A Spectral–Entropy Framework for the Goldbach Conjecture (v4.3): Prime-Band Transfer Principle and Most-Window Mixing

# Overview (v4.3) This record releases a **journal-cut v4.3** set for a *spectral–entropy “Goldbach flow”* program. **Main paper (proved layer):**- *A Spectral–Entropy Framework for the Goldbach Conjecture: Mellin–Trace Bridge, Prime-Band Conductance, and Polylog-Uniform Mixing on Most Windows* (v4.3). It proves a reusable **prime-band transfer principle** (TP0) and derives **polylog conductance / gap / LSI floors** for a windowed Goldbach sum-chain on a **density-1 set of windows**, yielding a classical dyadic “almost-all Goldbach” exceptional-set bound. **Companion (Density0):**- *Density–0 Closure for the Goldbach Flow: A Self-Contained Variational Proof* (v4.3). It is self-contained except for a single external analytic input **(EG)** (a per-window energy gap on a lifted exceptional region). Under (EG), it eliminates weakly mixed windows up to **logarithmic density 0**. **Referee ledger (1 page):**- A scope/claims dashboard separating the proved layer from optional certificate interfaces. **Optional certificate artifacts (interface only):**- `Goldbach_H3star_Artifact_v4.3.zip`: an auditable interface for a certificate-grade all-window mixing acceptance test (H3⋆), used only for optional hybrid closure.- `Goldbach_EGprime_Artifact_v4.3.zip`: an auditable interface for a sufficient condition (EG′) implying (EG), supporting certificate-grade runs with coverage and independent cross-checks. --- # What is proved in v4.3 (proved layer) Let \(I_T=[T,\,T+T^α]\) be a short window, \(S_T=I_T∩ 2N\), and let \(P_Tˢᵘᵐ\) be the reversible (lazy) Goldbach sum-chain on \(S_T\). ## TP0 (Transfer principle)A deterministic Markov/entropy spine turns window-averaged **prime-band statistics** (TB1–TB4) into:\[Φ(P_Tˢᵘᵐ) (log T)⁻¹,γ(P_Tˢᵘᵐ),\ αLSI(P_Tˢᵘᵐ) (log T)⁻²\]on a **density-1 set of windows** (most-window regime). ## E0 (Most-window closure)On most windows, KL divergence decays at rate \(αLSI\) and the windowed flow yields “almost-all positivity” on a typical subset of \(S_T\). ## C0 (Dyadic exceptional-set bound)On each dyadic interval \([X,2X]\),\[#\{2n∈[X,2X]: W(2n)=0\}\ ≤\ ε X + O\!(X/(log X)^C)+O(X^α),\]so Goldbach exceptions have **natural density 0** on dyadic scales (hence also logarithmic density 0). ## M0 (Mellin–trace bridge on \(Re(s)>2\))A Mellin–trace identity represents the Goldbach Dirichlet series in the absolutely convergent region \(Re(s)>2\), together with trace/HS criteria and Tonelli–Fubini justifications in that safe half-plane. --- # Inputs used by the proved layer (scope note) The proved layer is **RH/GRH-free**, but it is **conditional only on the stated short-interval mean-square prime inputs** in Appendix A″: - **(BV–SI)/(BDH–SI)**: short-interval mean-square inputs in an explicitly displayed integrated \(Λ/ψ\) format at the declared cutoff \(Q(T)\).- A scope note is included: global BV/BDH statements do **not** automatically imply these short-interval forms without additional short-interval technology. No numerical claims are used as proof inputs in the proved layer. --- # Companion note (Density0): what it proves Assuming a per-window energy gap on the lifted exceptional region \(E_T⊂Ω_T\), \[(EG): {G}_G[ρ]\ ≥\ δ (log T)-B for all densities ρ supported on E_T,\] the companion proves: **weakly mixed windows form a set of logarithmic density 0** (Module D0). It does **not** claim a global threshold \(K_0\) and does **not** claim an unconditional all-even theorem. --- # Optional hybrid closure (interface only; not part of proved layer) The optional hybrid-closure interface combines:1) a verified finite range \(N≤ Bᵥₑᵣ\), and2) an eventual analytic implication above \(Kₐₙ\), so that \(K_0=max\{Bᵥₑᵣ,Kₐₙ\}\). A certificate artifact (H3⋆) is provided as an **auditable acceptance-test interface** for all-window mixing bounds; it is not used anywhere in TP0/E0/C0/M0. --- # Optional EG′ artifact (interface only) `Goldbach_EGprime_Artifact_v4.3.zip` defines a certificate-grade interface for a sufficient condition (EG′) implying (EG), phrased in terms of:- polylog low-representation counts (tail), and- polylog fiber maxima on exceptional indices (fiber), with **coverage_slack = 0**, **raw JSON + SHA256 bindings** (raw_mode=required), and an **independent Verifier2 cross-check**. --- # Files in this record - `Goldbach_Main_v4.3.pdf`- `Goldbach_Density0_v4.3.pdf`- `Goldbach_Ledger_v4.3.pdf`- `Goldbach_H3star_Artifact_v4.3.zip` (optional interface)- `Goldbach_EGprime_Artifact_v4.3.zip` (optional interface) (If you publish a flat bundle ZIP and SHA256SUMS, list them here as well.) --- # How to cite Lee Byoungwoo, *A Spectral–Entropy Framework for the Goldbach Conjecture (v4.3): Prime-Band Transfer Principle and Most-Window Mixing*, Zenodo (v4.3), 2026.