Counterexample · E-GLOBAL-GEOMETRY-FAILS

Global E optima can violate their own rule

Counterexamplepublication: internal

Meaning

An exact finite optimum need not satisfy its rank-one nearest-cell rule.

Scientific context

The smallest-eigenvalue criterion is the natural choice when a guarantee is needed for every parameter direction rather than for the volume, but it has no exchange-implies-geometry theorem: an exhaustively searched finite example with a simple minimum eigenvalue has a global optimum whose own rank-one nearest-cell rule disagrees with a training label. This is why there is no E-optimality criterion implemented; the omission traces to this counterexample rather than to a gap in coverage.

The witness is floating-point rather than exact rational, unlike most of the other counterexamples discussed here, and no prior-art search is recorded for it, so it stands as a witness rather than an attribution claim.

Statement

An exhaustive finite example with a simple minimum eigenvalue has a global E optimum whose vv^T nearest-cell rule disagrees with a training label.

Where it stops

Refuted by

CE-E-GEOMETRY-001
At a finite global E optimum with a simple minimum eigenvalue, the rank-one metric vv^T built from the minimum eigenvector assigns every training row to its own cell (nearest-cell rule).

Relationships

Enables

Raises

Open next door

Local graph

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

The example

E3. Finite global E geometry fails even with simple minimum eigenvalue

Exhaustive $N=8,d=2,K=3$ search produced a global E optimum whose own rank-one $vv^\top$ nearest-cell rule disagrees with a training label; reported margin $\approx0.06796$ .

From E-optimality · chapter contents

Evidence: COUNTEREXAMPLES/CE-E-GEOMETRY-001.json. Measured, never theorem authority.