Radial background: one mechanism for stills and rotation, decided per run on empty apertures (D5)

Wherever the correction is set for the whole run its curve is the frame's peak-excluded ring
background (the adaptive spot finder's, whole frame) and the ring's clip follows it - as stills have
done since integ-pal - now rotation too. One empty-aperture measurement (BackgroundAperture) decides
it for both workflows:

- The apertures' corrected difference is now taken as the integrator applies the correction: the
  clip follows the curve (each ring pixel's threshold raised by the curve's rise from the centre, in
  whole counts) as well as the kernel correction. Before, the tally clipped flat, so it measured a
  correction the engine never applies.
- Rotation measures the stencils the spot width can give it (r1 = 4, 5, 6 with the ring-area rule,
  RotationStencil) and decides on the one it gets, and separately on the pre-adaptive stencil the
  geometry pre-pass and the starvation guard return to.
- Stills: on wherever the corrected ring predicts the empty disk better (unchanged rule).
  Rotation: the frame's curve on every image only where it takes >= 10% of the apertures' mean
  square error away (ROTATION_RADIAL_MIN_GAIN); otherwise the existing per-image ice gate with the
  reflections' curve is left exactly as it was. Why a margin on rotation only: a still has no other
  correction, a sweep has one. Measured over 18 rotation sets (forced on every image): photon-counting
  sweeps gain 0.1-5% on the apertures (z 3-4) and lose ISa/anomalous signal when corrected
  (insulin, lysozyme, cytochrome c; 9yzk R-free +0.009 at 1.2%), as the earlier stills-rule trial
  found; the home-source CCD and smooth-ice runs gain 13-31%.

Result (CPU A/B vs 87684012f, 18 rotation sets): 12 .hkl byte-identical (every photon-counting
control without ice), 5uth R-free 0.210 -> 0.201 / ISa 10.4 -> 10.8, the ice set neutral; stills:
lcls242_s turns the correction on (the clip-following tally), R-free 0.2399 -> 0.2406.
No integration kernel changed; the tally is host code. Tests: BackgroundAperture_CorrectedClipFollowsTheCurve,
RadialHelps margins.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_015CeEGgpKdfkCTJb3YNMJcu
(cherry picked from commit dd3f3f2da39f5338e7dda1874bbdd535229bce98)

docs/CPU_DATA_ANALYSIS_INTEGRATION.md

@@ -145,13 +145,20 @@ The clip cuts only the high tail, which matters: the **symmetric** trimmed mean
 
 Both estimators are computed in the shared background pass, but only the trim reaches plain box summation: the high-side clip is skipped for `--integrator boxsum`, which therefore uses the plain ring mean unless `--background-trim` is given.
 
-**Radial background correction (opt-in).** A ring mean estimates the background *under* the signal disk correctly only if the background is flat there. The signal disk and the ring are concentric, so for a background that is **linear** in position $\langle B\rangle_\mathrm{ring}=\langle B\rangle_\mathrm{disk}$ identically — a plane or gradient fit buys exactly nothing. The leading error is the **curvature** of the radial background, which is negligible on a smooth background but reaches tens of counts on a single reflection sitting on a sharp powder ring. That error is a kernel over radial offset,
+**Radial background correction.** A ring mean estimates the background *under* the signal disk correctly only if the background is flat there. The signal disk and the ring are concentric, so for a background that is **linear** in position $\langle B\rangle_\mathrm{ring}=\langle B\rangle_\mathrm{disk}$ identically — a plane or gradient fit buys exactly nothing. The leading error is the **curvature** of the radial background, which is negligible on a smooth background but reaches tens of counts on a single reflection sitting on a sharp powder ring. That error is a kernel over radial offset,
 $
 \delta \hat b \;=\; \textstyle\sum_k \kappa_k\, \bar B(r_0+k),
 $
-with $\kappa$ the annulus-minus-disk histogram of the stencil over radial offset, averaged over azimuth, and $\bar B(r)$ the image's own radial background curve. With the fixed circular stencil ($k=0$, §9.1) that single kernel serves every reflection. An elongated ring does not: its radial-offset histogram depends on how far that particular reflection's ring was grown, so $\kappa$ becomes a small table of kernels, indexed by the growth rounded to whole pixels. The azimuthal average survives the change unaltered, because the stencil is rebuilt in the reflection's own frame at each azimuth and so stays radially aligned: what is averaged over is the sub-pixel phase of the detector grid against the radius, which is what genuinely differs between reflections. Applying it costs one short dot product per reflection and no extra pixel reads; correcting the background *scalar* means the box sum, the profile fit and the variance all pick it up. The curve is accumulated from the same annulus pixels the background pass already reads (a pixel's radius is the reflection's radius plus the pixel's projection on the beam→reflection direction, so no per-pixel square root is needed) and specifically from the **clipped** pixels, or it would carry neighbour tails and zingers — which is why the correction is inert under `--integrator boxsum`, that path having no clip pass.
+with $\kappa$ the annulus-minus-disk histogram of the stencil over radial offset, averaged over azimuth, and $\bar B(r)$ the image's own radial background curve. With the fixed circular stencil ($k=0$, §9.1) that single kernel serves every reflection. An elongated ring does not: its radial-offset histogram depends on how far that particular reflection's ring was grown, so $\kappa$ becomes a small table of kernels, indexed by the growth rounded to whole pixels. The azimuthal average survives the change unaltered, because the stencil is rebuilt in the reflection's own frame at each azimuth and so stays radially aligned: what is averaged over is the sub-pixel phase of the detector grid against the radius, which is what genuinely differs between reflections. Applying it costs one short dot product per reflection and no extra pixel reads; correcting the background *scalar* means the box sum, the profile fit and the non-signal variance all pick it up. The variance carried to the merge, $\mathrm{var}_\mathrm{bkg} = g\,(n_S\hat b + n_S^2\,\mathrm{var}(\hat b))$, moves with the corrected $\hat b$ in its first term — the Poisson share of the background *under the disk* — while $\mathrm{var}(\hat b)$, the noise of the ring estimate, and $\sigma$, taken from the counts in the disk, do not depend on a smooth curve and stay.
 
-The model is a function of **radius alone**, so it is applied only where that is true of the background. `--background-radial` takes `on`, `off` or `auto`. In Rugnux it is **`auto` by default** (the broker keeps it off); under `auto` each image's peak-excluded ice score (§3.3) is taken after spot detection and before integration, and the correction is applied to that image when the score reaches the same `--ice-min-score` gate. Smooth powder ice *is* a radial feature and is corrected; ice made of discrete crystallite spots — which the profile channel is blind to and the spot channel catches — leaves no smooth ring to model, and correcting it makes matters worse. Measured against a fixed atomic model, comparing ice bands with resolution-matched decoy bands carrying no ice: on a crystal with pure smooth ice the correction removes **43 % of the bands' excess amplitude**, and the improvement is **7× larger inside the bands than outside**, which is its stated mechanism; on a crystal whose ice is textured the same correction *increased* the excess amplitude by half; on a clean crystal it is inert to four decimal places. Auto engages only where a peak-excluded score exists (adaptive spot finding, §3.2) — a plain azimuthal profile carries the Bragg peaks and cannot support an absolute threshold, so without one auto leaves the correction off.
+The curve $\bar B(r)$ is the adaptive spot finder's peak-excluded background per resolution ring of the **whole frame** (§3.2), one value per pixel of radius: every pixel of the frame, against the few dozen to few thousand reflections whose own rings would otherwise sample it. Where that curve is in use the ring's **high-side clip follows it**: a ring pixel at radial offset $u$ is clipped against the threshold plus the curve's rise $\bar B(r_0+u)-\bar B(r_0)$, in whole counts. On a background falling steeply away from the beam a flat threshold cuts the beam-side half of the ring as outliers and the background reads low — up to 150× too strong a low-resolution reflection on a binned CCD near the beam stop. Without a frame curve (fixed-threshold spot finding) the curve is accumulated from the clipped annulus pixels of the predicted reflections themselves, and the clip stays flat.
+
+**When it is applied.** `--background-radial` takes `on`, `off` or `auto`; in Rugnux it is **`auto` by default** (the broker keeps it off), and auto is decided **per run, in the pre-scan, on empty apertures** — the same measurement that, on stills, also chooses how far the ring reaches (the $r_3$, up to the one the run starts from, whose ring predicts the empty disk best). On a grid of apertures away from every found spot, the ring's prediction of the empty disk is taken twice: plain, and with the correction applied exactly as the integrator applies it (the clip following the curve, the kernel's correction). With nothing in the aperture each difference is the error the ring makes in a reflection's background, noise plus bias. Rotation measures every stencil the spot width can give it (and the one it starts from, which the geometry pre-pass and the starvation guard return to) and reads the one it gets.
+
+- **Stills** have no other correction, so it is applied to every image of a run where it lowers the apertures' mean square error at all.
+- **Rotation** has one: under auto, each image whose peak-excluded ice score (§3.3) reaches `--ice-min-score` is corrected with the curve of its reflections' rings. The frame's curve on every image replaces that only where it takes **at least 10 %** of the apertures' mean square error away. On photon-counting sweeps the apertures gain 0.1–5 % (statistically significant, $z\approx3$–4), and correcting every image there was measured to cost ISa and anomalous signal; where it pays — the rings of home-source CCD frames, a smooth ice ring — they gain 13–31 %. A run below the margin keeps the per-image ice correction, bit for bit.
+
+The model is a function of **radius alone**, so it is applied only where that is true of the background. The per-image ice gate rests on this: smooth powder ice *is* a radial feature and is corrected; ice made of discrete crystallite spots — which the profile channel is blind to and the spot channel catches — leaves no smooth ring to model, and correcting it makes matters worse. Measured against a fixed atomic model, comparing ice bands with resolution-matched decoy bands carrying no ice: on a crystal with pure smooth ice the correction removes **43 % of the bands' excess amplitude**, and the improvement is **7× larger inside the bands than outside**, which is its stated mechanism; on a crystal whose ice is textured the same correction *increased* the excess amplitude by half; on a clean crystal it is inert to four decimal places. A strongly polarised beam or panels of different response make the background non-radial too; the apertures see that as a ring the correction predicts *worse*, and a run where they do keeps it off. Auto engages only where a peak-excluded background exists (adaptive spot finding, §3.2) — a plain azimuthal profile carries the Bragg peaks and cannot support an absolute threshold, so without one auto leaves the correction off.
 
 ### 9.3 Profile-fitted extraction (default)