Article: the similarity in the step is earned, not scaffolding; literature in place.

S-ablation of A' (server): A' with each of the nine similarity functions in the step - tent 4443 on the paper protocol (13032 over 855 problems), the nearest other function e^-|z| 4988 (+12%; +9% over six corpora), quadratic ones 18410-18422 (+315%); ranking as the contraction law predicts; guarantee held with every S. The audit of section 7 rerun on the regenerated sweeps (APS unchanged: triple by shape -493 with the tent, +250 with 1/sqrt(1+z^2); defect -283 vs +215).

Text: abstract and section 3 no longer call the similarity scaffolding - the tent making the step Newton's is a property of the similarity chosen; a paragraph in section 7 on the audit; Table 9a; new 8.6 'What is known, and what is new' - Lagouanelle (the order estimate of 7.1 / 10.5), Rall and Hansen-Patrick (8.1), the log-log slope / COC / ACOC (8.2, formula 6), Schroeder and Traub (the split 7 = false position on sign(f)|f|^(1/m)), Brent's acceptance condition (half), Dekker/Brent (tolerance step); what we found no counterpart for is the decision layer by similarity and analogy and the step set by a similarity whose form is decisive. References 13-19. verify_article: Table 9a, the percentages and the audit numbers added; server 446 OK, 0 BAD; regression x2 1226 passed, 0 failed.

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

verify_article.py

@@ -800,6 +800,34 @@ def verify_flat_regime_and_oos():
     check("applied, variant carried: totals", (sum(cost(v) for v in app["A"].values()),
                                                sum(cost(v) for v in app["chandrupatla"].values())),
           (1580, 1382))
+    print("\nTable 9a and section 7: the similarity in the step (fresh audit run):")
+    aps_ = {r["spec"][1]: r for r in _load("sweep_aprime_s.json")}
+    abs_ = _load("sweep_ablation_s.json")
+    t9a = {"ratio": (4443, 13032), "exp_abs": (4988, 14142), "rational": (5486, 14843),
+           "ratio_soft": (8360, 28500), "sqrt": (12718, 45552), "cbrt": (17207, 63031),
+           "exp_sq": (18410, 67701), "cosine": (18412, 67854), "rat_sq": (18422, 67394)}
+    for s_, (two, six) in t9a.items():
+        r = aps_[s_]
+        check(f"Table 9a {s_} APS+CHA", r["APS"]["total"] + r["CHA"]["total"], two)
+        check(f"Table 9a {s_} six corpora", sum(tot(abs_[f"S:{s_}"], s) for s in sets), six)
+        check(f"Table 9a {s_} guarantee", r["APS"]["over"] + r["CHA"]["over"]
+              + sum(v[2] for s in sets for v in abs_[f"S:{s_}"][s].values()), 0)
+    check("nearest other S costs more, protocol %", round(100 * (4988 / 4443 - 1)), 12)
+    check("worst S costs more, protocol %", round(100 * (18422 / 4443 - 1)), 315)
+    check("nearest other S costs more, six corpora %", round(100 * (14142 / 13032 - 1)), 9)
+    check("quadratic S: four to five times",
+          (int(18410 / 4443), int(67854 / 13032)), (4, 5))
+    import audit_similarity_places
+    buf = io.StringIO()
+    with contextlib.redirect_stdout(buf):
+        audit_similarity_places.main()
+    rows_a = {ln.split("  ")[0].strip(): re.findall(r"([+-]\d+) \|", ln)
+              for ln in buf.getvalue().splitlines() if " | " in ln}
+    check("audit: shape gain with the tent (APS)", int(rows_a["tent 1-|z|"][1]), -493)
+    check("audit: shape gain with 1/sqrt(1+z^2) (APS)", int(rows_a["1/sqrt(1+z^2)"][1]), 250)
+    check("audit: defect gain with the tent (APS)", int(rows_a["tent 1-|z|"][3]), -283)
+    check("audit: defect gain with 1-|z|^1/3 (APS)", int(rows_a["1-|z|^1/3"][3]), 215)
+
     # every sign-test p the paper prints in Section 9, at the precision printed
     for (w, l), printed in (((103, 31), "3e-10"), ((31, 10), "0.001"), ((108, 65), "0.001"),
                             ((44, 20), "0.004"), ((48, 32), "0.09"), ((177, 28), "1e-27"),