diff --git a/PREREG_close_sub_2026-09-21.md b/PREREG_close_sub_2026-09-21.md
index 1a8a899..5767c56 100644
--- a/PREREG_close_sub_2026-09-21.md
+++ b/PREREG_close_sub_2026-09-21.md
@@ -48,3 +48,16 @@ arcsinh) — целевой для `close`**. Успех — смена знак
 
 «A′ переносится» — F1, F2, F3 одновременно. Сообщается также вывод **без целевых классов**
 (D1, D2, D4, D5), против Брента и TOMS748, и A′ на всех увиденных корпусах (описательно).
+
+## Поправка из разработки (до прогона на корпусе 4)
+
+1. **`close` только для достоверного обещания:** шаг на xtol/2 делается, лишь если предыдущий шаг
+   модели сдержал своё обещание (|f| упала ниже порога флага ¼). Без этого на x¹⁹ из увиденного
+   набора в начале прогона наклон, в котором доминирует f(100) = 10³⁸, обещает корень в x = −10, и
+   закрытие заменяло середину почти повторными точками (23 → 32). Новой константы нет — тот же ¼.
+2. ~~**Выход снимает и пологую память**~~ — **ОТМЕНЕНО** той же разработкой: правило выхода («ближе к
+   прямой, чем к нулю») отличает плоский корень от простого, но у корня порядка m < 1 f далеко от
+   нуля с обеих сторон — выход срабатывает ложно; сохранённая пологая память возвращает режим на
+   ближайшем перелёте (√|x|·sgn: 19 с памятью, 65 без). Пологая память выходом НЕ снимается.
+
+Обе поправки сделаны на увиденных корпусах; корпус 4 ещё не запускался.
diff --git a/results_fourth_seen.txt b/results_fourth_seen.txt
new file mode 100644
index 0000000..e3ea563
--- /dev/null
+++ b/results_fourth_seen.txt
@@ -0,0 +1,7 @@
+SEEN corpora (development, descriptive): totals; A' vs Chandrupatla w/t/l
+  APS  base 2634  A 2380  Aprime 2259  Aprime-close 2389  Aprime-sub 2253  chandrupatla 2595  brent 2707  toms748 2635   A' vs Ch 103/20/31 p=3.2e-10
+  CHA  base 2357  A 1109  Aprime 1074  Aprime-close 1109  Aprime-sub 1074  chandrupatla 1505  brent 3238  toms748 3111   A' vs Ch 31/4/10 p=0.0015
+  NEW  base 5388  A 3380  Aprime 3289  Aprime-close 3387  Aprime-sub 3282  chandrupatla 3717  brent 6391  toms748 5727   A' vs Ch 109/32/65 p=0.0011
+  RJL  base 1888  A 1464  Aprime 1371  Aprime-close 1449  Aprime-sub 1387  chandrupatla 1566  brent 2455  toms748 2338   A' vs Ch 44/14/20 p=0.0037
+  APP  base 1588  A 1577  Aprime 1382  Aprime-close 1512  Aprime-sub 1447  chandrupatla 1380  brent 1502  toms748 1757   A' vs Ch 48/52/32 p=0.093
+fourth done
diff --git a/simroot/pure.py b/simroot/pure.py
index 57371aa..6b8b578 100644
--- a/simroot/pure.py
+++ b/simroot/pure.py
@@ -764,19 +764,21 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
     # category variables below then carry the same string the old tests compared.
     if "+" in prestep:
         parts = prestep.split("+")
-        cats = ({"defect"}, {"contra"}, {"exp"}, {"half"}, {"flatanalog", "flatanalogmem", "flatanalogpt",
+        cats = ({"defect"}, {"contra"}, {"exp"}, {"half"}, {"close"}, {"sub"},
+                {"flatanalog", "flatanalogmem", "flatanalogpt",
                                           "flatanalogmempt"})
         if (len(parts) != len(set(parts)) or any(not any(p in c for c in cats) for p in parts)
                 or any(sum(p in c for p in parts) > 1 for c in cats)):
             raise ValueError(f"unknown composite prestep {prestep!r}")
-        if "exp" in parts and not any(p.startswith("flatanalog") for p in parts):
+        if (("exp" in parts or "sub" in parts)
+                and not any(p.startswith("flatanalog") for p in parts)):
             # the translation limit changes how the analogy reads its order; without the
             # flat regime there is no analogy to read (PREREG_flat_exp_limit)
             raise ValueError("prestep component 'exp' needs a flatanalog* component")
     elif prestep not in ("none", "progress", "slope", "band", "defect", "simpair", "flat",
                          "flat2", "flatcreep", "flatstick", "flatstickmid", "flatstickexit",
                          "flatjudge", "flatanalog", "flatanalogmem", "contra",
-                         "flatanalogpt", "flatanalogmempt", "half"):
+                         "flatanalogpt", "flatanalogmempt", "half", "close"):
         raise ValueError(f"unknown prestep {prestep!r}")
     parts = prestep.split("+")
     pre_dec = next((p for p in parts if p in ("band", "defect", "simpair")), "none")
@@ -788,6 +790,8 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
     pre_contra = "contra" in parts
     pre_exp = "exp" in parts            # PREREG_flat_exp_limit: translation similarity, m = inf
     pre_half = "half" in parts          # PREREG_half_contra: the half promised vs the bracket
+    pre_close = "close" in parts        # PREREG_close_sub: the tolerance step (borrowed)
+    pre_sub = "sub" in parts            # PREREG_close_sub: the overshoot branch, m <= 2/3
     if pre_dec != "none" and scale not in (
             "derived3", "derived3_cap", "derived3_m", "bracket", "far"):
         # the decision reads a TRIPLE; under the two-point scale there is no third point
@@ -847,6 +851,8 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
         analogy_seen = False     # flatanalogmem: the run has been judged analogous to its past
         analogy_m = math.inf     # flatanalog*pt: the order read from the latest analogous pair
         regime_target = None     # flatanalog*pt: the point the regime step aimed at
+        shallow_seen = False     # sub: the run has been judged analogous with m <= 2/3
+        shallow_m = math.inf     # sub: the order of that pair
         if stats is not None:
             stats['forced'] = 0
             stats['pre_refused'] = 0
@@ -1079,15 +1085,32 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                                 and fx / f_prev >= 0.25)
                         m_pair = _analogy_order(cur, configs, translation=pre_exp)
                         steep = m_pair is not None and m_pair >= 2.0
+                        # PREREG_close_sub 2026-09-21 (sub): the tangent step loses to a halving
+                        # when |1 - 1/m| > 1/2 - for m > 2 (undershoot, the branch above) AND
+                        # for m < 2/3 (overshoot).  The overshoot shows as an unkept promise WITH
+                        # a sign change; the pair's order says which branch it is.
+                        shallow = (pre_sub and m_pair is not None
+                                   and 0.0 < m_pair <= 2.0 / 3.0)
+                        flag_over = (pre_sub and sticky_m is None and last_model
+                                     and f_prev != 0.0 and fx / f_prev <= -0.25)
                         if flat_mem:
                             if steep:
                                 analogy_seen = True
                                 analogy_m = m_pair
+                            if shallow:
+                                shallow_seen = True
+                                shallow_m = m_pair
                             enter = flag and analogy_seen
+                            if not enter and flag_over and shallow_seen:
+                                enter = True
+                                analogy_m = shallow_m
                         else:
                             enter = flag and steep
                             if enter:
                                 analogy_m = m_pair
+                            elif flag_over and shallow:
+                                enter = True
+                                analogy_m = m_pair
                         if enter:
                             sticky_m = math.inf
                             if stats is not None:
@@ -1207,6 +1230,19 @@ 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))
+                        if (pre_close and abs(step) < 0.5 * xtol and last_model
+                                and f_prev != 0.0 and abs(fx / f_prev) < 0.25):
+                            # Only a CREDIBLE promise is closed on: the model's previous step
+                            # kept its own promise (|f| fell below the flag's 1/4).  Measured in
+                            # development on x^19: at the start, a slope dominated by f(100) =
+                            # 1e38 promises the root at x = -10, and closing on that promise
+                            # replaced the midpoint by near-duplicate points (23 -> 32).
+                            # PREREG_close_sub 2026-09-21 (close) - BORROWED: Dekker (1969) and
+                            # Brent (1973) step at least the tolerance.  A model promising the
+                            # root closer than xtol/2 is taken at its word one step further: the
+                            # new point lands past the root and the bracket closes to the
+                            # tolerance, instead of being halved down from the far end.
+                            step = math.copysign(0.5 * xtol, far - x)
                         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
@@ -1272,6 +1308,11 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                 # flatanalogmem: "linear" refutes "flat" - the remembered analogy is
                 # cleared and must be established anew (PREREG_flat_analogy_mem)
                 analogy_seen = False
+                # NOT shallow_seen (PREREG_close_sub, amendment): the exit judgment - f nearer
+                # the line than 0 - tells a flat root from a simple one; at a root of order
+                # m < 1 f is far from 0 on both sides, the exit fires falsely, and keeping the
+                # shallow memory lets the regime resume at the next overshoot (development:
+                # sqrt|x| sgn 19 with it, 65 without)
                 if stats is not None:
                     stats['stick_exits'] = stats.get('stick_exits', 0) + 1
             if prestep == "flatjudge":
diff --git a/sweep_fourth.py b/sweep_fourth.py
new file mode 100644
index 0000000..4c72408
--- /dev/null
+++ b/sweep_fourth.py
@@ -0,0 +1,152 @@
+"""PREREG_close_sub_2026-09-21: A' = A + close + sub against A, its two ablations and the
+library rivals.
+
+    python -u sweep_fourth.py --seen     development: the five SEEN corpora (descriptive)
+    python -u sweep_fourth.py --fourth   the ONE check on corpus_fourth.py -> sweep_fourth.json
+    python -u sweep_fourth.py --report   verdicts from sweep_fourth.json
+"""
+from __future__ import annotations
+
+import json
+import math
+import sys
+from multiprocessing import Pool
+
+from corpus_roots_jl import solved
+from sweep_full import MAXFEV, XTOL, load_problems, make_method
+from sweep_oos import CONFIGS as OOS
+from sweep_oos import sign_p
+
+
+def spec(pre):
+    return ("pure", "ratio", "derived3", "project", "x", pre, "recent", "distance", "none",
+            "pattern")
+
+
+CONFIGS = {
+    "base": OOS["base"],
+    "A": OOS["A"],
+    "Aprime": spec("contra+flatanalogmempt+exp+half+close+sub"),
+    "Aprime-close": spec("contra+flatanalogmempt+exp+half+sub"),
+    "Aprime-sub": spec("contra+flatanalogmempt+exp+half+close"),
+    "chandrupatla": OOS["chandrupatla"],
+    "brent": OOS["brent"],
+    "toms748": OOS["toms748"],
+}
+
+
+def corpus(which):
+    """[(set, id, f, a, b)] - success is always a sign change within 1e-6."""
+    out = []
+    if which == "fourth":
+        from corpus_fourth import build
+        out += [("D", pid, f, a, b) for _c, pid, f, a, b, _r in build()[0]]
+        return out
+    out += [(s, pid, f, a, b) for s, pid, f, a, b, _r in load_problems()]
+    from corpus_new import build as b1
+    out += [("NEW", pid, f, a, b) for _c, pid, f, a, b, _r in b1()[0]]
+    from corpus_roots_jl import build as b2
+    out += [("RJL", pid, f, a, b) for pid, f, a, b, ins in b2()[0] if not ins]
+    from corpus_applied import build as b3
+    out += [("APP", pid, f, a, b) for pid, f, a, b in b3()[0]]
+    return out
+
+
+def evaluate(job):
+    name, which = job
+    import random
+
+    import scipy.optimize._tstutils as T
+    method = make_method(CONFIGS[name])
+    res = {}
+    for s, pid, f, a, b in corpus(which):
+        T._f6_cache.clear()
+        random.seed(20260921)
+        try:
+            x, n, conv = method(f, a, b, XTOL, MAXFEV)
+            x = float(x)
+        except Exception:
+            x, n, conv = float("nan"), MAXFEV, False
+        ok = bool(conv and solved(f, x))
+        over = ok and int(n) > 2 * math.ceil(math.log2(abs(b - a) / XTOL)) + 4
+        res.setdefault(s, {})[pid] = [int(n), ok, bool(over)]
+    return name, res
+
+
+def cost(v):
+    return v[0] if v[1] else MAXFEV
+
+
+def h2h(r, o, s, keep=lambda p: True):
+    ps = [p for p in r[s] if keep(p)]
+    w = sum(cost(r[s][p]) < cost(o[s][p]) for p in ps)
+    l = sum(cost(r[s][p]) > cost(o[s][p]) for p in ps)
+    return w, len(ps) - w - l, l
+
+
+def tot(r, s, keep=lambda p: True):
+    return sum(cost(v) for p, v in r[s].items() if keep(p))
+
+
+def report_seen(R):
+    print("SEEN corpora (development, descriptive): totals; A' vs Chandrupatla w/t/l")
+    for s in ("APS", "CHA", "NEW", "RJL", "APP"):
+        w, t, l = h2h(R["Aprime"], R["chandrupatla"], s)
+        print(f"  {s:<4} " + "  ".join(f"{k} {tot(R[k], s)}" for k in CONFIGS)
+              + f"   A' vs Ch {w}/{t}/{l} p={sign_p(w, l):.2g}")
+
+
+def report_fourth(R):
+    s, ch, ap = "D", R["chandrupatla"], R["Aprime"]
+    print("FOURTH corpus (the one check)")
+    for k in CONFIGS:
+        r = R[k]
+        print(f"  {k:<14} total {tot(r, s):6d}  solved {sum(v[1] for v in r[s].values())}/"
+              f"{len(r[s])}  violations {sum(v[2] for v in r[s].values())}")
+    w, t, l = h2h(ap, ch, s)
+    f1 = tot(ap, s) < tot(ch, s)
+    f2 = w > l and sign_p(w, l) < 0.05
+    f3 = (sum(v[2] for v in ap[s].values()) == 0
+          and sum(v[1] for v in ap[s].values()) >= sum(v[1] for v in ch[s].values()))
+    f4 = tot(ap, s) < tot(R["A"], s)
+    print(f"  F1 {'✓' if f1 else '✗'} ({tot(ap, s)} vs {tot(ch, s)}, "
+          f"{100 * (tot(ap, s) - tot(ch, s)) / tot(ch, s):+.1f}%)  F2 {'✓' if f2 else '✗'} "
+          f"({w}/{t}/{l} p={sign_p(w, l):.2g})  F3 {'✓' if f3 else '✗'}  F4 {'✓' if f4 else '✗'} "
+          f"(A' {tot(ap, s)} vs A {tot(R['A'], s)})  -> "
+          + ("A' TRANSFERS" if f1 and f2 and f3 else "A' does NOT transfer"))
+    for comp in ("close", "sub"):
+        d = tot(R[f"Aprime-{comp}"], s) - tot(ap, s)
+        print(f"  ablation - {comp}: {d:+d} ({'earned' if d > 0 else 'NOT earned'})")
+    nt = lambda p: p[:2] in ("D1", "D2", "D4", "D5")
+    w, t, l = h2h(ap, ch, s, nt)
+    print(f"  without the targeted classes (D1 D2 D4 D5): A' {tot(ap, s, nt)} vs Ch {tot(ch, s, nt)}"
+          f"  {w}/{t}/{l} p={sign_p(w, l):.2g}")
+    for rn in ("brent", "toms748"):
+        w, t, l = h2h(ap, R[rn], s)
+        print(f"  vs {rn:<8} {tot(ap, s)} vs {tot(R[rn], s)}  {w}/{t}/{l} p={sign_p(w, l):.2g}")
+    print("  per class: A' / A / Chandrupatla / Brent / TOMS748   (A' vs Ch w/t/l)")
+    for cls in ("D1", "D2", "D3", "D4", "D5", "D6"):
+        k = lambda p, c=cls: p.startswith(c)
+        w, t, l = h2h(ap, ch, s, k)
+        print(f"    {cls}: " + " / ".join(f"{tot(R[n], s, k):5d}" for n in
+                                         ("Aprime", "A", "chandrupatla", "brent", "toms748"))
+              + f"   {w}/{t}/{l}")
+
+
+def main():
+    which = "fourth" if "--fourth" in sys.argv else ("seen" if "--seen" in sys.argv else None)
+    if which is not None:
+        with Pool(len(CONFIGS)) as pool:
+            R = dict(pool.map(evaluate, [(n, which) for n in CONFIGS]))
+        with open(f"sweep_fourth_{which}.json", "w", encoding="utf8") as fh:
+            json.dump(R, fh)
+    else:
+        which = "fourth"
+    with open(f"sweep_fourth_{which}.json", encoding="utf8") as fh:
+        R = json.load(fh)
+    (report_fourth if which == "fourth" else report_seen)(R)
+    print("fourth done")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/sweep_fourth_seen.json b/sweep_fourth_seen.json
new file mode 100644
index 0000000..a43597b
--- /dev/null
+++ b/sweep_fourth_seen.json
@@ -0,0 +1 @@
+{"base": {"APS": {"aps.01.00": [8, true, false], "aps.02.00": [12, true, false], "aps.02.01": [13, true, false], "aps.02.02": [11, true, false], "aps.02.03": [24, true, false], "aps.02.04": [23, true, false], "aps.02.05": [26, true, false], "aps.02.06": [23, true, false], "aps.02.07": [28, true, false], "aps.02.08": [12, true, false], "aps.02.09": [12, true, false], "aps.03.00": [19, true, false], "aps.03.01": [22, true, false], "aps.03.02": [22, true, false], "aps.04.00": [17, true, false], "aps.04.01": [22, true, false], "aps.04.02": [14, true, false], "aps.04.03": [18, true, false], "aps.04.04": [22, true, false], "aps.04.05": [14, true, false], "aps.04.06": [12, true, false], "aps.04.07": [12, true, false], "aps.04.08": [21, true, false], "aps.04.09": [15, true, false], "aps.04.10": [15, true, false], "aps.04.11": [15, true, false], "aps.04.12": [13, true, false], "aps.04.13": [15, true, false], "aps.05.00": [8, true, false], "aps.06.00": [7, true, false], "aps.06.01": [8, true, false], "aps.06.02": [9, true, false], "aps.06.03": [9, true, false], "aps.06.04": [9, true, false], "aps.06.05": [10, true, false], "aps.06.06": [10, true, false], "aps.06.07": [10, true, false], "aps.06.08": [13, true, false], "aps.06.09": [13, true, false], "aps.07.00": [10, true, false], "aps.07.01": [11, true, false], "aps.07.02": [11, true, false], "aps.08.00": [3, true, false], "aps.08.01": [10, true, false], "aps.08.02": [12, true, false], "aps.08.03": [15, true, false], "aps.08.04": [11, true, false], "aps.09.00": [8, true, false], "aps.09.01": [10, true, false], "aps.09.02": [12, true, false], "aps.09.03": [10, true, false], "aps.09.04": [10, true, false], "aps.09.05": [11, true, false], "aps.09.06": [10, true, false], "aps.10.00": [9, true, false], "aps.10.01": [12, true, false], "aps.10.02": [15, true, false], "aps.10.03": [21, true, false], "aps.10.04": [16, true, false], "aps.11.00": [5, true, false], "aps.11.01": [5, true, false], "aps.11.02": [5, true, false], "aps.11.03": [5, true, false], "aps.12.00": [9, true, false], "aps.12.01": [9, true, false], "aps.12.02": [10, true, false], "aps.12.03": [10, true, false], "aps.12.04": [10, true, false], "aps.12.05": [10, true, false], "aps.12.06": [10, true, false], "aps.12.07": [10, true, false], "aps.12.08": [10, true, false], "aps.12.09": [10, true, false], "aps.12.10": [10, true, false], "aps.12.11": [10, true, false], "aps.12.12": [10, true, false], "aps.12.13": [9, true, false], "aps.12.14": [9, true, false], "aps.12.15": [9, true, false], "aps.12.16": [9, true, false], "aps.12.17": [9, true, false], "aps.12.18": [9, true, false], "aps.13.00": [28, true, false], "aps.14.00": [15, true, false], "aps.14.01": [15, true, false], "aps.14.02": [15, true, false], "aps.14.03": [15, true, false], "aps.14.04": [15, true, false], "aps.14.05": [15, true, false], "aps.14.06": [15, true, false], "aps.14.07": [15, true, false], "aps.14.08": [15, true, false], "aps.14.09": [15, true, false], "aps.14.10": [15, true, false], "aps.14.11": [15, true, false], "aps.14.12": [15, true, false], "aps.14.13": [15, true, false], "aps.14.14": [15, true, false], "aps.14.15": [15, true, false], "aps.14.16": [15, true, false], "aps.14.17": [15, true, false], "aps.14.18": [15, true, false], "aps.14.19": [15, true, false], "aps.14.20": [15, true, false], "aps.14.21": [15, true, false], "aps.14.22": [15, true, false], "aps.14.23": [15, true, false], "aps.14.24": [15, true, false], "aps.14.25": [15, true, false], "aps.14.26": [15, true, false], "aps.14.27": [15, true, false], "aps.14.28": [15, true, false], "aps.14.29": [15, true, false], "aps.14.30": [15, true, false], "aps.14.31": [15, true, false], "aps.14.32": [15, true, false], "aps.14.33": [15, true, false], "aps.14.34": [15, true, false], "aps.14.35": [15, true, false], "aps.14.36": [15, true, false], "aps.14.37": [15, true, false], "aps.14.38": [15, true, false], "aps.14.39": [15, true, false], "aps.15.00": [31, true, false], "aps.15.01": [30, true, 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false], "P.well.z020": [11, true, false]}}}
\ No newline at end of file
diff --git a/tests/test_flat_gate.py b/tests/test_flat_gate.py
index 310ac0b..f169879 100644
--- a/tests/test_flat_gate.py
+++ b/tests/test_flat_gate.py
@@ -449,3 +449,41 @@ def test_half_is_silent_on_a_line_and_breaks_the_fun9_flank():
                    guard="project", prestep=p)[1]
          for p in ("defect+flatanalogmempt+exp", "defect+flatanalogmempt+exp+half")]
     assert n[1] < n[0], n
+
+
+# ---- PREREG_close_sub_2026-09-21: bracket closing (borrowed) and the overshoot branch ----
+APRIME = "contra+flatanalogmempt+exp+half+close+sub"
+
+
+@pytest.mark.parametrize("bad", ("sub", "defect+sub", "contra+sub"))
+def test_sub_needs_the_flat_regime(bad):
+    with pytest.raises(ValueError):
+        pure_root(lambda t: t - 1.0, 0.0, 3.0, XTOL, MAXFEV, similarity="ratio",
+                  scale="derived3", guard="project", prestep=bad)
+
+
+@pytest.mark.parametrize("f,a,b", PROBS)
+@pytest.mark.parametrize("guard", ("project", "credit"))
+@pytest.mark.parametrize("pre", ("close", APRIME))
+def test_the_guarantee_holds_with_close_and_sub(f, a, b, guard, pre):
+    x, n, ok = pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                         guard=guard, prestep=pre, extrapolate="pattern")
+    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 _n(f, a, b, pre):
+    return pure_root(f, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
+                     guard="project", prestep=pre, extrapolate="pattern")[1]
+
+
+def test_close_and_sub_measured_in_development():
+    """Measured before pinning: sin[3,4] 24 -> 7, sqrt|x| sgn 84 -> 19, x^19 unchanged."""
+    a_ = "contra+flatanalogmempt+exp+half"
+    assert _n(math.sin, 3.0, 4.0, APRIME) < _n(math.sin, 3.0, 4.0, a_)
+    sq = lambda x: math.sqrt(abs(x)) * (0.0 if x == 0 else math.copysign(1.0, x))  # noqa: E731
+    assert _n(sq, -1 / 3, 1.0, APRIME) < _n(sq, -1 / 3, 1.0, a_)
+    p19 = lambda x: x ** 19  # noqa: E731
+    assert _n(p19, -10.0, 100.0, APRIME) == _n(p19, -10.0, 100.0, a_)
+    assert _n(lambda x: 3.0 * x - 1.0, -2.0, 5.0, APRIME) == 4