbf{1}_{\{2^p\equiv 1\ (\mathrm{mod}\ q)\}}, \qquad S_T^{(\mathrm{ms})}(p):=\exp\!\Big(-\sum_j w_j H_{T,j}(p)\Big)\in(0,1]. \] The window-level diagnostic is \[ \mathrm{Diag}^{(\mathrm{ms})}(T):=\int_{\Omega_T} S_T^{(\mathrm{ms})}(p)\,dm_T(p). \]- The multi-scale Mask A is the deficit set \[ E_T^{(A,\mathrm{ms})}:=\{p\in\Omega_T:\ S_T^{(\mathrm{ms})}(p)\le \tau\}. \] 5) **Entropy–rigidity functional + density–0 closure**- A rigidity functional on densities \(\rho\) on \(\Omega_T\) is introduced: \[ \widetilde G^{(\mathrm{ms})}_{M,T}[\rho] := H(\rho\mid m_T) +\lambda_1\|\widetilde P_{M,T}\rho-\rho\|_{L^2(m_T)}^2 +\lambda_2\int_{\Omega_T}\rho^2\,\mathbf{1}_{E_T^{(A,\mathrm{ms})}}\,dm_T. \]- Once (i) a mixing diagnostic (e.g. LSI) and (ii) a uniform “energy-gap” lower bound are available, a self-contained argument yields \(\delta(E)=0\). --- ## What is new in v1.1 (relative to earlier drafts) - **Multi-scale factor bands** and a softened survival diagnostic \(S_T^{(\mathrm{ms})}\), designed to prevent “escape” by shifting factor behavior outside a single cutoff.- A **self-contained second-moment / anti-concentration module** for the multi-scale score, including: - a bounded-hit-count route (Hoeffding + Jensen), - and a more realistic “degree control with tail truncation” route.- A key structural upgrade: **the energy-gap premise is no longer kept as a black box once transduction is granted**. Concretely: - (Empty window) ⇒ (nontrivial mask mass) via a Markov inequality argument, - (mask mass) ⇒ (explicit variational energy gap) via an information-theoretic lower bound with a closed-form constant, e.g. \[ \varepsilon_*(\beta,\lambda_2)=\frac{2\lambda_2\beta^2}{2\beta+\lambda_2}>0. \]- Expanded “reader map” and an included **graphical abstract** summarizing the pipeline and the explicit transduction barrier. --- ## Scope, status, and the explicit bottleneck (important) This paper **does not claim an unconditional proof** of infinitely many Mersenne primes from classical sieve bounds alone. All downstream steps (Mask A ⇒ energy gap ⇒ density–0 closure) are rigorous **once** the following is supplied: ### (A) Mixing diagnostic (via bridge)A polylog-scale spectral/LSI bound for \(\widetilde P_{M,T}\), intended to be derived from incidence degree/overlap controls (the conditional sieve-to-mixing bridge). ### (B) Transduction (the genuinely Mersenne-specific obstruction)A quantitative implication that rules out Mersenne-empty windows whenever \(\mathrm{Diag}^{(\mathrm{ms})}(T)\) exceeds a threshold. The paper records this as explicit “program targets” (Gate–T, TR-avg, TR-moment) and provides a heuristic reduction template showing why such targets are natural under standard LPW/Wagstaff-type heuristics plus weak decorrelation. --- ## Empirical calibration (supporting, non-proof) Appendix B provides an empirical calibration protocol for a conservative transduction threshold \(\theta_{\mathrm{emp}}\) against the published catalogue of known Mersenne-prime exponents, together with:- a reproducible Monte–Carlo estimator for \(\mathrm{Diag}^{(\mathrm{ms})}(T)\),- ROC/PR and calibration (reliability) diagnostics on a bounded proxy range (e.g. \(p\le 10^7\)),- robustness grids, negative controls (permuted labels / wrong congruence / shifted windows),- logged RNG seeds and acceptance rates for reproducibility. (As recorded in the manuscript: the catalogue comparison is performed against the known GIMPS exponent list as of 2026-01-04.) --- ## Keywords Mersenne primes; infinitude problem; logarithmic density; Markov chains; conductance; spectral gap; logarithmic Sobolev inequality; entropy dissipation; variational methods; sieve methods; primes in arithmetic progressions; empirical calibration; ROC/AUC; reliability diagrams. --- ## Citation Lee Byoungwoo, *A Spectral–Entropy Framework for the Mersenne Infinitude Problem (v1.1): Mellin–Trace Bookkeeping, Sieve-to-Mixing Bridge, and Variational Density–0 Closure*, 2026.