Counterexamples

A fixture is a small table of exact rational scores and weights, together with the labels and objective values that follow from it, verified by enumerating every labelling or by an exact formula rather than by a run. Each one is cited by at least one statement: it either refutes that statement in the generality someone was tempted to give it, or sits on the edge of its hypotheses, showing that a condition cannot be dropped.

A fixture marked refusal is one the library names out loud when it declines a call; the calls that trigger them are listed in the library.

Fixture	Criterion	Level	Rows	d	K	What it settles
CE-O6-ETA-ZERO-MULTIATOM-VARIANCE-001	Criterion-independent	Information accounting	6	1	2	bounds Independent audit of the frozen-rule scalar retention plug-in CLT (O6) bounds Frozen-rule scalar retention: plug-in CLT with a consistent influence-function variance
CE-AUC-INVARIANT-PROXY-RETENTION-001	D-optimality	Information accounting	4	1	2	bounds Posterior error to score error for mixture-fraction scores: one Lipschitz inequality, and the excess Brier score as the classifier requirement bounds Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score error
CE-CLASSIFIER-CALIBRATION-CHART-001	D-optimality	Information accounting	2	1	2	bounds Posterior error to score error for mixture-fraction scores: one Lipschitz inequality, and the excess Brier score as the classifier requirement
CE-CLASSIFIER-CALIBRATION-RETENTION-001	D-optimality	Information accounting	2	1	2	bounds Posterior error to score error for mixture-fraction scores: one Lipschitz inequality, and the excess Brier score as the classifier requirement
CE-D-COMPILE-BATCH-TOLERANCE-001	D-optimality	Finite assignment	4	1	2	bounds Positive solver tolerance gives only a tolerance-stamped compile guarantee
CE-D-COMPILE-SINGLETON-TIE-001	D-optimality	Finite assignment	3	1	2	bounds Positive solver tolerance gives only a tolerance-stamped compile guarantee
CE-D-LLOYD-001	D-optimality	Finite assignment	8	2	3	refutes Adaptive Mahalanobis Lloyd need not increase D
CE-D-UNMERGED-DUPLICATES-001refusal	D-optimality	Finite assignment	3	1	3	refutes Publication-grade audit of the D exchange theorem bounds Finite D exchange stability implies strict self-consistent Voronoi bounds Exact finite inductive closure / compiler refutes Split duplicate atoms defeat strict D-Voronoi closure
CE-D-VORONOI-CONVERSE-001refusal	D-optimality	Finite assignment	4	1	2	refutes D-Voronoi fixed point does not imply exchange stability
CE-O7-ELLIPSOID-ZERO-VARIANCE-001	D-optimality	Information accounting	8	2	4	bounds Frozen-rule vector retention: plug-in CLT for the geometric-mean D-retention with a consistent influence-function variance
CE-O7-UNIT-RETENTION-SINGULAR-SAMPLE-001	D-optimality	Information accounting	2	2	2	bounds Independent audit of the frozen-rule vector retention plug-in CLT (O7) bounds Frozen-rule vector retention: plug-in CLT for the geometric-mean D-retention with a consistent influence-function variance
CE-SCORE-ERROR-ALIGNMENT-ORDER-001	D-optimality	Information accounting	4	2	3	bounds Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score error
CE-SCORE-ERROR-BOUNDARY-ATOM-001	D-optimality	Information accounting	3	1	2	bounds Applying a frozen rule to the proxy instead of the truth: mislabel mass under a margin condition and an exact Loewner sandwich on the retained information
CE-SCORE-ERROR-LOG-LOWER-001	D-optimality	Information accounting	2	1	2	bounds Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score error
CE-SCORE-ERROR-RHO-MIN-NECESSARY-001	D-optimality	Information accounting	5	2	3	bounds Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score error
CE-SCORE-ERROR-SINGULAR-LS-001	D-optimality	Information accounting	2	1	1	bounds Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score error
CE-SCORE-ERROR-TRANSLATION-001	D-optimality	Information accounting	3	1	2	bounds Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score error
CE-DS-DEGENERATE-GLOBAL-TIE-001	Profiled Ds-optimality	Finite assignment	8	2	3	refutes A finite global Ds optimum can be a degenerate exact tie class bounds Conditional finite-to-population Ds bridge
CE-DS-GLOBAL-GEOMETRY-001refusal	Profiled Ds-optimality	Finite assignment	8	2	3	refutes D-style finite exchange-to-geometry theorem fails for Ds refutes A finite global Ds optimum can be non-geometric bounds Population stationary geometry for profiled Ds
CE-DS-GLOBAL-GEOMETRY-002	Profiled Ds-optimality	Finite assignment	8	2	3	refutes A finite global Ds optimum can be non-geometric
CE-DS-INTERVAL-SEED-UNSTABLE-001	Profiled Ds-optimality	Finite assignment	8	2	3	refutes Exchange-stable Ds states span both margin regimes; ascent selects the funnel at scale bounds The beta-zero strip DP is value-consistent on the exact DS18 law bounds Best practical Ds solver with theorem-backed certificate
CE-DS-LCM-SIGNSPLIT-MARGIN-001	Profiled Ds-optimality	Population design	8	2	3	bounds Conditional centering empties the margin-certified stable branch bounds Merged-branch margins on LCM laws live only in wasted cells and never compile
CE-DS-LCM-SIGNSPLIT-MINIMAL-001	Profiled Ds-optimality	Population design	3	2	3	bounds Conditional centering empties the margin-certified stable branch bounds Measured gate scans: eight (L)-laws empty, mix3 inhabited free, sign-split family at v_2 bounds Merged-branch margins on LCM laws live only in wasted cells and never compile
CE-DS-MARGINS-RANK-VACUITY-001refusal	Profiled Ds-optimality	Finite assignment	4	3	3	bounds Margins dichotomy at global finite Ds optima (conditionally centered laws, scalar nuisance) bounds Vector-parameter margins dichotomy beyond the scalar DS15 class
CE-DS-MATRIX-TILT-NONQUASICONVEX-001	Profiled Ds-optimality	Finite assignment	8	4	8	bounds Publication-grade audit of the DS19 tilt-DP certificate complex refutes The multivariate-POI matrix-tilt outer ceiling need not be quasiconvex bounds Profiled compile verdict is selected by an observable certificate state refutes Best practical Ds solver with theorem-backed certificate
CE-DS-NONCENTERED-POPULATION-CUT-UNSTABLE-001	Profiled Ds-optimality	Finite assignment	4	2	3	bounds Exact off-class global basin transfers to empirical exchange-stable optima bounds The beta-zero strip DP is value-consistent on the exact DS18 law bounds Best practical Ds solver with theorem-backed certificate
CE-DS-NONCENTERED-SINGULAR-DESTINATION-001	Profiled Ds-optimality	Finite assignment	4	2	3	bounds Publication-grade audit of the DS18 exact off-(L) basin and empirical transfer bounds Exact off-class global basin transfers to empirical exchange-stable optima bounds Profiled compile verdict is selected by an observable certificate state bounds Scalar-POI tilt-DP gives a valid two-sided certificate with an exact saddle closure gate bounds Best practical Ds solver with theorem-backed certificate
CE-DS-POP-WASTED-CELLS-001	Profiled Ds-optimality	Population design	8	2	4	refutes Population-stationary Ds partitions can defeat every efficient-semimetric rule bounds Merged-branch margins on LCM laws live only in wasted cells and never compile bounds Population stationary geometry for profiled Ds
CE-DS-STABLE-MARGIN-RETAINING-001	Profiled Ds-optimality	Finite assignment	8	2	3	bounds Conditional centering empties the margin-certified stable branch bounds Margin price and value funnel at arbitrary finite Ds labelings
CE-DS-TILT-DUAL-GAP-001	Profiled Ds-optimality	Finite assignment	4	2	3	bounds Publication-grade audit of the DS19 tilt-DP certificate complex bounds Profiled compile verdict is selected by an observable certificate state bounds Scalar-POI tilt-DP gives a valid two-sided certificate with an exact saddle closure gate refutes The scalar-POI tilt dual can have an order-one strong-duality gap refutes Best practical Ds solver with theorem-backed certificate
CE-DS-TILT-DUAL-GAP-002	Profiled Ds-optimality	Finite assignment	3	2	2	refutes Publication-grade audit of the DS19 tilt-DP certificate complex bounds Profiled compile verdict is selected by an observable certificate state bounds Scalar-POI tilt-DP gives a valid two-sided certificate with an exact saddle closure gate refutes The scalar-POI tilt dual can have an order-one strong-duality gap refutes Best practical Ds solver with theorem-backed certificate
CE-DS-TILT-DUAL-TIE-MASK-001	Profiled Ds-optimality	Finite assignment	3	2	2	refutes Publication-grade audit of the DS19 tilt-DP certificate complex bounds Profiled compile verdict is selected by an observable certificate state bounds Scalar-POI tilt-DP gives a valid two-sided certificate with an exact saddle closure gate bounds Best practical Ds solver with theorem-backed certificate
CE-A-DSTYLE-001	A-optimality	Finite assignment	6	2	3	refutes D-style finite exchange-to-geometry theorem fails for A
CE-E-GEOMETRY-001	E-optimality	Finite assignment	8	2	3	refutes Positive first-order E margin need not imply exact finite improvement refutes Finite global E optimum can violate its own rank-one nearest-cell rule