PREREG_flat_stick outcome: NOT accepted. Sticky regime under the projection: CHA 3220 -> 3019 (W1 holds), flat fun3-6 1823 -> 1057 (W4, near pure bisection 1044), guarantee holds (W3), but APS 2806 -> 6781 (W2 fails): the unkept-promise flag enters on 134/154 APS problems and 20/25 non-flat CHA problems, and a false entry now sticks to the end of the run. W6 fails: the order read from the promise is m = inf in 163/174 entries (r >= 1/e; our step is not a pure tangent), so the derived split equals the midpoint on CHA (3019 both). Audit bug fixed and pinned: an infinite |f| at an end made the split nan. Server regression x2: 999 passed, 1 failed = the expected paper-number check.

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

diag_flat_stick.py

@@ -0,0 +1,62 @@
+"""Plausibility check of PREREG_flat_stick before the full grid: which problems enter
+the sticky regime, what order the failed promise reads at entry, and where the cost goes.
+
+    python -u diag_flat_stick.py
+"""
+from __future__ import annotations
+
+import math
+from collections import Counter as Tally
+
+import simroot.pure as P
+from sweep_full import ACC, MAXFEV, XTOL, load_problems
+
+FLAT = ("fun3", "fun4", "fun5", "fun6")
+_orig = P._order_from_promise
+ENTRIES = []
+
+
+def _spy(r):
+    m = _orig(r)
+    ENTRIES.append((r, m))
+    return m
+
+
+def run(gate, f, a, b):
+    st = {}
+    x, n, conv = P.pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                             guard="project", prestep=gate, stats=st)
+    return x, n, conv, st["pre_refused"]
+
+
+def main():
+    P._order_from_promise = _spy
+    probs = load_problems()
+    for s in ("APS", "CHA"):
+        groups = {"flat": [0, 0, 0, 0], "other": [0, 0, 0, 0]}   # problems, entered, base, stick
+        ms = Tally()
+        for st_, pid, f, a, b, root in probs:
+            if st_ != s:
+                continue
+            g = "flat" if s == "CHA" and pid.split(".")[0] in FLAT else "other"
+            _x, n0, c0, _ = run("none", f, a, b)
+            ENTRIES.clear()
+            x1, n1, c1, k = run("flatstick", f, a, b)
+            ok0 = c0 and abs(_x - root) <= ACC
+            ok1 = c1 and math.isfinite(x1) and abs(x1 - root) <= ACC
+            groups[g][0] += 1
+            groups[g][1] += 1 if ENTRIES else 0
+            groups[g][2] += n0 if ok0 else MAXFEV
+            groups[g][3] += n1 if ok1 else MAXFEV
+            for r, m in ENTRIES:
+                ms["inf" if math.isinf(m) else ("2-3" if m < 3 else "3-10" if m < 10 else ">=10")] += 1
+        for g, (np_, ent, c0, c1) in groups.items():
+            if np_:
+                print(f"{s} {g:<5}: {np_:3d} problems, entered sticky {ent:3d}, "
+                      f"cost none {c0:5d} -> flatstick {c1:5d}")
+        print(f"{s} order read at entry: {dict(ms)}")
+    print("diag done")
+
+
+if __name__ == "__main__":
+    main()

results_analyze_flatstick.txt

@@ -0,0 +1,62 @@
+===== flatstick
+tent, 3-point, pairs (no gate -> flat gate)      APS          CHA        both      flat fun3-6
+  none           2806->6781    4104->3019    6910->9800    2707->1057  solved 193 over 0
+  none     +G2   2782->6802    3863->3055    6645->9857    2474->1057  solved 193 over 0
+  step           3036->6544    3524->3019    6560->9563    2111->1057  solved 193 over 0
+  step     +G2   3089->6590    3529->3163    6618->9753    2106->1165  solved 193 over 1
+  bracket        7260->7173    3604->3604   10864->10777   1730->1730  solved 193 over 0
+  bracket  +G2   6837->6845    3650->3639   10487->10484   1717->1717  solved 193 over 0
+  both           7239->7183    3515->3515   10754->10698   1529->1529  solved 193 over 0
+  both     +G2   7130->7151    3463->3452   10593->10603   1504->1504  solved 193 over 0
+  project        2806->6781    3220->3019    6026->9800    1823->1057  solved 193 over 0
+  project  +G2   2782->6802    3169->3055    5951->9857    1780->1057  solved 193 over 0
+  safe           3036->6544    3390->3019    6426->9563    1977->1057  solved 193 over 0
+  safe     +G2   3089->6590    3392->3129    6481->9719    1969->1131  solved 193 over 0
+  credit         3568->6740    3020->3002    6588->9742    1044->951   solved 193 over 0
+  credit   +G2   3799->6838    3064->3002    6863->9840    1044->951   solved 193 over 0
+
+VERDICTS
+  proposed rule        APS 2806->6781 (Z2 ✗)  CHA 3220->3019 (Z1 ✓)  Z3 ✓ (0 viol, 0 cells <193)  Z4 ✓  both 9800  vs Chandrupatla APS 9/4/140 p=2.3e-31   not accepted
+  proposed rule + G2   APS 2782->6802 (Z2 ✗)  CHA 3169->3055 (Z1 ✓)  Z3 ✓ (0 viol, 0 cells <193)  Z4 ✓  both 9857  vs Chandrupatla APS 9/4/140 p=2.3e-31   not accepted
+
+nine functions, projection, no gate -> flat gate, both sets:
+  cbrt         18003 ->  12787
+  cosine       19741 ->  10974
+  exp_abs       6652 ->  10606
+  exp_sq       19786 ->  10777
+  rat_sq       19793 ->  10752
+  ratio         6026 ->   9800
+  ratio_soft    9352 ->  10297
+  rational      6823 ->  10639
+  sqrt         13388 ->  11541
+===== flatstickmid
+tent, 3-point, pairs (no gate -> flat gate)      APS          CHA        both      flat fun3-6
+  none           2806->6897    4104->3019    6910->9916    2707->1057  solved 193 over 0
+  none     +G2   2782->6918    3863->3055    6645->9973    2474->1057  solved 193 over 0
+  step           3036->6663    3524->3019    6560->9682    2111->1057  solved 193 over 0
+  step     +G2   3089->6709    3529->3055    6618->9764    2106->1057  solved 193 over 0
+  bracket        7260->7214    3604->3604   10864->10818   1730->1730  solved 193 over 0
+  bracket  +G2   6837->6886    3650->3650   10487->10536   1717->1717  solved 193 over 0
+  both           7239->7241    3515->3515   10754->10756   1529->1529  solved 193 over 0
+  both     +G2   7130->7196    3463->3463   10593->10659   1504->1504  solved 193 over 0
+  project        2806->6897    3220->3019    6026->9916    1823->1057  solved 193 over 0
+  project  +G2   2782->6918    3169->3055    5951->9973    1780->1057  solved 193 over 0
+  safe           3036->6663    3390->3019    6426->9682    1977->1057  solved 193 over 0
+  safe     +G2   3089->6709    3392->3055    6481->9764    1969->1057  solved 193 over 0
+  credit         3568->6837    3020->3002    6588->9839    1044->951   solved 193 over 0
+  credit   +G2   3799->6953    3064->3002    6863->9955    1044->951   solved 193 over 0
+
+VERDICTS
+  proposed rule        APS 2806->6897 (Z2 ✗)  CHA 3220->3019 (Z1 ✓)  Z3 ✓ (0 viol, 0 cells <193)  Z4 ✓  both 9916  vs Chandrupatla APS 9/4/140 p=2.3e-31   not accepted
+  proposed rule + G2   APS 2782->6918 (Z2 ✗)  CHA 3169->3055 (Z1 ✓)  Z3 ✓ (0 viol, 0 cells <193)  Z4 ✓  both 9973  vs Chandrupatla APS 9/4/140 p=2.3e-31   not accepted
+
+nine functions, projection, no gate -> flat gate, both sets:
+  cbrt         18003 ->  12694
+  cosine       19741 ->  10825
+  exp_abs       6652 ->  10621
+  exp_sq       19786 ->  10810
+  rat_sq       19793 ->  10793
+  ratio         6026 ->   9916
+  ratio_soft    9352 ->  10447
+  rational      6823 ->  10634
+  sqrt         13388 ->  11332

simroot/pure.py

@@ -549,6 +549,50 @@ def _slope3(x2, f2, x1, f1, x0, f0):
     return s if math.isfinite(s) else d12
 
 
+def _order_from_promise(r):
+    """The order m that a tangent step's unkept promise r = |f_new/f_old| implies.
+
+    PREREG_flat_stick 2026-09-21.  On an order-m approach a tangent step leaves
+    r = (1 - 1/m)^m, which rises monotonically from 1/4 at m = 2 to 1/e as m -> inf.
+    Inverted on u = 1/m by halving until the float interval stops shrinking - a
+    precision, not a chosen number.  r >= 1/e has no finite solution: m = inf.
+    Called only with r >= 1/4 (the flag's own condition); below it m = 2 is returned,
+    the end of the range the flag admits."""
+    if r >= math.exp(-1.0):
+        return math.inf
+    target = math.log(r)
+    lo, hi = 0.0, 0.5            # u = 1/m in (0, 1/2]; h(u) = ln(1-u)/u falls with u
+    while True:
+        mid = 0.5 * (lo + hi)
+        if mid == lo or mid == hi:
+            break
+        if math.log1p(-mid) / mid > target:
+            lo = mid
+        else:
+            hi = mid
+    return 1.0 / hi
+
+
+def _order_split(A, FA, B, FB, m):
+    """The point where |f| ~ |x - r|^m puts the root: the ends are at distances
+    dA : dB = |FA|^(1/m) : |FB|^(1/m) from it.  m = inf is the midpoint.  Worked in
+    logarithms so neither ratio can overflow; a point that lands on an end is caught
+    by the loop's own fallback to the midpoint."""
+    if math.isinf(m):
+        return 0.5 * (A + B)
+    lq = (math.log(abs(FA)) - math.log(abs(FB))) / m     # ln (|FA|/|FB|)^(1/m)
+    if not math.isfinite(lq):
+        # an infinite |f| at an end: no power law to read, and a nan step would reach
+        # the safeguard's bookkeeping before the loop's own finiteness check
+        return 0.5 * (A + B)
+    if lq > 0.0:
+        t = 1.0 / (1.0 + math.exp(-lq))
+    else:
+        e = math.exp(lq)
+        t = e / (1.0 + e)
+    return A + t * (B - A)
+
+
 def _local_order(x2, f2, x1, f1, x0, f0):
     """The local order m of f, from three evaluations and nothing else.
 
@@ -666,7 +710,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
     if space not in SPACES:
         raise ValueError(f"unknown space {space!r}")
     if prestep not in ("none", "progress", "slope", "band", "defect", "simpair", "flat",
-                       "flat2", "flatcreep"):
+                       "flat2", "flatcreep", "flatstick", "flatstickmid"):
         raise ValueError(f"unknown prestep {prestep!r}")
     if prestep in ("band", "defect", "simpair") and scale not in (
             "derived3", "derived3_cap", "derived3_m", "bracket", "far"):
@@ -721,6 +765,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
         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)
+        sticky_m = None          # order read at entry to the sticky flat regime (flatstick*)
         if stats is not None:
             stats['forced'] = 0
             stats['pre_refused'] = 0
@@ -914,6 +959,22 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                         # late, before the crossing (flat2's failure).
                         step = 0.5 * (A + B) - x
                         pre_refused += 1
+                    if step is None and prestep in ("flatstick", "flatstickmid"):
+                        # PREREG_flat_stick 2026-09-21.  The same entry as "flat", but the
+                        # regime is a property of the root, not of one step: once the
+                        # promise has failed, every later step splits the bracket.  The
+                        # single flag replaced one step and the model undershot again.
+                        # flatstick splits where |f| ~ |x-r|^m puts the root, m read from
+                        # the failed promise itself; flatstickmid halves (the ablation).
+                        if (sticky_m is None and last_model and f_prev != 0.0
+                                and fx / f_prev >= 0.25):
+                            sticky_m = _order_from_promise(fx / f_prev)
+                        if sticky_m is not None:
+                            if prestep == "flatstick":
+                                step = _order_split(A, FA, B, FB, sticky_m) - x
+                            else:
+                                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

sweep_full.py

@@ -221,22 +221,26 @@ def _preflight_extrap_mode():
 
 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')."""
+    on the proposed rule and on the proposed rule with G2 (extrapolate='pattern').
+    `gate` may be a tuple: every gate in it gets its own cell next to the same 'none'
+    (PREREG_flat_stick: flatstick and its ablation flatstickmid in one grid)."""
+    gates = gate if isinstance(gate, tuple) else (gate,)
+    for gt in gates:
         x, _n, ok = pure_root(lambda t: t - 1.0, 0.0, 3.0, 1e-10, 100, similarity="ratio",
-                          scale="derived3", guard="project", prestep=gate)
+                              scale="derived3", guard="project", prestep=gt)
         if not (ok and abs(x - 1.0) < 1e-8):
-        raise SystemExit(f"prestep={gate} preflight FAILED: refusing to report zeros")
+            raise SystemExit(f"prestep={gt} 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", gate):
+            for pre in ("none",) + gates:
                 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", gate):
+            for pre in ("none",) + gates:
                 specs.append(("pure", sim, "derived3", gu, "x", pre, "recent",
                               "distance", "none", "none"))
     return specs
@@ -414,10 +418,12 @@ def main():
     flatmode = "--flat" in sys.argv        # the flat-regime gate
     flat2mode = "--flat2" in sys.argv      # the flat gate, promise failed twice
     creepmode = "--flatcreep" in sys.argv  # the flat gate AND a creep
+    stickmode = "--flatstick" in sys.argv  # the sticky flat regime + its midpoint ablation
     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, "flatcreep") if creepmode else
+    specs = (build_grid_flat(smoke, ("flatstick", "flatstickmid")) if stickmode else
+             build_grid_flat(smoke, "flatcreep") if creepmode else
              build_grid_flat(smoke, "flat2") if flat2mode else
              build_grid_flat(smoke) if flatmode else
              build_grid_extrap2(smoke) if ext2mode else
@@ -446,8 +452,9 @@ 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 creepmode or flat2mode or flatmode or ext2mode or extmode or decmode or ordermode or capmode or combomode or pointsmode or dirmode:
+    if stickmode or creepmode or flat2mode or flatmode or ext2mode or extmode or decmode or ordermode or capmode or combomode or pointsmode or dirmode:
-        base = ("sweep_flatcreep" if creepmode else
+        base = ("sweep_flatstick" if stickmode else
+                "sweep_flatcreep" if creepmode else
                 "sweep_flat2" if flat2mode else
                 "sweep_flat" if flatmode else
                 "sweep_extrap2" if ext2mode else

tests/test_flat_gate.py

@@ -97,3 +97,64 @@ def test_the_gate_fires_on_a_pure_power():
     pure_root(lambda t: t ** 9, -1.0, 2.0, XTOL, MAXFEV, similarity="ratio",
               scale="derived3", guard="project", prestep="flat", stats=st)
     assert st["pre_refused"] > 0
+
+
+# ---- PREREG_flat_stick_2026-09-21: the sticky flat regime and its midpoint ablation ----
+from simroot.pure import _order_from_promise, _order_split  # noqa: E402
+
+
+@pytest.mark.parametrize("m", (2.0, 3.0, 5.0, 9.0, 19.0, 100.0))
+def test_the_order_is_read_back_from_its_own_promise(m):
+    r = (1.0 - 1.0 / m) ** m
+    assert _order_from_promise(r) == pytest.approx(m, rel=1e-9)
+
+
+def test_no_finite_order_at_or_above_one_over_e_gives_the_midpoint():
+    assert math.isinf(_order_from_promise(math.exp(-1.0)))
+    assert math.isinf(_order_from_promise(0.9))
+    assert _order_split(-1.0, -3.0, 2.0, 5.0, math.inf) == 0.5
+
+
+@pytest.mark.parametrize("m", (1, 3, 5, 9))
+def test_the_split_lands_on_the_root_of_an_exact_power(m):
+    f = lambda t: (t - 0.3) ** m  # noqa: E731
+    assert _order_split(-1.0, f(-1.0), 2.0, f(2.0), float(m)) == pytest.approx(0.3, abs=1e-12)
+
+
+def test_the_split_does_not_overflow_on_extreme_ends():
+    x = _order_split(0.0, -1e-300, 1.0, 1e300, 2.0)
+    assert 0.0 <= x < 1e-100
+    x = _order_split(0.0, -1e300, 1.0, 1e-300, 2.0)
+    assert 1.0 - x < 1e-100 and x <= 1.0
+
+
+@pytest.mark.parametrize("f,a,b", PROBS)
+@pytest.mark.parametrize("guard", ("project", "credit"))
+@pytest.mark.parametrize("ext", ("none", "pattern"))
+@pytest.mark.parametrize("gate", ("flatstick", "flatstickmid"))
+def test_the_guarantee_holds_with_the_sticky_regime(f, a, b, guard, ext, gate):
+    x, n, ok = pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                         guard=guard, prestep=gate, 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
+
+
+@pytest.mark.parametrize("gate", ("flatstick", "flatstickmid"))
+def test_the_sticky_regime_is_silent_on_a_line_and_holds_on_a_power(gate):
+    st = {}
+    pure_root(lambda t: 3.0 * t - 1.0, -2.0, 5.0, XTOL, MAXFEV, similarity="ratio",
+              scale="derived3", guard="project", prestep=gate, stats=st)
+    assert st["pre_refused"] == 0
+    st = {}
+    _x, n, _ok = pure_root(lambda t: t ** 9, -1.0, 2.0, XTOL, MAXFEV, similarity="ratio",
+                           scale="derived3", guard="project", prestep=gate, stats=st)
+    # once entered, every later non-forced step is a split: many, not one
+    assert st["pre_refused"] >= 3, st
+
+
+def test_an_infinite_end_gives_the_midpoint_not_nan():
+    """Audit bug 2026-09-21: an infinite |f| made lq nan, and the nan step reached the
+    safeguard before the loop checked finiteness."""
+    assert _order_split(0.0, -math.inf, 2.0, 1.0, 3.0) == 1.0
+    assert _order_split(0.0, -math.inf, 2.0, math.inf, 3.0) == 1.0