Falsification of a Golden-Ratio Seismic Precursor Hypothesis: A Pre-Registered Negative Result and a Re-examination of the Depth-2 Cascade Premise

We report a pre-registered test of the hypothesis that the seismic energy cascade is governed by a depth-2 recurrence S(n) = S(n−1) + S(n−2), which would make the golden ratio φ ≈ 1.618 the cascade’s stable spectral attractor and would motivate a Seismic Precursor Index SPI(t,k) = |α(k,t) − φ| / φ. The framework’s most directly testable prediction — that the global seismicity correlation fractal dimension D converges to φ — is falsified. A global meta-analysis (USGS catalog, ~2×10⁵ events) yields D ≈ 1.23–1.30, a fractional deviation δ ≈ 0.20–0.23 that exceeds the pre-registered falsification threshold (δ > 0.10). We further show that the depth-2 premise is not derivable from the physics it invokes: elastic Coulomb stress transfer is a long-range, distance-weighted superposition (Okada-type half-space Green’s functions, ~r⁻³ far field), not an equal-weight two-term recurrence, and the general two-term recurrence admits φ only at the measure-zero point of unit coefficients, which the elasticity does not select. The φ interpretation of the seismic cascade is therefore unsupported both empirically and structurally. We separate this from the narrower, still-open empirical question — whether the seismic spectral slope deviates systematically before large events — which can be tested against archival waveform data without reference to φ, and we specify those tests. This is presented as a pre-registered negative result, reported in full.