From Bernoulli Numbers to the Critical Line: Analytic Continuation, Curvature Measures, Joint Exclusions, Four-Body Dynamics, and Algebraic Phase Transitions

This paper presents an audit architecture whose central thesis is that the completed zeta function is an adelic product of local factors; the symmetry centre 1/2, the smooth zero-counting constant 7/8, the normalization of any completed spectral determinant, the low-energy positivity of any Weil-type form, the curvature unit π2/12, the four-body collision law, the spacing lower bound for the imaginary parts, and the algebraic phase transition of the Bost–Connes system are internal invariants of the Gamma–Theta factorization and of the arithmetic local factors. A valid proof architecture preserves these invariants along the logical spine master equation =⇒ solution datum =⇒ identification with ξ =⇒ self-adjointness or algebraic uniqueness =⇒ Re ρ = 1 2 . The architecture is organized around three parallel load-bearing paths: the curvature path, the Ledger–Stiffness path, and the Bost–Connes KMS path; any one of them, if its load-bearing input is verified, closes the logical spine for the corresponding framework. Joint exclusion is organized algebraically: if the set of off-axis zeros is contained in the intersection of the surviving sets of the exclusion mechanisms, and if for some non-empty subfamily the intersection is empty or its counting measure is zero, then the set of off-axis zeros is empty.