Exact counts and fixed-weight nesting

These are the fixed-weight identities recorded in Psy01-A CheungPsy01A. Training between operations can change the ranking. For a frozen late-cutoff gap $\Gamma >0$, a perturbation with $2\text{norm}{\delta w}_{\infty }<\Gamma$ preserves the selected support, because every cross-cutoff magnitude ordering is preserved.

For corrected regrowth, we use

This makes $r=0$ exactly the identity structural operation and $r=1$ restore all inactive positions. Integer discontinuities remain; repairing the zero endpoint does not make hard regrowth smooth.