Counterexample CE-AUC-INVARIANT-PROXY-RETENTION-001
What it falsifies
A proxy score with the same ranking as the true score (hence the same ROC curve and AUC for every threshold) reports the true retention. Refuted: two strictly monotone distortions of a four-atom scalar law leave the labels of corresponding rank-cut threshold rules (including the fixed zero cut here) and the true retention 9/10 unchanged while the reported (proxy) retention is 100/101 for one and 5105/10006 for the other; ranking quality says nothing about the reported number, the Fisher-whitened score error does (SCORE-ERROR-RETENTION-BUDGET), and it is not observable from a ROC curve.
- Cell 0
- Cell 1
| Atom | Weight | Score 1 | Cell before |
|---|---|---|---|
| 1 | 1/4 | -2 | 0 |
| 2 | 1/4 | -1 | 0 |
| 3 | 1/4 | 1 | 1 |
| 4 | 1/4 | 2 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | true retention 9/10 (V = 5/2, I_Z = 9/4) |
| Labels after | reported retention 100/101 (compress) and 5105/10006 (stretch) with the same labels |
How it was verified
Method: exact_formula.
fractions.Fraction; the distortions are strictly increasing, so the ROC curve of any threshold classifier built on the proxy equals that of the truth; the B1/B2 brackets of SCORE-ERROR-RETENTION-BUDGET hold on both (checks_all_true) - the stretch has eps_R^2 = 4802/9 > 1, outside the lower bracket's domain, and the reported number is indeed far from the truth.
Claims that cite it
- Posterior error to score error for mixture-fraction scores: one Lipschitz inequality, and the excess Brier score as the classifier requirementCLASSIFIER-CALIBRATION-SCORE-LIPSCHITZ
- Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score errorSCORE-ERROR-RETENTION-BUDGET
Recorded 8 September 2026. The fixture as JSON.