Space-group search: decide a screw from the evidence, not from a count of absences

A screw's predicted-absent class was required to hold min_absent_observed = 8 reflections before the
screw could be claimed. That count is the wrong measure of evidence, and it is wrong in both
directions.

A screw extinguishes one row of reciprocal space, and that row is often the one a rotation sweep
records least: it lies near the spindle, where the blind cusp maps onto itself and symmetry cannot
fill it in. Counting it measures the geometry of the sweep. A monoclinic crystal whose 2-fold sits
7.6 deg from the spindle contributed six 0k0-odd reflections, every one of them measured between
-0.013 and 4e-5 of the shell mean with zero violations, against a 0k0 row averaging 1.44x the shell
mean - and was refused its 2_1 for being six rather than eight. XDS's own integration of the same
images finds seventeen of those reflections and every one of them is likewise dead.

The count is equally wrong the other way: a uniformly weak axial row produces no violations at all,
so with enough reflections on it a screw is claimed from no evidence whatsoever. The second new test
section demonstrates exactly that on the old gate.

Judge the class by how unlikely it would be if the screw did not exist. Under "no screw" the absent
class and the rest of its row are both Wilson-distributed with the same mean, so with each absent
intensity taken in units of its row's control mean, sum_u/(sum_u + n_control) follows Beta(n_absent,
n_control) exactly; the reported evidence is -log of that lower tail. The row's own strength cancels,
which is the property the count lacks, and the scale is set by the number of reflections, so
few-but-decisive and many-but-marginal are told apart. It is sigma-free by design: the merged sigma
carries the error model's intensity-proportional term and so shrinks with I, reading much the same on
an absent reflection as on a present one.

This follows POINTLESS (Evans, Acta Cryst D67, 282-292 (2011), Appendix A3), which likewise scores an
absence against the rest of its own axial row rather than against a global mean or a fixed cut, and
likewise lets confidence fall away with the number of axial reflections instead of refusing outright
below a count. POINTLESS calibrates its null width from control transforms of non-axial reflections;
the Beta tail here is an analytic null in its place. XDS is not a reference for this: it "deliberately
avoids any test for the presence of screw axes as these tests would depend strongly on the
completeness of the data" (Kabsch, Acta Cryst D66, 133-144 (2010), section 6), so a screw axis in a
CORRECT.LP was supplied to it, not determined by it.

Measured over five probe crystals, genuine screw conditions read 34-800 nats and false ones - the
4_1/4_3 conditions of a cubic crystal that has no screw, whose predicted-absent class is STRONGER
than its control row - read -7 to -8.5. The bound is set at 20, in the gap, at p <= 2e-9: three
well-measured dead axial reflections clear it and two do not.

min_absent_observed keeps its job for CENTERING, where a count is a fair measure - that class is a
third to a half of every reflection in the data set and the bound is never binding on a centering
that exists.

The candidate table now prints the screw-absent count and this evidence in place of the two E^2
medians that were its raw ingredients, so a refusal can be read off the log.

Measured on the five probes: the monoclinic crystal above returns to P2_1 with every merge statistic
unchanged (R_meas 58.8 -> 58.7%, CC1/2 49.1 -> 49.3%, ISa 6.61 -> 6.59 - P2 and P2_1 share a point
group, so only the symbol and the absent reflections differ). The other four are untouched, space
group included, and the two-pass branch fingerprint (indexed frames, distance, mosaicity) is
identical on all five. The full battery has not been run.

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

tests/SearchSpaceGroupTest.cpp

@@ -234,4 +234,63 @@ TEST_CASE("SearchSpaceGroup finds a screw axis whose absent class is weak only w
     for (const auto& alt : result.alternatives)
         accepted.push_back(alt.short_name());
     CHECK(std::find(accepted.begin(), accepted.end(), "P43212") != accepted.end());
-}
+}
+// Regression: a screw's predicted-absent class is one row of reciprocal space, and that row is often
+// the one a rotation sweep records least - it lies near the spindle, where the blind cusp maps onto
+// itself and symmetry cannot fill it in. Counting the class therefore measures the geometry of the
+// sweep, not the strength of the evidence, and a count gate refused a monoclinic crystal its 2_1 for
+// having six 0k0-odd reflections rather than eight, every one of them measured at a thousandth of the
+// row beside them. The class is judged by ScrewAbsenceEvidence instead, which reads the contrast
+// against the row - so few-but-decisive is accepted and many-but-marginal is not.
+TEST_CASE("SearchSpaceGroup weighs a screw's absences by evidence, not by how many were recorded") {
+    const gemmi::SpaceGroup& sg = gemmi::get_spacegroup_by_name("P 1 21 1");
+    const gemmi::GroupOps gops = sg.operations();
+
+    SearchSpaceGroupOptions opt;
+    opt.merge_friedel = true;
+
+    SECTION("five decisive absences, below min_absent_observed: the screw is still found") {
+        auto merged = GenerateMergedReflectionsForSpaceGroup(sg, 18);
+        // Keep five of the 0k0-odd reflections, at a thousandth of their row, and drop the rest - as a
+        // sweep along the 2-fold does, leaving too few to satisfy a count but plenty to decide.
+        int kept = 0;
+        std::erase_if(merged, [&](MergedReflection& r) {
+            if (!gops.is_systematically_absent(gemmi::Op::Miller{{r.h, r.k, r.l}}))
+                return false;
+            if (kept >= 5)
+                return true;
+            ++kept;
+            r.I = 0.5;
+            return false;
+        });
+        REQUIRE(kept == 5);
+        REQUIRE(kept < opt.min_absent_observed);
+
+        const auto result = SearchSpaceGroup(merged, opt);
+        INFO(SearchSpaceGroupResultToText(result));
+        REQUIRE(result.best_space_group.has_value());
+        CHECK(result.best_space_group->short_name() == "P21");
+    }
+
+    SECTION("a uniformly weak axial row decides nothing, however many absences it holds") {
+        // The whole 0k0 row badly measured: the predicted-absent reflections are weak, but so is the
+        // rest of their row, so there is no contrast and no screw to claim. A violation count cannot
+        // see this - nothing on the row clears an absolute cut, so it reads zero violations and, with
+        // enough reflections to satisfy the count, would claim the 2_1 from no evidence at all.
+        auto merged = GenerateMergedReflectionsForSpaceGroup(sg, 18);
+        int absent_on_row = 0;
+        for (auto& r : merged) {
+            if (r.h != 0 || r.l != 0)
+                continue;
+            const bool absent = gops.is_systematically_absent(gemmi::Op::Miller{{r.h, r.k, r.l}});
+            r.I = absent ? 4.0 : 5.0;
+            absent_on_row += absent ? 1 : 0;
+        }
+        REQUIRE(absent_on_row >= opt.min_absent_observed);
+
+        const auto result = SearchSpaceGroup(merged, opt);
+        INFO(SearchSpaceGroupResultToText(result));
+        REQUIRE(result.best_space_group.has_value());
+        CHECK(result.best_space_group->short_name() == "P2");
+    }
+}