Split/quotient catalogues, gateway and singleton coordinates checked by literal enumeration.
Configured n=16, lambda=1/10000:
  sp: singleton lower 2.16238345e-07; RAF upper 4.56826817e-05; p=1/1146882500.
  qt: singleton lower 2.0405514e-07; RAF upper 4.29469382e-05; p=1/1146748750.
Both profiles: exact S(1e-20) < 1e-629. These are upper bounds, not the profile values.
Interior evaluation at a=1/2 stops at an explicit resource limit; it returns no probability estimate.
One shared channel mark gives survival a, showing why marginal probabilities alone do not justify history powers.