Open question · OPEN-CRITERION-CHARACTERIZATION
Which criteria force geometry?
Openpublication: open
Meaning
Characterize matrix criteria for which finite exchange stability implies first-order geometric consistency.
Scientific context
The determinant criterion has an exchange-implies-geometry theorem (D-EXCHANGE-IMPLIES-VORONOI); the profiled and smallest-eigenvalue criteria demonstrably do not. Why is unknown: there is no characterization of which concave matrix criteria admit such an implication. Nothing proves that the determinant is the unique criterion with this property.
Statement
Find necessary/sufficient conditions on concave F for one-point exchange stability to imply pointwise assignment under a common first-order metric/supergradient.
Relationships
Related settled results
- Finite D exchange stability implies strict self-consistent VoronoiD-EXCHANGE-IMPLIES-VORONOI
- D-style finite exchange-to-geometry theorem fails for DsDS-FINITE-GEOMETRY-FAILS
- D-style finite exchange-to-geometry theorem fails for AA-FINITE-GEOMETRY-FAILS
- Finite global E optimum can violate its own rank-one nearest-cell ruleE-GLOBAL-GEOMETRY-FAILS
The full chain beneath it
- Exact finite D log-determinant gainD-LOGDET-GAIN
- D leverage inequalityD-LEVERAGE
- Ds first-order gradient is an efficient semimetricDS-GRADIENT-EFFICIENT-SEMIMETRIC
- Exact weighted rank-two relocationD-RANK2-MOVE
- Quantized Fisher-information identityFI-QUANT-IDENTITY
- Infinitesimal cell-transfer identityGENERAL-FIRST-VARIATION
- Ds objective is the logdet of efficient/profiled Schur informationDS-SCHUR
Where it comes from
Raised by
- D-style finite exchange-to-geometry theorem fails for AA-FINITE-GEOMETRY-FAILS
- Finite D exchange stability implies strict self-consistent VoronoiD-EXCHANGE-IMPLIES-VORONOI
- D-style finite exchange-to-geometry theorem fails for DsDS-FINITE-GEOMETRY-FAILS
- Finite global E optimum can violate its own rank-one nearest-cell ruleE-GLOBAL-GEOMETRY-FAILS
Local graph
The question in full
Which concave matrix criteria have finite exchange ⇒ first-order geometry?
For concave , characterize when one-point exchange stability implies pointwise first-order assignment under a common . Anchors: true for full D; false for A, for , and naively for E; the screening direction follows from concavity for all four. Desired: a curvature or operator inequality, a useful subclass, or an impossibility theorem showing log-det is exceptional.
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