In the library

The log determinant of the binned information. Chosen partly because it is invariant under invertible reparameterization

The log determinant of the Schur complement for the declared parameters of interest. The interest argument is the parameter block; at least one nuisance column must remain

Retained normalized trace, which after Fisher whitening is exactly weighted k-means

Exact positive-gain one-point exchange: the closed-form relocation gain on the rank-two update, accepted only on strictly positive exact gains so the search terminates

Batch reassignment as a proposal, accepted on the exact objective. The guard exists because the unguarded step is not monotone; guard="exchange" hands the labels to the exact engine so the reported state stays compilable

Weighted k-means on whitened scores, which is the trace criterion exactly

Optimization of a randomized quantizer's exact information, which exists because the hard empirical objective is piecewise constant and has no usable gradient. Its guarantees are those of smooth nonconvex optimization and no more

Exact interval dynamic programming on one score coordinate — the optimal cells are intervals, so the program returns a global optimum rather than a local one

A labelling of one fixed sample — the first of the three questions. It deliberately has no predict method, because a finite labelling does not determine its own extension

The one exception: an exchange-stable, nonsingular DOptimality result compiles into a Mahalanobis rule. It is bookkeeping, not a second fit

Measures both sides of the exchange-implies-Voronoi theorem on the terminal state instead of assuming them. guaranteed_violation_gain is the theorem's lower bound; maximum_violation_gain is the exact gain the theorem bounds; maximum_separation_residual checks the leverage inequality, so a positive value indicates numerical trouble rather than a better partition

One complete exact scan certifying that no single relocation improves a supplied labelling, at a stated gain_tolerance. A labelling certified at one tolerance is not certified at a smaller one

The outcome of a bounded global search using the singleton-completion ceiling. status is either "optimal" or "budget_exhausted"; the second proves nothing about the incumbent, and incumbent_was_optimal is False whenever it occurs

The full-data efficient-score ceiling on profiled information, attained for one parameter of interest by the exact interval program. Its labels also serve as a strong initializer

Diagnoses the finite efficient-semimetric gap — the quantity that the profiled case has instead of a geometry theorem

The normalized retention spectrum, every eigenvalue in the unit interval

Determinant efficiency against unbinned inference

Direction-resolved diagnostics. They are reported separately because optimizing the determinant does not equalize retention across directions — a caution, not a theorem

The rank the binned information actually reached, against the ceiling min(d, K-1)

Profiled efficiency, undefined where the Schur complement is singular

The scores are supplied directly; provenance is exact and every identity on these pages applies as written

Component densities give the local score of a linear mixture

The local score is the gradient of a log density ratio, so normalized densities are unnecessary

Calibrated classifier posterior odds recover component density ratios when the training priors are known

The record of which of the above applies. Estimated ratios never claim exact Fisher semantics, because the retained information is governed by the true score under the estimated rule and no theorem yet bounds the difference

A necessary check that model ratios integrate to one under the reference measure. It is necessary and not sufficient: a small residual never upgrades estimated provenance to exact