A Spectral–Entropy Framework for the Goldbach Conjecture (v4.2): Most-Window Mixing via Prime-Band Transfer

# Overview (v4. 2) This record releases **Goldbach v4. 2**, a two-paper set plus a 1-page referee ledger and an optional verification-only artifact. - **Main paper (proved layer): ** a reusable *prime-band intersection transfer principle* turning BV/BDH-type short-interval mean-square inputs into polylog lower bounds for conductance, spectral gap, and log-Sobolev constants of a natural Goldbach sum-chain on **most windows**. This yields a standard dyadic “exceptions have natural density 0” corollary. - **Companion note (Density0): ** a self-contained variational closure proving that *weakly-mixed windows have logarithmic-density 0*, conditional on a per-window energy-gap hypothesis (EG). - **Ledger: ** a one-page “proved vs optional” summary for referees. - **Optional artifact (ZIP): ** a verification-only skeleton implementing the *interface* for an external all-window certificate (H3⋆) used only in the optional hybrid-closure module. # Files in this record 1. `GoldbachMainᵥ4. 2. pdf` 2. `GoldbachDensity0ᵥ4. 2. pdf` 3. `GoldbachLedgerᵥ4. 2. pdf` 4. `GoldbachH3starArtifactᵥ4. 2. zip` (optional; interface only) # Claims (scope / hygiene) **Proved (unconditional in-paper): **- TP0: Transfer principle (TB1–TB4 ⇒ polylog conductance/gap/LSI ⇒ KL decay ⇒ windowed positivity). - E0: Most-window polylog mixing/entropy closure. - C0: Dyadic exceptional-set bound (natural density 0 on dyadic scales). - M0: Mellin–trace identity in the absolutely convergent region Re (s) > 2. **Not proved / not claimed in the proved layer: **- No unconditional global threshold K0 (“all even N ≥ K0”) is claimed. - Any all-window / eventual closure is treated as an **optional external acceptance test**. # Companion note (Density0) The companion note proves module D0 (density–0 closure for weakly-mixed windows) **conditional on (EG) **. It does not claim a global Goldbach theorem or an unconditional threshold. # Optional hybrid closure (interface only) The main paper includes an optional “hybrid closure” statement: if Goldbach is verified for all even N ≤ Bᵥer and if an external certificate supplies all-window bounds implying W (N) > 0 for all even N ≥ Kₐn, then Goldbach holds globally with K0 = maxBᵥer, Kₐn. This is an interface specification only; it is not part of the proved layer. # Reproducibility / artifact (ZIP) `GoldbachH3starArtifactᵥ4. 2. zip` is a verification-only skeleton: - It defines a certificate schema and an auditor that recomputes recorded minima and checks file digests and coverage rules. - The certificate must match `paperᵢd = v4. 2`. # Suggested citation Lee Byoungwoo, “A Spectral–Entropy Framework for the Goldbach Conjecture: Mellin–Trace Bridge, Prime-Band Conductance, and Polylog-Uniform Mixing on Most Windows”, v4. 2 (Zenodo record).