When Certainty Emerges from Stochasticity: Hidden Attractor of Deterministic Motion

Abstract

Macroscopic deterministic motion is traditionally interpreted as a result of statistical averaging. In this paper, we show that it is a strict geometric attractor of the contact flow. We reveal a contact constraint mechanism where the exponential amplification of probability gradients is exactly counterbalanced by the decay of second-order contact stiffness, forcing the macroscopic-microscopic coupling to vanish. This coupling acts as a Jacobi field, which decays in dissipative systems to enable deterministic focusing. We construct the contact potential via an invariant-measure construction, unifying the treatment of point attractors, limit cycles, and chaotic systems. Unlike the Mori-Zwanzig projection, this approach strictly conserves information, showing that determinism arises from the geometric reorganisation of information rather than its loss.