The Resonance Hilbert Space and the Herglotz Property of the Scattering Determinant: Closing the Hard Core of the Riemann Hypothesis

Papers 79 of the TIC/CIT series established that the Core Inequality implies no W-III modes, that the Riemann zeros appear as scattering resonances of the MukhanovSasaki Hamiltonian HˆMS on the modular surface MTIC = SL(2, Z)\H2, and that the GibbonsHawking lattice is exactly SL(2, Z). The residual gapthe hard core of the problemis that self-adjointness of HˆMS on L2 (MTIC) constrains only discrete eigenvalues, not scattering resonances (poles of the resolvent in the continuous spectrum). The present paper closes this gap via two independent constructions. Construction I (Resonance Hilbert Space): We build the LaxPhillips resonance space Hres = K ⊖ (L2 (MTIC)) from the compressed scattering semigroup and prove that the extended Core Inequality on HeTIC = L2 (MTIC) ⊕ Hres implies positive semi-deniteness of the extended Hamiltonian HeMS, forcing all resonant eigenvalues to be real. Construction II (Herglotz Extension): We prove that the scattering determinant φ(s) = ξ(2s − 1)/ξ(2s) is a HerglotzNevanlinna function (mapping the upper half-plane to the closed upper half-plane) if and only if the Core Inequality holds on Hres, and that the Herglotz property of φ(s) is equivalent to all its poles lying on the real axis, which is equivalent to the Riemann Hypothesis. Together, Constructions I and II close the TIC/CIT proof of RH without residual hypotheses.