docs: correct CPU_DATA_ANALYSIS against the code it describes

Ten places where the document and the implementation had drifted apart.
Each was checked against the source before rewriting:

  * 7.5 rotation post-refinement: it is TWO separate cross-validated
    steps (cell+axis from the angles, then distance+beam from the
    positions with the cell fixed), not one joint fit against the merged
    fulls; the held-out split is an hkl hash, not a frame split; the
    bounds are +-5% on distance and +-15 px on the beam, not "under
    ~1%"; and only the distance and beam centre reach pass 2, which
    re-indexes de novo.
  * 9.2 the trimmed-mean background: it is computed in the shared
    background pass, so it DOES apply to --integrator boxsum. Only the
    broadband sigma-clip is excluded. The section previously said both,
    contradicting itself two paragraphs apart.
  * 9.3 per-reflection profile rebuild, sub-pixel centring and radial
    elongation are gaussian-only; the empirical profile keeps the fixed
    per-shell grid and is accumulated on rounded predicted positions,
    not centroids.
  * 10.5 the asymptotic ISa and the b_ISa sigma floor are rotation-only;
    stills report 1/b and floor with the whole-range b.
  * 8.4 centering absences are applied only when the user fixes the
    space group - de novo, prediction runs in P so the search can
    confirm the centering from the intensities.
  * 10.6 per-batch relative-B cross-validates on ASU-group parity, not
    the frame parity the other surfaces use.
  * 13 the resolution cutoff sits one reported-shell width PAST the
    CC1/2 = 0.30 crossing, so data below 0.30 are kept.
  * 7.1 the orientation-only prior penalises all three components of the
    angle-axis vector.
  * 10.1 rlp is the RECIPROCAL Lorentz factor, L = 1/rlp.
  * 14.3 the model scaling is fitted over work and free reflections
    alike, so R-free is free of refinement, not of the scaling fit.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

docs/CPU_DATA_ANALYSIS.md

@@ -342,7 +342,7 @@ The refinement jointly optimizes, depending on mode and constraints:
 - crystal orientation (a global rotation),
 - unit-cell parameters, with constraints determined by inferred crystal system.
 
-By default only the beam center, unit cell and crystal orientation are refined; the detector distance, tilt angles and rotation-axis direction are held fixed unless explicitly enabled. A lighter **orientation-only** mode refines just the crystal orientation (with a weak small-rotation prior on the poorly-determined out-of-plane component), for stills whose geometry is already trusted.
+By default only the beam center, unit cell and crystal orientation are refined; the detector distance, tilt angles and rotation-axis direction are held fixed unless explicitly enabled. A lighter **orientation-only** mode refines just the crystal orientation, for stills whose geometry is already trusted. It carries a weak small-rotation prior penalising the whole angle-axis vector (all three components, at a low weight); what it is there for is the poorly-determined out-of-plane component, which is the one the data barely constrain.
 
 For higher symmetries, constraints are enforced, e.g.
 - cubic: $a=b=c,\ \alpha=\beta=\gamma=90^\circ$,
@@ -383,8 +383,12 @@ The loose first stage necessarily admits some spots that are not reflections of
 The refinement above (§7.2) runs per image against that image's spots. For rotation data an additional **post-refinement** (on by default; `--rotation-no-postrefine` disables it) improves the detector distance, beam centre and crystal cell/axis using **all** frames at once, then re-integrates:
 
 1. **Pass 1** integrates, scales and merges at the header geometry.
-2. Against the merged fulls, the shared detector distance and beam centre and the crystal cell / goniometer-axis are refined jointly over all frames (Ceres, robust loss). The fit is **cross-validated** on a deterministic frame split — refined on one half, gated on the held-out half — and the geometry is **committed only for a small move** (a distance/beam shift under ~1 %); a larger move is treated as a fit instability and rejected. The beam centre is gauge-weak in a rotation series (it trades off against the crystal orientation), so it is restrained toward the header value.
-3. **Pass 2** re-indexes de novo and re-integrates at the committed geometry, reusing pass-1's space group for the merge only.
+2. From pass-1's integrated reflections, the geometry is refined over all frames (Ceres, robust loss) in **two separate steps**, not one joint fit — the same lesson as integration, where refining the profile width and the scale together fails and separately works:
+   - **Step A**: crystal cell scale + goniometer-axis direction, from the observed rotation angles (a distance-independent excitation residual).
+   - **Step B**: shared detector distance + beam centre, from the observed spot positions, with the cell held at step A — so the positional residual is no longer degenerate with the cell scale.
+
+   Each step is **cross-validated** on a deterministic split of the *reflections* (an avalanche-mixed $hkl$ hash, not a frame split and not an $h+k+l$ parity, which would collide with a centering condition and leave the held-out half empty): fitted on one half, committed only if it lowers the held-out residual, otherwise left at nominal. The solver bounds the move — distance within ±5 %, beam centre within ±15 px — and detector tilt is held fixed, being gauge-coupled to the crystal orientation on a single crystal.
+3. **Pass 2** re-indexes de novo and re-integrates at the committed geometry, reusing pass-1's space group for the merge only. Only the **detector distance and beam centre** carry over: the refined cell and axis are used to make step B well-posed, but pass 2 re-indexes from scratch, so they are not propagated.
 
 The refined pass is written as the canonical `<prefix>_*` output; the pass-1 (header-geometry) result is kept alongside as `<prefix>_01_*` for comparison.
 
@@ -436,7 +440,7 @@ Reflections are predicted if they meet minimum $\zeta$ and mosaicity-window crit
 
 ### 8.4 Systematic absences (centering)
 
-Systematic absences are applied at least at the centering level (prior to full space-group symmetry). For centering symbol $C$:
+Systematic absences are applied at the centering level (prior to full space-group symmetry) **when the space group is supplied by the user**. With no user-fixed space group, prediction runs in $P$ regardless of the centering the lattice search inferred: the centering-absent reflections are integrated so that the space-group search (§13) can confirm or disprove the centering from the measured intensities, and so that a missed superstructure shows up. For centering symbol $C$:
 
 - $I$: absent if $h+k+l$ odd,
 - $A$: absent if $k+l$ odd,
@@ -484,18 +488,18 @@ with a Poisson-like uncertainty $\sigma(\hat{I})=\max\!\big(1,\ r_\sigma\hat{I},
 
 **Trimmed-mean background (monochromatic, default on).** On monochromatic data — rotation *and* still (the discriminator is the beam, not the acquisition mode) — the ring mean $\hat{b}$ is by default replaced with a **symmetric trimmed mean**: the ring pixels are sorted, the lowest and highest fraction $f$ are dropped, and the central $(1-2f)$ are averaged ($f=0.10$ by default, `--background-trim`; $f=0$ restores the plain mean). Because $\hat{I}=S-n_S\hat{b}$ is a small difference of large numbers for weak reflections, a per-pixel background bias $\delta\hat{b}$ becomes a *fractional* intensity bias $\approx n_S\,\delta\hat{b}/\hat{I}$ that grows as $\hat{I}$ shrinks — worst at the resolution edge. The plain mean reads high there because neighbour-spot wings that survive the signal-disk mask, tails and zingers are one-sided (positive) contaminants; dropping the extreme ring pixels removes that bias while a clean Poisson ring is essentially unchanged. In practice this lowers the resolution-edge $R_\text{meas}$ several-fold, raises $\langle I/\sigma\rangle$, and rescues the high-resolution CC$_{1/2}$ on data where the plain mean had collapsed it (at a small CC$_{1/2}$ cost in already-clean shells, from the slightly higher variance of the trimmed estimate). It is applied to the shared background used by both the box sum and the profile fit.
 
-For the **profile-fit path on broadband (non-zero bandwidth: pink-beam / DMM) data**, the trimmed mean is *not* used; instead the background mean is computed with a single high-outlier reject (drop ring pixels above $\hat{b}+3\sqrt{\hat{b}}$, then recompute): a bandwidth-streaked high-resolution spot or a close neighbour can leak into the ring and bias the mean high, over-subtracting and driving weak high-resolution intensities negative. A clean Poisson background is essentially unchanged by the cut. Neither robustification is applied to plain box summation (`--integrator boxsum`).
+For the **profile-fit path on broadband (non-zero bandwidth: pink-beam / DMM) data**, the trimmed mean is *not* used; instead the background mean is computed with a single high-outlier reject (drop ring pixels above $\hat{b}+3\sqrt{\hat{b}}$, then recompute): a bandwidth-streaked high-resolution spot or a close neighbour can leak into the ring and bias the mean high, over-subtracting and driving weak high-resolution intensities negative. A clean Poisson background is essentially unchanged by the cut. This $\sigma$-clip is the one robustification that is *not* applied to plain box summation (`--integrator boxsum`); the trimmed mean above is computed in the shared background pass and so applies to box summation too.
 
 ### 9.3 Profile-fitted extraction (default)
 
 A fixed signal disk captures a *width-dependent* fraction of each spot, which puts a multiplicative floor on the per-observation precision of strong reflections and weights weak reflections poorly. Profile fitting removes this by extracting each intensity against a fitted spot shape, without needing reference intensities. Per frame:
 
 1. **Seed.** Box-sum every reflection (§9.2) to get a rough intensity and observed centroid, and select strong spots (significance $\ge 5$).
-2. **Build the profile.** For `gaussian` (the default) the width is taken **per resolution shell** from the measured second moment of the strong spots (shell-dependent because spot size grows with resolution); the intrinsic spot is essentially round in the detector plane (per-detector-region and crystal-anisotropy profiles were evaluated and add nothing — the real crystal anisotropy lives in the discarded rocking direction). For `empirical` the profile is instead the averaged, centroid-aligned, background-subtracted pixel grid of the shell's strong spots. Either way the profile is then **rebuilt for each reflection**, centred on its **sub-pixel predicted position** (the noise-free geometric centre, not the observed centroid) and, where needed, **elongated only along the radial direction** (away from the beam centre) — because two effects stretch a spot radially but not tangentially:
+2. **Build the profile.** For `gaussian` (the default) the width is taken **per resolution shell** from the measured second moment of the strong spots (shell-dependent because spot size grows with resolution); the intrinsic spot is essentially round in the detector plane (per-detector-region and crystal-anisotropy profiles were evaluated and add nothing — the real crystal anisotropy lives in the discarded rocking direction). For `empirical` the profile is instead the averaged, background-subtracted pixel grid of the shell's strong spots, accumulated on their **rounded predicted** positions. For `gaussian` only, the profile is then **rebuilt for each reflection**, centred on its **sub-pixel predicted position** (the noise-free geometric centre, not the observed centroid) and, where needed, **elongated only along the radial direction** (away from the beam centre) — because two effects stretch a spot radially but not tangentially:
    - a finite energy **bandwidth** smears each spot by $\sigma_\mathrm{bw}=\text{bandwidth}\cdot R_\mathrm{px}$ ($R_\mathrm{px}$ = distance from the beam centre, large at high resolution), and
    - sensor **parallax** — the depth over which a photon converts in a thick Si/CdTe sensor — adds a term $\propto\tan^2(2\theta)$ (material- and energy-dependent), plus, on the monochromatic path, a small fixed weak-spot capture term.
 
-   These combine as $\sigma^2_\mathrm{radial}=\sigma^2_\mathrm{intrinsic}+\sigma_\mathrm{bw}^2+c_\mathrm{par}\tan^2(2\theta)$ (tangential unchanged), on a grid grown to hold the streak — capturing it without the tangential background an isotropic widening would add.
+   These combine as $\sigma^2_\mathrm{radial}=\sigma^2_\mathrm{intrinsic}+\sigma_\mathrm{bw}^2+c_\mathrm{par}\tan^2(2\theta)$ (tangential unchanged), on a grid grown to hold the streak — capturing it without the tangential background an isotropic widening would add. The `empirical` profile keeps the fixed per-shell grid and gets none of this.
 3. **Fit (Kabsch).** With profile $P$, background $B$ and the shell variance model, the intensity and its uncertainty are
 $
 I = \frac{\sum P\,(c-B)/v}{\sum P^2/v},\qquad
@@ -525,7 +529,7 @@ $
 where:
 
 - $G_i$ is the image scale factor,
-- $L_{ij}$ is a Lorentz-like / geometry factor (stored as `rlp` or derived),
+- $L_{ij}$ is a Lorentz-like / geometry factor; predictions carry its **reciprocal** as `rlp`, so $L = 1/\texttt{rlp}$ and the correction below is applied as a multiplication by `rlp`,
 - $P_{ij}$ is a partiality term (model-dependent),
 - $I_h$ is the merged (true) intensity parameter for that unique reflection.
 
@@ -550,7 +554,7 @@ The partiality applied is fixed by the data type and scaling stage, not chosen f
 
 2. **Unity** ($P_{ij}=1$): used for the scale-on-fulls refit (§10.6), where each observation is already a complete reflection.
 
-3. **Fixed**: use the per-reflection partiality carried from prediction. Still/serial images are predicted with $P=1$, so a single-pass stills scale is effectively unity/fixed — which is exactly what `--simple-stills` keeps. By default the stills path instead **post-refines a physical partiality**: a small per-crystal orientation tilt $(\delta\psi_x,\delta\psi_y)$ about the two axes perpendicular to the beam is refined against the running merge, and every reflection's partiality is then recomputed analytically from the refined lattice through its excitation error $\Delta_\mathrm{Ewald}=\big|\,|\mathbf{q}+\mathbf{S}_0|-1/\lambda\,\big|$ and an angular width. A tilt moves reflections on opposite sides of the Ewald sphere in opposite directions, so it reshapes the *spatial* pattern of partialities in a way the per-image scale $G$ cannot mimic — which is why it succeeds where a freely-fitted scalar partiality width simply collapses into $G$. Nothing is re-integrated (the integrated intensities are fixed); the tilt is hard-bounded at about 1° and held by a soft prior, so it stays inert on sparse or weak crystals. The cycle is merge → per-crystal tilt refinement (with $G$ profiled out by the same robust Cauchy IRLS used for the per-frame scales, §10.3) → recompute $P$ → re-merge, repeated a few times.
+3. **Fixed**: use the per-reflection partiality carried from prediction. Still/serial images are predicted with $P=1$, so a single-pass stills scale is effectively unity/fixed — which is exactly what `--simple-stills` keeps. By default the stills path instead **post-refines a physical partiality**: a small per-crystal orientation tilt $(\delta\psi_x,\delta\psi_y)$ about the two axes perpendicular to the beam is refined against the running merge, and every reflection's partiality is then recomputed analytically from the refined lattice through its excitation error $\Delta_\mathrm{Ewald}=\big|\,|\mathbf{q}+\mathbf{S}_0|-1/\lambda\,\big|$ and a Gaussian width $\sigma^2=\gamma_0^2+(\gamma_e d^*)^2+(\mathrm{bw}\,|q_z|)^2$ — the reciprocal-lattice point's own radius (resolution-independent), the mosaic/divergence spread, and the bandwidth smear along the beam, in quadrature. In practice the fit drives $\gamma_e\to0$, so the width is essentially the resolution-independent $\gamma_0$. A tilt moves reflections on opposite sides of the Ewald sphere in opposite directions, so it reshapes the *spatial* pattern of partialities in a way the per-image scale $G$ cannot mimic — which is why it succeeds where a freely-fitted scalar partiality width simply collapses into $G$. Nothing is re-integrated (the integrated intensities are fixed); the tilt is hard-bounded at about 1° and held by a soft prior, so it stays inert on sparse or weak crystals. The cycle is merge → per-crystal tilt refinement (with $G$ profiled out by the same robust Cauchy IRLS used for the per-frame scales, §10.3) → recompute $P$ → re-merge, repeated a few times.
 
 Reflections below a minimum partiality can be rejected from merging to avoid unstable corrections.
 
@@ -588,7 +592,7 @@ Per-shell and overall merging statistics are computed on corrected intensities,
 - completeness against the enumerated reflections for the cell and symmetry,
 - the anomalous signal-to-noise $\mathrm{SigAno}$ (below).
 
-The error model is refined as $\sigma_\mathrm{corr}^2 = a\,\sigma^2 + (b\,\langle I\rangle)^2$, with $a$ set by the scatter of weak (counting-limited) reflections and $b$ the intensity-proportional systematic scatter of the strong ones. **ISa** is the asymptotic ($I\to\infty$) signal-to-noise — by definition the reproducibility limit of the strongest reflections (Diederichs, *Acta Cryst.* **D66** (2010) 733) — and is read directly from the strong symmetry equivalents as the counting-subtracted fractional scatter of well-measured reflection groups (a robust median over strong groups; the $I/\sigma$ threshold is relaxed on weak or radiation-damaged data that has few strong reflections), rather than as $1/b$ of the whole-range fit, whose $b$ is raised slightly by an intermediate-intensity excess and so understates the limit. The reported ISa and the merged-intensity systematic floor $\sigma \ge b_\mathrm{ISa}\,|I|$ both use this asymptotic value, so a high-multiplicity merged $I/\sigma$ approaches ISa; the per-observation $\sigma_\mathrm{corr}$ (the merge weights) uses the whole-range $a,b$ and is unchanged.
+The error model is refined as $\sigma_\mathrm{corr}^2 = a\,\sigma^2 + (b\,\langle I\rangle)^2$, with $a$ set by the scatter of weak (counting-limited) reflections and $b$ the intensity-proportional systematic scatter of the strong ones. On the **rotation** path, **ISa** is the asymptotic ($I\to\infty$) signal-to-noise — by definition the reproducibility limit of the strongest reflections (Diederichs, *Acta Cryst.* **D66** (2010) 733) — and is read directly from the strong symmetry equivalents as the counting-subtracted fractional scatter of well-measured reflection groups (a robust median over strong groups; the $I/\sigma$ threshold is relaxed on weak or radiation-damaged data that has few strong reflections), rather than as $1/b$ of the whole-range fit, whose $b$ is raised slightly by an intermediate-intensity excess and so understates the limit. The reported ISa and the merged-intensity systematic floor $\sigma \ge b_\mathrm{ISa}\,|I|$ both use this asymptotic value, so a high-multiplicity merged $I/\sigma$ approaches ISa; the per-observation $\sigma_\mathrm{corr}$ (the merge weights) uses the whole-range $a,b$ and is unchanged. The **stills** path has no asymptotic estimate: it reports $\mathrm{ISa}=1/b$ and floors the merged $\sigma$ with the same whole-range $b$.
 
 **Anomalous signal-to-noise (SigAno).** The strength of the anomalous signal is reported per shell and overall as $\mathrm{SigAno}=\langle|\Delta I|\rangle / \langle\sigma(\Delta I)\rangle$, where $\Delta I = I(+)-I(-)$ over acentric reflections measured in both Bijvoet hands and $\sigma(\Delta I)=\sqrt{\sigma_+^2+\sigma_-^2}$. It is computed from the **full-multiplicity** inverse-variance $I(+)/I(-)$ split (the same one written to the output), i.e. from all observations rather than a half-set. For pure noise $\mathrm{SigAno}$ approaches the half-normal value $\sqrt{2/\pi}\approx0.8$, and it rises above $1$ once a real anomalous difference is present. A half-set anomalous correlation ("$\mathrm{CC}_\mathrm{anom}$") is deliberately **not** used: its two half estimates $\Delta I_0,\Delta I_1$ are complementary partitions of one observation pool ($\Delta I_0+\Delta I_1=2\,\Delta I_\mathrm{full}$), so subtracting the two Bijvoet hands cancels the large common intensity that keeps $\mathrm{CC}_{1/2}$ non-negative and leaves only the small anomalous signal against the per-half split noise; once the anomalous signal-to-noise per half drops below $1$ that correlation is driven towards $-1$ rather than $0$, misrepresenting a weak-but-real signal, whereas $\mathrm{SigAno}$ has no such floor. It is emitted only when an anomalous split was made, using the standard PDBx items `_reflns.pdbx_absDiff_over_sigma_anomalous` (overall) and `_reflns_shell.pdbx_absDiff_over_sigma_anomalous` (per shell), and appears as the `SigAno` column of the printed merge-statistics table.
 
@@ -607,7 +611,7 @@ The fulls are then re-scaled in the XDS sense — a per-image scale refit direct
 
 After scale-fulls, three **correction surfaces** are fitted on the combined fulls (rotation path, **on by default**; disable all with `--no-scaling-corrections`), each an alternating multiplicative refinement of the per-full scale against the merged reference:
 
-- **Decay.** Radiation damage weakens later frames more at higher resolution — a resolution×time (Debye–Waller) systematic the resolution-flat per-image scale cannot capture. A single global relative-$B$ rate is fitted, $\ln(I_\mathrm{ref}/I_\mathrm{obs}) = 2\,(\mathrm{d}B/\mathrm{d}n)\,(n-\bar n)\,s^2$ (frame $n$, $s^2 = 1/4d^2$), and folded into the scale. It engages only when the total relative-$B$ over the run exceeds a physical floor (2 Ų); below that the decay is negligible and "correcting" it only spreads symmetry equivalents (same $s^2$, different frames). An optional **per-batch relative-$B$** (`--relative-b[=deg]`, off unless requested; 10°-of-rotation batches by default) extends the single global rate to a smooth $B(n)$ curve — the same $s^2$-weighted decay fit solved independently over short frame batches, curvature-penalized so it cannot over-fit and cross-validated like the surfaces below — for crystals whose decay is non-linear in dose.
+- **Decay.** Radiation damage weakens later frames more at higher resolution — a resolution×time (Debye–Waller) systematic the resolution-flat per-image scale cannot capture. A single global relative-$B$ rate is fitted, $\ln(I_\mathrm{ref}/I_\mathrm{obs}) = 2\,(\mathrm{d}B/\mathrm{d}n)\,(n-\bar n)\,s^2$ (frame $n$, $s^2 = 1/4d^2$), and folded into the scale. It engages only when the total relative-$B$ over the run exceeds a physical floor (2 Ų); below that the decay is negligible and "correcting" it only spreads symmetry equivalents (same $s^2$, different frames). An optional **per-batch relative-$B$** (`--relative-b[=deg]`, off unless requested; 10°-of-rotation batches by default) extends the single global rate to a smooth $B(n)$ curve — the same $s^2$-weighted decay fit solved independently over short frame batches, curvature-penalized so it cannot over-fit and cross-validated like the surfaces below — for crystals whose decay is non-linear in dose. Its cross-validation splits on **ASU-group parity**, not the frame parity the surfaces below use: a per-batch parameter owns whole frames and so cannot be scored on a held-out frame, whereas splitting the symmetry equivalents tests whether a batch's $B$ generalises to reflections it was not fitted on.
 - **Absorption.** A smooth multiplicative factor over the diffracted-beam direction expressed in the goniometer (crystal) frame: each full's predicted detector position gives the lab diffracted direction, de-rotated by the spindle so a fixed crystal-frame direction is sampled at many rotation angles and its grid cell is well-determined. Negligible at hard X-rays / thin crystals; it matters at low photon energy.
 - **Modulation** (detector-plane flat-field). A smooth multiplicative factor over where each reflection lands on the detector (predicted $x,y$): symmetry-equivalents land at different positions as the crystal rotates, over-determining the surface. It absorbs detector-response and geometric systematics that inflate $R_\mathrm{meas}$.
 
@@ -708,7 +712,7 @@ A **dataset-wide** Wilson $B$ is also estimated over the merged reflections —
 - **Space-group determination.** When no space group is supplied, a POINTLESS-like search scores Laue-group symmetry (CC of $I(h)$ vs $I(Rh)$ plus merge self-consistency) and detects screw/centering absences from the $P1$-merged intensities. The self-consistency test is calibrated so a merohedral twin — whose twin law forces non-equivalent reflections together and inflates the merged $\chi^2$ — stays in its true lower symmetry rather than being over-promoted to the holohedral group. Because a partial twin's within-orbit $\chi^2$ can nonetheless look self-consistent, a chi²-passing promotion is additionally **vetoed** when merging its extra operator balloons the error-model $b$ (the intensity-proportional systematic) relative to the confirmed subgroup: a genuine symmetry step gains multiplicity without inflating $b$, whereas a twin forces non-equivalent reflections together and $b$ balloons. A $\chi^2$ test alone cannot decide this, because it is a ratio to an error model that moves with the *amount* of data: the parent's systematic term grows as $\sigma$ shrinks with $1/\sqrt{N}$ while a twin's is already saturated, so the same crystal is promoted or not depending only on how much of it the search saw. The promotion is therefore also gated on a sigma-free **operator disagreement** $H=\mathrm{median}\,|I_1-I_2|/(I_1+I_2)$, taken as the ratio of the operators a promotion *adds* to the parent's own, measured on the same reflections; the parent normalisation divides out the systematic floor that symmetry mates always carry on real data, which varies by crystal *and* by operator, so no absolute bound on such a statistic can work. A median, because a twin perturbs every pair while a badly-measured minority perturbs only the tail; and judged against the most damning parent when a candidate has several of the same order, since a rival subgroup can itself contain the twin laws. The Lorentz factor $\zeta$ (§8.3) governs how well a reflection can be measured, so when the spindle lies in a plane of the lattice an operator permuting the two in-plane axes samples a different mixture of measurement qualities than one that only flips signs. The search is therefore repeated on a merge of only the well-measured observations (`--search-min-zeta`, rotation default 0.85) and **whichever search found more symmetry is kept** — one-way safe, because discarding observations can starve an operator correlation but never invent one. The filtered merge decides the point group only; absences always come from the full merge, since they live in the weak reflections the filter removes. An unbreakable tie (same order, different symmetry) is reported with both candidates named, for trying in molecular replacement. **Centering** is accepted when the systematically-absent class is weak relative to the present one by *either* of two floor-independent tests — its mean signed $I/\sigma$ well below the present mean, *or* its rate of individually-significant reflections well below the present class's own significant rate. The second test matters on weak / low-energy data, where a positive intensity floor (background/profile leakage) lifts the absent class's mean $I/\sigma$ to $\sim1.5$–$2.3$ instead of $\sim0$ and, when the present class is itself weak, inflates the plain mean ratio past its bound and hides a real centering (an $I$-centred cubic recorded at 5 keV was otherwise kept primitive); a false centering fails both tests because its absent class is as strong as the present one. When several centerings pass, they are ranked by their **net** systematic absences (absent minus violating), not the gross absent count, so a super-centering (e.g. $F$ over a true $C$) whose extra, only-half-populated absent class merely dilutes the strength ratio does not out-rank the correct lower centering.
 - **Twinning check.** A Padilla–Yeates $L$-test ($\langle|L|\rangle$, $\langle L^2\rangle$) and the second moment $\langle I^2\rangle/\langle I\rangle^2$ (taken per resolution shell with noise-only shells skipped and Wilson outliers rejected, so a single strong reflection in a collapsed-mean shell cannot skew it) are written to the merged mmCIF as a twinning diagnostic. Twinning is only flagged in Laue classes where a merohedral twin law can exist; the holohedral high-symmetry classes ($4/mmm$, $6/mmm$, $m\bar{3}m$, and $\bar{3}m$ on a rhombohedral lattice) are exempt, so a low $\langle|L|\rangle$ there is reported as a statistical artefact rather than twinning.
 - **Outlier rejection.** Merging applies an optional per-observation median-based $N\sigma$ cut (`--reject-outliers`, default 6σ for `rot3d`, off otherwise). The same $N\sigma$ cut is fed back into the error model: after an initial $a,b$ fit the parameters are re-fit once on the reflections that survive rejection (dropping any whose squared deviation exceeds $N\sigma^2\,[a\,\sigma^2 + (b\,\langle I\rangle)^2]$), so the calibrated errors describe the reflections that actually enter the merge rather than the pre-rejection pool.
-- **Automatic resolution cutoff.** By default the reported/written high-resolution limit is trimmed where $\mathrm{CC}_{1/2}$ falls off (logistic, target 0.30); `--scaling-high-resolution` overrides it and `--resolution-cutoff off` disables it.
+- **Automatic resolution cutoff.** By default the reported/written high-resolution limit is trimmed where $\mathrm{CC}_{1/2}$ falls off: a logistic is fitted to $\mathrm{CC}_{1/2}(s)$, and the limit is set **one reported-shell width past** the point where the fit crosses 0.30 — deliberately "one shell too far", so weak-but-real data below the crossing are kept rather than discarded. The extension is measured over the range that is actually kept, not the full measured range, so a detector reaching far past where the crystal diffracts cannot inflate it. `--scaling-high-resolution` overrides the limit and `--resolution-cutoff off` disables it.
 - **Amplitudes and intensities.** The merged output carries both intensities (mmCIF `intensity_meas`, MTZ `IMEAN`/`SIGIMEAN`) and French–Wilson amplitudes (mmCIF `F_meas_au`, MTZ `F`/`SIGF`; §10.8), so a downstream program can refine against either.
 
 ---
@@ -739,7 +743,7 @@ $
 R = \frac{\sum \big|\,|F_o| - |F_\mathrm{model}|\,\big|}{\sum |F_o|},
 $
 
-with R-free the same sum restricted to the free set — an unbiased measure of how well the model explains data it was not scaled against.
+with R-free the same sum restricted to the free set. Note that the scaling of §14.2 is fitted over **all** reflections, work and free alike — its few parameters ($k$, an anisotropic $B$, $k_\mathrm{sol}$, $B_\mathrm{sol}$) are far too few to absorb individual reflections, but R-free here is strictly "free of refinement", not free of the scaling fit.
 
 ### 14.4 Electron-density maps