(1) identity check: |finite-difference dH/dt - formula|, 1000 random (z, r) pairs
   vanilla: max error 3.04e-10
   natural: max error 5.08e-10

(2) entropy (nats) and P(rewarded action) along the flows; log 20 = 2.996
   pi_0(c) = 0.4   vanilla  max entropy 2.440 at t = 0;  t=0: H=2.440 p=0.400  t=20: H=0.437 p=0.934  t=100: H=0.089 p=0.990  t=200: H=0.046 p=0.995  t=400: H=0.024 p=0.998
   pi_0(c) = 0.4   natural  max entropy 2.440 at t = 0;  t=0: H=2.440 p=0.400  t=2: H=0.951 p=0.831  t=4: H=0.202 p=0.973  t=6: H=0.035 p=0.996  t=12: H=0.000 p=1.000
   pi_0(c) = 0.01  vanilla  max entropy 2.996 at t = 79.2;  t=0: H=2.971 p=0.010  t=20: H=2.975 p=0.013  t=100: H=2.606 p=0.330  t=200: H=0.090 p=0.989  t=400: H=0.032 p=0.997
   pi_0(c) = 0.01  natural  max entropy 2.996 at t = 1.65;  t=0: H=2.971 p=0.010  t=2: H=2.992 p=0.069  t=4: H=2.549 p=0.355  t=6: H=1.076 p=0.803  t=12: H=0.007 p=0.999

natural flow closed form: P(c) at t = 6 from pi_0 = 0.01: simulated 0.80296, predicted sigma(logit(0.01) + 6) = 0.80296
