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 \(M_p = 2^p-1\), 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 \(Ω_T\),- Markovize to a lazy reversible chain \( PM,T\),- Quantify mixing via **conductance \(Φ_T\)** and the **spectral gap \(λ_2\)** (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 \(m_T\) to be the reversible stationary measure of the Markovized kernel \( PM,T\).Writing the one-step incidence kernel as \(K⁽¹⁾M,T(p,r)\) on \(Ω_T\), define the degree\[dM,T(p) := ∑r∈Ω_T K⁽¹⁾M,T(p,r),\]and the stationary measure\[m_T(p) := {dM,T(p)}{∑u∈Ω_T dM,T(u)}.\]Then \( PM,T\) is reversible with respect to \(m_T\). ## (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 \(E⊂Ω_T\) has stationary mass \(m_T(E)≥ β\), and \(λ_2>0\) is the spectral gap, then one obtains an explicit lower bound\[ε_(β,λ_2) \;=\; 2λ_2\,β^2/λ_2+2β\]for the combined entropy/energy functional on \(E\).The proof keeps \(v:=m_T(E)\) throughout and closes via the monotonicity of\[g(v)=2λ_2 v^2/λ_2+2v, v≥ β ⇒ g(v)≥ g(β),\]avoiding any shortcut substitution of \(β\) 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:** \(m_T\) is explicitly fixed as the reversible stationary measure of \( PM,T\) (degree-weighted normalization), removing any “uniform/reference” ambiguity.- **Numbered definitions:** the definitions of \(dM,T(p)\) and \(m_T(p)\) 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 “\(m_T(E)≥β\)” step is now fully rigorous by keeping \(v=m_T(E)\) 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/graphical_abstract_mersenne_v0_8.png`, `figures/roc_p_le_1e7_.png`, `figures/calibration_p_le_1e7.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.)