Result · D-GLOBAL-GEOMETRIC-REALIZABILITY

Global D optima are geometric

Proved herepublication: internal

Meaning

Global optimality implies exchange stability, and hence geometric realizability under the structural theorem’s hypotheses.

Statement

Every finite global D optimum is exchange stable and therefore strict self-consistent D-Voronoi; unrestricted finite D optimization and realizable affine/Voronoi labeling optimization have the same optimum value.

Relationships

Rests on

Finite D exchange stability implies strict self-consistent VoronoiD-EXCHANGE-IMPLIES-VORONOI

The full chain beneath it

Enables

Raises

Unrestricted global D consistencyOPEN-D-UNRESTRICTED-CONSISTENCY

Open next door

Local graph

Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

D7. Every finite global D optimum is geometrically realizable

A finite global optimum is exchange stable, hence strict D-Voronoi. Therefore unrestricted finite D assignment optimization and global optimization over the corresponding realizable D-Voronoi/affine-max labelings have the same optimum value.

This does not say every D-Voronoi fixed point is globally optimal.

The realizability half is machine-checked as ScoreQuantFormal.globalOptimum_strictVoronoi in formal/ScoreQuantFormal/Corollaries.lean. The equal-optimum-value half is not formalized, so this claim carries no formal_proof field.

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