Critical-exponent stratification and inverse realization on biregular trees

For free type-preserving discrete actions on the biregular tree $\mathcal T_{r+1,s+1}$, we stratify the critical-exponent spectrum by quotient complexity.

The unrestricted spectrum is the full interval $[0,\frac12\log(rs)]$, whereas the finitely generated spectrum is countable and dense and is encoded by the Hashimoto radii of finite typed cores.

At fixed rank, finitely many typed kernels parametrize all values, and every nonzero accumulation belongs to a lower-rank stratum.

Rank two admits a complete effective inverse classification through the figure-eight, theta, and dumbbell polynomial families.