d=3 saturated invalidation check
p=0.1
  alpha theory/sim = 0.271000 / 0.270261
  cov1  theory/sim = 0.124659 / 0.124463
  cov2  theory/sim = 0.059049 / 0.058937
  cov3  sim         = 0.000009 (theory 0)
p=0.3
  alpha theory/sim = 0.657000 / 0.657755
  cov1  theory/sim = 0.122451 / 0.122222
  cov2  theory/sim = 0.050421 / 0.050275
  cov3  sim         = 0.000068 (theory 0)
p=0.5
  alpha theory/sim = 0.875000 / 0.876105
  cov1  theory/sim = 0.046875 / 0.046194
  cov2  theory/sim = 0.015625 / 0.015212
  cov3  sim         = -0.000160 (theory 0)
p=0.8
  alpha theory/sim = 0.992000 / 0.992059
  cov1  theory/sim = 0.001536 / 0.001530
  cov2  theory/sim = 0.000256 / 0.000255
  cov3  sim         = -0.000001 (theory 0)
