P(R=1) = 0.0550;  P(last token = 0) = 0.2321;  P(R=1 and last token = 0) = 0.0034
|g| = 0.04294

(i)  constant a=0.9, f=0.1                   |gv|/|g| =  0.8000   cosine =  1.0000   predicted ratio 0.800
(ii) constant a=0.7, f=0.3                   |gv|/|g| =  0.4000   cosine =  1.0000   predicted ratio 0.400
(ii') constant a=0.5, f=0.5 (uninformative)  |gv|/|g| =  0.0000   cosine =     n/a   predicted ratio 0.000
(iii) f(y)=0.5 if last token 0 else 0.02     |gv|/|g| =  2.8875   cosine = -0.6194

decomposition of the verifier gradient in (iii): gv = (true-positive part) + (false-positive part)
   |true-positive part| = 0.04010   |false-positive part| = 0.15267
   cosine(false-positive part, g) = -0.7703
   cosine(false-positive part, grad P(wrong and last token 0)) = 1.0000
