Volume I (ai.viXra:2610.0045v2) showed that the Riemann Hypothesis (RH) is equivalent to unitarity (the absence of negative-norm ("ghost") directions) of the logarithmic scaling flow for the Weil form Q; Volume II measured that obstruction on the Weil form itself. This volume asks which structure could supply the positivity. For curves over F_q, Weil's proof needs a commutative algebra containing Frobenius, an involution with F F^dagger = q, a trace for which the involution is positive, and Albert's algebraic step. Zeta has the first, second and fourth canonically; the third is axiom G5: a positivity of the zero-side trace that holds for a geometric reason, built without the zeros. We then test candidate sources of G5: thermal, modular and Kepler structures; Osterwalder-Schrader reflection positivity; causality, Kramers-Kronig and passivity; de Branges/Krein lines from the prime side, with G5 stated as a passive transmission line; Galois symmetry, Iwasawa theory and Bost-Connes; polarization, Krein unitarity and the two stages of ghost birth; half-sided modular inclusions; holomorphic flows, cavities, boson/fermion second quantizations; and the Verschiebung and the reflection N -> Z. One conclusion recurs as a theorem: a reflection gives the functional equation; positivity needs a polarization. Algebra gives Krein unitarity; a positive polarization gives Hilbert unitarity. An editorial section (from Lemma 50) recasts this in spinor language: in the abstract model, RH for curves is equivalent to a Frobenius-invariant positive compatible complex structure (a positive fermionic vacuum), and second quantization never repairs an indefinite signature. Every statement carries a claim label (Theorem, Restatement, Analogy, Conjecture, Open, Numerical). RH is not proved; no candidate tested supplies G5. Computations still running at submission are marked as placeholders.