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Counterexample · A-FINITE-GEOMETRY-FAILS

A stability does not force geometry

Counterexamplepublication: internal

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Meaning

A counterexample prevents transferring the finite D exchange-to-geometry theorem to A.

Statement

Finite exchange stability does not generally collapse to the first-order I^{-2} Voronoi assignment rule.

Where it stops

Refuted by

  • Counterexample fixtureCE-A-DSTYLE-001

    The D exchange-to-geometry mechanism transfers to the A criterion: a move whose first-order I_q^{-2} Mahalanobis margin is positive (row strictly closer to the destination centroid in the I^{-2} metric) must not decrease the exact A objective -tr(I_q^{-1}).

Relationships

Enables

Raises

Open next door

Local graph
Local map of D-style finite exchange-to-geometry theorem fails for A0 prerequisites to the left, 2 results it enables or questions it raises to the right, 1 boundary cases below.Quantitative A finite-geometry necessity boundQuantitative A finite…Characterize criteria with finite exchange => first-order geometryCharacterize criteria…D-style finite exchange-to-geometry theorem fails for AD-style finite exchan…CE-A-DSTYLE-001CE-A-DSTYLE-001
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The example

A2. D-style finite geometry theorem fails

The reported search found 443 A moves violating the would-be D-style implication. Therefore finite exchange stability does not generally collapse to the first-order I−2I^{-2} Voronoi rule.

From A-optimality · chapter contents

Evidence: COUNTEREXAMPLES/CE-A-DSTYLE-001.json. Measured, never theorem authority.