This volume collects TAA Lemmas 86-118 of the programme Thermodynamics of Arithmetic Action. Volumes I-IV (ai.viXra:2610.0045v2, 2610.0054, 2610.0052, 2610.0056, 2610.0055) recast the Riemann Hypothesis (RH) as the absence of negative-norm ("ghost") directions for the Weil form, and traced the missing positivity to the Hodge index theorem on C x C, which zeta lacks. Over finite fields Frobenius is a rotation in the Rosati metric; a ghost would be a boost. Volume V asks what a surface Spec Z x Spec Z would have to supply. Central-value positivity (Gross-Zagier) turns out to be a parity statement about real zeros. Tested next: the standard conjectures and correspondence rings, the Connes-Consani self-intersection with synthetic ghost zeros, ample classes, two scalings and weight gradings. Certified (Arb) results: positivity of the Weil form of zeta on explicit spline spaces up to support a = 8, and certified detection of hypothetical off-line zeros. A compatible complex structure commuting with Frobenius exists exactly when the Frobenius torus is compact; the volume then compares Krein, Toeplitz and Newton-over-Hodge signatures, the modular cusp, BRST and thermodynamic geometry, and ends with Lax-Phillips passivity: the doubled modular system is passive unconditionally. A statement at the start lists exactly what is claimed, and where. Three proposed errata await the author's decision; they are listed, not applied. Formalization and computations were AI-assisted, and this is disclosed. RH is NOT proved: structures built from the zeros restate Weil positivity, those built without them are blind to off-line zeros, and an RH-independent ample class for Spec Z x Spec Z is still missing.