Measurement-Induced Phase Transitions in Modal Triplet Theory

Measurement-induced phase transitions in monitored many-body systems and collapse-like irreversibility in measurement contexts are often treated as distinct phenomena. We show that both arise as effective four-dimensional shadows of the same reduced-dynamical mechanism in Modal Triplet Theory: noninvertible projection onto an admissible coherent sector together with basin stabilization. We formulate a general projected evolution map for reduced states, define admissible basins via contractivity margins, and prove that both measurement-induced phase transitions and collapse thresholds correspond to the same loss of contractivity at basin boundaries. Using a local Ornstein–Uhlenbeck/Kramers reduction near basin boundaries, we derive a finite-strength, nonanalytic crossover (“knee”) and establish generic protocol dependence, including Zeno and anti-Zeno regimes. We further show that linear measurement-only or decoherence-only models cannot reproduce this combined structure without introducing state-dependent stabilization equivalent to basin dynamics. All results are slab-local and admissibility-conditioned, providing a unifying reduced-dynamical bridge between monitored-circuit transitions, continuous measurement, and irreversibility.