Counterexample CE-DS-POP-WASTED-CELLS-001
What it falsifies
A first-order stationary population profiled-Ds quantizer has pairwise-distinct efficient-projected centroids, so its cells can always be recovered from a common efficient-semimetric nearest-cell rule.
- Cell 0
- Cell 1
- Cell 2
- Cell 3
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/8 | -3 | 1 | 0 | 0 |
| 2 | 1/8 | -3 | -1 | 1 | 1 |
| 3 | 1/8 | -1 | 2 | 0 | 0 |
| 4 | 1/8 | -1 | -2 | 1 | 1 |
| 5 | 1/8 | 1 | 1 | 2 | 2 |
| 6 | 1/8 | 1 | -1 | 3 | 3 |
| 7 | 1/8 | 3 | 2 | 2 | 2 |
| 8 | 1/8 | 3 | -2 | 3 | 3 |
Consequences
population stationarity does not force pairwise-distinct projected centroids: the partition satisfies its own efficient-semimetric nearest-cell rule everywhere (with ties) yet no such rule can separate the coincident-centroid cell pairs
stationary Ds partitions can carry profiled-information-free cells; the deployability characterization must merge coincident projected centroids
the same symmetry argument applies verbatim to any nuisance-sign-symmetric atomless law: the fixture is the exact quadrature verification of the population construction
How it was verified
Method: exact rational computation over the 8-atom symmetric quadrature law.
Reproduced by py/ds_population_bridge.py degenerate. Regression: tests/test_research_claims.py::test_symmetric_wasted_cells_defeat_the_efficient_semimetric_rule.
Claims that cite it
Recorded 28 August 2026. The fixture as JSON.