A Spectral–Entropy Framework for the Goldbach Conjecture (v3.8)

# Summary (v3. 8) This record releases a two-paper set for a programmatic entropy–spectral approach to the Goldbach problem. - **Main paper: ** *Goldbach Conjecture Proof v3. 8*- **Companion note: ** *Goldbach Density0 v3. 8* The set is organized around a strict **layering contract**: - **Proved layer (unconditional): ** most-window mixing invariants (conductance / spectral gap / log–Sobolev) for a canonical **sum-chain** on dyadic windows, yielding a dyadic density–0 exceptional-set corollary. - **Conditional / roadmap layers: ** optional upgrades that require clearly labeled hypotheses (HT), (EG), (H3*). No unconditional global threshold \ (K₀\) is claimed. --- # Overview We define a reversible Markov chain that encodes Goldbach representations on dyadic windows and convert standard prime-distribution inputs into **mixing invariants**. The proofs are packaged as a reusable transfer pipeline: -band control conductance T gap T LSI ₋ₒ₈ (PT) entropy decay positivity on most windows. \ The companion note isolates a variational “density–0 closure” module under an explicitly stated energy-gap hypothesis (EG), without mixing it into the proved layer. --- # Files in this record - `GoldbachConjectureProofᵥ3. 8. pdf`- `GoldbachDensity0ᵥ3. 8. pdf`--- # What is new in v3. 8 - **Notation cross-walk and single-source-of-truth rules** are enforced (sum-chain vs pair-chain), reducing ambiguity about which kernel each theorem refers to. - The **proved vs conditional** boundary is tightened: pointwise/all-window statements are explicitly relegated to scenario modules. - The literature-positioning paragraph and table are strengthened to emphasize that the contribution is a **transfer principle / mixing-invariant packaging**, not an optimization of exceptional-set exponents. --- # Closed results (proved layer; unconditional) All unconditional results are stated for **most windows** on dyadic scales and do not require analytic continuation / functional equations. 1. **Sum-chain and mixing invariants. ** Let \ (ST: =\2n 2Z: T 2n T+T^\\) and let \ (P^sumT\) be the reversible Markov chain on \ (ST\) (with lazy version \ (P^sumT\) ). For all sufficiently large dyadic scales, for all but at most \ (X/ (X) C\) values of \ (T, 2X\), one obtains polylogarithmic lower bounds of the form \ T 1 T, (P^sumT) 1² T, ₋ₒ₈ (P^sumT) 1² T. \ 2. **Entropy dissipation on good windows. ** On those windows, the KL divergence to stationarity decays exponentially at rate \ (₋ₒ₈\), giving a quantitative mixing/regularization statement. 3. **Dyadic exceptional-set corollary. ** The mixing bounds imply that Goldbach positivity fails on a set of even integers of **density 0** on dyadic intervals, yielding a “most-window / dyadic density–0” Goldbach consequence. --- # Conditional / roadmap layers (explicitly isolated) These layers are included for completeness and as targets for future closure. They are **not used** in the unconditional proved layer. - ** (HT) Mellin–trace / analytic continuation layer. ** A heat-trace hypothesis enabling analytic continuation / functional equation statements for the Mellin–trace bridge. - ** (EG) Density–0 closure energy gap (Companion). ** An energy-gap hypothesis used to eliminate weakly mixed windows in a variational \ (\) -limit framework. - ** (H3*) Window-uniform mixing (global closure scenario). ** A uniform (all-window) mixing package that would imply an eventual all-even threshold \ (K₀\). This record does **not** claim such a threshold unconditionally. --- # Main–Companion interface (Inputs / Outputs) - The **Main paper (proved layer) ** outputs most-window mixing invariants and dyadic density–0 exceptional-set results for the **sum-chain** \ (P^sumT\). - The **Companion** outputs “weakly mixed windows have log-density 0” conditional on ** (EG) **. - Any upgrade from “window statements” to “integer statements” must pass through the explicitly stated bridge modules; no hidden implication is assumed. --- # Scope & non-toy status - This record should be read as a **programmatic, layer-disciplined** contribution: it supplies a rigorous transfer mechanism from prime-distribution inputs to Markov mixing invariants that are directly relevant to additive representation positivity. - The record **does not** claim an unconditional global Goldbach threshold \ (K₀\). - Numerical material (if present) is **exploratory diagnostics** and is not used as a proof input. --- # How to cite Please cite as: - *Goldbach Conjecture Proof v3. 8* (main paper) and- *Goldbach Density0 v3. 8* (companion note), Zenodo record (this entry), version v3. 8. --- # Keywords / MSC (suggested) **Keywords: ** Goldbach conjecture, exceptional set, short intervals, Markov chains, conductance, spectral gap, log–Sobolev inequality, entropy dissipation, variational methods. **MSC 2020: ** 11P32, 11N36, 60J10, 60J27.