A Finite Refinement Kernel for Quantum-Like Interference in the Finite Relational Closure Framework

This manuscript introduces a finite candidate generator for contribution transport within the Finite Relational Closure Framework (FRCF). Prior work formulated the refinement-closure generator problem as the question of how a contribution-bearing admissible structure at one context should be related to a corresponding structure at a refined or transformed context. The present manuscript provides a restricted constructive example by defining a normalized phase-weighted refinement kernel supported only on admissible refinement pairs. For a one-step refinement, the kernel transports a coarse contribution to its admissible refinements using equal square-root normalization and a finite phase factor. The normalization preserves local squared contribution under equal splitting, while the phase factors allow refined assignments to interfere when later grouped into common measurement-induced outcome classes. Two finite toy models illustrate the construction. In the two-branch case, unresolved refined assignments produce a phase-dependent intensity term, while distinguishable assignments produce separate intensities without a cross term. A multi-outcome example shows how different finite measurement partitions produce different interference structures from the same transported contribution data. The model is not proposed as a complete quantum dynamics. It does not derive the Schrödinger equation, a Hamiltonian, a path integral, full unitarity, decoherence, field theory, or empirical predictions. Its purpose is narrower: to show that the FRCF generator requirements are constructively satisfiable in a finite toy setting and that admissible refinement, phase-weighted contribution transport, and measurement aggregation can produce quantum-like interference.