Counterexample CE-E-GEOMETRY-001
What it falsifies
At a finite global E optimum with a simple minimum eigenvalue, the rank-one metric vv^T built from the minimum eigenvector assigns every training row to its own cell (nearest-cell rule).
- Cell 0
- Cell 1
- Cell 2
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/8 | -0.226534 | 0.428773 | 0 | 0 |
| 2 | 1/8 | -0.629944 | -1.223406 | 1 | 1 |
| 3 | 1/8 | 1.253439 | -0.109445 | 1 | 1 |
| 4 | 1/8 | 1.807897 | 0.734952 | 2 | 2 |
| 5 | 1/8 | -1.520937 | -0.061786 | 0 | 0 |
| 6 | 1/8 | -0.488606 | -0.002247 | 0 | 0 |
| 7 | 1/8 | 0.710355 | 1.154412 | 0 | 0 |
| 8 | 1/8 | -0.905669 | -0.921253 | 1 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | 0.2805833814266841 |
| Labels after | 0.2805833814266841 |
How it was verified
Method: high_precision.
Exhaustive enumeration of all canonical 8-row/3-cell partitions; global E optimum (max lambda_min) is unique with spectral gap ~ 0.27480 (so the minimum eigenvector is well-defined), yet row 7 is strictly closer to cell 2 than to its own cell 1 under the vv^T rank-one metric, margin ~ 0.06796. Regression: tests/test_research_claims.py::test_global_e_partition_can_violate_simple_eigenvector_rule.
Claims that cite it
Recorded 28 August 2026. The fixture as JSON.