Goldbach via Spectral–Entropy Mixing: Exact Core, Density–0 Closure Companion, and Conditional Global Completion Interface (v5.0)

Title: A Spectral–Entropy Framework for the Goldbach Conjecture: Mellin–Trace Bridge, Prime-Band Conductance, and Polylog-Uniform Mixing on Most Windows This is the main proof carrier for the exact core package: TP0 / E0 / C0 / M0 Its proved layer establishes: A reusable transfer principle from prime-band statistics to polylogarithmic conductance / spectral gap / log-Sobolev lower bounds on most windows. A most-window spectral–entropy mixing closure. A dyadic exceptional-set corollary (classical almost-all Goldbach on dyadic scales). A Mellin–trace identity for the Goldbach Dirichlet series in the safe region: (s) > 2. Beyond the exact proved layer, the main paper also isolates the remaining all-even obstruction into a single explicit residual divisor-type short-interval source input. Under this input, the spectral–entropy spine upgrades to a conditional global Goldbach closure theorem. 2. Density–0 companion Title: Density–0 Closure for the Goldbach Flow: A Self–Contained Variational Proof This note should be read as the Gate B closure note of the program. Its sole proved output is: (EG) D0 Key findings: A quantitative per-window energy gap on the lifted exceptional region implies logarithmic-density–0 elimination of weakly mixed windows. (Note: It does not prove the flagship transfer theorem, establish most-window mixing, or produce a finite global threshold K₀. ) 3. Referee ledger A compact reading aid summarizing the bundle-level status, including: The exact proved core of the main paper. The optional threshold / conditional scenario layer. The logical role of the Density–0 companion. Bundle-level status summary The v5. 0 bundle should be read in three distinct layers: Layer 1: Exact core (proved in the main paper) TP0 / E0 / C0 / M0 Layer 2: Gate B closure layer (companion) (EG) D0 With an optional sufficient interface: (EG'-tail) + (EG'-fiber) (EG) D0 Layer 3: Optional conditional global layer (main paper) A conditional completion interface under a residual divisor-type source input. Scope and claim discipline The exact proved layer of the bundle does not claim an unconditional finite global threshold (K₀), and does not claim a fully unconditional global proof of the even Goldbach conjecture. Any threshold-level or all-even completion statement belongs only to the conditional scenario layer of the main paper. What is new in v5. 0 Front matter and reading flow streamlined for both main paper and companion. Clearer separation between: Exact proved core. Density–0 closure companion. Conditional global completion interface. Explicit formulation of the residual divisor-type source-input bottleneck. Stronger bundle-level synchronization across the main paper, companion, and referee ledger.