Counterexample CE-D-LLOYD-001
What it falsifies
Reassigning all rows to their nearest centroid under the current I_q^{-1} Mahalanobis metric, then recomputing centroids, monotonically increases log det I_q.
- Cell 0
- Cell 1
- Cell 2
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/8 | 0.1116 | 0.4427 | 1 | 0 |
| 2 | 1/8 | -0.2932 | 0.6537 | 0 | 0 |
| 3 | 1/8 | -0.5995 | -1.2685 | 0 | 1 |
| 4 | 1/8 | -0.6848 | -1.5456 | 1 | 1 |
| 5 | 1/8 | 0.4810 | 0.9521 | 2 | 2 |
| 6 | 1/8 | 1.6707 | 0.9370 | 2 | 1 |
| 7 | 1/8 | 0.1689 | 1.7090 | 2 | 0 |
| 8 | 1/8 | -0.8548 | -1.8805 | 1 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | -3.810643191735313 |
| Labels after | -3.9471643803321044 |
How it was verified
Method: exact_formula.
Scores are exact decimals (rationals). The batch adaptive-Mahalanobis Lloyd step (nearest centroid under the exact inverse of the current binned information, argmin decided in Fraction arithmetic) produces labels_after_or_optimum, and the exact determinant strictly decreases: det_before > det_after. logdet drop ~ 0.136521 nat. Regression: tests/test_research_claims.py::test_adaptive_mahalanobis_lloyd_step_can_decrease_d_objective.
Claims that cite it
Recorded 28 August 2026. The fixture as JSON.