Emergent Temporal Compatibility in Unitary Dynamics

Abstract We investigate whether an operational temporal-compatibility scale, Δτ_compat, can emerge in a finite-dimensional quantum system whose global evolution remains strictly unitary. For two conditional interferometric branches separated by a relative evolution parameter δτ, Δτ_compat is defined as the first value at which the visibility falls below the reference level e^−1. This first-passage scale is distinguished from the persistence of visibility suppression, which is assessed independently through the prevalence of threshold crossings, the statistical stability of Δτ_compat, and the amplitude of late-time revivals.A systematic sweep of 192 parameter combinations, comprising 4800 independent unitary realisations, produces 21 reversible, 141 crossover, and 30 post-compatibility groups. In the post-compatibility groups, the threshold is crossed in all realisations, the extracted values of Δτ_compat are statistically stable, and late-time visibility recovery remains weak throughout the 5 ms observation interval. The 5 ms duration is a fixed observation window rather than a fitted dynamical scale, and the regime classification is conditional on that interval. This regime occurs only at the two largest tested internal dimensions, although dimension alone is insufficient to determine the classification.A second analysis reconstructs the energetic quantities associated with the same realisations and tests which scale organises Δτ_compat. The strongest relation is obtained from the differential energy between the two conditional branch evolutions. A free power-law fit gives an exponent of 1.123, with a 95% bootstrap interval of 0.955-1.296 and R2 = 0.771, consistent with an approximate inverse energy-time organisation of Δτ_compat. Control scales based on the common internal generator and on the complete branch Hamiltonians provide substantially weaker organisation.The dimensionless product formed from the differential energy and Δτ_compat is not universal and varies systematically between internal dynamical families. The results therefore identify Δτ_compat as a system-dependent operational scale whose dominant timing dependence is set by the differential branch energy, while the internal unitary dynamics supply trajectory-dependent corrections and govern the later recoverability of interferometric visibility.