Counterexample · DS-GLOBAL-NONGEOMETRIC

A profiled global optimum can violate geometry

Counterexamplepublication: internal

Meaning

An exhaustive exact witness has a global optimum that violates its own first-order rule.

Scientific context

Worse than the first-order failure above, a globally optimal profiled partition can violate its own first-order rule. The witness is exact: eight rows, two score dimensions, one parameter of interest, three cells, enumerated exhaustively, with the unique global optimum violating its own rule by margins of 2862/3239 and 618/3239 (CE-DS-GLOBAL-GEOMETRY-001).

The consequence is that a profiled result has no canonical rule to compile into. Calling compile_quantizer() on one is refused, citing this fixture, and the remedy is to fit an explicit quantizer rather than convert a fixed-sample profiled partition into one. Verifying this result authorized no change to the library's existing refusal behaviour — it confirmed the behaviour was already correct.

Statement

An exact centered equal-weight N=8,d=2,d_ψ=1,K=3 exhaustive example has a unique global Ds optimum that violates its own efficient-semimetric nearest-cell rule.

Where it stops

Refuted by

CE-DS-GLOBAL-GEOMETRY-001
Every finite globally optimal Ds assignment is self-consistent under its own first-order efficient-score/common-semimetric nearest-cell rule.
CE-DS-GLOBAL-GEOMETRY-002
A finite globally optimal profiled Ds partition assigns each row to its nearest cell under its own efficient-score semimetric.

Relationships

Enables

Raises

Unrestricted Ds/E finite-to-population consistencyOPEN-DS-E-UNRESTRICTED-CONSISTENCY

Open next door

In the library

Enforced by the refusal CE-DS-GLOBAL-GEOMETRY-001: the library declines the operation and names the fixture.

Every object and refusal

Local graph

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

The example

DS5. A global finite $D_s$ optimum can be non-geometric

There is a centered equal-weight $N=8,d=2,d_\psi=1,K=3$ example for which exhaustive enumeration of all 966 unlabeled nonempty partitions gives a unique global $D_s$ optimum that violates its own efficient-semi-metric nearest-cell rule.

Canonical fixture is stored in COUNTEREXAMPLES/CE-DS-GLOBAL-GEOMETRY-001.json.

This proves that unrestricted finite $D_s$ assignment and deployable self-consistent geometric $D_s$ fitting are genuinely different finite problems.

From Profiled Ds-optimality · chapter contents

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