Narrow scalar clocks, not level repulsion: a fourth spurious-GOE hazard and a re-audit of metallogenic timing statistics

A short audit note. The consecutive spacing ratio ⟨r⟩ is increasingly used to claim Wigner–Dyson (GOE/GUE) level repulsion in geophysical timing data. This note adds a fourth spurious-GOE hazard to the three of a companion note (declustering, seasonal gating, the Wishart null): a narrow scalar clock. A charge-and-release reservoir is a relaxation oscillator, i.e. a renewal process with independent intervals, so it has no spectral rigidity and cannot produce genuine level repulsion; yet a renewal clock with a narrow interval distribution reproduces any value of ⟨r⟩, sweeping from Poisson (0.386) through GOE (0.531) and GUE (0.603) to 0.88 purely as a function of the interval coefficient of variation. Re-auditing a published four-province metallogenic-timing study, we exactly reproduce its reported orogenic-gold ⟨r⟩ = 0.678 and show that it (i) measures within-deposit scatter among geochronometers rather than province-scale timing; (ii) is reproduced by a zero-repulsion scalar clock of CV = 0.38; (iii) has an apparent rigidity (shuffle deficit −0.031) that is an artifact of within-deposit (compositional/closed-data) normalisation, reproduced identically by a zero-repulsion null; and (iv) is unresolvable, since jittering the ages within their ±4–6 Myr uncertainties moves ⟨r⟩ across the entire Poisson-to-GUE range. The organising principle is restated: a single isolated relaxation-oscillator source yields a scalar renewal clock, not GOE level repulsion, while superposition yields Poisson; genuine Wigner–Dyson statistics require a chaotic, strongly-mixed spectrum that a low-dimensional dissipative reservoir does not possess. The superposition→Poisson result is unaffected.