Establish single-setting richness for the full register

Establish single-setting richness, or lower-inclusion diversity, for the full-register implementation of the squeezed-light reservoir computer, thereby proving that the character-reachability results established for the reduced model remain valid without relying on the designed mask-sweep construction.

Background

The paper proves that the gauge symmetry, charge-selection rule, ordering law, and per-slot frequency bound of the reduced single-loop model extend to the full register containing the delay-line bins, passive interferometers, losses, ancilla reset, and nonlinear transducer. These results provide the upper inclusion: full-register readout features cannot contain characters outside the admissible charge-graded class.

However, the proof explicitly distinguishes this transported upper inclusion from the lower-inclusion or richness direction, which would require showing that the full register can actually generate a sufficiently diverse set of admissible characters under a single operating setting. The reduced model obtains such reachability constructively through designed mask sweeps, but the full-register argument leaves the corresponding single-setting diversity as an unproved hypothesis.