Counterexample `CE-D-VORONOI-CONVERSE-001`

What it falsifies

Every strictly self-consistent I_q^{-1}-Voronoi finite partition (each row strictly nearest its own centroid) is one-point exchange stable.

This is the boundary fixture for D-VORONOI-NOT-EXCHANGE: a four-row, one-dimensional, two-cell example of a partition that is a fixed point of its own nearest-centre rule while still admitting a single relocation with a strictly positive exact information gain. compile_quantizer() refuses to compile a partition whose stability certificate is False, citing this fixture, and reports the remaining gain so the size of the violation is visible.

Labels before

Labels after / optimum

Cell 0
Cell 1

Atoms of CE-D-VORONOI-CONVERSE-001
Atom	Weight	Score 1	Cell before	Cell after
1	`1/4`	`-3/4`	0	0
2	`1/4`	`-3/4`	0	0
3	`1/4`	`1/4`	0	1
4	`1/4`	`5/4`	1	1

Objective

The criterion value at each labelling, exactly as the fixture certifies it.

Labelling	Objective
Labels before	`-0.6523251860396901`
Labels after	`-0.5753641449035618`

How it was verified

Method: exact_formula.

Scalar scores, uniform weights, weighted mean zero, all arithmetic in Fraction. Every row is strictly nearest its own cell centroid (in 1-D the I^{-1} metric is a positive scalar, so Voronoi = Euclidean), yet relocating moved_row to destination_cell strictly increases I_q. Hence Voronoi fixed does not imply exchange stable; the audited implication is strictly one-directional.

Claims that cite it

D-Voronoi fixed point does not imply exchange stabilityD-VORONOI-NOT-EXCHANGErefuted by this fixture

What the library refuses

Trigger: compile_quantizer() on a partition whose stability certificate is False
Reason: looking geometric is strictly weaker than being exchange-stable
Remedy: Inspect best_remaining_gain; raise max_scans, or set guard="exchange" on the guarded batch solver