Counterexample CE-SCORE-ERROR-BOUNDARY-ATOM-001
What it falsifies
The mislabel mass P(q(s_hat) != q(s)) of a frozen rule tends to zero with the score error epsilon without a margin condition on the law at the cell boundaries. Refuted: an atom of mass 1/2 sits on the boundary of the threshold rule Z = 1{s > 0}; every proxy that lifts it by kappa > 0 relabels it, so pi = 1/2 for every epsilon > 0 and the true retention of the transferred labels jumps from 4/5 to 1/2. The margin function M(t) = P(dist_V(s, boundary) <= t) is >= 1/2 for all t > 0, so the SCORE-ERROR-RULE-TRANSFER bound pi <= inf_t {M(t) + epsilon^2/t^2} is vacuous, as it must be.
- Cell 0
- Cell 1
| Atom | Weight | Score 1 | Cell before | Cell after |
|---|---|---|---|---|
| 1 | 1/3 | -1 | 0 | 0 |
| 2 | 1/2 | 0 | 0 | 1 |
| 3 | 1/6 | 2 | 1 | 1 |
Objective
The criterion value at each labelling, exactly as the fixture certifies it.
| Labelling | Objective |
|---|---|
| Labels before | eta = 4/5 under the rule applied to the truth |
| Labels after | eta = 1/2 under the rule applied to the proxy (any kappa > 0) |
How it was verified
Method: exact_formula.
fractions.Fraction throughout; E[s] = 0; the Loewner sandwich of SCORE-ERROR-RULE-TRANSFER holds with Gamma of order one, which is the point: the budget is a function of the mislabel mass, and nothing makes the mass small here.
Claims that cite it
Recorded 8 September 2026. The fixture as JSON.