[sanity] Kingman a=1    levels 2..8 max|residual|=0  Ewens mismatches=0
[sanity] Kingman a=3/2  levels 2..8 max|residual|=0  Ewens mismatches=0
[sanity] star           levels 2..8 max|residual|=0  n/a
[sanity] star c=2       levels 2..8 max|residual|=0  n/a
[consistency] Xi-rates (2 atoms, a=1/3), levels <= 8: violations = 0
[Lambda] Lambda=delta_1/2   level-4 residual at (2,2) = -1/430 ; other residuals zero: True
[Lambda] Kingman+star       level-4 residual at (2,2) = -2/225 ; other residuals zero: True
[Lambda] two one-box atoms  level-4 residual at (2,2) = -8/1197 ; other residuals zero: True
[ThmA] level-4: 16/16 trials consistent (residual(2,2)=0 iff C4=0 iff sum q=1; hook eqs exact; q>=0); 8 trials at rho=rho* (regenerative at level 4)
[Lemma4] Xi-realizable rates: 15/15 levels exact (n=4..8)
[Lemma4] arbitrary consistent arrays (top level 8): 15/15 levels exact
[wrong-input] inconsistent array: Lemma-4 identity at n=8 holds? False (expected False)
[wrong-input] perturbed formula (W(n) for W(n-1)) at n=6 matches? False (expected False); true formula matches? True
[wrong-input] certificate with rho+1/5120: level-4 residual (2,2) nonzero? True
