Counterexample CE-DS-INTERVAL-SEED-UNSTABLE-001
What it falsifies
The exact efficient-score interval labeling - the DP optimum of the full-sample efficient score, the documented initializer for profiled exchange and the finite analogue of the DS15 degenerate attainer J* - is itself one-point exchange-stable for the in-bin profiled-Ds objective. FALSE: on this exactly centered N=8 sample from the centered06 grid law, relocating row 7 to cell 1 raises the profiled value by 0.447 exactly. The improving move works by growing the binned nuisance block 27-fold (0.0042 -> 0.111): near the interval seed the projection tax is the dominant term and the profiled objective climbs by buying back nuisance information, the finite mechanism behind DS15 Proposition 6's steering.
- Cell 0
- Cell 1
- Cell 2
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/8 | -1613/4096 | 1517/20480 | 0 | 0 |
| 2 | 1/8 | 4875/4096 | -3003/20480 | 1 | 1 |
| 3 | 1/8 | 2563/4096 | 30109/20480 | 0 | 0 |
| 4 | 1/8 | -9421/4096 | -44131/20480 | 0 | 0 |
| 5 | 1/8 | -1657/4096 | -15751/20480 | 2 | 2 |
| 6 | 1/8 | 1571/4096 | -5379/20480 | 2 | 2 |
| 7 | 1/8 | -1949/4096 | 16717/20480 | 0 | 0 |
| 8 | 1/8 | 5631/4096 | 19921/20480 | 2 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | 4644025/112623616 |
| Labels after | 724740241/1483132192 |
Consequences
the documented profiled initializer is generally not a terminal state: seeding inside the DS15 degeneracy does not mean returning it, so the optimizer's output is a genuinely different object from the projected rule (study blocker 'selection effects')
the finite projection tax at the interval labeling is Theta(1) and one-point moves reduce it, the mechanism DS15's achievability construction exploits at scale
How it was verified
Method: exact_formula.
exact rational arithmetic; gain verified by from-scratch recomputation of both binned information matrices; regression test tests/test_research_claims.py::test_ds16_efficient_score_interval_seed_is_not_exchange_stable
Claims that cite it
Recorded 30 August 2026. The fixture as JSON.