A Spectral–Entropy Framework for the Goldbach Conjecture (v3.7r6)

# 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* (v3.7r6)- **Companion note:** *Density–0 Closure for the Goldbach Flow* (v3.7r6) A referee-facing claim ledger is placed on the first page of each document.The release enforces strict **claim discipline**: the proved layer is separated from explicitly conditional layers(heat-trace asymptotics (HT), strong mixing (H3*), and an energy-gap assumption (EG) for the density–0 module). # What is new in v3.7r6 - **Claim-discipline hardening:** all statements are aligned with the proved vs conditional ledger. In particular, any all-window or all-even language has been removed unless explicitly marked conditional.- **Front-matter optimization:** the classical dyadic exceptional-set bound and the claim ledger are visible on the first 1–2 pages.- **Roadmap isolation:** Section 9 is explicitly programmatic; it is not part of the proved layer. # Closed results (proved layer) ## (C0) Classical dyadic exceptional-set bound (proved) Fix \(0<θ<α<1\) and a window scale \(W(T)=T^α\).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 \([T,T+T^α]\) (equivalently, on each dyadic scale \([X,2X]\) up to at most \(O(X/(log X)^C)\) exceptional windows),\[γ(P_T),\ αLSI(P_T)\ \ 1/log^2 T,\]and one obtains exponential entropy decay on those windows. ## (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 \( s>2\)(with 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 only under an explicit per-window energy-gap assumption (EG). # Companion note (Density–0 module) The companion note isolates the density–0 elimination mechanism for weakly mixed windows.It is intentionally short and explicitly conditional on (EG), and it does not claim a global all-even threshold. # 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 (HT/H3*/EG and roadmap hypotheses).