Density–0 Rigidity Program v3.8: Analytic–Spectral Closure toward the Riemann Hypothesis, Generalized RH, and the Selberg Class Extension
Creators
Description
Description
Overview
This record collects version v3.8 of the Density–0 Rigidity Program, version v3.0 of the RH Circle A framework, version v1.1 of the Guide Book to the Rigidity Program, and version v1.0 of the Selberg Class Extension via Density–0 Flow.
Together they form a unified analytic–spectral proof system addressing the Riemann Hypothesis (RH), the Generalized RH (GRH), and the Selberg–class extension within the entropy–rigidity formalism.
1. Density–0 Proof v3.8
This paper completes the analytic closure of the rigidity functional
$$
\widetilde{\mathrm{G}} = E_{\mathrm{LSI}} + E_{\mathrm{EF}} + E_{W_2},
$$
on Selberg–class strata equipped with the Wasserstein geometry.
The argument establishes:
(H1) Well–posedness via two–parameter renormalization, counterterm subtraction, and full $\Gamma$–convergence.
(H2) Displacement convexity of $\widetilde{\mathrm{G}}$ along $W_2$–geodesics with uniform log–Sobolev constant $\lambda_{\mathrm{LSI}} > 0$.
(H3) Explicit–formula and large–sieve (EF+LS) cancellation upgraded from mean–square to pointwise.
The closure condition
$$
\widetilde{\mathrm{G}}(F) = 0 \;\Longleftrightarrow\; \text{RH/GRH for all } F
$$
is proven for Dirichlet and $\mathrm{GL}(2)$ families and conditionally extended to the Selberg class.
Appendix O introduces the Selberg extension window $S_{\mathrm{ext}}$ and formulates a global LSI condition across families.
2. RH Circle A v3.0
Develops the geometric counterpart of the analytic chain.
It shows that the R–Circle Criterion—a geometric constraint on zero angles—is equivalent to the entropy–flow law derived from the Density–0 proof:
$$
\text{Circular rigidity} \;\Longleftrightarrow\; \text{Entropy dissipation on } W_2.
$$
3. Guide Book v1.1
An expository companion that reorganizes the full Rigidity Program for accessibility. It explains:
– The proof roadmap (H1–H3 → Master Uniformity → RH/GRH).
– Toolkits (explicit formula, large sieve, log–Sobolev, displacement convexity).
– Assumption maps ensuring non–circularity.
– Worked examples and numerical illustrations.
The Guide Book lowers the entry barrier for readers in number theory, analysis, and probability while maintaining full mathematical precision.
4. Selberg Class Extension v1.0
Extends the Density–0 framework to the entire Selberg class $S$.
Under a generalized Bombieri–Vinogradov condition $(\mathrm{BV}_S)$, the family–averaged Kullback–Leibler entropy of zero distributions satisfies
$$
\mathrm{KL}_f(x; Q(x)) \;\le\; \frac{C_S}{\log x},
\qquad Q(x) = x^{1/2} (\log x)^{-B}.
$$
Consequently,
$$
(\mathrm{BV}_S) \;\Rightarrow\; \chi^2 \;\Rightarrow\; \mathrm{KL}
\;\Rightarrow\; \text{Spectral Rigidity} \;\Rightarrow\; \text{Selberg–RH (Density–0 form)}.
$$
This generalizes the Dirichlet and $\mathrm{GL}(2)$ results to all admissible Selberg families and demonstrates a global entropy–spectral closure on the moduli space $\mathcal{M}_L$.
Key Consequences
– RH and GRH proved for zeta, Dirichlet, and $\mathrm{GL}(2)$ families.
– Selberg–RH (density–0 form) established under $(\mathrm{BV}_S)$.
– Unified analytic–geometric chain
$$
(\mathrm{BV}/\mathrm{BDH}) \;\Rightarrow\; \chi^2 \;\Rightarrow\; \mathrm{KL}\text{ decay}
\;\Rightarrow\; \text{Spectral Rigidity}
\;\Rightarrow\; \text{RH/GRH/Selberg–RH}.
$$
– Bridges between entropy, Fisher information, and random–matrix statistics (GUE law).
– Provides the functional–analytic backbone for the MUGS (Modular Uniqueness of GUE Statistics) theorem.
Author
Byoungwoo Lee (leeclinic@protonmail.com)