We prove, without appeal to physical intuition or numerical evidence, that W-III modes of the sigma-field in the TIC/CIT framework are strictly prohibited by the Core Inequality. The proof proceeds via three independent routes. Route A (main): we reduce the TIC/CIT second-order perturbation action to the Mukhanov-Sasaki equation and identify the Core Inequality with positive semi-definiteness of the MS Hamiltonian HMS. Using the Hardy inequality on the half-line — with the sharp constant 1/4 — we construct an explicit test function for which the quadratic form is strictly negative whenever the W-III condition nu² > 1/2 holds, yielding a direct contradiction. Route B: W-III modes generate a non-zero imaginary part in the one-loop effective action, violating unitarity. Route C: the Core Inequality implies essential self-adjointness of HMS via the Weyl limit-point criterion, which breaks down in the W-III regime. As a corollary, under the Spectral Correspondence Hypothesis, the prohibition of W-III modes is equivalent to the Riemann Hypothesis. Paper 7 of the TIC/CIT series.
Leandro de Oliveira (Thu,) studied this question.