docs: why the spots half a turn apart are Friedel mates, and how many there are

The Laue condition fixes the component along the beam; a half turn about the spindle negates the
two perpendicular to it and taking -h negates all three, so the composition leaves the beam
component alone and the mate is on the sphere exactly, not nearly. Says that they are separated
by half a turn rather than recorded together precisely because the Ewald sphere is curved - a
near-flat one would excite both at once - and that the positions need only the reciprocal lattice
to be centrosymmetric, Friedel's law entering as the test of a pairing rather than its basis.

Also states the yield: a sweep of S degrees gives S-180 degrees' worth of pairs, so half a turn
gives none and a little over half a turn gives few.

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

docs/CPU_DATA_ANALYSIS.md

@@ -156,18 +156,33 @@ Jungfraujoch uses $|\Delta_\mathrm{Ewald}|$ as an operational proxy for excitati
 Two exact facts about a rotation sweep let the direct beam be measured from spot positions alone,
 before anything is indexed (`--estimate-beam-center`).
 
-**Friedel mates half a turn apart.** Rotating 180° about the spindle $\mathbf{m}$ and taking $-h$
-negates a reflection's component along $\mathbf{m}$ and leaves the rest. With the spindle
-perpendicular to the beam this preserves $\mathbf{q}\cdot\mathbf{S}_0$, so $-h$ satisfies the Laue
-condition at $\varphi+180°$ exactly where $h$ satisfies it at $\varphi$, and the spots recorded half a
-turn apart are mirror images along the spindle direction. This gives the beam coordinate **along**
-the spindle. Note these are Friedel mates, not the same reflection.
+**Friedel mates half a turn apart.** The Laue condition fixes one component of $\mathbf{q}$, the one
+along the beam: $q_\parallel = -\lVert\mathbf{q}\rVert^2\lambda/2$. Rotating 180° about the spindle
+$\mathbf{m}$ negates the two components perpendicular to $\mathbf{m}$; taking $-h$ negates all three.
+Composing the two therefore negates **only** the component along $\mathbf{m}$ and leaves the other
+two, including $q_\parallel$ — so when the spindle is perpendicular to the beam, $-h$ satisfies the
+Laue condition at $\varphi+180°$ **exactly** where $h$ satisfies it at $\varphi$, and its spot sits at
+the mirror image of $h$'s along the spindle direction. This gives the beam coordinate **along** the
+spindle.
+
+These are Friedel mates, not the same reflection, and they are *not* recorded simultaneously: the
+curvature of the Ewald sphere is what separates them by half a turn instead of exciting both at once
+as a near-flat sphere would. Note also that only the *positions* are used here, and those follow from
+the reciprocal lattice being centrosymmetric — $-h$ is a lattice point whenever $h$ is, for every
+crystal whatever its point group. Friedel's law $|F(h)| = |F(-h)|$ is not needed for the geometry; it
+is used only to tell a true pairing from an accidental one, and it is what anomalous scattering
+slightly violates.
+
+Because a pair needs both $\varphi$ and $\varphi+180°$ to have been recorded, a sweep of $S°$ yields
+only $S-180$ degrees' worth of pairs. **A sweep of exactly half a turn yields none**, and one a few
+degrees longer yields correspondingly few — which is why the useful range begins well above 180° and
+why a full turn, where every reflection has its mate, is the comfortable case.
 
 **The second crossing.** The same reflection meets the Ewald sphere twice, at two angles that are
 generally *not* 180° apart, differing only in the sign of the lab component perpendicular to both
-$\mathbf{m}$ and the beam. This gives the remaining coordinate. The two crossings are separated by a
-sweep angle fixed by the reflection's own position, so genuine pairs are identified without a cell or
-an orientation matrix.
+$\mathbf{m}$ and the beam. This gives the remaining coordinate, and it is available within a sweep
+shorter than half a turn. The two crossings are separated by a sweep angle fixed by the reflection's
+own position, so genuine pairs are identified without a cell or an orientation matrix.
 
 Neither observable requires the reflections to be indexed: each candidate pairing votes for a beam
 coordinate, and the true value accumulates while wrong pairings scatter.