Counterexample CE-DS-DEGENERATE-GLOBAL-TIE-001
What it falsifies
An exact finite global profiled-Ds optimum is unique up to label permutation, keeps its efficient-projected centroids pairwise separated, and (when it satisfies its own first-order efficient-semimetric rule with zero violations) is reproducible from that rule.
- Cell 0
- Cell 1
- Cell 2
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/8 | -47/64 | -3/8 | 0 | 0 |
| 2 | 1/8 | 1/64 | 1/4 | 0 | 0 |
| 3 | 1/8 | 33/64 | 7/4 | 0 | 0 |
| 4 | 1/8 | 49/64 | -3/8 | 1 | 1 |
| 5 | 1/8 | -15/64 | -3/4 | 0 | 0 |
| 6 | 1/8 | 65/64 | 3/8 | 1 | 1 |
| 7 | 1/8 | -47/64 | 1/8 | 0 | 0 |
| 8 | 1/8 | -39/64 | -1 | 2 | 2 |
Consequences
the finite global optimizer set can be a large exact tie class whose members differ arbitrarily on Phi-neutral splits
zero first-order violations do not make an optimum compilable: coincident projected centroids leave the efficient-semimetric rule unable to separate the tied cells
only the reduced (merged) e-space configuration is identified by the objective; a projected-centroid separation margin is not automatic at exact finite global optima
How it was verified
Method: exact rational exhaustive enumeration of all canonical labelings in Fraction arithmetic.
Instance drawn from a correlated-Gaussian law (rho=0.6), coordinates rounded to exact multiples of 1/8 and centered by the exact sample mean (seed 3764241046 in py/ds_population_bridge.py law gauss06, N=8, rep 0); the fixture is self-contained given the exact scores. Regression: tests/test_research_claims.py::test_global_profiled_ds_optimum_can_be_a_degenerate_tie_class.
Claims that cite it
Recorded 28 August 2026. The fixture as JSON.