Counterexample CE-SCORE-ERROR-RHO-MIN-NECESSARY-001
What it falsifies
The relative/log reporting gap between the reported (proxy) geometric-mean retention and the true retention of the same frozen label map admits a bound that depends only on an upper bound on the Fisher-whitened score error epsilon, uniformly over laws and rules. Refuted: in the family below epsilon^2 = kappa^2 / (10 V_22) = kappa^2/(9+delta^2) stays within [0.026, 0.028] while the proxy-to-true retention ratio grows without bound as the least retained eigenvalue rho_min -> 0 (delta -> 0). The retention-weighted error epsilon_R^2 = E[e_Z^T I_Z^{-1} e_Z] <= epsilon_Z^2 / rho_min of SCORE-ERROR-RETENTION-BUDGET is the right scale.
- Cell 0
- Cell 1
- Cell 2
| Atom | Weight | Score 1 | Score 2 | Cell before |
|---|---|---|---|---|
| 1 | 1/4 | 1 | 1 | 0 |
| 2 | 1/4 | 1 | -1 | 0 |
| 3 | 7/40 | -1 | 1 | 1 |
| 4 | 9/40 | -1 | -1 | 1 |
| 5 | 1/10 | -1 | 1/2 | 2 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | true det ratio 5/148 (eta_D = 0.183804) |
| Labels after | proxy det ratio 83/798 (eta_tilde_D = 0.322506) |
How it was verified
Method: exact_formula.
All moments in fractions.Fraction; E[s] = 0 exactly in every member; eta_D and the ratio are square roots of exact rationals (d = 2). The B1/B2 inequalities of SCORE-ERROR-RETENTION-BUDGET hold on every member (checks_all_true); only their epsilon-alone reading fails. For fixed kappa>0 and delta->0, I_Z,22=delta^2/8, V_22=(9+delta^2)/10; det I_tilde_Z tends to 9 kappa^2/100 and det V_tilde tends to 9/10+9 kappa^2/100. Thus the ratio diverges while eps<=kappa/3. This refutes relative/log control, not the universal absolute bound |eta_tilde-eta|<=1.
Claims that cite it
Recorded 8 September 2026. The fixture as JSON.