Counterexample CE-DS-TILT-DUAL-GAP-002
What it falsifies
The scalar-POI tilt-dual duality gap needs N>=4 (support minimality of CE-DS-TILT-DUAL-GAP-001 across all K). FALSE: at N=3, K=2 the exact dual minimum strictly exceeds the exact global profiled value, and N=3 is the smallest support at which any exact-K>=2 problem has more than one labeling.
- Cell 0
- Cell 1
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/3 | -1 | 0 | 0 | 0 |
| 2 | 1/3 | 0 | -1 | 1 | 0 |
| 3 | 1/3 | 1 | 0 | 1 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | 1/3 |
| Labels after | 1/3 |
How it was verified
Method: exhaustive.
All arithmetic is fractions.Fraction. The dual minimum is exact (a rational tilt certified by the one-sided subgradient condition D-(beta*) <= 0 <= D+(beta*)); the rational convex-mixture certificate independently lower-bounds it.
Claims that cite it
- Publication-grade audit of the DS19 tilt-DP certificate complexAUDIT-DS-PRACTICAL-CERTIFIED-SOLVER
- Profiled compile verdict is selected by an observable certificate stateDS-PROFILED-COMPILE-CERTIFICATE
- Scalar-POI tilt-DP gives a valid two-sided certificate with an exact saddle closure gateDS-TILT-DUAL-CERTIFICATE
- The scalar-POI tilt dual can have an order-one strong-duality gapDS-TILT-DUAL-STRONG-DUALITY-FAILS
- Best practical Ds solver with theorem-backed certificateOPEN-DS-PRACTICAL-CERTIFIED-SOLVER
Recorded 2 September 2026. The fixture as JSON.