The Mathematical Genesis of Time

We adopt a geometric-operational stance: the phase-induced refractive indexN(x,t) = exp(βₙ·δ(x,t)) is treated as the optical avatar of a weak-field metric.Under local Lorentz invariance the light-cone speed c₀ is fixed; gravity enters optics via the weak-field lawsg = (c₀²/2) · ∇ln N, and Δt = (1/c₀) · ∫ (N − 1) ds.We test this phase→metric chain using true Riemann zeros to build a smoothed phase lattice, perform a single boundary-safe lock of t* in a fixed K ≈ 2 band, and freeze both βₙ (a priori) and the independent structural scale λ_str via spatial medians at t*. No retuning is allowed. Two falsifiable checks are reported with the same frozen N on blinded segments and Monte Carlo noise scans.(i) c₀-link closure:Π(t*) = [ (c₀²/2) · median_x(∂_x ln N) ] / [ median_x(g_x^str) ] ≈ 1holds within ~10⁻³ for weak noise (e.g., 1.000000, 1.000166, 0.997957, 1.002357, 1.012819 for noise std 0, 10⁻³, 10⁻², 10⁻¹, 1).(ii) Group delay: an independent 1D Yee (Mur) solver gives Δt_sim = 6.675284e−8 s versus the eikonal prediction Δt_pred = 6.696047e−8 s, a 3.10e−3 relative error. Thus the local gradient law and the path-delay law co-hold under the same frozen N, establishing weak-field geometric–electromagnetic consistency. A unitful structural estimator for G is tracked as a diagnostic but not used to tune the closure.