Counterexample CE-O7-UNIT-RETENTION-SINGULAR-SAMPLE-001
What it falsifies
At eta_D = 1 (S = c_Z almost surely) the frozen-rule geometric-mean retention plug-in eta_hat_D = (det I_hat_Z / det V_hat)^(1/d) equals 1 on every sample.
- Cell 0
| Atom | Weight | Score 1 | Score 2 | Cell before |
|---|---|---|---|---|
| 1 | 1/2 | 1 | 0 | 0 |
| 2 | 1/2 | 1 | 0 | 0 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | exact plug-in eta_hat_D = 0 (det V_hat = 0, phi := 0); library geometric_mean_retention = 1 on the retained rank-1 subspace |
How it was verified
Method: exact_formula.
V_hat = diag(1, 0) is singular, so the estimator's own everywhere-defined functional phi returns 0 by its det V_hat = 0 convention while the population value is 1; the library projects the null direction out (rank_rtol) and reports the ratio on the retained subspace, which is 1. Corrected statement: at eta_D = 1 the plug-in is 1 on every sample with V_hat positive definite, i.e. whenever the occupied cells' means span R^d; the singular event has probability at most sum_b (1 - p_b)^n ... P(fewer than d spanning cells occupied) -> 0 exponentially under A1, so the CLT statement is unaffected, only the 'every sample' sentence.
Claims that cite it
Recorded 6 September 2026. The fixture as JSON.