Introduction, provenance, and the Enter distinction

IX-A separates three mathematical tasks: admitting an exact distributional functional, choosing a topology on its carrier, and constructing a quantum product on that carrier CheungIXA. The present paper addresses the first two on a supported regular presentation of the conical product used in I-B CheungIB. It then supplies a finite clock–evaluation–transport interface. The latter does not solve the third task.

The operational background is the Enter discipline. Quantum names specify access to declared tensor factors; destructive readout consumes the measured names and returns a classical record with specified surviving factors CheungEnter,CheungPhilosophy. This interpretation distinguishes a readout from recovery of an unknown quantum state. It also distinguishes a channel identity from a thermodynamic reset law and a mathematical analogy from a physical identification.

We specify finite Born instruments directly. We do not rederive the measurement-characterization theorem of the Enter calculus, enlarge its hypotheses to continuum fields, or infer a spacetime metric from its operational conditions.