The background is estimated from the r2..r3 ring and then subtracted from every
pixel of the r1 disk, so the ring mean's own error enters the intensity n_inner
times over: var(I) carries n_inner^2 * bkg / n_B. That term is first-order in
sigma, and it is set by how many pixels the ring holds - not by anything about
the reflection. At r3 = 10 the ring holds about 200 px against the disk's 50.
Widening it to 13 roughly doubles that. The signal disk is untouched, and the
pixels gained lie further from the reflection rather than nearer, so nothing is
traded for them.
The effect is not subtle once looked for. Matched observation by observation on
one crystal, halving the ring's pixel count leaves the intensity alone and
inflates sigma by 4.7%, and the inflation rank-orders with the ring collapse
across the battery.
Over the whole rotation battery, against the same binary at r3 = 10: ISa better
on 14 crystals and worse on 4, the summed shortfall against the reference
164.7 -> 155.9, the summed low-resolution R_meas excess 69.7 -> 59.1 percentage
points, and one more crystal reaching the reference space group (33/37 -> 34/37,
a trigonal case that was over-promoting). Largest gains where the ring was
starved worst; the four losses are 0.25 to 2.16 in ISa and none of them changes
a space group.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
leonarski_fandClaude Opus 5
06a5a118bf