A Spectral–Entropy Framework for the Mersenne Infinitude Problem (Version v1.0)

# Summary (v1.0) ## OverviewThis preprint develops a **windowed operator–variational program** for the infinitude of Mersenne primes\(M_p = 2^p - 1\).For exponent windows \(I_T = [T,\,T+T^\alpha]\) we build **factor-admissible incidence kernels**on the finite prime set \(\Omega_T\), and use them to define a lazy reversible Markov chain \(P_{M,T}\)and an associated energy functional \(G_{M,T}\).The pipeline is modular: \[\text{factor-admissible incidence kernels}\Rightarrow\text{(conditional) sieve-to-mixing bridge}\Rightarrow\text{spectral / entropy diagnostics}\Rightarrow\text{density–0 elimination of empty windows}.\] The architecture isolates the genuinely Mersenne-specific upgrades into two explicit targets:**(i) transduction**, and **(ii) an energy-gap mechanism (closed once transduction is fixed)**. ## Closed results- **Kernel and chain layer (closed):** a reversible reference measure \(m_T\), a lazy chain \(P_{M,T}\), and Cheeger/spectral-gap diagnostics at polylogarithmic scale. - **Energy-gap mechanism (closed conditional on transduction):** the multi-scale “Mask A” construction yields a uniform lower bound \(\inf_\rho G_{M,T}[\rho] \ge \varepsilon_\star > 0\) on Mersenne-empty windows once transduction is granted. - **Density–0 closure theorem (conditional):** assuming an LSI-type diagnostic (entropy dissipation for \(P_{M,T}\)) and transduction, the set of Mersenne-empty windows has logarithmic density \(0\). Consequently, if only finitely many Mersenne primes existed, then all large windows would be empty, contradicting the density–0 closure; hence infinitude follows **conditionally**. - **Mellin–trace bookkeeping clarified:** windowed Dirichlet/Mellin traces are used only to package diagonal observables into finite-dimensional trace identities; they do **not** encode Mersenne primality by themselves. ## What is new in v1.0- The **program boundary is made explicit**: the remaining open technical targets are concentrated in (i) establishing the LSI diagnostic via the sieve-to-mixing bridge, and (ii) proving (or weakening/replacing) the transduction hypothesis. - A strengthened **empirical calibration module** for the transduction threshold: Appendix “Empirical calibration of the transduction threshold” provides a conservative, catalogue-conditioned procedure to calibrate \(\theta_{\mathrm{emp}}\) and to report: ROC/AUC, PR/Average Precision (under severe imbalance), and calibration / reliability diagrams. The procedure includes negative controls (permuted labels, wrong congruences, shifted windows). - An explicit **contrast with the public GIMPS catalogue** (data recency stated in the manuscript): as of access date 2026-01-04, the list contains 52 known Mersenne-prime exponents and the most recent exponent is \(p = 136{,}279{,}841\) (announced 2024-10-21). A reproducible proxy score is reported for the catalogue exponents under the stated budget. ## Reproducibility and artifactsAll Monte–Carlo runs are designed to be reproducible:- fixed pseudorandom seeds and run identifiers,- logged acceptance rates for candidate prime factors,- logged primality tests and modular exponentiation implementation choices,- per-window sampling counts and “zero retained primes” events. **Budget notation (used throughout):**for window-level estimation, define the per-window sampling budget\[B_T := N_p \, J \, N_q,\]and for the small-range proxy experiment use\[q = 2pk + 1,\quad 1 \le k \le k_{\max},\quad \widehat{S}(p) := \exp(-\#\mathrm{hits}),\]with the default \(k_{\max}=12000\) in the included pilot figures. Included figure artifacts:- ROC curve for the \(p \le 10^7\) proxy experiment,- an illustrative calibration / reliability diagram. ## Scope & non-claims (important)- This work does **not** claim an unconditional proof of infinitude in v1.0.- The empirical section is **not** a proof of transduction; it is a stress-test and calibration aid.- The “trace” layer is **bookkeeping**, not a standalone primality-certificate method.- The point of the framework is to make the dependence on genuinely arithmetic input explicit, and to separate “closed analytic modules” from the remaining arithmetic gates. ## Program closure and targetsTo upgrade the result from conditional to unconditional, the core remaining tasks are: 1) **LSI diagnostic (mixing) from arithmetic overlap control** via the sieve-to-mixing bridge (conductance, spectral profile, or comparable mixing certificates). 2) **Transduction** (prove, weaken, or replace the transduction hypothesis by a verifiable arithmetic certificate). Once transduction is fixed, the energy-gap mechanism is already organized to close. ## Suggested keywordsMersenne primes; prime exponents; sieve methods; Markov chains; spectral gap; log-Sobolev inequality;entropy dissipation; density–0 arguments; Monte Carlo calibration; ROC; calibration. ## Related preprints (by the author)- Byoungwoo Lee, “Collatz Final Gate v3.9b,” Zenodo (2025), DOI: 10.5281/zenodo.17581829.- B. Lee, “Goldbach Flow Program v3.6,” Zenodo (2025), DOI: 10.5281/zenodo.17836952. ## How to citePlease cite this Zenodo record as:Byoungwoo Lee (2026). *A Spectral–Entropy Framework for the Mersenne Infinitude Problem* (Version v1.0). Zenodo.(Add the Zenodo DOI assigned to this upload.)