Number Theory

Arithmetic Unitarity II: The Light Cone of the Weil Form

Authors: J. W. McGreevy

Volume I (ai.viXra:2610.0045v2) showed that the Riemann Hypothesis (RH) is equivalent to unitarity, meaning the absence of negative-norm ("ghost") directions, of the logarithmic scaling flow for the Weil form Q. This volume measures that obstruction directly in one geometric language: the Weil form is the transverse part of a Lorentzian form. For curves over F_q, the Hodge index theorem makes NS(C x C) Lorentzian, the graph of Frobenius is the diagonal boosted by rapidity (1/2) log q, and Weil's bound is Cauchy-Schwarz on the transverse part. For zeta, the pole plane minus Q is a form L computable without zeros, and RH is equivalent to L having Lorentzian index <= 1 on every finite-dimensional subspace (a restatement of Weil's criterion). We certify positivity of Q on every window of length <= 1.6 (computer-assisted, Arb ball arithmetic); give the arithmetic action a Lagrangian, Noether, Ostrogradsky and transmission-line reading; make the damped (sigma-shifted) window index exact; separate trace bounds from Rosati positivity and read T_p as two boosts; prove no-go theorems for zero-side certificates, prolate detectors, tropicalization and hbar-deformation; correct explicit zero-counting constants and certify zero-sampling inequalities on Platt's zeros; read each zero as a telegrapher line in a Krein space, with a passivity ladder from the functional equation to RH; show that ghost birth is a two-stage event (Krein-space degeneration, then a Krein collision); and prove pole dominance for damped forms with an exact closed-form threshold. Every statement carries a claim label (Theorem, Restatement, Analogy, Conjecture, Open, Numerical). RH is not proved. The positivity source that geometry supplies for curves is not found for Spec Z x Spec Z; several natural candidates are proved not to supply it. Computations still running at submission are marked as placeholders.

Comments: 173 Pages. Thermodynamics of Arithmetic Action, Lemmas 14-29 (compiled volume II). Sequel to ai.viXra:2610.0045.

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[v1] 2026-10-22 04:55:27

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