Counterexample CE-O6-ETA-ZERO-MULTIATOM-VARIANCE-001
What it falsifies
The influence variance sigma^2 of the frozen-rule scalar retention plug-in vanishes only when S | Z = b is supported on at most two atoms per cell; in particular an atomless (or many-atom) cell of positive probability implies sigma^2 > 0 (O6.4 wording and the A4 parenthetical of RETENTION-PLUGIN-CLT-FROZEN-SCALAR as recorded on 5 September 2026).
- Cell 0
- Cell 1
| Atom | Weight | Score 1 | Cell before |
|---|---|---|---|
| 1 | 1/8 | -3 | 0 |
| 2 | 1/8 | -1 | 0 |
| 3 | 1/8 | 1 | 0 |
| 4 | 1/8 | 3 | 0 |
| 5 | 1/4 | -2 | 1 |
| 6 | 1/4 | 2 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | eta = 0 |
How it was verified
Method: exact_formula.
Every cell mean is zero, so eta = 0 and psi(s, b) = ((1 - 0) s^2 - (s - 0)^2)/v = 0 for every s: the root equation (s - c_b)^2 = (1 - eta) s^2 is the identity at eta = 0 and has no two-atom restriction. Cell 0 carries four atoms. The same holds for any atomless S with a partition whose cells all have conditional mean zero (e.g. cells symmetric about 0), see AUDITS/artifacts/AUDIT-SCORE-ORACLE-ROBUSTNESS-001/coverage.json law atomless_eta_zero. Correct statement: for 0 < eta <= 1 the support is at most two atoms per cell; for eta = 0 every law has sigma^2 = 0; (A4) is automatic iff eta > 0 and some positive-probability cell is atomless.
Claims that cite it
Recorded 5 September 2026. The fixture as JSON.