Counterexample CE-O7-ELLIPSOID-ZERO-VARIANCE-001
What it falsifies
The scalar characterisation of sigma^2 = 0 in RETENTION-PLUGIN-CLT-FROZEN-SCALAR (A4) lifts to vector scores: for 0 < eta_D < 1 the influence variance of the frozen-rule geometric-mean retention plug-in vanishes only for laws with at most two atoms per cell, so an atomless cell of positive probability implies sigma^2 > 0 whenever eta_D > 0.
- Cell 0
- Cell 1
- Cell 2
- Cell 3
| Atom | Weight | Score 1 | Score 2 | Cell before |
|---|---|---|---|---|
| 1 | 1/8 | 3 | 4 | 0 |
| 2 | 1/8 | 3 | -4 | 0 |
| 3 | 1/8 | -4 | 3 | 1 |
| 4 | 1/8 | 4 | 3 | 1 |
| 5 | 1/8 | -3 | -4 | 2 |
| 6 | 1/8 | -3 | 4 | 2 |
| 7 | 1/8 | 4 | -3 | 3 |
| 8 | 1/8 | -4 | -3 | 3 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | eta_D = 9/25 (det ratio 81/625) |
How it was verified
Method: exact_formula.
E[S] = 0, p_b = 1/4, V = (25/2) I, I_Z = (9/2) I, det ratio 81/625, eta_D = 9/25. psi = (eta_D/2)[2 s^T I_Z^{-1} c_b - c_b^T I_Z^{-1} c_b - s^T V^{-1} s] = 0 at every atom, and as a polynomial in s it is a multiple of the circle |s - (25/3, 0)|^2 = (20/3)^2, the sigma^2 = 0 ellipsoid of O7.4. Hence every law on that circle with mean (3, 0) - the two-atom law and the atomless arc law alike - has sigma^2 = 0 at eta_D = 9/25. Correct statement: sigma^2 = 0 iff S | Z = b is supported on the cell's ellipsoid for every cell; absolutely continuous laws are excluded, atomless singular laws are not.
Claims that cite it
Recorded 6 September 2026. The fixture as JSON.