Counterexample CE-DS-MARGINS-RANK-VACUITY-001
What it falsifies
DS15's margins dichotomy as first registered for general nuisance dimension under the bare cardinality assumption K >= 3: for d_lambda >= 2 with K = d_lambda + 1, conclusion (i) (value convergence to the positive unrestricted supremum v_K) is false, because exact centering forces rank(I_z) <= K - 1 = d_lambda, so by Schur-complement rank additivity EVERY feasible labeling (nonsingular binned nuisance block) has profiled value exactly 0 on every sample, while v_K > 0.
This is the boundary fixture behind OPEN-DS-MARGINS-AT-OPTIMA, and behind it the rank ceiling FI-RANK-CEILING: with too few cells, exact centring forces every feasible labelling's profiled value to exactly zero — a rank effect, not a data problem, that no sample size can fix. The library refuses such a configuration up front rather than blaming the nuisance parameterization, citing this fixture; the remedy is to raise the cell count above the score dimension, since on an exactly centred sample no sample size helps.
| Atom | Weight | Score 1 | Score 2 | Score 3 | Cell before | Cell after |
|---|---|---|---|---|---|---|
| 1 | 1/4 | 3 | 1 | 0 | 0 | 0 |
| 2 | 1/4 | -1 | -2 | 1 | 0 | 0 |
| 3 | 1/4 | -1 | 1 | -2 | 1 | 1 |
| 4 | 1/4 | -1 | 0 | 1 | 2 | 2 |
The fixture lives in 3 score dimensions, so its atoms are listed rather than drawn.
Consequences
the DS15 dichotomy needs K >= d_lambda + 2, not K >= 3; at d_lambda = 1 the two coincide, which is why the theorem's own K >= 3 is exactly right there
for d_lambda >= 2 with K >= d_lambda + 2 the dichotomy remains open: Proposition 6's steering construction is one-dimensional in the nuisance and (R) is stated for a scalar nuisance sign
the same rank ceiling shows the DS15 mechanism at K = 2, d_lambda = 1: the profiled value is identically zero there as well, so K >= 3 is load-bearing at d_lambda = 1 too
How it was verified
Method: exact rational exhaustive enumeration of all canonical labelings in Fraction arithmetic.
Hand-minimized integer witness with exact zero column sums (no centering arithmetic needed), nonsingular 3x3 full second-moment matrix (determinant > 0), and a strictly positive efficient-score interval optimum. Regression: tests/test_research_claims.py::test_ds15_rank_deficiency_zeroes_every_feasible_profiled_value. Independent scan: py/audit_ds_margins_at_optima.py vacuity (random N=6,7,8 instances all confirm).
Claims that cite it
What the library refuses
- Trigger
A profiled fit or partition whose cell count cannot generate nonsingular binned information
- Reason
The cardinality boundary of
OPEN-DS-MARGINS-AT-OPTIMA, and behind it the rank ceilingFI-RANK-CEILING- Remedy
Raise
n_binsabove the score dimension. On an exactly centred sample no sample size helps — this is arithmetic, not a data problem
Recorded 30 August 2026. The fixture as JSON.