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 \ (00\) there exist \ (C>0\) and \ (X₀ () \) such that for all \ (X X₀ () \), \\#\\, 2n[X, 2X: W (2n) =0\, \ X + O\! (X/ (X) C) + O (X^). \]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^\), \ (PT), \ ₋ₒ₈ (PT) \ \ 1² T, from prime-band conductance on auxiliary chains and comparison estimates. ## (M0) Mellin–trace bridge for \ (s>2\) (proved) For \ (G (s) =₍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 \ (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 \ (K₀\) (i. e. , \ (W (2n) 1\) for all \ (2n K₀\) ) 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.