Two audits, both on existing data or traces, method untouched. (1) Where S is evaluated: only in the step length (and simpair, and two refuted modes); the slope gate, the triple by shape, the far end and 'defect' decide by comparing - which is a similarity judgment of the ordinal kind ('which is more similar'), where every decreasing S gives the same decision. Their gains depend strongly on the S in the STEP: triple by shape -493 with the tent, up to +250 with others; defect -283 with the tent, up to +215. The records are the pairing of the tent step with those judgments.

(2) Valeri: 'similarity should solve the flat-family problem'. The similarity of two successive contraction ratios, min/max(rho_n, rho_n-1), was pre-declared as the detector; it FAILS and is inverted - AUC 0.28-0.42 flat against the rest, and 'the contraction did not speed up' fires more on APS (37.6%) than on the flat families (28.2%). The rho sequence shows why: on x^m the method runs a period-4 cycle - two Newton undershoots at exactly (1-1/m)^m (0.328 for m=5, 0.346 for m=9), one crossing, one nearly empty step (rho ~ 1) - so successive ratios are dissimilar, not constant. The safeguard is not the cause: the steps are as proposed. Candidate detectors suggested by the cycle are new hypotheses, not built.

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

diag_contraction_sequence.py

@@ -0,0 +1,62 @@
+"""Why the contraction-ratio similarity came out inverted: the rho sequence itself.
+
+diag_contraction_similarity.py falsified "successive contraction ratios are similar on
+the flat families": they are LESS similar there than on the canonical set.  The picture
+behind the hypothesis - rho constant at about (1 - 1/m)^m - holds for pure Newton steps on
+x^m.  The explanation to check is that our safeguarded run is not pure: the projection
+interleaves forced moves, so rho alternates.  Printed per step for one flat problem and
+one canonical problem: rho_n, and whether the step was the similarity step as proposed or
+was changed by the safeguard (x_new != x + raw).
+
+    python -u diag_contraction_sequence.py
+"""
+from __future__ import annotations
+
+from scipy.optimize._tstutils import _APS_TESTS_DICTS, _CHANDRUPATLA_TESTS_DICTS
+
+from simroot.pure import pure_root
+
+CFG = dict(similarity="ratio", scale="derived3", guard="project")
+
+
+def show(name, f, a, b):
+    evals = []
+
+    def g(t):
+        v = f(t)
+        evals.append((t, v))
+        return v
+    st = {"trace": []}
+    pure_root(g, a, b, 1e-12, 200, stats=st, **CFG)
+    # map each traced state to the evaluation that followed it
+    order = [e[0] for e in evals]
+    print(f"\n== {name}: {len(evals)} evaluations")
+    print(f"   {'k':>3} {'rho_n=|f_n/f_(n-1)|':>22} {'step as proposed?':>18}")
+    for k, s in enumerate(st["trace"][:40]):
+        proposed = s["x"] + s["raw"]
+        try:
+            i = order.index(s["x"])
+        except ValueError:
+            continue
+        if i + 1 >= len(evals):
+            break
+        x_next, f_next = evals[i + 1]
+        rho = abs(f_next / s["fx"]) if s["fx"] != 0.0 else float("nan")
+        kept = "yes" if x_next == proposed else "CHANGED"
+        print(f"   {k:3d} {rho:22.3e} {kept:>18}")
+
+
+def main():
+    cha = {str(d["ID"]): d for d in _CHANDRUPATLA_TESTS_DICTS}
+    d = cha["fun5.1"]
+    show("fun5.1 = x^9 (flat)", d["f"], *(float(v) for v in d["bracket"]))
+    d = cha["fun4.3"]
+    show("fun4.3 = 6(x-2)^5 on a wide bracket (flat)", d["f"],
+         *(float(v) for v in d["bracket"]))
+    d = _APS_TESTS_DICTS[10]
+    f = (lambda ff, aa: lambda t: ff(t, *aa))(d["f"], d.get("args", ()))
+    show(f"{d['ID']} (canonical)", f, *(float(v) for v in d["bracket"]))
+
+
+if __name__ == "__main__":
+    main()