An Endpoint Law for Rate-Controlled Temporal Seriation in Quantum Collision Models

Sequential environments can store information about a system's past, but pairwise similarities between the resulting records need not encode temporal adjacency. For a qubit system interacting successively with fresh qubit records through any excitation-preserving two-qubit unitary, the record-pair exchange correlator obeys an exact law, K_ij = ε ημ r^(i−1) t^(j−i−1), and the pairwise mutual information follows the endpoint law I(R_i:R_j) = C ε² α^(i−1) β^(j−1) + O(ε³). The collision order is non-Robinsonian for α < 1, endpoint-degenerate for α = 1 and strictly Robinsonian for α > 1. At q = 1/2 the regime is switched by the conditional phase alone, and dephasing favors seriation. With a shared phase reference, the correlator phases recover the collision order in both regimes, including N = 3, where the moduli do not identify it. We tested the law on a superconducting processor (Rigetti Cepheus-1-108Q) with three records, two operating points and decision rules specified before data acquisition. At conditional phase φ = 1.0 we measure α = 0.33 [0.22, 0.51] (95% interval; ideal 0.50), and at φ = 2.2 we measure α = 2.72 [1.96, 3.95] (ideal 1.88). At φ = 1.0 the moduli misplace the middle record in every bootstrap replicate, whereas the correlator phases identify it correctly in 99.6%. At φ = 2.2 phase recovery fails, as expected from a near-violation of the phase-separation condition. Data and code: https://doi.org/10.5281/zenodo.23356098