The Harmonic Early-Warning Theorem: Tipping-Point Monitoring as a Thermodynamic Detection Problem

Paper supported by machine-verified Lean 4. The Harmonic Early-Warning Theorem (HEWT) recasts tipping-point monitoring as a thermodynamic detection problem and explains, from physics, a notorious failure of the ecological early-warning-signal (EWS) literature: spurious alarms exactly near the threshold. Its results: (i) a fluctuation–dissipation identity making the monitored variance equal to the system's susceptibility; (ii) a detection dissipation floor of 2 k_B ln 2 per bit of early warning; (iii) the headline observability wall — the detection statistic peaks not at the threshold but at an interior optimal detection distance ε_opt=ε₀·3^(−1/m), and the signal-to-noise ratio degrades as the threshold is approached (the physical origin of the spurious-alarm problem); (iv) a wu-wei active↔passive sensing crossover; and (v) detection efficiency equal to cos²Θ. This version (1.0.0) deposits the clean core (11 machine-verified theorems). Part of the S2 Monitoring series; derived from the Viridis Intelligence-Bound canon, concept DOI 10.5281/zenodo.19317982.