Goldbach Program v4.0 (Main + Companion): Prime-Band Transfer Theorem, Most-Window Polylog Mixing, and Density–0 Closure (EG)

# Overview This record releases a **two-paper set (v4.0)** developing a spectral–entropy / Markov-chain formulation of the binary Goldbach problem on short windows. - **Main paper:** *A Spectral–Entropy Framework for the Goldbach Conjecture* (v4.0)- **Companion note:** *Density–0 Closure for the Goldbach Flow: A Self–Contained Variational Proof* (v4.0) The set enforces a strict **scope ledger**: the proved layer is *most-window* and produces a classical dyadic density–0 exceptional-set corollary, while any eventual all-even threshold \(K_0\) is **scenario-level** only (under explicitly stated hypotheses). The companion note is **conditional on (EG)** and is logically independent of the main transfer theorem. --- # Closed results (proved layer; unconditional) ## (TP0) Flagship transfer theorem (reusable)On each window \(I_T=[T,\,T+T^α]\) with \(0<θ<α<1\), let\(S_T := I_T ∩ 2N\).Given two (possibly equal) prime bands \(B_T⁽¹⁾,B_T⁽²⁾\), define the band-limited representation neighborhood\[G_T(x) := \{\,p∈ B_T⁽¹⁾ : x-p∈ B_T⁽²⁾\,\}, _T(x):=|G_T(x)|.\]Under window-averaged band statistics (TB1–TB4), the theorem yields polylogarithmic mixing invariants for the lazy sum-chain \( P_Tˢᵘᵐ\) on **most windows**:\[Φ( P_Tˢᵘᵐ) 1/log T,γ( P_Tˢᵘᵐ) 1/log^2 T,αLSI( P_Tˢᵘᵐ) 1/log^2 T,\]and forces windowed almost-all positivity (vanishing exceptional subset inside each good window). ## (E0) Most-window spectral / LSI closureSpecializing TP0 to Goldbach prime bands yields polylog-uniform gap/LSI on a density-1 set of windows on each dyadic scale. ## (C0) Classical dyadic exceptional-set corollaryFor every \(ε>0\) there exist \(C>0\) and \(X_0(ε)\) such that for all \(X≥ X_0(ε)\),\[#\{2n∈[X,2X]: W(2n)=0\}\ ≤\ ε X + O\!(X/(log X)^C) + O(X^α).\]In particular, Goldbach exceptions have natural density \(0\) on dyadic scales. ## (M0) Mellin–trace bridge on \((s)>2\)A Mellin–weighted trace identity is proved in the safe half-plane \((s)>2\), together with trace/HS criteria and Tonelli/Fubini justifications. --- # Second application (explicit reuse beyond unconstrained Goldbach) To make reusability explicit, the main paper also treats a **congruence-restricted** variant where the two primes are constrained to residue classes modulo \(q\) (two constrained prime bands). This is presented as a second application of the same transfer theorem to a modified band family. --- # Companion note (Density0): what it proves (and what it does not) **Companion output (D0; conditional on EG):**Assuming a quantitative **per-window energy gap (EG)** on the exceptional region, weakly mixed windows form a **logarithmic-density-0** set. **Not claimed:**The companion does not imply an unconditional global Goldbach theorem, does not produce an unconditional threshold \(K_0\), and does not upgrade window statements to a pointwise all-even theorem without additional bridge modules. **Logical independence:**The companion note does **not** use the main paper’s flagship transfer theorem; it is self-contained given (EG). --- # Numerics policy (important for scope) Any numerical material is labeled **diagnostic/illustrative** and is never used as proof input. Scenario-based thresholds (if discussed) are explicitly conditional and separated from the proved layer. --- # Files in this record - `Goldbach_Conjecture_Proof_v4.0.pdf`- `Goldbach_Density0_v4.0.pdf` --- # Suggested keywords / MSC (2020) **Keywords:** Goldbach conjecture, exceptional set, short windows, Markov chains, conductance, spectral gap, log–Sobolev inequality, entropy dissipation, Mellin trace, Γ–convergence. **MSC 2020:** 11P32, 11N36, 60J10, 60J27.