URCM–Riemann: Curvature Stability, Symmetric Pair Splitting, and the Non‑Splitting Functional for the Completed Zeta Function

This preprint consolidates the URCM–Riemann programme into a unified analytical framework. The completed zeta function ξ(s) satisfies the functional equation ξ(s)=ξ(1−s), forcing first‑order stationarity of the log‑amplitude F(σ,t)=log|ξ(σ+it)| at σ=1/2. This symmetry makes all first‑order stability criteria tautological and therefore uninformative for the Riemann Hypothesis (RH). The first non‑tautological object is the second‑order curvature Kc(t)=∂σ2F(1/2,t). However, curvature positivity alone cannot distinguish a zero on the critical line from a symmetric off‑axis pair, since both configurations produce identical local curvature sign at σ=1/2. To overcome this, we introduce the non‑splitting functional S(t)=∫w(σ) [∂σ2F(σ,t)−∂σ2F(1/2,t)]2 dσ, which measures the transverse spread of curvature across the critical strip. We show analytically, using explicit local models, that: S(t)=0 for a zero located exactly on the critical line. S(t)>0 for any symmetric off‑axis zero pair. Symmetric weights cannot trivially annihilate S(t). A regularised functional Sε(t) remains strictly positive for off‑axis pairs uniformly as ε→0. The supplement provides full derivations, regularisation, and sensitivity proofs, closing the analytical gaps in the original definition of S(t). No claim of a proof of RH is made. The contribution is structural: RH is precisely the condition S(t)=0 for all t, i.e., the absence of transverse splitting of curvature wells in the completed zeta function.