Collision Operators and Sector Transitions: A Quantum Mechanical Formalism for Hamiltonian Collisions

The φ-locked framework describes the emergence of stable physical laws via inflationary selection on a direct-sum Hilbert space of incompatible Hamiltonian sectors. While the framework includes a decoherence formalism, ensemble flow, and a gradient flow for couplings, the fundamental collision event — the transition from two parent sectors to a daughter sector — has lacked an explicit operator. This paper supplies that missing operator formalism. We define an inter-sector coupling Hamiltonian Ĥcoll that directly couples distinct sectors when their worldvolumes overlap, with coupling strength derived from the known decoherence rate. We promote the gradient flow's incompatibility term to an operator Ĉᵢⱼ = (ĝ_i - ĝ_j)^2 and build a Fock space over the direct-sum Hilbert space, interpreting a_i^ as creating a new disconnected causal patch. The collision amplitude is written as a time-ordered matrix element, and the form factor is identified from the influence functional's decoherence weight. The complete collision operator recovers the gradient flow in the semi-classical limit. Unitarity is proved. A worked example demonstrates the formalism. Geometric embedding of the overlap state is deferred to future work.