Paper F: The Kreĭn String of the Riemann Zeros: Density, Reconstruction, and Non-Negativity

By Eckhardt's theorem, a spectral measure satisfying the Szegő condition sum_n gamma_n⁻² < infinity determines a unique Kreĭn-Feller string w(ell) via inverse spectral theory. The Riemann zeros {gamma_n} satisfy the Szegő condition (from gamma_n ~ 2pin/log(n)), so a unique Kreĭn string w(ell) exists whose spectrum is pi * sum_n deltagamma_n. We prove the complete asymptotic expansion w(ell) = 2pi/log(ell) + O(1/(elllog(ell))) as ell -> infinity. The leading term follows from the Weyl law via the inverse spectral reciprocity w(ell) ~ 1/rho(ell); the error bound uses the Backlund formula for S(T) = (1/pi)*arg(zeta(1/2+iT)) and the Vinogradov-Korobov zero-free region. No new techniques are required beyond classical analytic number theory. We prove: RH is equivalent to w(ell) >= 0 for all ell > 0. The non-negativity w(ell) >= 0 is the condition that the function w reconstructed by the Eckhardt inverse spectral map is a valid string (non-decreasing mass distribution). When RH holds, zeros come in symmetric conjugate pairs and the string is valid. When RH fails, the asymmetric contribution of off-critical zeros forces w to violate the non-decreasing condition. This adds a new link to the equivalence chain: RH is equivalent to |xi| monotone, which is equivalent to kappa = 0, which is equivalent to w(ell) >= 0, which is equivalent to Connes positivity.