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 \ (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^\) (equivalently, on each dyadic scale \ (X, 2X\) up to at most \ (O (X/ (X) C) \) exceptional windows), \ (PT), \ ₋ₒ₈ (PT) \ \ 1² T, one obtains exponential entropy decay on those windows. ## (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\) (with 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 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).