Theorem: Capacity crossover before a positive cutoff

For the supplied laws above:

1. $A(M(t))$ increases strictly and $B(M(t))$ decreases strictly;

2. there is exactly one crossover in the retained clock interval if and only if

   $$A(M_{\mathrm{cut}})\ge B(M_{\mathrm{cut}});$$

3. a strict interior turnover occurs if and only if that inequality is strict;

4. if $A(M_{\mathrm{cut}})<B(M_{\mathrm{cut}})$, the envelope is increasing throughout the interval.

The crossover is a capacity-envelope time, not automatically a physical recovery transition or the exact maximum of a finite Page mean.

Proof. Differentiating gives

The difference $A-B$ starts at $-B(M_0)$, increases strictly in time, and is continuous. The intermediate value theorem proves existence and uniqueness exactly under the endpoint condition. The minimum selects the increasing branch before the crossing and the decreasing branch afterward.