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>

results_diag_contraction_similarity.txt

@@ -0,0 +1,24 @@
+
+== proposed (projection)
+   group               states   rho_n >= rho_(n-1)
+   APS                   1429                37.6%
+   CHA fun1                83                28.9%
+   CHA flat fun3-6       1763                28.2%
+   CHA other              237                25.3%
+   AUC of S, flat states against:
+     APS            0.422
+     CHA fun1       0.409
+     CHA other      0.386
+
+== reading (3), as for m>=2
+   group               states   rho_n >= rho_(n-1)
+   APS                    990                27.5%
+   CHA fun1                36                27.8%
+   CHA flat fun3-6        659                31.3%
+   CHA other              204                16.7%
+   AUC of S, flat states against:
+     APS            0.284
+     CHA fun1       0.345
+     CHA other      0.229
+
+bar to beat (three-point order estimate, m >= 2): 0.2% canonical, 5.8% flat