PREREG_flat_twice outcome: stage 1 passed (twice-in-a-row fires on 27.8% of model steps on the flat families, 8.7% on APS, 4.3% on fun1; flat/APS ratio 3.20 against the single flag's 2.47), stage 2 FAILED: under the projection APS 2806 -> 2803 (W2 holds, the per-problem edge over Chandrupatla kept at 83/28/42) but CHA 3220 -> 3341 and the flat families 1823 -> 1975 (W1, W4 fail); with G2 the same. Under safe/step the twice gate is inert (3390 -> 3388) where the single one gave 3037.

Mechanism: the flat cycle is undershoot, undershoot, crossing, near-empty step. The single flag fires after the first undershoot and replaces the second - a useless step; the twice flag fires after the second and replaces the CROSSING, the one step of the cycle that collapses the bracket. The second undershoot's r is ~0.26, next to the 1/4 threshold, so the pair often does not even form. Again a detectability gain that did not carry to run cost. Next candidate (not pre-registered): the unkept promise AND a creep - the same step moving the same way as the previous one.

Removed a vacuous pin I had written (s2 <= max(s1, s2)). Regression on the branch: 935 pass twice; the only red test is the paper-number check.

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

analyze_flat.py

@@ -9,8 +9,8 @@ import json
 from analyze_extrap import FAMILY, FLAT, fam_sum, h2h, load, sign_p
 
 
-def main():
+def main(path="sweep_flat.json", gate="flat"):
-    rows = load("sweep_flat.json")
+    rows = load(path)
     cha = {r["label"]: r for r in load("sweep_full.json")}["chandrupatla"]
     cell = {}
     for r in rows:
@@ -19,7 +19,7 @@ def main():
     print("tent, 3-point, pairs (no gate -> flat gate)      APS          CHA        both      flat fun3-6")
     for gu in ("none", "step", "bracket", "both", "project", "safe", "credit"):
         for ext in ("none", "pattern"):
-            a, b = cell[("ratio", gu, ext, "none")], cell[("ratio", gu, ext, "flat")]
+            a, b = cell[("ratio", gu, ext, "none")], cell[("ratio", gu, ext, gate)]
             print(f"  {gu:<8} {'+G2' if ext == 'pattern' else '   '}  "
                   f"{a['APS']['total']:5d}->{b['APS']['total']:<5d}  {a['CHA']['total']:5d}->"
                   f"{b['CHA']['total']:<5d}  {a['APS']['total'] + a['CHA']['total']:5d}->"
@@ -28,8 +28,8 @@ def main():
                   f"over {b['APS']['over'] + b['CHA']['over']}")
     print("\nVERDICTS")
     for ext, name in (("none", "proposed rule"), ("pattern", "proposed rule + G2")):
-        a, b = cell[("ratio", "project", ext, "none")], cell[("ratio", "project", ext, "flat")]
+        a, b = cell[("ratio", "project", ext, "none")], cell[("ratio", "project", ext, gate)]
-        fam = [r for k, r in cell.items() if k[3] == "flat" and k[1] in FAMILY]
+        fam = [r for k, r in cell.items() if k[3] == gate and k[1] in FAMILY]
         viol = sum(r["APS"]["over"] + r["CHA"]["over"] for r in fam)
         low = sum(1 for r in fam if r["APS"]["solved"] + r["CHA"]["solved"] < 193)
         z1 = b["CHA"]["total"] < a["CHA"]["total"]
@@ -44,10 +44,11 @@ def main():
               f"p={sign_p(w, l):.2g}" + ("   ACCEPTED" if z1 and z2 and z3 else "   not accepted"))
     print("\nnine functions, projection, no gate -> flat gate, both sets:")
     for sim in sorted({k[0] for k in cell}):
-        a, b = cell[(sim, "project", "none", "none")], cell[(sim, "project", "none", "flat")]
+        a, b = cell[(sim, "project", "none", "none")], cell[(sim, "project", "none", gate)]
         print(f"  {sim:<11} {a['APS']['total'] + a['CHA']['total']:6d} -> "
               f"{b['APS']['total'] + b['CHA']['total']:6d}")
 
 
 if __name__ == "__main__":
-    main()
+    import sys
+    main(*sys.argv[1:3]) if len(sys.argv) > 2 else main()

results_analyze_flat2.txt

@@ -0,0 +1,89 @@
+sweep: 140 configurations x 199 problems (FULL)
+wrote sweep_flat2.json  (8s total)
+====================================================================================
+APS: 154 problems   (top 20 of 140)
+====================================================================================
+configuration                              solved   total fev  over 2x
+------------------------------------------------------------------------------------
+pure+extP+flat2.ratio.derived3.project     153/154        2781        0
+pure+extP+flat2.ratio.derived3.none        153/154        2781        0
+pure+extP.ratio.derived3.project           153/154        2782        0
+pure+extP.ratio.derived3.none              153/154        2782        0
+flat2.ratio.derived3.none                  153/154        2803        0
+flat2.ratio.derived3.project               153/154        2803        0
+pure.ratio.derived3.none                   153/154        2806        0
+pure.ratio.derived3.project                153/154        2806        0
+pure.ratio.derived3.step                   153/154        3036        0
+pure.ratio.derived3.safe                   153/154        3036        0
+flat2.ratio.derived3.step                  153/154        3042        0
+flat2.ratio.derived3.safe                  153/154        3042        0
+pure+extP.ratio.derived3.step              153/154        3089        0
+pure+extP.ratio.derived3.safe              153/154        3089        0
+pure+extP+flat2.ratio.derived3.safe        153/154        3097        0
+pure+extP+flat2.ratio.derived3.step        153/154        3097        0
+flat2.exp_abs.derived3.none                153/154        3257        0
+flat2.exp_abs.derived3.project             153/154        3257        0
+pure.exp_abs.derived3.none                 153/154        3264        0
+pure.exp_abs.derived3.project              153/154        3264        0
+
+====================================================================================
+CHA: 45 problems   (top 20 of 140)
+====================================================================================
+configuration                              solved   total fev  over 2x
+------------------------------------------------------------------------------------
+flat2.cbrt.derived3.credit                  40/45         3016        0
+pure.cbrt.derived3.credit                   40/45         3016        0
+pure.ratio.derived3.credit                  40/45         3020        0
+flat2.ratio.derived3.credit                 40/45         3040        0
+pure.exp_abs.derived3.credit                40/45         3049        0
+pure.ratio_soft.derived3.credit             40/45         3054        0
+flat2.ratio_soft.derived3.credit            40/45         3054        0
+pure+extP.ratio.derived3.credit             40/45         3064        0
+flat2.exp_abs.derived3.credit               40/45         3066        0
+pure+extP+flat2.ratio.derived3.credit       40/45         3084        0
+pure.cosine.derived3.credit                 40/45         3129        0
+flat2.cosine.derived3.credit                40/45         3129        0
+pure.exp_sq.derived3.credit                 40/45         3130        0
+flat2.exp_sq.derived3.credit                40/45         3130        0
+pure.rat_sq.derived3.credit                 40/45         3130        0
+flat2.rat_sq.derived3.credit                40/45         3130        0
+flat2.rational.derived3.credit              40/45         3130        0
+pure.rational.derived3.credit               40/45         3130        0
+pure.sqrt.derived3.credit                   40/45         3130        0
+flat2.sqrt.derived3.credit                  40/45         3130        0
+
+====================================================================================
+THE STAGE-0C LAW ACROSS THE GRID
+====================================================================================
+  scale=derived3  slope c=1 (linear)         mean total      6829   mean solved  193.0/199
+  scale=derived3  slope c=0 (quadratic)      mean total     19620   mean solved  193.0/199
+tent, 3-point, pairs (no gate -> flat gate)      APS          CHA        both      flat fun3-6
+  none           2806->2803    4104->3571    6910->6374    2707->2205  solved 193 over 15
+  none     +G2   2782->2781    3863->3585    6645->6366    2474->2217  solved 193 over 15
+  step           3036->3042    3524->3522    6560->6564    2111->2111  solved 193 over 15
+  step     +G2   3089->3097    3529->3528    6618->6625    2106->2106  solved 193 over 15
+  bracket        7260->7260    3604->3604   10864->10864   1730->1730  solved 193 over 0
+  bracket  +G2   6837->6837    3650->3650   10487->10487   1717->1717  solved 193 over 0
+  both           7239->7239    3515->3515   10754->10754   1529->1529  solved 193 over 0
+  both     +G2   7130->7130    3463->3463   10593->10593   1504->1504  solved 193 over 0
+  project        2806->2803    3220->3341    6026->6144    1823->1975  solved 193 over 0
+  project  +G2   2782->2781    3169->3345    5951->6126    1780->1977  solved 193 over 0
+  safe           3036->3042    3390->3388    6426->6430    1977->1977  solved 193 over 0
+  safe     +G2   3089->3097    3392->3386    6481->6483    1969->1964  solved 193 over 0
+  credit         3568->3603    3020->3040    6588->6643    1044->1064  solved 193 over 0
+  credit   +G2   3799->3800    3064->3084    6863->6884    1044->1064  solved 193 over 0
+
+VERDICTS
+  proposed rule        APS 2806->2803 (Z2 ✓)  CHA 3220->3341 (Z1 ✗)  Z3 ✓ (0 viol, 0 cells <193)  Z4 ✗  both 6144  vs Chandrupatla APS 83/28/42 p=0.00031   not accepted
+  proposed rule + G2   APS 2782->2781 (Z2 ✓)  CHA 3169->3345 (Z1 ✗)  Z3 ✓ (0 viol, 0 cells <193)  Z4 ✗  both 6126  vs Chandrupatla APS 92/21/40 p=7e-06   not accepted
+
+nine functions, projection, no gate -> flat gate, both sets:
+  cbrt         18003 ->  18006
+  cosine       19741 ->  18984
+  exp_abs       6652 ->   6858
+  exp_sq       19786 ->  16993
+  rat_sq       19793 ->  17677
+  ratio         6026 ->   6144
+  ratio_soft    9352 ->   9299
+  rational      6823 ->   6908
+  sqrt         13388 ->  13416

results_diag_flat_twice.txt

@@ -0,0 +1,9 @@
+group               model steps   single    twice
+APS                        1275    21.6%     8.7%
+CHA fun1                     47    17.0%     4.3%
+CHA flat fun3-6            1251    53.6%    27.8%
+CHA fun7                      0     0.0%     0.0%
+CHA other                   174    36.2%    20.7%
+
+T1 twice: flat 27.8% > APS 8.7% and > fun1 4.3%  ->  PASS
+T2 flat/APS ratio: twice 3.20 vs single 2.47  ->  PASS

simroot/pure.py

@@ -665,7 +665,8 @@ 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", "flat"):
+    if prestep not in ("none", "progress", "slope", "band", "defect", "simpair", "flat",
+                       "flat2"):
         raise ValueError(f"unknown prestep {prestep!r}")
     if prestep in ("band", "defect", "simpair") and scale not in (
             "derived3", "derived3_cap", "derived3_m", "bracket", "far"):
@@ -719,6 +720,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
         slope = 0.0
         slope_prev = None
         last_model = False       # was the step just evaluated the model's own? (flat gate)
+        unkept_run = 0           # consecutive model steps whose promise failed (flat2)
         if stats is not None:
             stats['forced'] = 0
             stats['pre_refused'] = 0
@@ -900,6 +902,14 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                         # 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 == "flat2" and unkept_run >= 2:
+                        # PREREG_flat_twice 2026-09-21.  The same unkept promise as
+                        # "flat" (model step, no crossing, r >= 1/4), but only after it has
+                        # failed TWICE IN A ROW: on a flat function the undershoot repeats,
+                        # at a simple root a single failure is usually followed by fast
+                        # convergence.  Two is the fewest at which "repeats" is defined.
+                        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
@@ -998,6 +1008,11 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
             last_model = model_target is not None and x_new == model_target
             w_before = abs(B - A)
             fnew = g(x_new)
+            # the run of unkept promises (flat2): x, fx are still the point the step left
+            if last_model and fx != 0.0 and fnew / fx >= 0.25:
+                unkept_run += 1
+            else:
+                unkept_run = 0
             sg.j += 1          # one step of the bisection budget is spent
             sg.allow *= 0.5
             x_prev2, f_prev2 = x_prev, f_prev

sweep_full.py

@@ -219,24 +219,24 @@ def _preflight_extrap_mode():
         raise SystemExit("extrapolate preflight FAILED: refusing to report zeros")
 
 
-def build_grid_flat(smoke):
+def build_grid_flat(smoke, gate="flat"):
     """PREREG_flat_regime_2026-09-21 stage 2: the flat gate (prestep='flat') in pairs,
     on the proposed rule and on the proposed rule with G2 (extrapolate='pattern')."""
     x, _n, ok = pure_root(lambda t: t - 1.0, 0.0, 3.0, 1e-10, 100, similarity="ratio",
-                          scale="derived3", guard="project", prestep="flat")
+                          scale="derived3", guard="project", prestep=gate)
     if not (ok and abs(x - 1.0) < 1e-8):
-        raise SystemExit("prestep=flat preflight FAILED: refusing to report zeros")
+        raise SystemExit(f"prestep={gate} preflight FAILED: refusing to report zeros")
     guards = ("project", "credit") if smoke else GUARD_MODES
     specs = []
     for gu in guards:
         for ext in ("none", "pattern"):
-            for pre in ("none", "flat"):
+            for pre in ("none", gate):
                 specs.append(("pure", "ratio", "derived3", gu, "x", pre, "recent",
                               "distance", "none", ext))
     sims = ["sqrt"] if smoke else [s for s in sorted(PURE_SIMS) if s != "ratio"]
     for sim in sims:
         for gu in guards:
-            for pre in ("none", "flat"):
+            for pre in ("none", gate):
                 specs.append(("pure", sim, "derived3", gu, "x", pre, "recent",
                               "distance", "none", "none"))
     return specs
@@ -412,10 +412,12 @@ def main():
     extmode = "--extrap" in sys.argv       # A in the method: step extrapolation
     ext2mode = "--extrap2" in sys.argv     # gates G1-G3 on the lengthening
     flatmode = "--flat" in sys.argv        # the flat-regime gate
+    flat2mode = "--flat2" in sys.argv      # the flat gate, promise failed twice
     combomode = "--combo" in sys.argv      # modes A and B together
     pointsmode = "--points" in sys.argv    # mode A: triple by shape
     dirmode = "--direction" in sys.argv    # mode B: far end by dissimilarity
-    specs = (build_grid_flat(smoke) if flatmode else
+    specs = (build_grid_flat(smoke, "flat2") if flat2mode else
+             build_grid_flat(smoke) if flatmode else
              build_grid_extrap2(smoke) if ext2mode else
              build_grid_extrap(smoke) if extmode else
              build_grid_decision(smoke) if decmode else
@@ -442,8 +444,9 @@ def main():
 
     # the published sweep_full.json is what verify_article.py checks the paper
     # against, so the log grid never writes into it
-    if flatmode or ext2mode or extmode or decmode or ordermode or capmode or combomode or pointsmode or dirmode:
+    if flat2mode or flatmode or ext2mode or extmode or decmode or ordermode or capmode or combomode or pointsmode or dirmode:
-        base = ("sweep_flat" if flatmode else
+        base = ("sweep_flat2" if flat2mode else
+                "sweep_flat" if flatmode else
                 "sweep_extrap2" if ext2mode else
                 "sweep_extrap" if extmode else
                 "sweep_decision" if decmode else

tests/test_flat_gate.py

@@ -36,6 +36,28 @@ def test_the_guarantee_holds_with_the_gate(f, a, b, guard, ext):
     assert n <= (2 * half + 3 if guard == "project" else half + 4), n
 
 
+@pytest.mark.parametrize("f,a,b", PROBS)
+@pytest.mark.parametrize("guard", ("project", "credit"))
+@pytest.mark.parametrize("ext", ("none", "pattern"))
+def test_the_guarantee_holds_with_the_twice_gate(f, a, b, guard, ext):
+    x, n, ok = pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                         guard=guard, prestep="flat2", extrapolate=ext)
+    assert ok and min(a, b) <= x <= max(a, b)
+    half = math.ceil(math.log2(abs(b - a) / XTOL))
+    assert n <= (2 * half + 3 if guard == "project" else half + 4), n
+
+
+def test_the_twice_gate_is_silent_where_the_model_is_exact_and_fires_on_a_power():
+    st = {}
+    pure_root(lambda t: 3.0 * t - 1.0, -2.0, 5.0, XTOL, MAXFEV, similarity="ratio",
+              scale="derived3", guard="project", prestep="flat2", stats=st)
+    assert st["pre_refused"] == 0
+    st = {}
+    pure_root(lambda t: t ** 9, -1.0, 2.0, XTOL, MAXFEV, similarity="ratio",
+              scale="derived3", guard="project", prestep="flat2", stats=st)
+    assert st["pre_refused"] > 0
+
+
 def test_the_gate_never_fires_where_the_model_is_exact():
     """On a straight line the model's step lands on the root: r = 0, never >= 1/4."""
     st = {}