Sign tests by product underflowed: FIXED everywhere (branch, not master - the paper is not yet updated). FA * fnew < 0.0 with FA ~ -1e-200, fnew ~ 1e-150 is -0.0 and '-0.0 < 0.0' is False, so on flat functions (x^19) the bracket update moved the wrong end, the root left the bracket and the run still reported convergence - the proposed rule returned 4.9e-9, bisection 3.5e-9, ITP 1.6e-8 with a claimed 1e-12 bracket; the 1e-6 harness scored them solved. Same defect in our bisection, false-position family, Ridders, Brent, ITP and twelve copies in stage scripts, simroot/sim.py and one test; all replaced by _opposite / _same_sign, pinned by test_sign_underflow.

Impact, every sweep regenerated: the paper's key configurations are unchanged to the evaluation (proposed 2806/3220, credit 3568/3020, Chandrupatla, TOMS 748, Brent, bisection); 46 stated numbers move (ITP CHA 2857 -> 2896, ITP oracle 2611 -> 2647, record CHA figures by 14-18, the grid means by ~1%) and one finding weakens: the measured-order exponent of section 7.1 was cheaper in 29 of 60 far cells and 7 of 7 safeguards; with the fix 12 of 61 and 6 of 7, and on the wide-bracket set it now loses in every safeguard and drops 5 problems.

PREREG G1-G3 outcome on the fixed code: G2 (repeating step pattern, period from the data) PASSES Y1-Y4 and is ACCEPTED by the stop rule - proposed rule APS 2806 -> 2782 (below Chandrupatla's 2787), CHA 3220 -> 3169, both 6026 -> 5951, per problem vs Chandrupatla on APS 95/20/38 p = 8e-7. G1 fails Y1/Y2 (its Y3 miss was the sign bug), G3 fails Y2, A fails Y2. Regression 887 pass; the two red tests are the paper-number check (awaiting the paper update) and test_blend (fixed since, copy in stage16).

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

tests/test_extrapolate.py

@@ -73,6 +73,46 @@ def test_on_a_budget_bound_run_extrapolation_changes_the_path_not_the_count():
     assert ev["steps"][1] != ev["none"][1]
 
 
+GATES = ("confirmed", "pattern", "steps+results")
+
+
+@pytest.mark.parametrize("mode", GATES)
+@pytest.mark.parametrize("f,a,b", SMOOTH + FLAT)
+@pytest.mark.parametrize("guard", ("project", "credit"))
+def test_every_gate_keeps_the_guarantee(mode, f, a, b, guard):
+    x, n, ok = pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                         guard=guard, extrapolate=mode)
+    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), (mode, n)
+
+
+def test_the_gates_decline_without_history():
+    """Each gate needs history to confirm anything; without it the step is untouched."""
+    from simroot.pure import _extrapolation_s
+    # opposite directions -> nothing to confirm, in every mode
+    for mode in ("steps",) + GATES:
+        assert _extrapolation_s(mode, 0.1, 1.0, 1.5, None, 1.0, 2.0, None, []) == 0.0
+    # same direction, but G1 has no third point, G3 no third value, G2 no history
+    assert _extrapolation_s("confirmed", 0.1, 1.0, 0.8, None, 1.0, 2.0, None, []) == 0.0
+    assert _extrapolation_s("steps+results", 0.1, 1.0, 0.8, None, 1.0, 2.0, None, []) == 0.0
+    assert _extrapolation_s("pattern", 0.1, 1.0, 0.8, None, 1.0, 2.0, None, [0.2]) == 0.0
+
+
+def test_results_after_a_crossing_confirm_nothing():
+    """G3: a sign change of f between the last results is a crossing, not a creep."""
+    from simroot.pure import _extrapolation_s
+    assert _extrapolation_s("steps+results", 0.1, 1.0, 0.9, 0.8, 1.0, -2.0, 4.0, []) == 0.0
+    assert _extrapolation_s("steps+results", 0.1, 1.0, 0.9, 0.8, 1.0, 2.0, 4.0, []) > 0.0
+
+
+def test_a_confirmed_geometric_series_is_read_exactly():
+    """G1 on three moves in ratio 0.5, 0.5: fully confirmed, s = q2 = 0.5."""
+    from simroot.pure import _extrapolation_s
+    s = _extrapolation_s("confirmed", 0.025, 1.0, 0.95, 0.85, 1.0, 2.0, 4.0, [])
+    assert s == pytest.approx(0.5)
+
+
 def test_unknown_mode_is_refused():
     with pytest.raises(ValueError):
         pure_root(lambda t: t, -1.0, 1.0, XTOL, MAXFEV, extrapolate="aitken")