== part e: equality in (c) at single atoms K = delta_x, 1 <= x <= sqrt 2
PASS E|Z_delta_x| = g(1/x^2) at x = 1 (|diff| < 2^-200; E1 agrees within its tail)
PASS E|Z_delta_x| = g(1/x^2) at x = 1.1 (|diff| < 2^-200; E1 agrees within its tail)
PASS E|Z_delta_x| = g(1/x^2) at x = 1.2 (|diff| < 2^-200; E1 agrees within its tail)
PASS E|Z_delta_x| = g(1/x^2) at x = 1.3 (|diff| < 2^-200; E1 agrees within its tail)
PASS E|Z_delta_x| = g(1/x^2) at x = 1.4142 (|diff| < 2^-200; E1 agrees within its tail)
PASS E|Z_delta_x| = g(1/x^2) at x = 1.41421356 (|diff| < 2^-200; E1 agrees within its tail)
PASS x = sqrt 2: only N = 0 has S < m = 1/sqrt 2, E|Z| = 2 e^(-1/2)/sqrt 2 = g(1/2) = sqrt(2/e)
PASS delta_0.9: m = 1/0.9 > 0.9, so lambda+_m = 0 and (c) does not apply (x < 1)
== part n: negative tests
PASS theta = 0.5 < m = 0.94872 (atom 0.6 in [theta, m)): (a) fails, E|Z| = 0.816600 vs 0.678124, so theta >= m is needed
PASS scope (i): K = 0.95 d1.4 + 0.05 d0.05: m = 1.67857 > 1.4, lambda+_m = 0, E|Z_K| = 0.836111 > g(0) = 0
INFO RQ-A J2 maximiser (rounded): K = 0.99 d1.49 + 0.01 d0.68: Lambda = 0.46755 < 1/2, E|Z_K| = 0.8510029, B = 0.8568202, g(Lambda) = 0.8568202 (outside (c))
== part x: cross-validation E1 (sign-free, tail) = E2 (finite sum)
PASS E1 = E2 on 200 random K with <= 3 atoms (agreement within the E1 tail, < 2^-99)
== part a: identity (a), both forms, against the sign-free E1, for random theta >= m
PASS (a) both forms equal E|Z_K| on 300 random (K, theta), theta in {m, an atom above m, m + U(0, 0.5)}
== part b/c: (b) at theta = m and a random theta >= m, (c) at theta = m when lambda+_m >= 1/2 (20000 K, E2); equality is required exactly when K is one point mass with rho^2 = 0, strict inequality otherwise
PASS (b): 40000 (K, theta) pairs: equality to 2^-200 in the 5440 one-point cases, B - E|Z_K| > 0 in the other 34560; smallest 3.64e-10 at 0.9997 d1.166 + 0.0003 d1.164, theta - m = 0 (Lambda 0.7355, rho^2 0)
PASS (c): 12074 K with lambda+_m >= 1/2: equality to 2^-200 in the 2304 one-point cases, g(lambda+_m) - E|Z_K| > 0 in the other 9770; smallest 3.64e-10 at 0.9997 d1.166 + 0.0003 d1.164 (Lambda 0.7355, rho^2 0)
== part s: scope (ii): B(Lambda, rho^2) > max(g(Lambda), C(q5)); B is concave in rho^2 (sum of square roots of affine functions), so the two endpoints decide each stated interval
INFO C(q5) = g(2 ln(5/4)) = 0.855100455470819
PASS Lambda = 0.49: claimed [0.001, 0.078]; B > max at the left end: True, at the right end: True; exact set {B > max} = (0.000000, 0.078400); grid points (step 0.001) 0.001 .. 0.078
PASS Lambda = 0.49: max = g(Lambda) and {B > g(Lambda)} = (0, 8 Lambda (1 - 2 Lambda)) = (0, 0.0784); B = g(Lambda) exactly at the right end (to 2^-200)
REMARK Lambda = 0.45: claimed [0.001, 0.36]; B > max at the left end: True, at the right end: False; exact set {B > max} = (0.000000, 0.360000); grid points (step 0.001) 0.001 .. 0.359 -- the stated interval is rounded outward
PASS Lambda = 0.45: max = g(Lambda) and {B > g(Lambda)} = (0, 8 Lambda (1 - 2 Lambda)) = (0, 0.36); B = g(Lambda) exactly at the right end (to 2^-200)
PASS Lambda = 0.3: claimed [0.21, 0.86]; B > max at the left end: True, at the right end: True; exact set {B > max} = (0.204736, 0.861126); grid points (step 0.001) 0.205 .. 0.861
PASS Lambda = 0.005: claimed [0.67, 1.0]; B > max at the left end: True, at the right end: True; exact set {B > max} = (0.668578, 1.000000); grid points (step 0.001) 0.669 .. 1.0
== part h: hostile float search (Nelder-Mead from 60 feasible starts per case), certified re-evaluation at the rationalised maximiser
PASS hostile (c) J=2 rho^2 >= 0.01 (60 starts): float sup of E|Z| - bound = -3.652e-03; certified bound - E|Z| = 0.00365 at 0.01 d0.6936 + 0.99 d1.002 (Lambda 0.98561, rho^2 0.01)
PASS hostile (c) J=3 rho^2 >= 0.01 (60 starts): float sup of E|Z| - bound = -3.650e-03; certified bound - E|Z| = 0.00365 at 0.003272 d0.2978 + 0.006728 d0.9786 + 0.99 d1.004 (Lambda 0.98221, rho^2 0.01)
PASS hostile (c) J=2 rho^2 = 0, two separated atoms (60 starts): float sup of E|Z| - bound = -2.391e-05; certified bound - E|Z| = 2.39e-5 at 0.95 d1.412 + 0.05 d1.462 (Lambda 0.50000, rho^2 0)
PASS hostile (b) J=2 rho^2 >= 0.01 (60 starts): float sup of E|Z| - bound = -1.111e-05; certified bound - E|Z| = 1.11e-5 at 0.01 d1.401 + 0.99 d1.42, theta - m = 0.6963 (Lambda 0.49083, rho^2 0.01)
PASS hostile (b) J=3 rho^2 >= 0.01 (60 starts): float sup of E|Z| - bound = -1.134e-05; certified bound - E|Z| = 1.13e-5 at 0.001852 d1.387 + 0.99 d1.43 + 0.008148 d1.399, theta - m = 0.7311 (Lambda 0.48387, rho^2 0.01)
PASS hostile (b) J=2 rho^2 = 0, two separated atoms (60 starts): float sup of E|Z| - bound = -6.369e-06; certified bound - E|Z| = 6.37e-6 at 0.95 d2.48 + 0.05 d2.53, theta - m = 0.7362 (Lambda 0.16233, rho^2 0)
time 33 s
ALL CHECKS PASSED
