The Triadic Coherence Condition

Abstract

We establish the Triadic Coherence Condition (TCC): any information system that remains coherent under continuous drift must implement three irreducible roles - Potentiality (P), Actuation (A), and Convergence (Φ). The necessity result follows from the Belief Update Lemma, which shows that coherence-restoring updates require the current belief state, the latest action, and the incoming environmental signal. Sufficiency is obtained through a conditional Borromean cascade stability theorem under four explicit ISS hypotheses. We introduce the Triadic Coherence Index (TCI), a computable bottleneck-sensitive diagnostic based on component coherence scores and debiased Sinkhorn tracking. The resulting coherence-delay trade-off yields a cube-root law, E_min ∝ ζ^(1/3), showing that perfect coherence is impossible under continuous drift when estimators have finite effective memory. We further define an effort-corrected TCI, TCI^E, to capture coercive masking in action-effective systems. Empirically, the Triadic Gaussian Tracker validates the ablation structure and the slow-drift scaling law, while the Triadic Particle Tracker extends the result to non-Gaussian, non-linear environments and exposes coercive masking. On passive ELEC2 and Bikes benchmarks, ADWIN remains a strong generic detector on residual streams. On the KuaiRand logged benchmark, TCI^E improves bubble detection from 45.8% to 69.2% and collapse detection from 54.8% to 70.0% on 367 eligible users, while ADWIN remains near zero on the raw watch-ratio stream. Together, these results position TCC as a structural theory of drift-coherent systems and TCI^E as a practical diagnostic for control-induced coherence under finite memory.

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