Goldbach Conjecture Proof v3.5 (Main + Density–0 Companion)

# Overview This record releases **Goldbach Conjecture Proof v3. 5** as a **two-PDF set**: - **Main paper: ** *Goldbach Conjecture Proof v3. 5* - **Companion note (closure module): ** *Goldbach Density–0 v3. 5* The program is **explicitly scope-managed**: it separates an **unconditional “most-window” layer** from **scenario-based global closure**, and isolates the **density–0 exceptional-window closure** into an independent variational module. Date: **February 21, 2026** Version: **v3. 5** # Contents This record contains two PDF files: 1) `GoldbachConjectureProofᵥ3. 5final. pdf` 2) `GoldbachDensity0ᵥ3. 5final. pdf` # Main claims (what is proved here) ## (A) Unconditional layer (proved) The main paper establishes a **most-window / density-1** Goldbach validity statement in the calibrated window framework. Formally, for “good windows” (in the sense of the program’s acceptance tests), the set of even integers \ (2n\) in the windowthat admit a Goldbach representation \ (2n=p+q\) has **density \ (1\) ** inside the window. This layer is designed to be robust under: - window rescaling within the admissible policy class, - renormalized entropy/energy comparisons, - and a log-regularized boundary analysis at \ (s=2\) (no simple-pole claim). ## (B) Scenario layer (optional, clearly marked) A **global eventual Goldbach** statement (existence of a finite threshold \ (K₀\) such that every even \ (2n K₀\) is Goldbach) is derived only under an explicit **strong-mixing scenario** (uniform minorization / gap / quantitative inputs), presented as a separate, assumption-labeled module. This scenario layer is *not* part of the unconditional claims unless the stated hypotheses are invoked. # Companion note: Density–0 closure (what it does) The companion note proves that the set of **exceptional windows** can be controlled in **logarithmic density**: it provides a variational closure mechanism showing that weakly mixed or “bad” windows form a set of (logarithmic) density \ (0\), under the note’s stated hypotheses. Key features: - a modern variational structure (including \ (\) -convergence style arguments), - equicoercivity driven by an entropy term \ (\), - and a quantitative upgrade from total-variation control to pointwise positivity on the relevant “good” set. ## What is not claimed (explicit) The companion note **does not claim** a fully unconditional global proof of the Goldbach conjecture. Its role is a **closure module** controlling exceptional windows at logarithmic density \ (0\). # Technical highlights (high-level) - **Entropy–spectral framework: ** a Markov/variational control of Goldbach representations via an entropy functional and a calibrated “global energy” proxy on admissible windows. - **Band-limited comparison discipline: ** kernel comparisons are performed on matched support / typical sets, avoiding polynomial collapse in entrywise band comparisons. - **Log-regularized boundary at \ (s=2\): ** the Mellin–trace interface is treated with logarithmic regularization; no claim of a simple pole at \ (s=2\) is used in v3. 5. # Relation to the RH/rigidity proof set (external references) Some background “rigidity program” inputs are cited from the following Zenodo record: - **Four-paper set (Published February 18, 2026 | Version v4. 0r) ** Record DOI: **10. 5281/zenodo. 18684367** Included PDFs: *Density-0 Proof v4. 0*, *RH Circle A v4. 0*, *RH Guide v1. 3r*, *Selberg Class Extension v1. 1*. This Goldbach program does not require the RH set for its unconditional “most-window” layer, but references it for structural analogies and calibrated constants where explicitly noted. # Suggested citation B. Lee, *Goldbach Conjecture Proof v3. 5 (Main + Density–0 Companion) *, Zenodo, 2026, Version v3. 5. # Keywords Goldbach conjecture; additive prime problems; entropy method; spectral gap; Markov chains; variational closure;\ (\) -convergence; logarithmic density; exceptional sets; Tauberian regularization; sieve heuristics.