Direct proof of vanishing interval widths for nearly good numbers in the infinite-particle limit

Prove that, for the two-species spin-1 boson system with an odd interspecies channel leading to non-commutable spin-dependent terms, the widths of the intervals defining the nearly good real numbers that specify the ground-state averages of the combined spins (\overline{S_A} and \overline{S_B}) tend to zero as the particle numbers N_A and N_B tend to infinity, thereby rigorously justifying the use of nearly good real numbers to replace good quantum numbers in specific parameter regions.

Background

The authors numerically observe that the averages \overline{S_A} and \overline{S_B} vary stepwise with interaction parameters and reside within very narrow intervals between critical points. Increasing particle numbers substantially reduces these interval widths, suggesting they vanish in the infinite-particle limit.

Based on these observations, the authors infer that specific parameter regions permit eigenstate specification by nearly good real numbers instead of conserved quantum numbers. However, they emphasize that this inference requires a direct mathematical proof.