Operational Bounds for a Ten-Coordinate Phase-Echo Benchmark

This technical note presents a four-qubit implementation of a ten-coordinate phase-echo benchmark. Two coherent preparations have identical intermediate basis populations but yield different receiver probabilities under the same continuation. We verify 36 OpenQASM circuits against rational reference predictions and an independent matrix simulation. We provide a finite-sample criterion for testing a population-only model and illustrate sensitivity to gate noise. The accompanying archive contains the circuits, code, and simulation outputs. All results are mathematical or simulated.