PREREG_half_contra outcome: ACCEPTED (server = smoke in every projection cell; 0 violations in the guaranteed family).

'half': the step promises the root in the half near x (S >= 1/2) while the bracket line puts it in the far half (smaller |f| at the far end) -> halve.
  half alone                                  APS 2806 -> 2564 (!)  CHA 3220 -> 3202
  G2 + contra + rem + point + exp + half      APS 2773 -> 2559  CHA 2073 -> 2047  both 4606 (all of H1-H4; goal kept)
     per problem vs Chandrupatla: APS 98/17/38 p=2.7e-7, CHA 27/6/7 p=8e-4
  contra + rem + point + exp + half (no G2)   APS 2551  CHA 2044  both 4595 (H2 fails by 24)
  with defect it does not combine (APS 2531 -> 2651): the two refuse overlapping steps.
fun9: 77 -> 68 (Chandrupatla 55); fun9.5 24 -> 16.
Server regression x2: 1204 passed, 1 failed = the paper-number check.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

tests/test_flat_gate.py

@@ -424,3 +424,28 @@ def test_the_exp_limit_leaves_simple_roots_alone():
                        guard="project", prestep=p)[1]
              for p in ("defect+flatanalogmempt", "defect+flatanalogmempt+exp")]
         assert n[0] == n[1], n
+
+
+# ---- PREREG_half_contra_2026-09-21: the half promised by the step vs the bracket ----
+@pytest.mark.parametrize("f,a,b", PROBS)
+@pytest.mark.parametrize("guard", ("project", "credit"))
+@pytest.mark.parametrize("pre", ("half", "defect+flatanalogmempt+exp+half",
+                                 "contra+flatanalogmempt+exp+half"))
+def test_the_guarantee_holds_with_half(f, a, b, guard, pre):
+    x, n, ok = pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                         guard=guard, prestep=pre)
+    assert ok and min(a, b) <= x <= max(a, b)
+    half = math.ceil(math.log2(abs(b - a) / XTOL))
+    assert n <= (2 * half + 3 if guard == "project" else half + 4), n
+
+
+def test_half_is_silent_on_a_line_and_breaks_the_fun9_flank():
+    """Measured before pinning: on [2e-4, 81] the flank costs 24 -> 16 in the best cell."""
+    st = {}
+    pure_root(lambda t: 3.0 * t - 1.0, -2.0, 5.0, XTOL, MAXFEV, similarity="ratio",
+              scale="derived3", guard="project", prestep="half", stats=st)
+    assert st.get("half_refused", 0) == 0
+    n = [pure_root(_fun9, 0.0002, 81.0, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                   guard="project", prestep=p)[1]
+         for p in ("defect+flatanalogmempt+exp", "defect+flatanalogmempt+exp+half")]
+    assert n[1] < n[0], n