Toward a Proof of the Spectral Factorization Theorem

The String-EFT Hilbert-Pólya program proposes that the non-trivial zeros of the Riemann zeta function are the eigenvalues of a Hamiltonian H = -Δ + βV(z), where the potential V(z) is derived from the Type IIB D²ᵏR⁴ higher-derivative tower and the perturbation parameter β = -1/(2√2) is topologically locked by the D7-brane matter coupling. This note identifies the four analytic components required to elevate this conjecture to a formal proof: (1) the derivation of the perturbed scattering matrix via the Lippmann-Schwinger equation on the modular surface, (2) the geodesic-prime correspondence in the Selberg trace formula, (3) the factorization of the Fredholm determinant as Z(s) = ζ(s)·exp(P(s)), and (4) the ground-state positivity bound ensuring eigenvalues remain above 1/4. This roadmap serves as the connective framework between the Hilbert-Pólya candidate (doi:10.5281/zenodo.19432229) and the Triple Product Obstacle reduction (doi:10.5281/zenodo.19432627), completing the five-paper String-EFT program initiated in Kanhai (2026a), doi:10.5281/zenodo.19431956.