A Spectral–Variational Z-Route at K = 1: From the WG Conjecture to HP⇒RH

We formulate a two-part hypothesis—the WG Conjecture—consisting of a Weyl–Riemann– von Mangoldt calibration (W) and a Gaussian-family identity (G) for the one-dimensional Schr¨odinger operator H=−∂2 u + VS(u) arising at the variational balance K = 1. Assuming (W)+(G), we prove a full heat-trace identity SH(τ) = Sζ(τ; k∞) + M(τ; k∞) for all τ >0; this yields equality of spectral zeta-regularised determinants and hence multiset equality {λn}= {(k∞γn)2}, so the rescaled operator H′ = √H/k∞ has spectrum {γn}: HP holds and therefore RH. Our analytic pipeline (the Z-route) is unconditional given (W)+(G). We also record an equivalent R-route via a resolvent-trace identity (Birman–Krein) that is not used in the main proof but provides an alternative closure. Appendices supply sector analyticity, short-time bounds, a uniform WKB derivation of (W) under exponential tails VS(u)∼C2e8|u| with the calibration C = 4π, and a uniqueness principle upgrading Gaussian data to even-Schwartz test functions.