Mersenne Infinity v1.4.1 — A Spectral–Entropy Framework for the Mersenne Infinitude Problem

# Overview This record releases **Mersenne Infinity v1. 4. 1**, a spectral–entropy / Markov-mixing framework for organizing *density–0 rarity* statements around Mersenne numbers \ (Mₚ = 2ᵖ-1\), with an explicit and isolated **transduction barrier** (the only place where genuinely arithmetic input beyond the internal mixing engine is required). Author: **Lee Byoungwoo** Core idea: - Build a family of *factor-admissible incidence kernels* on a windowed exponent state space \ (T\), - Markovize to a lazy reversible chain \ (P₌, ₓ\), - Quantify mixing via **conductance \ (T\) ** and the **spectral gap \ (₂\) ** (LSI is treated as optional), - Push “most-window” mixing into **density–0 closure** statements, - Clearly separate the remaining arithmetic obstacle as **Transduction (Gate–T) **. This is a *methods/program* paper: it does **not** claim an unconditional proof of Mersenne infinitude. --- # Closed results (internal, self-contained) ## (1) Stationary-measure normalization (reversibility convention) We fix the measure \ (mT\) to be the reversible stationary measure of the Markovized kernel \ (P₌, ₓ\). Writing the one-step incidence kernel as \ (K^ (1) ₌, ₓ (p, r) \) on \ (T\), define the degree₌, ₓ (p): = ₑₓ K^ (1) ₌, ₓ (p, r), the stationary measureT (p): = d₌, ₓ (₏) ₔₓ d₌, ₓ (u). \ (P₌, ₓ\) is reversible with respect to \ (mT\). ## (2) Conductance \ (\) gap (primary diagnostic) For a finite reversible Markov chain, conductance controls the spectral gap via standard Cheeger-type inequalities. Accordingly, the paper treats **spectral gap** as the primary mixing diagnostic imported from incidence bounds. (Logarithmic-Sobolev / entropy-dissipation diagnostics are presented as **optional** upgrades under standard finite-chain comparison inputs. ) ## (3) Energy-gap from mask mass (no shortcut substitution) A key lemma (“energy-gap from mask mass”) is proved in a fully self-contained form and is used as a deterministic closure step inside the mixing engine. If \ (ET\) has stationary mass \ (mT (E) \), and \ (₂>0\) is the spectral gap, then one obtains an explicit lower bound\_ (, ₂) \;=\; 2₂\, ²₂+2 the combined entropy/energy functional on \ (E\). The proof keeps \ (v: =mT (E) \) throughout and closes via the monotonicity of (v) =2₂ v²₂+2v, v g (v) g (), any shortcut substitution of \ (\) into the KL reference argument. --- # What is new in v1. 4. 1 This is a rigor + hygiene upgrade (no new external arithmetic inputs): - **Measure consistency: ** \ (mT\) is explicitly fixed as the reversible stationary measure of \ (P₌, ₓ\) (degree-weighted normalization), removing any “uniform/reference” ambiguity. - **Numbered definitions: ** the definitions of \ (d₌, ₓ (p) \) and \ (mT (p) \) are given with equation numbers/labels for stable cross-referencing. - **Cheeger vs LSI phrasing: ** “conductance \ (\) spectral gap” is treated as primary; LSI is explicitly labeled as *optional* and requiring additional comparison structure. - **Energy-gap lemma proof fix: ** the “\ (mT (E) \) ” step is now fully rigorous by keeping \ (v=mT (E) \) and using monotonicity \ (g (v) \). - **Duplicate statement removal: ** repeated proposition/corollary blocks are removed; only one canonical statement remains, with a pointer remark where needed. - **PDF robustness: ** math in headings is made bookmark-safe (via PDF-string fallbacks). --- # Scope & non-claims (critical) - This work does **not** claim an unconditional proof that there are infinitely many Mersenne primes. - The internal engine yields *most-window / density–0 style* statements **conditional on a clearly stated transduction target** (Gate–T), which represents the remaining arithmetic obstacle beyond the mixing/entropy machinery. - Pilot numerics (ROC / calibration) are presented as *diagnostics only* and are **not** used as proof inputs. --- # Reproducibility / build notes - If the record includes the LaTeX source, place the image files under: `figures/graphicalₐbstractₘersenneᵥ0₈. png`, `figures/rocₚₗe₁e7_. png`, `figures/calibrationₚₗe₁e7. png`. - Compile with a standard LaTeX toolchain (e. g. , `pdflatex` twice, or `latexmk -pdf`). --- # Suggested citation Lee Byoungwoo, *Mersenne Infinity v1. 4. 1 — A Spectral–Entropy Framework for the Mersenne Infinitude Problem*, Zenodo record (v1. 4. 1), 2026. (Insert the Zenodo DOI assigned to this upload here after publication. )