A screw's control class is the complement of its absent class on one axial row, so the two move
together: a candidate that predicts more of the row absent leaves fewer reflections to be judged
against. The Beta tail AbsenceEvidence computes grows with that control count, so the stricter
candidate was charged for the very reflections it correctly called extinct, and a group whose
predicted-absent class is a strict superset of another's - with the extra reflections equally dead -
could score LOWER than the group that explains only part of the row.
Two mechanisms, one dataset each.
* The control-count minimum gated the row MEAN, which is the scale the zone's evidence is stated
in, on the same count as the row MEDIAN, which is a violation threshold. A candidate could
therefore forfeit a whole zone by being right: on a tetragonal 4_1/4_3 wedge, thirteen 00l
reflections measured at 0.1% of the two l = 4n beside them scored ZERO because only two control
reflections were left, while the nine of them a 4_2 also predicts absent scored 38.7. The mean
now stands on two, the median still on three.
* The Beta tail is replaced, for a SCREW zone only, by its b -> infinity limit - the same
statistic with the control count dropped. A p-value computed against each candidate's own null
is not one scale across candidates; what is left is the likelihood ratio of the absent class
against Wilson at the row's own mean, which is a sum over reflections and therefore comparable.
Asymptotically it is n_absent * (log(1/ubar) - 1), so an equally dead superset can no longer
score lower. Measured on a tetragonal 4_1/4_3 crystal: 29 dead 00l against 8 control read 47.7
nats where a subset of 19 of them against 18 control read 55.5.
The centring statistic is untouched: its control is the whole present population, not the complement
of a claim on one row, so neither mechanism applies to it.
Measured. Analytic superset monotonicity, on a 176368-point grid with no calibration and no corpus:
1.29% violations -> 0.00%, exactly monotone wherever both candidates keep a control class. Over 112
stored merges (open, in-house and private arms) exactly ONE decision changes and it is a gain, a
4_2 -> 4_1/4_3 swap on a deposited 4_3; the screw column over the deposited arm goes 33/41 -> 34/41
with zero screw over-calls before and after, and the four centring outcomes are bit-identical. An
eight-wedge reproducer whose 00l row a strict candidate eats goes from two mis-calls to none,
end to end. The refused side moves the safe way: 103 zones the old statistic scored negative go
lower, none turns positive, and the minimum evidence of an adopted screw rises 20.4 -> 23.8 against
an unchanged bound of 20.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01EFEJG6WBQv8th4UJFNe53N
leonarski_fandClaude Opus 5
71cca91ec1