Counterexample CE-DS-LCM-SIGNSPLIT-MARGIN-001
What it falsifies
The DS17 linear-conditional-mean obstruction extends to the (M5)-free margin pair: on an LCM law every bounded-study stationary K=3 configuration with positive cell masses has a rank-deficient binned information matrix, so the conditioning margin (M3) alone already empties the stationary class.
- Cell 0
- Cell 1
- Cell 2
| Atom | Weight | Score 1 | Score 2 | Cell before | Cell after |
|---|---|---|---|---|---|
| 1 | 1/8 | -3 | 1 | 0 | 0 |
| 2 | 1/8 | -3 | -1 | 1 | 1 |
| 3 | 1/8 | -1 | 2 | 0 | 0 |
| 4 | 1/8 | -1 | -2 | 1 | 1 |
| 5 | 1/8 | 1 | 1 | 2 | 2 |
| 6 | 1/8 | 1 | -1 | 2 | 2 |
| 7 | 1/8 | 3 | 2 | 2 | 2 |
| 8 | 1/8 | 3 | -2 | 2 | 2 |
Consequences
(M5) is load-bearing for the DS17(a) obstruction: dropping the separation margin, LCM laws do admit bounded-study stationary configurations with full-rank binned information and macroscopic (M2)+(M3) margins
every such configuration is a wasted-cell structure: its projected centroids coincide pairwise and its reduced (mergeable) rule is a nuisance-degenerate strip rule, so the DS14 companion object compiled from it carries a singular nuisance block regardless
on N(0, I_2) the analogous family (any zero-mean measurable nuisance split of one threshold cell) is stationary with profiled value exactly v_2 = 2/pi and lambda_min up to 1/pi: the population constraint class {lambda_min(I_q) >= kappa} of DS16's v*(kappa) is nonempty for kappa <= 1/pi, while remaining compile-dead
How it was verified
Method: exact rational computation over the 8-atom symmetric quadrature law.
Reproduced by py/ds_stable_basins.py selftest (signsplit_eight_atom_exact). Regression: tests/test_research_claims.py::test_ds17_signsplit_stationary_state_retains_margins_without_separation. The atomic law verifies the algebra exactly; the population statement lives on the atomless Gaussian counterpart (DS12 necessity is vacuous on finite atoms, as for CE-DS-POP-WASTED-CELLS-001).
Claims that cite it
Recorded 31 August 2026. The fixture as JSON.