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Counterexample · DS-FINITE-GEOMETRY-FAILS

Profiled stability does not force geometry

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Meaning

Finite profiled exchange stability does not imply the strict geometric conclusion available for D.

Scientific context

For the profiled objective — some parameters of interest, the rest nuisance — the argument that works for the full determinant criterion fails at its first step: a positive first-order margin in the natural profiled semimetric does not imply a positive exact gain, so exchange stability does not force nearest-cell assignment. This is the finite efficient-semimetric gap that the profiled case has in place of a geometry theorem.

Statement

Positive first-order efficient-semimetric margin need not imply positive exact finite Ds gain; exchange stability does not force exact finite efficient-Voronoi assignment.

Where it stops

Refuted by

Relationships

Rests on

The full chain beneath it

Enables

Raises

Open next door

In the library

Carried by ProfiledGeometryReport. Reference.

Every object and refusal

Local graph
Local map of D-style finite exchange-to-geometry theorem fails for Ds1 prerequisites to the left, 1 results it enables or questions it raises to the right, 1 boundary cases below.Ds first-order gradient is an efficient semimetricDs first-order gradie…Characterize criteria with finite exchange => first-order geometryCharacterize criteria…D-style finite exchange-to-geometry theorem fails for DsD-style finite exchan…CE-DS-GLOBAL-GEOMETRY-001CE-DS-GLOBAL-GEOMETRY…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The example

DS4. D-style finite geometry theorem fails

A positive first-order efficient-Voronoi margin need not imply positive exact finite DsD_s gain. Therefore

exchange stable⇏exact Gs-Voronoi\text{exchange stable}\not\Rightarrow\text{exact }G_s\text{-Voronoi}

in general.

From Profiled Ds-optimality · chapter contents