Minimal Causal Order in a Coupled Binary Recurrent System

This preprint establishes a constructive existence result for a strict pairwise clamp-constitutive causal-order relation in a finite deterministic binary product-coordinate system. The primary witness is T(a,b) = (a XOR b, a). Under explicitly defined local actual-context causal efficacy, time-unrolled phase-wise recurrent closure, and constant-coordinate clamp interventions, the A organization is necessary for recurrence of B while B is not reciprocally necessary for recurrence of A. An analytic lemma excludes qualifying joint orbits of period one or two, and exhaustive enumeration of all 256 deterministic maps on the four-state binary product domain identifies exactly four qualifying maps. The deposit includes the main manuscript, a technical verification dossier that freezes the predicates and audit criteria, and the exact executable Python enumeration script used for the reported computational check. The result is class-relative and does not claim universal physical minimality, closure-specific necessity under replacement drive, exact trajectory identity across regimes, or endogenous generation of a new causal order.