Docs: repair sigma_rho / sigma_tau in the spot-footprint paragraph

The backslash escapes \r and \t had been turned into a line break and a tab,
which broke the LaTeX and split the paragraph.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01K5K8jvPPbmCrbqnWkddTuB

docs/CPU_DATA_ANALYSIS_INTEGRATION.md

@@ -106,9 +106,7 @@ Only the ring moves. The signal disk $r_1$ stays circular, deliberately: it sets
 
 What a circular $r_1$ loses is flux, and that loss is **not** a function of resolution alone: measured per reflection, it carries a directional component worth several Ų with a definite principal axis, on top of the isotropic part. Nor is there anything in the merge to absorb it. There is **no per-shell scale**, and there cannot usefully be one: every scale in §10 is fitted against a reference built from a reflection's own symmetry equivalents, and equivalents share $s^2$ exactly, so any function of $s^2$ lies in the exact null space of the whole scaling model — a per-shell parameter would have zero residual to fit against. (XDS and DIALS have the same null space, for the same reason.) The isotropic part of the loss is instead degenerate with the overall Wilson $B$ and is silently reported as part of it, so **the reported `WILSON_B` / `_reflns.B_iso_Wilson_estimate` carries an $r_1$-dependent contribution**: measured across a constant-ring-area radius sweep it falls monotonically as the disk grows, by 0.5 Ų on sharp strong data and by up to ~10 Ų on weak wide-spot data. What this costs the *data* is much less than what it costs the flux, because most of the loss is matched by a proportional $\sigma$: it moves no CC$_{1/2}$ and no $R_\text{meas}$, and — to within a few hundredths of an ångström — no resolution cut.
 
-**Measured spot footprint (automatic).** The radii above are chosen from spots at 5 Å, which at high X-ray energy sit close to the beam. Away from it a spot can grow several times wider — radially from the sensor's parallax and the obliquity of the incidence, tangentially from the crystal's azimuthal spread, which rotates the diffracted beam about the incident one and smears the spot along its ring. On small-molecule data at 20–25 keV the standard deviation grows from ~1 px near the beam to ~5 px at the detector edge: the $r_1 = 4$ disk holds a quarter of the flux there, the $6\ldots13$ px ring a third of it, and the profile widths learned inside $r_1$ (§9.3) saturate near $r_1^2/4$. So the pre-scan measures every spot it finds with a window that follows the spot — three of its own standard deviations, iterated and re-centred — separately along and across the radius, and tabulates the median widths $\sigma_
-ho,\sigma_	au$ against the distance from the beam. Wherever $3\max(\sigma_
-ho,\sigma_	au)>r_1$ the integrator then (i) starts the background ring at $4\sigma$ along each axis, (ii) sums the reflection over the $r_1$ disk **and** the $4\sigma$ footprint ellipse, so the summation — the profile fit's seed and its fallback — holds the spot rather than its core, and (iii) builds the per-reflection Gaussian at the measured widths on a grid grown to hold them. Where every spot fits the disk nothing is installed and the integration is unchanged bit for bit, which is the case for compact protein spots; like the measured radius, the footprint applies to the canonical pass and not to the geometry pre-pass, and a canonical pass whose wider rings the neighbours starve falls back to the settings without it. The reach is $4\sigma$ rather than the $3\sigma$ that decides whether a spot outgrew the disk because wide spots are not Gaussian: mosaic streaks and diffuse halos carry flux past $3\sigma$, which a ring starting there reads as background. Judged by refining the published structures with SHELXL, it removes the intensity loss that grew with resolution on the small-molecule sets (rugnux/model intensity in the outermost shell 0.81–0.91 → 0.98–1.02).
+**Measured spot footprint (automatic).** The radii above are chosen from spots at 5 Å, which at high X-ray energy sit close to the beam. Away from it a spot can grow several times wider — radially from the sensor's parallax and the obliquity of the incidence, tangentially from the crystal's azimuthal spread, which rotates the diffracted beam about the incident one and smears the spot along its ring. On small-molecule data at 20–25 keV the standard deviation grows from ~1 px near the beam to ~5 px at the detector edge: the $r_1 = 4$ disk holds a quarter of the flux there, the $6\ldots13$ px ring a third of it, and the profile widths learned inside $r_1$ (§9.3) saturate near $r_1^2/4$. So the pre-scan measures every spot it finds with a window that follows the spot — three of its own standard deviations, iterated and re-centred — separately along and across the radius, and tabulates the median widths $\sigma_\rho,\sigma_\tau$ against the distance from the beam. Wherever $3\max(\sigma_\rho,\sigma_\tau)>r_1$ the integrator then (i) starts the background ring at $4\sigma$ along each axis, (ii) sums the reflection over the $r_1$ disk **and** the $4\sigma$ footprint ellipse, so the summation — the profile fit's seed and its fallback — holds the spot rather than its core, and (iii) builds the per-reflection Gaussian at the measured widths on a grid grown to hold them. Where every spot fits the disk nothing is installed and the integration is unchanged bit for bit, which is the case for compact protein spots; like the measured radius, the footprint applies to the canonical pass and not to the geometry pre-pass, and a canonical pass whose wider rings the neighbours starve falls back to the settings without it. The reach is $4\sigma$ rather than the $3\sigma$ that decides whether a spot outgrew the disk because wide spots are not Gaussian: mosaic streaks and diffuse halos carry flux past $3\sigma$, which a ring starting there reads as background. Judged by refining the published structures with SHELXL, it removes the intensity loss that grew with resolution on the small-molecule sets (rugnux/model intensity in the outermost shell 0.81–0.91 → 0.98–1.02).
 
 **Split reflections (automatic).** A crystal made of slightly misaligned domains — the ferroelastic domains a crystal forms below a phase transition, or a cracked or split crystal — records each reflection as two or more compact spots on either side of the position the averaged lattice predicts, moving apart with resolution. The widths above are measured about each spot and so see compact spots; the $r_1$ disk then holds the gap between them and the background ring lands on them, and the loss grows to almost everything at the detector edge. Once the geometry pre-pass has a lattice, every spot it indexes is compared with the predicted position of its own reflection on the same frame, and the mean square of that offset, along and across the radius and in the same distance bins, is added to the pre-scan widths. The table then describes the *reflection* rather than the spot, and the canonical pass integrates with it exactly as above. Where spots sit on their predictions it moves the widths by the prediction error alone (0.2–1 px on protein data, where no reflection then outgrows $r_1$); on an inorganic crystal measured below its ferroelectric transition, where every reflection off one zone is a doublet, the offsets reach 8–16 px at the edge, and SHELXL refinement of the published structure goes from $R_1 = 0.27$ with a spurious extinction parameter to $R_1 = 0.05$.