The Uniqueness of Period-2 Dynamics: Deterministic Constraints on Binary Partitioned Systems

The organizing term ξ introduced in the companion paper defines a binary state space {0,1}for any physically distinguishable system. In this manuscript we ask: given this binary state space, what are the possible deterministic dynamical rules? We prove by complete enumeration that there are exactly four deterministic maps on {0,1}, two of which are fixed points (physically corresponding to Heat Death and Static Eternity) and one of which is the identity (trivial). The fourth and only non-trivial map is the Period-2 alternation f(0) = 1, f(1) = 0, corresponding to the 0 →1 →0 →1 oscillation. We demonstrate that this Period-2 map is the unique deterministic evolution consistent with (i) non-triviality (the system changes state), (ii) persistence (the system does not collapse permanently to either state), and (iii) information preservation (the map is invertible). We further show that this Period-2 structure corresponds physically to the slow oscillation of the scalar-imprint field Φ between the organizing and dissolving phases, and provides the foundational justification for treating cosmological time as the sequenced realization of the binary partition’s persistence and termination. The Period-2 motif is identified as the structural origin of wave-like behaviour and periodic recurrence throughout fundamental physics.