Family I: all requirements at p=1/2 = 9/16; at least one = 15/16.
Family II: all requirements at p=1/2 = 1/2; at least one = 7/8.
Exact polynomial difference: q^3 - q^4 = p(1-p)^3.
Its maximum is 27/256 at p=1/4 (differentiate p(1-p)^3).
Dedicated channels: Bernoulli union = 63/64; at fixed Q=2, union = 17/22, product of fixed-Q marginals would give 7273/10648 (not equal).
Both inclusion-exclusion levels agree with literal enumeration; fixed-Q coefficients agree for every Q in each checked catalogue.
Channel counts, n=2..10: 4, 18, 64, 188, 502, 1264, 3048, 7130, 16334
Food-accessible counts, n=2..10: 4, 18, 34, 34, 34, 34, 34, 34, 34
Restricted n=4 system: checked all 65,536 relevant assignments; RAF definition, catalogue activation, and pruning agree.
Its exact RAF probability at p=1/2 is 65279/65536; all 61 fixed-Q coefficients on the 60 ambient coordinates agree.
A catalysed entry without a RAF, and a two-channel RAF, both verified.
Gateway bound at f=1.3:
  n=   6: 0.999985014
  n=  10: 0.996060576
  n=  64: 0.509781087
  n= 128: 0.295870965
  n= 512: 0.083017294
  n=1024: 0.042326652
  n=4096: 0.010738217
Linear-scale example λ=0.03:
  n=   6: 0.784839513
  n=  64: 0.651076726
  n=4096: 0.639584696
  limiting gateway bound: 0.639405060
All exact finite checks passed. General asymptotic claims rely on the paper.
