Flat regime (PREREG_flat_regime): the unkept-promise flag - after a model step that did not cross the root, r = f_new/f_old >= 1/4, the tangent-versus-halving break-even at m = 2 - RECOGNISES the regime (stage 1: 53.6% of model steps flagged on the flat families against 21.6% on APS and 17.0% on fun1; recall ten times the m>=2 estimator's 5.8%) and cuts the wide-bracket set by 350-490 wherever the budget does not already bind: safe 3390 -> 3037, step 3524 -> 3037, unguarded 4104 -> 3198.

NOT accepted into the proposed rule by the stop rule: under the projection the flat families are already budget-bound (CHA 3220 -> 3198 only) while the false flags on APS cost 2806 -> 3029 (Z2 fails); with G2 both sets fail. The per-problem edge over Chandrupatla on APS turns into 26/72/55. Guarantee intact in every cell. Next candidate, not pre-registered: require the promise to fail twice in a row.

Regression on the branch: 918 pass (twice); the one red test is the paper-number check awaiting the paper update.

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

simroot/pure.py

@@ -665,7 +665,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                          "{'derived3', 'derived3_cap', 'bracket', 'far'}")
     if space not in SPACES:
         raise ValueError(f"unknown space {space!r}")
-    if prestep not in ("none", "progress", "slope", "band", "defect", "simpair"):
+    if prestep not in ("none", "progress", "slope", "band", "defect", "simpair", "flat"):
         raise ValueError(f"unknown prestep {prestep!r}")
     if prestep in ("band", "defect", "simpair") and scale not in (
             "derived3", "derived3_cap", "derived3_m", "bracket", "far"):
@@ -718,6 +718,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
         dir_changed = 0
         slope = 0.0
         slope_prev = None
+        last_model = False       # was the step just evaluated the model's own? (flat gate)
         if stats is not None:
             stats['forced'] = 0
             stats['pre_refused'] = 0
@@ -725,6 +726,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
             stats['dir_changed'] = 0
 
         while abs(B - A) > xtol:
+            model_target = None
             if sg.bisect_before_step(abs(B - A)):
                 x_new = 0.5 * (A + B)
                 forced += 1
@@ -886,6 +888,18 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                                           x_prev2, f_prev2):
                             step = 0.5 * (A + B) - x
                             pre_refused += 1
+                    if (step is None and prestep == "flat" and last_model
+                            and f_prev != 0.0 and fx / f_prev >= 0.25):
+                        # PREREG_flat_regime 2026-09-21.  The last step was the model's
+                        # own, it did not cross the root (r > 0), and it left r = f/f_prev
+                        # >= 1/4 of |f|.  Under a tangent step on an order-m approach
+                        # r = (1 - 1/m)^m, and the step loses to a halving from m = 2 on,
+                        # where r = (1 - 1/2)^2 = 1/4: bisection's 1/2, carried through
+                        # the same formula, not a chosen number.  So the model is losing
+                        # here, and this evaluation is spent halving; the next step is
+                        # the model's again and is judged by its own result.
+                        step = 0.5 * (A + B) - x
+                        pre_refused += 1
                     if step is None and prestep == "slope":
                         # PRE-STEP GATE ON SHAPE (PREREG 2026-09-21).  The step trusts the
                         # local slope, so the question of trust is whether that slope has
@@ -941,6 +955,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                                 reach = abs(far - x)
                                 step = (longer if abs(longer) <= reach
                                         else math.copysign(reach, step))
+                        model_target = x + step      # the model's own proposal
                     if stats is not None and "trace" in stats:
                         # DIAGNOSTIC HOOK, behaviour-neutral: records the state each
                         # step was taken from, so another model can be evaluated on the
@@ -978,6 +993,9 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                         stats['forced'] = forced; stats['pre_refused'] = pre_refused; stats['triple_changed'] = triple_changed; stats['dir_changed'] = dir_changed
                     return x_new, g.n, abs(B - A) <= xtol
 
+            # the flat gate judges only the model's own steps: a midpoint from a gate or a
+            # step moved by the safeguard says nothing about the model
+            last_model = model_target is not None and x_new == model_target
             w_before = abs(B - A)
             fnew = g(x_new)
             sg.j += 1          # one step of the bisection budget is spent