A Spectral–Entropy Framework for the Goldbach Conjecture (v3.7r4)

# Overview This record releases a two-paper set for a windowed spectral–entropy program toward Goldbach: - **Main paper:** *A Spectral–Entropy Framework for the Goldbach Conjecture* (v3.7r4)- **Companion note:** *Density–0 Closure for the Goldbach Flow* (v3.7r4) The program separates **proved, window-averaged (most-window) statements** from **explicitly conditional layers** (heat-trace asymptotics, strong mixing, and an energy-gap hypothesis for the density–0 module). A referee-facing claim ledger is placed on the first page of each document. # What is new in v3.7r4 - **Front-matter optimization:** the first 1–2 pages now expose (i) the classical dyadic exceptional-set corollary and (ii) a compact referee ledger without scope ambiguity.- **Scope hygiene:** all “proved vs conditional” layers are aligned across Abstract / Introduction / Tables.- **Companion alignment:** references between the main paper and the density–0 note are version-consistent. # Closed results (proved layer) ## (C0) Classical dyadic exceptional-set bound (proved)Fix \(0<θ<α<1\) and \(W(T)=T^α\).For every \(ε>0\) there exist \(C>0\) and \(X_0(ε)\) such that for all \(X≥ X_0(ε)\),\[#\{\,2n∈[X,2X]: W(2n)=0\,\}≤ ε X + O\!(X/(log X)^C) + O(X^α).\]Hence Goldbach exceptions have natural density \(0\) on dyadic scales. ## (E0) Most-window spectral / log-Sobolev closure (proved)On a density-1 set of short windows \([T,T+T^α]\),\[γ(P_T),\ αLSI(P_T)\ \ 1/log^2 T,\]derived from prime-band conductance on auxiliary chains and comparison estimates. ## (M0) Mellin–trace bridge for \( s>2\) (proved)For \(ζ_G(s)=∑n≥1 W(2n)/(2n)^s\), a rigorous Mellin–trace identity and trace/HS criteria are proved on \( s>2\) (Tonelli/Fubini justified). # Conditional layers (explicitly isolated) - **(HT) Heat-trace hypothesis:** analytic continuation of \(ζ_G(s)\) to \( s>1\) and a functional equation are derived only under an explicit short-time heat-trace asymptotic hypothesis.- **(H3*) Strong mixing hypothesis:** an eventual all-even threshold \(K_0\) (i.e., \(W(2n)≥1\) for all \(2n≥ K_0\)) is derived only under the optional strong-mixing assumption.- **(EG) Energy-gap hypothesis (companion):** the density–0 elimination of weakly mixed windows is proved under an explicit per-window energy-gap assumption (EG). # Files - Main paper: PDF + TeX (v3.7r4)- Companion density–0 note: PDF + TeX (v3.7r4) # Reproducibility To build PDFs from TeX:- Run `pdflatex` twice on each `.tex` file. # Scope (non-claim) This release does **not** claim an unconditional global all-even Goldbach theorem.The proved layer establishes a most-window closure and a classical dyadic exceptional-set bound; any promotion to an eventual all-even threshold is isolated as conditional.