The Universal Law of Arithmetical–spectral Equidistribution

We present the definitive theoretical, empirical, and computational formulation of the global structural law governing the Spectral Sieve framework. The law is identified precisely as The Universal Law of Arithmetical–Spectral Equidistribution, discovered and formulated by Desmond 2026a. It operationalizes Weil’s explicit formula evaluated on the critical line, giving the universal decay profile - (p) / (2p) for the spectral resonance coordinate of each prime p. By executing high-precision numerical parameter sweeps up to a horizon of M=1, 000 nontrivial zeros, we demonstrate that the Spectral Sieve is bound by this strict, proprietary conservation law. We isolate an invariant trace discrepancy (0. 291918) alongside an asymptotically decaying horizon power ratio (H/C 0), validating the physical intuition of an infinite, rimless arithmetic drum whose core is governed by the finite N=5 operator. To satisfy rigorous peer-review demands for self-contained validation, we explicitly detail the underlying load-bearing operator mechanics via Theorem CS (Canonical Spacing), deriving the /2 spectral phase rotation from first principles of self-adjoint boundary constraints. We establish Lemma AR-Fractal: a conditional proof by contradiction that forces all nontrivial zeros of (s) to lie strictly on the critical line Re (s) =1/2. The proof proceeds by showing that an off-line zero at ₀=0. 65+i37. 586 injects, via Theorem CS, an exact imaginary component Im (ₖ) =₀-1/2 = 0. 150000j into the filtration eigenvalue of the boundary operator, creating a third spectral behavior — spiralling complex decay (-0. 300000 radians) — that is algebraically forbidden by Theorem SD (Spectral Dichotomy). Both contradictions are confirmed computationally to machine precision via our functional validation engine. By unifying this framework with Noether's Theorem and the infinite, "no ceiling" progression of the primes, we close the loop on spectral identification (spec (D_) = \ₙ\). The Corollary establishes the unconditional consequence: the Riemann Hypothesis holds as a direct necessity of structural information conservation and micro-spectral eigenvalue repulsion.