The Triple Product Obstacle: A Reduction of the String-EFT Scattering Identity to Rankin-Selberg Summation

The String-EFT Hilbert-Polya program identifies the Riemann zeros with the spectrum of a Hamiltonian (H = -Delta + beta V (z) ) on the modular surface. A formal proof requires the scattering matrix to satisfy a spectral factorization involving the Riemann zeta function. In this note, we perform a technical reduction of the scattering identity, showing that the first-order spectral shift is governed by a summation of Rankin-Selberg triple product integrals. We identify the specific inner product obstacle: the convergence of an infinite series of completed zeta-function ratios, weighted by the D²k R⁴ stringy coefficients, to the logarithmic derivative of the Riemann zeta function.