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>

apply_article_audit_step.py

@@ -0,0 +1,50 @@
+"""Section 7 gets the audit that shows the similarity in the step decides whether the readings
+pay (audit_similarity_places.py, rerun on the regenerated server sweeps); section 3 and the
+abstract refer to it correctly; 314% -> 315% (18422/4443 = 4.146).  Exact anchors.
+
+    python -u apply_article_audit_step.py
+"""
+from __future__ import annotations
+
+PATH = "article/sim_numerics.md"
+
+AUDIT_PARAGRAPH = (
+    "\n\n**Which similarity stands in the step decides whether a reading pays.** The readings above, "
+    "and the decision *defect* of §7.2, compare quantities directly — an ordinal judgment of which is "
+    "more similar, in which every decreasing $S$ would decide alike. Their gains nevertheless depend "
+    "on the $S$ that sets the step (`audit_similarity_places.py`, on the same sweeps): on the canonical "
+    "set the triple chosen by shape gains 493 evaluations with the tent in the step and loses 250 with "
+    "$1/\\sqrt{1+z^2}$; *defect* gains 283 with the tent and loses 215 with $1-|z|^{1/3}$. The records "
+    "of this section are the pairing of the tent step with these judgments, not the judgments alone."
+)
+
+REPL = [
+    ("the readings of §7 pay only with the tent in the step (§7, the audit of the gains), and in the "
+     "proposed method replacing the tent by any other function costs 12% to 314% (Table 9a).",
+     "the readings of §7 pay most with the tent in the step and some of them lose with other functions "
+     "(§7), and in the proposed method replacing the tent by any other function costs 12% to 315% "
+     "(Table 9a)."),
+    ("and the similarity that sets the step, whose form is decisive (replacing it costs 12% to 314%).",
+     "and the similarity that sets the step, whose form is decisive (replacing it costs 12% to 315%)."),
+]
+
+ANCHOR = ("Two selections that are each an improvement interfere, which is a reason to measure "
+          "composition rather than assume it.")
+
+
+def main():
+    with open(PATH, encoding="utf8") as fh:
+        text = fh.read()
+    for old, _new in REPL + [(ANCHOR, "")]:
+        if text.count(old) != 1:
+            raise SystemExit(f"found {text.count(old)}x: {old[:80]}")
+    for old, new in REPL:
+        text = text.replace(old, new)
+    text = text.replace(ANCHOR, ANCHOR + AUDIT_PARAGRAPH)
+    with open(PATH, "w", encoding="utf8") as fh:
+        fh.write(text)
+    print("audit paragraph written")
+
+
+if __name__ == "__main__":
+    main()