docs: disambiguate the reused symbols and the bandwidth definition

sigma_bw is one physical smear written in reciprocal units in 8.2/11.1 and in
pixels in 9; the two Delta-phi of the partiality formula are named; 13.5's
|s| = 1/d is reconciled with the s = sin(theta)/lambda of 10.6/14.2; bandwidth
is the rms spread, with the FWHM-input conversion (/2.355,
BraggIntegrationEngine.cpp) stated. The ice-extinction clause now covers (104),
not only (00l), and the AIMLESS <I/sigma> = 2 constant is named as AIMLESS's
default rather than pointed at a criterion this project does not use.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01EFEJG6WBQv8th4UJFNe53N

docs/CPU_DATA_ANALYSIS.md

@@ -413,7 +413,7 @@ Both channels are used offline as a **gate** on ice handling: unless the run rea
 
 **Where the ring positions come from.** The eleven bands from 3.895 to 1.522 Å are the *measured* positions of Moreau, Atakisi & Thorne (Acta Cryst D77, 2021, 540–554). That list ends at 1.522 Å by its own scope — the paper states that hexagonal ice "has 11 diffraction rings between 4 and 1.5 Å resolution", and its subject was detecting ice in deposited data rather than masking it — not because ice stops there. On a detector that reaches further, the rings it does not list are the ones left in the data.
 
-The eight bands below it are **calculated**, since past that paper there is nothing measured to copy. Enumerating $hkl$ from the ice Ih cell is not enough: ice Ih is $P6_3/mmc$ with oxygen on $4f$, and most of what enumeration emits is extinguished by the *oxygen sublattice* rather than by the space group — which is why (004) at 1.830 Å and (104) at 1.657 Å are missing from the measured list even though they lie inside its range and its reflection conditions allow them (for $(00l)$ the structure factor goes as $\cos 2\pi l z$, and $z \approx 1/16$ kills $l=4$). Structure factors are computed instead — oxygen only, the hydrogens being half-occupancy disordered and weak to X-rays — and the lines kept are those reaching 3 % of the strongest. That rule **reproduces the measured eleven exactly**, and every line it drops inside their range computes to zero, which is what makes it trustworthy below 1.522 Å. The cell is that of Röttger *et al.* (Acta Cryst B50, 1994, 644–648). The list stops at 1.170 Å because below it the calculated real lines fall to 2–3 % while the extinct ones rise to about 1 %, and an oxygen-only calculation cannot separate them any further.
+The eight bands below it are **calculated**, since past that paper there is nothing measured to copy. Enumerating $hkl$ from the ice Ih cell is not enough: ice Ih is $P6_3/mmc$ with oxygen on $4f$, and most of what enumeration emits is extinguished by the *oxygen sublattice* rather than by the space group — which is why (004) at 1.830 Å and (104) at 1.657 Å are missing from the measured list even though they lie inside its range and its reflection conditions allow them (for even $l$ the $4f$ structure factor carries a factor $\cos 2\pi l z$ at every $hk$, and $z \approx 1/16$ kills $l=4$ — $(104)$ along with $(004)$). Structure factors are computed instead — oxygen only, the hydrogens being half-occupancy disordered and weak to X-rays — and the lines kept are those reaching 3 % of the strongest. That rule **reproduces the measured eleven exactly**, and every line it drops inside their range computes to zero, which is what makes it trustworthy below 1.522 Å. The cell is that of Röttger *et al.* (Acta Cryst B50, 1994, 644–648). The list stops at 1.170 Å because below it the calculated real lines fall to 2–3 % while the extinct ones rise to about 1 %, and an oxygen-only calculation cannot separate them any further.
 
 One consequence is worth stating: the profile score is the **strongest** ring's ratio, a maximum over the bands, so a longer list can only raise it. The gate at 1.5 is therefore read against a list of this length, and lengthening it again would need the gate re-checked.
 
@@ -756,7 +756,7 @@ Pixels are classified by their squared distance $r^2=(x-x_p)^2+(y-y_p)^2$:
 
 Invalid pixels (masked/bad/saturated) are excluded from both sums. In addition, pixels lying inside the signal disk ($r<r_2$) of any *other* predicted reflection are removed from this reflection's background annulus, so a neighbouring spot cannot leak into the background estimate. (Both the annulus and that exclusion become ellipses when the option below is used; with it off, which is the default, they are the circles just described.)
 
-**Radially elongated background ring (opt-in, `--integration-stencil <k>`, default 0).** The three radii above are one triple for the whole run, identical for every reflection at every resolution. A reflection is not round, though: a finite bandwidth streaks it radially by $\sigma_\mathrm{bw}=\text{bandwidth}\cdot R_\mathrm{px}$. On a radially smeared spot the fixed $6\ldots13$ px ring therefore sits only $\approx1.3$–$2.2$ radial $\sigma$ from the centre — on the reflection's own tails, which it then measures as background.
+**Radially elongated background ring (opt-in, `--integration-stencil <k>`, default 0).** The three radii above are one triple for the whole run, identical for every reflection at every resolution. A reflection is not round, though: a finite bandwidth streaks it radially by $\sigma_\mathrm{bw}=\text{bandwidth}\cdot R_\mathrm{px}$, with $R_\mathrm{px}$ the distance from the beam centre — the same physical smear as §8.2's and §11.1's $\sigma_\mathrm{bw}$, expressed here in detector pixels where those sections use reciprocal units; the two forms are never mixed in one formula. Throughout, $\text{bandwidth}$ is the **rms** relative energy spread: the user-facing `--bandwidth` takes a FWHM (a DMM's usual specification) and it is divided by 2.355 on input. On a radially smeared spot the fixed $6\ldots13$ px ring therefore sits only $\approx1.3$–$2.2$ radial $\sigma$ from the centre — on the reflection's own tails, which it then measures as background.
 
 With $k>0$ the **background ring becomes an ellipse**, elongated along the beam→reflection direction by $k\sigma_\mathrm{bw}$. The **radial** semi-axes become $r_2+k\sigma_\mathrm{bw}$ and $r_3+k\sigma_\mathrm{bw}$; the **tangential** half-widths stay $r_2$ and $r_3$; and the growth is capped at $2r_3$, which bounds what a mis-declared bandwidth can do to the bounding box. Pixels are then classified as
 
@@ -892,7 +892,7 @@ The partiality applied is fixed by the data type and scaling stage, not chosen f
    \mathrm{erf}\!\left(\frac{\Delta\phi_{ij}-\Delta\phi/2}{\sqrt{2}\,\sigma_{M,i}/\zeta_{ij}}\right)
    \right].
    $
-   The mosaicity $\sigma_{M,i}$ is **measured once per image at indexing** (MLE, §11.2) and held fixed during scaling — only smoothed in frame order (§10.3), never re-refined (it is degenerate with the scale $G$; §11.2).
+   Here $\Delta\phi_{ij}$ is observation $j$'s rocking offset from its exact Bragg angle on image $i$, and the unsubscripted $\Delta\phi$ is the oscillation width per frame — two different quantities that share a letter. The mosaicity $\sigma_{M,i}$ is **measured once per image at indexing** (MLE, §11.2) and held fixed during scaling — only smoothed in frame order (§10.3), never re-refined (it is degenerate with the scale $G$; §11.2).
 
 2. **Unity** ($P_{ij}=1$): used for the scale-on-fulls refit (§10.6), where each observation is already a complete reflection.
 
@@ -1116,11 +1116,13 @@ How fast the intensity falls off with resolution can depend on direction. rugnux
 
 $$\ln \langle I(\mathbf{s})\rangle = c(\text{shell}) - \tfrac{1}{2}\,\mathbf{s}^\mathsf{T} B\, \mathbf{s},\qquad \mathbf{s} = \text{reciprocal-space vector},\ |\mathbf{s}| = 1/d$$
 
+(§10.6 and §14.2 write $s$ for $\sin\theta/\lambda = 1/2d$, so their $s^2 = 1/4d^2$; the two conventions give the same exponent $-(B/2)(1/d^2)$, and $B$ is the same $B$.)
+
 with one free constant per resolution shell, so every isotropic feature — the Wilson curve, an ice ring, a noise floor, a scaling error — is absorbed exactly and only the $\ell = 2$ angular part drives the tensor. For an isotropic $B$ this reduces to the ordinary Wilson plot, so $B$ here is the ordinary crystallographic ($B = 8\pi^2 U$) $B$, directly comparable with phenix.xtriage's `B_cart`, ctruncate's anisotropic $B$ eigenvalues and AIMLESS's anisotropic $\Delta B$. Only the deviatoric part is fitted: the isotropic part is degenerate with the overall scale. The tensor is constrained to the directions the Laue class allows — five free deviatoric parameters in triclinic, three in monoclinic, two in orthorhombic, one in tetragonal, trigonal and hexagonal, and **none at all in cubic**, where symmetry forces $\Delta B$ to be exactly zero.
 
 The fit is on **intensities, with no positivity cut**. Fitting amplitudes, or dropping non-positive intensities as an amplitude-based tool must, loses roughly 40% of the signal: in a direction that has died half the merged intensities are negative, so a positivity cut keeps only the positive noise excursions and flattens the fall-off exactly where the anisotropy is largest.
 
-**Two different quantities are reported, and they are not interchangeable.** $\Delta B$ (the range of the principal components) is a *rate*; the diffraction limit along each principal direction — where $\langle I/\sigma(I)\rangle$ in a 20° cone about that direction falls through 2, read by interpolation in $s^2$ over equal-count shells — is where the signal actually runs out. A crystal can have a large $\Delta B$ and almost no spread in directional limit, or the reverse. Where $\langle I/\sigma(I)\rangle$ never falls through 2 in a direction, the limit returned is the **edge of the measured data** rather than the crystal's own; such a direction is marked — with a `<` in the report, a 1 in `ANISOTROPY_D_MIN_CENSORED`, and a note in the mmCIF — so the spread is not read as a measurement when it is a lower bound. Both constants are AIMLESS's (cone half-angle 20°, the $\langle I/\sigma\rangle$ level this project states resolution at).
+**Two different quantities are reported, and they are not interchangeable.** $\Delta B$ (the range of the principal components) is a *rate*; the diffraction limit along each principal direction — where $\langle I/\sigma(I)\rangle$ in a 20° cone about that direction falls through 2, read by interpolation in $s^2$ over equal-count shells — is where the signal actually runs out. A crystal can have a large $\Delta B$ and almost no spread in directional limit, or the reverse. Where $\langle I/\sigma(I)\rangle$ never falls through 2 in a direction, the limit returned is the **edge of the measured data** rather than the crystal's own; such a direction is marked — with a `<` in the report, a 1 in `ANISOTROPY_D_MIN_CENSORED`, and a note in the mmCIF — so the spread is not read as a measurement when it is a lower bound. Both constants are AIMLESS's defaults (cone half-angle 20°, $\langle I/\sigma\rangle = 2$). The 2 is a per-direction diagnostic level only: the dataset-wide resolution cut (§13.4) is CC$_{1/2}$-based, and the two criteria are not interchangeable.
 
 **The resolution signature.** A genuine Debye–Waller $B$ makes the directional deficit a straight line through the origin in $s^2$. The per-shell $\ell = 2$ amplitude is therefore fitted against $s^2$ and the curve is classified: *linear* (a real $B$), *flat* (a deficit that does not follow $\exp(-\tfrac12 \mathbf{s}^\mathsf{T} B \mathbf{s})$ at all, so the fitted $\Delta B$ describes the data with the wrong functional form and may be an **under**-estimate), or *convex* (a deficit that grows faster than $s^2$, which a $B$ cannot do). The verdict is re-derived at 8 and at 16 shells, and reported as undetermined if it moves.