Abstract

We separate four closure questions for the fixed-regulator coefficient inherited from X-A–Convolution, the convolution manuscript. An encoding-independent orbit lemma proves that every invariant readout of that coefficient alone is constant: its inherited gauge orbit covers the entire real input domain. The retained amplitude encoding gives an explicit finite Enter witness, including its selective-record formula and second-order obstruction.

Conditional on an independently supplied reference coefficient with a proved gauge charge, we construct a two-qubit relational readout. Its CNOT–Hadamard circuit has an invariant declared record after its even-parity outcomes are merged into a retained failure outcome. We prove the complete selective maps. Whole-algebra invariance requires an additional charge-matching hypothesis. A separately imposed bounded gauge segment gives quantitative readout bounds and an exact tolerance–visibility trade-off; no amplitude cutoff is inherited from the support condition. Sharp-regulator limits and coordinate counterterms remain excluded.

A separate comparison names the Ramp Prescription manuscript. Its classical invariant-observable identity is not an Enter theorem, and its excluded-edge defect is not the convolution coefficient $c_{\epsilon }$. No ramp identity is used as a premise of the four convolution releases.