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Revision step 2 into the paper: section 9.1 (Tables 9b, 9c) as PREREG_controls prescribes. The analogy detector beats Lagouanelle-with-memory, Chandrupatla's band and no detector on corpus 4 (3656 vs 4488, 6236, 6165); with the multiplicity given, the classical transformation is 2.4-6.8x cheaper on every multiple or fractional root - stated as the price of estimating the order, in the abstract, 8.3, 8.6, the conclusion and as the first future-work question. verify_controls(): 26 checks. Server: verify 526 OK / 0 BAD, regression x2 1232 passed, PDF 30 pages, 0 errors, 0 undefined, 0 overfull.

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Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
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2026-09-22 07:53:58 +01:00
co-authored by Claude Opus 5
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apply_article_controls.py
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"""PREREG_controls_2026-09-22 outcome into article/sim_numerics.md, as the pre-registration
prescribes: C1 passed for every detector (the claim of 8.6 stands, with the numbers), and K1 -
with the multiplicity known the classical transformation is much cheaper - stated plainly.
New section 9.1 (Tables 9b, 9c); a sentence each in the abstract, 8.3, 8.6, the conclusion, and
the future work. Exact anchors, each found exactly once, or nothing is written.
python -u apply_article_controls.py
"""
from __future__ import annotations
PATH = "article/sim_numerics.md"
REPL = []
def R(old, new):
REPL.append((old, new))
R("and without them the lead is 2403 against 3569, on the simple-root families alone 737 against 747.",
"and without them the lead is 2403 against 3569, on the simple-root families alone 737 against 747. "
"Against simpler detectors put in the place of the analogy — Lagouanelle's multiplicity estimate "
"with the same memory, Chandrupatla's acceptance band, or none — it wins on that corpus in total "
"and problem by problem (3656 against 4488, 6236 and 6165); against the classical transformation "
"$\\operatorname{sign}(f)|f|^{1/m}$ with the multiplicity given in advance it loses every multiple or "
"fractional root, by a factor of 2.4 to 6.8 — the price of estimating the order.")
R("classical since Schröder [16] and Traub [17]; what is new here is only where $m$ comes from.",
"classical since Schröder [16] and Traub [17]; what is new here is only where $m$ comes from. With "
"the multiplicity given in advance the transformation is far cheaper still (§9.1): the contribution "
"is the estimate, and its price is measured there.")
R("Roots.jl's `Order1B` and `Order2B` [12]); Chandrupatla takes the decision to halve by an empirical acceptance band [3].",
"Roots.jl's `Order1B` and `Order2B` [12]); Chandrupatla takes the decision to halve by an empirical "
"acceptance band [3]. §9.1 puts this claim against the detectors a reader would try first, and "
"against the transformation with the multiplicity known in advance.")
SEC91 = """
### 9.1 Controls: simpler detectors, and the multiplicity known in advance
Two controls ask whether the gain of §8 comes from the recognition by analogy or only from the split (7) with any reasonable detector. Both were pre-registered, with what the paper would say under every outcome, before the run (`sweep_controls.py`).
**Other detectors in A′.** A′ is run with only its detector swapped; the memory, the split, the exit and the five readings of §8.4 are unchanged. (a) *Lagouanelle's estimate* $m_L=1/(1-\\theta)$, $\\theta=ff''/f'^2$ from the parabola through the last three points ($m_L=\\infty$ for $\\theta\\ge1$), with the same memory — once seen, held until the exit, a hysteresis — entering at $m_L\\ge2$ and at $0<m_L\\le\\tfrac23$ and splitting at $m_L$. (b) *Chandrupatla's acceptance band*, entering when the band rejects the current triple and splitting at the current $m_L$. (c) *No detector*: the regime is entered at every unkept promise.
**Table 9b.** A′ against its detector swapped: totals on the six corpora of §9 (uniform criterion), and A′'s wins/ties/losses on corpus 4. No variant breaks the bound.
| detector | APS | CHA | new-1 | Roots.jl | applied | corpus 4 | all | A′ on corpus 4 |
|---|---|---|---|---|---|---|---|---|
| **analogy (A′)** | 2259 | **1074** | **3290** | **1371** | **1382** | **3656** | **13032** | — |
| Lagouanelle, with memory | **2235** | 1437 | 4026 | 1495 | 1490 | 4488 | 15171 | 84/121/35, $p=8\\cdot10^{-6}$ |
| Chandrupatla's band | 2682 | 2526 | 5485 | 2117 | 1622 | 6236 | 20668 | 110/96/34, $p=1.5\\cdot10^{-10}$ |
| none | 2693 | 2526 | 5569 | 2126 | 1670 | 6165 | 20749 | 113/87/40, $p=3\\cdot10^{-9}$ |
On corpus 4, the one test, A′ beats all three in total and problem by problem, as the pre-registered criterion required. The competent alternative is Lagouanelle's estimate: it is slightly better on the canonical set (2235 against 2259), and on corpus 4 A′'s lead over it lies in the multiple-root and fractional-power families (1163 against 1739, 924 against 1207); it is behind on $\\log(1+q^2)$ of a cubic (503 against 477) and level on the rest. The band and the regime without a detector are worse than no regime at all. The decision layer by analogy therefore earns its place against the detectors a reader would try first.
**The multiplicity known in advance.** The second control asks how good the classical transformation is when its multiplicity is given. Library Chandrupatla and Brent are run on $g=\\operatorname{sign}(f)\\,|f|^{1/m^*}$, with $m^*$ the order read at the known root — the slope of $\\ln|f|$ against $\\ln|x-r|$ on both sides at $h=10^{-5}$, rounded to 0.01, and 1 where either side shows no zero (a jump). It is not an achievable method but an upper bound. On the multiple and fractional roots, $m^*\\ge2$ or $m^*\\le\\tfrac23$ — the branches of §8.1 where a tangent step loses to a halving:
**Table 9c.** Multiple and fractional roots with the multiplicity given: totals, failures charged. A′ wins none of the 148 problems.
| corpus | problems | A′ | Chandrupatla on $g$ | Brent on $g$ | Chandrupatla on $f$ |
|---|---|---|---|---|---|
| Chandrupatla | 20 | 654 | 96 | 66 | 1068 |
| new corpus 1 | 40 | 1229 | 358 | 337 | 1792 |
| corpus 4 | 88 | 2159 | 901 | 844 | 4151 |
With the multiplicity known the transformation is 2.4 to 6.8 times cheaper than A′. A′ is far ahead of Chandrupatla on $f$ itself, which does not know the multiplicity either. On the remaining problems the transformed and untransformed runs differ little and A′ is close to them: ahead on the canonical sets (2259 against 2640, 420 against 437 for Chandrupatla on $g$), behind on corpora 1 and 4 (2061 against 1924, 1497 against 1427). So on multiple roots the contribution of A′ is the estimate of the order without knowing it, and the gap to Table 9c is the price of that estimate — large, and the clearest direction for improving the method."""
R("and the smooth equations of the applied corpus, where the two are level.\n",
"and the smooth equations of the applied corpus, where the two are level." + SEC91 + "\n")
R("and on a corpus fixed after its code was frozen it needs 36% fewer evaluations and wins 177 problems to 28.",
"and on a corpus fixed after its code was frozen it needs 36% fewer evaluations and wins 177 problems "
"to 28. Its decision layer beats Lagouanelle's estimate, Chandrupatla's band and no detector put in "
"its place; with the multiplicity known in advance the classical transformation is 2.4 to 6.8 times "
"cheaper on multiple roots, which measures what estimating the order costs.")
R("Two questions remain, both narrower than the one this section used to pose.\n\nThe first is the steep flank.",
"Three questions remain, all narrower than the one this section used to pose.\n\nThe first is the "
"price of estimating the order. With the multiplicity given, the transformation of §8.3 is 2.4 to 6.8 "
"times cheaper than A′ on multiple and fractional roots (Table 9c): the order read from the analogous "
"pair, or its use in the split, loses most of what the transformation could deliver.\n\nThe second is "
"the steep flank.")
R("The second is the one geometric choice of §8.2,",
"The third is the one geometric choice of §8.2,")
def main():
src = open(PATH, encoding="utf8").read()
bad = [old[:90] for old, _ in REPL if src.count(old) != 1]
if bad:
for b in bad:
print("ANCHOR NOT FOUND EXACTLY ONCE:", repr(b))
raise SystemExit(f"{len(bad)} anchors failed; nothing written")
for old, new in REPL:
src = src.replace(old, new)
open(PATH, "w", encoding="utf8").write(src)
print(f"applied {len(REPL)} replacements")
if __name__ == "__main__":
main()
apply_table9b_narrow.py
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"""Table 9b fits a page: the three seen corpora new-1, Roots.jl and applied stay in the six-corpus
total (their columns are quoted nowhere in the text); the numbers themselves are unchanged and
still checked row by row by verify_controls().
python -u apply_table9b_narrow.py
"""
PATH = "article/sim_numerics.md"
OLD = """| detector | APS | CHA | new-1 | Roots.jl | applied | corpus 4 | all | A′ on corpus 4 |
|---|---|---|---|---|---|---|---|---|
| **analogy (A′)** | 2259 | **1074** | **3290** | **1371** | **1382** | **3656** | **13032** | — |
| Lagouanelle, with memory | **2235** | 1437 | 4026 | 1495 | 1490 | 4488 | 15171 | 84/121/35, $p=8\\cdot10^{-6}$ |
| Chandrupatla's band | 2682 | 2526 | 5485 | 2117 | 1622 | 6236 | 20668 | 110/96/34, $p=1.5\\cdot10^{-10}$ |
| none | 2693 | 2526 | 5569 | 2126 | 1670 | 6165 | 20749 | 113/87/40, $p=3\\cdot10^{-9}$ |"""
NEW = """| detector | APS | CHA | corpus 4 | six corpora | A′ on corpus 4 | $p$ |
|---|---|---|---|---|---|---|
| **analogy (A′)** | 2259 | **1074** | **3656** | **13032** | — | |
| Lagouanelle, with memory | **2235** | 1437 | 4488 | 15171 | 84/121/35 | $8\\cdot10^{-6}$ |
| Chandrupatla's band | 2682 | 2526 | 6236 | 20668 | 110/96/34 | $1.5\\cdot10^{-10}$ |
| none | 2693 | 2526 | 6165 | 20749 | 113/87/40 | $3\\cdot10^{-9}$ |"""
src = open(PATH, encoding="utf8").read()
assert src.count(OLD) == 1, src.count(OLD)
src = src.replace(OLD, NEW)
src2 = src.replace("**Table 9b.** A′ against its detector swapped: totals on the six corpora of §9 (uniform criterion), and A′'s wins/ties/losses on corpus 4. No variant breaks the bound.",
"**Table 9b.** A′ against its detector swapped: totals on the two canonical sets, on corpus 4 and over the six corpora of §9 (uniform criterion), and A′'s wins/ties/losses on corpus 4 with the exact sign test. No variant breaks the bound.")
assert src2 != src
open(PATH, "w", encoding="utf8").write(src2)
print("Table 9b narrowed")
apply_verify_controls.py
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"""verify_article.py checks section 9.1 (Tables 9b, 9c and the sentences quoting them) against
sweep_controls.json, sweep_fourth_*.json and sweep_ablation_all.json. Adds verify_controls()
and its call; nothing else changes.
python -u apply_verify_controls.py
"""
from __future__ import annotations
PATH = "verify_article.py"
FUNC = '''
def verify_controls():
"""Section 9.1: the controls (PREREG_controls_2026-09-22, sweep_controls.json)."""
import math
print("\\nSection 9.1 (controls, sweep_controls.json):")
R = _load("sweep_controls.json")
seen, four = _load("sweep_fourth_seen.json"), _load("sweep_fourth_fourth.json")
ap = dict(seen["Aprime"])
ap.update(four["Aprime"])
chf = dict(seen["chandrupatla"])
chf.update(four["chandrupatla"])
sets = ("APS", "CHA", "NEW", "RJL", "APP", "D")
def cost(v):
return v[0] if v[1] else 200
def tot(r, keep=None):
return sum(cost(v) for p, v in r.items() if keep is None or keep(p))
def h2h(a, b, keep=None):
ps = [p for p in a if keep is None or keep(p)]
w = sum(cost(a[p]) < cost(b[p]) for p in ps)
l = sum(cost(a[p]) > cost(b[p]) for p in ps)
return w, len(ps) - w - l, l
def sign_p(w, l):
k, n = min(w, l), w + l
return min(1.0, 2.0 * sum(math.comb(n, i) for i in range(k + 1)) / 2.0 ** n)
rows = {"K2L": (2235, 1437, 4026, 1495, 1490, 4488, 15171),
"K2P": (2682, 2526, 5485, 2117, 1622, 6236, 20668),
"K2F": (2693, 2526, 5569, 2126, 1670, 6165, 20749)}
check("Table 9b A' row", tuple(tot(ap[s]) for s in sets) + (sum(tot(ap[s]) for s in sets),),
(2259, 1074, 3290, 1371, 1382, 3656, 13032))
for k, want in rows.items():
k2 = R["K2"][k]
check(f"Table 9b {k} row", tuple(tot(k2[s]) for s in sets) + (sum(tot(k2[s]) for s in sets),),
want)
for k, wtl, pp in (("K2L", (84, 121, 35), ("1g", "8e-06")), ("K2P", (110, 96, 34), ("2g", "1.5e-10")),
("K2F", (113, 87, 40), ("1g", "3e-09"))):
got = h2h(ap["D"], R["K2"][k]["D"])
check(f"Table 9b A' vs {k} on corpus 4", got, wtl)
check(f"Table 9b p vs {k}", format(sign_p(got[0], got[2]), "." + pp[0]), pp[1])
check("Table 9b violations", sum(v[2] for k in rows for s in sets for v in R["K2"][k][s].values()), 0)
for cls, want in (("D2", (1163, 1739)), ("D3", (924, 1207)), ("D5", (503, 477))):
keep = lambda p, c=cls: p.startswith(c) # noqa: E731
check(f"9.1 {cls}: A' vs Lagouanelle", (tot(ap["D"], keep), tot(R["K2"]["K2L"]["D"], keep)), want)
level = [(tot(ap["D"], lambda p, c=c: p.startswith(c)),
tot(R["K2"]["K2L"]["D"], lambda p, c=c: p.startswith(c))) for c in ("D1", "D4", "D6")]
check("9.1 level on the rest (within 1.5%)", all(abs(a - b) <= 0.015 * b for a, b in level), True)
ab = _load("sweep_ablation_all.json")["abl-regime"]
no_regime_all = sum(tot(ab[s]) for s in sets)
check("band and no-detector worse than no regime (six corpora)",
all(sum(tot(R["K2"][k][s]) for s in sets) > no_regime_all for k in ("K2P", "K2F")), True)
check("band and no-detector worse than no regime (corpus 4)",
all(tot(R["K2"][k]["D"]) > tot(ab["D"]) for k in ("K2P", "K2F")), True)
t9c = {"CHA": (20, 654, 96, 66, 1068), "NEW": (40, 1229, 358, 337, 1792),
"D": (88, 2159, 901, 844, 4151)}
nonsimple = lambda m: m >= 2.0 or m <= 2.0 / 3.0 # noqa: E731
wins = 0
for s, want in t9c.items():
k1 = R["K1"][s]
keep = lambda p, _m=k1["mstar"]: nonsimple(_m[p]) # noqa: E731
got = (sum(1 for p in k1["mstar"] if keep(p)), tot(ap[s], keep), tot(k1["chandrupatla_g"], keep),
tot(k1["brent_g"], keep), tot(chf[s], keep))
check(f"Table 9c {s}", got, want)
wins += h2h({p: v for p, v in ap[s].items() if keep(p)}, k1["chandrupatla_g"])[0]
check("Table 9c: A' wins none of the 148", wins, 0)
check("Table 9c: 2.4 to 6.8 times", (round(2159 / 901, 1), round(654 / 96, 1),
min(round(a / c, 1) for _n, a, c, _b, _f in t9c.values()),
max(round(a / c, 1) for _n, a, c, _b, _f in t9c.values())),
(2.4, 6.8, 2.4, 6.8))
APS_ = "APS"
for s, want in ((APS_, (2259, 2640)), ("CHA", (420, 437)), ("NEW", (2061, 1924)), ("D", (1497, 1427))):
k1 = R["K1"][s]
keep = lambda p, _m=k1["mstar"]: not nonsimple(_m[p]) # noqa: E731
check(f"9.1 rest {s}: A' vs Chandrupatla on g", (tot(ap[s], keep), tot(k1["chandrupatla_g"], keep)),
want)
'''
src = open(PATH, encoding="utf8").read()
call_old = " verify_revision1()\n\n print()"
call_new = " verify_revision1()\n verify_controls()\n\n print()"
marker = '\n\nif __name__ == "__main__":\n main()\n'
assert src.count(call_old) == 1 and src.count(marker) == 1
src = src.replace(call_old, call_new).replace(marker, FUNC + marker)
open(PATH, "w", encoding="utf8").write(src)
print("added verify_controls")
article/sim_numerics.md
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@@ -10,7 +10,7 @@ Bisection places the next iterate at the midpoint because it uses no information
The derivation fixes a *base rule* before any benchmark is consulted: a similarity whose approach to 1 is linear, the finest available data-derived scale, and a safeguard that projects the step onto the admissible interval rather than replacing it. It is, explicitly, a Newton step with a three-point divided-difference derivative, clipped to the bracket and projected onto the bisection budget. The similarity sets that step, and its form is decisive: with the tent the step is Newton's, with any of the other eight functions the proposed method costs from 19% to five times more. Its order is Muller's, 1.839. It carries a proven worst case of twice bisection plus one step and one free numerical constant, the multiplier of that budget, which we declare. On the 154 problems of Alefeld, Potra and Shi it needs 2634 evaluations of $f$ against 2636 for TOMS 748, 2707 for Brent and 2595 for Chandrupatla's method; it wins more problems than it loses against Chandrupatla, 87 to 43, and needs 1.5% more in total because its losses are the larger ones. On the 45 wide-bracket problems of Chandrupatla it beats Brent and TOMS 748 and loses to ITP and to Chandrupatla (2357 against 1946 and 1505). For every data-derived scale the three similarity functions whose $1-S$ has a non-zero slope at the origin come before the three whose slope is zero, as the derived law predicts. The derivation fixes a *base rule* before any benchmark is consulted: a similarity whose approach to 1 is linear, the finest available data-derived scale, and a safeguard that projects the step onto the admissible interval rather than replacing it. It is, explicitly, a Newton step with a three-point divided-difference derivative, clipped to the bracket and projected onto the bisection budget. The similarity sets that step, and its form is decisive: with the tent the step is Newton's, with any of the other eight functions the proposed method costs from 19% to five times more. Its order is Muller's, 1.839. It carries a proven worst case of twice bisection plus one step and one free numerical constant, the multiplier of that budget, which we declare. On the 154 problems of Alefeld, Potra and Shi it needs 2634 evaluations of $f$ against 2636 for TOMS 748, 2707 for Brent and 2595 for Chandrupatla's method; it wins more problems than it loses against Chandrupatla, 87 to 43, and needs 1.5% more in total because its losses are the larger ones. On the 45 wide-bracket problems of Chandrupatla it beats Brent and TOMS 748 and loses to ITP and to Chandrupatla (2357 against 1946 and 1505). For every data-derived scale the three similarity functions whose $1-S$ has a non-zero slope at the origin come before the three whose slope is zero, as the derived law predicts.
The wide-bracket gap turns out to be not the order of the model but a decision — when to give the model up — and the decision is a similarity after all, of an unfamiliar kind: the similarity of a run to its own past. A pure power is self-similar under zoom, so the dimensionless configuration of the last three points recurs, while at a simple root it tends to the linear prototype; the analogous pair yields the local order, $m=\ln L_f/\ln L_x$, and an exponential, self-similar by translation, is its limit $m=\infty$. Entering a halving regime when that analogy holds with $m\ge2$ — the condition under which a tangent step loses to a halving — splitting at the point the order predicts, and leaving when the split looks linear, together with four smaller readings of the same kind and one borrowed element, gives the method proposed here, A′. On the two sets it needs 2259 and 1074 evaluations against Chandrupatla's 2595 and 1505 and wins more problems than it loses on both, 103 to 31 and 31 to 10. The chain of decisions that produced it was selected on those sets; tested once, after its code was frozen, on 240 problems from six new families, it needs 3656 evaluations against Chandrupatla's 5717 and wins 177 problems to 28 — a corpus the author composed knowing where the method is strong: two of its six families have multiple or fractional roots, and without them the lead is 2403 against 3569, on the simple-root families alone 737 against 747. Over 855 problems in six corpora it is ahead of Brent, TOMS 748, ITP and the recent ModAB everywhere and stays within its own bound of twice bisection on every problem, a bound that Brent, TOMS 748 and ModAB, which do not promise it, exceed on 118, 84 and 9 problems; against Chandrupatla it is ahead in total on five corpora and level on the sixth. The lead is large on multiple, fractional and nearly flat roots and a matter of per cent on simple ones. Several of the resulting mechanisms coincide with classical ones — Lagouanelle's multiplicity estimate, false position on $|f|^{1/m}$ — reached here from similarity and analogy and recognised afterwards; what we found no counterpart for is the decision layer, where every choice — when to give the model up, where to split, when to return — is a judgment of similarity or of analogy, and the similarity that sets the step, whose form is decisive (replacing it costs 19% to 425%). The wide-bracket gap turns out to be not the order of the model but a decision — when to give the model up — and the decision is a similarity after all, of an unfamiliar kind: the similarity of a run to its own past. A pure power is self-similar under zoom, so the dimensionless configuration of the last three points recurs, while at a simple root it tends to the linear prototype; the analogous pair yields the local order, $m=\ln L_f/\ln L_x$, and an exponential, self-similar by translation, is its limit $m=\infty$. Entering a halving regime when that analogy holds with $m\ge2$ — the condition under which a tangent step loses to a halving — splitting at the point the order predicts, and leaving when the split looks linear, together with four smaller readings of the same kind and one borrowed element, gives the method proposed here, A′. On the two sets it needs 2259 and 1074 evaluations against Chandrupatla's 2595 and 1505 and wins more problems than it loses on both, 103 to 31 and 31 to 10. The chain of decisions that produced it was selected on those sets; tested once, after its code was frozen, on 240 problems from six new families, it needs 3656 evaluations against Chandrupatla's 5717 and wins 177 problems to 28 — a corpus the author composed knowing where the method is strong: two of its six families have multiple or fractional roots, and without them the lead is 2403 against 3569, on the simple-root families alone 737 against 747. Against simpler detectors put in the place of the analogy — Lagouanelle's multiplicity estimate with the same memory, Chandrupatla's acceptance band, or none — it wins on that corpus in total and problem by problem (3656 against 4488, 6236 and 6165); against the classical transformation $\operatorname{sign}(f)|f|^{1/m}$ with the multiplicity given in advance it loses every multiple or fractional root, by a factor of 2.4 to 6.8 — the price of estimating the order. Over 855 problems in six corpora it is ahead of Brent, TOMS 748, ITP and the recent ModAB everywhere and stays within its own bound of twice bisection on every problem, a bound that Brent, TOMS 748 and ModAB, which do not promise it, exceed on 118, 84 and 9 problems; against Chandrupatla it is ahead in total on five corpora and level on the sixth. The lead is large on multiple, fractional and nearly flat roots and a matter of per cent on simple ones. Several of the resulting mechanisms coincide with classical ones — Lagouanelle's multiplicity estimate, false position on $|f|^{1/m}$ — reached here from similarity and analogy and recognised afterwards; what we found no counterpart for is the decision layer, where every choice — when to give the model up, where to split, when to return — is a judgment of similarity or of analogy, and the similarity that sets the step, whose form is decisive (replacing it costs 19% to 425%).
--- ---
@@ -425,7 +425,7 @@ Halving is the answer when nothing is known about the shape; inside the regime t
$$t=\frac{|F_A|^{1/m}}{|F_A|^{1/m}+|F_B|^{1/m}},\tag{7}$$ $$t=\frac{|F_A|^{1/m}}{|F_A|^{1/m}+|F_B|^{1/m}},\tag{7}$$
which is exact for a pure power and becomes the midpoint as $m\to\infty$. It is false position applied to $\operatorname{sign}(f)|f|^{1/m}$ — the transformation that turns a root of multiplicity $m$ into a simple one, classical since Schröder [16] and Traub [17]; what is new here is only where $m$ comes from. The order read from a *failed promise*, $(1-1/m)^m=r$, was tried first and gave $m=\infty$ in 163 of 174 entries, because the base step is not a pure tangent; the order read from the *analogy* is the one that works. With it the flat families of the wide-bracket set cost 743–847 evaluations against 1044 for pure bisection and 1068 for Chandrupatla. which is exact for a pure power and becomes the midpoint as $m\to\infty$. It is false position applied to $\operatorname{sign}(f)|f|^{1/m}$ — the transformation that turns a root of multiplicity $m$ into a simple one, classical since Schröder [16] and Traub [17]; what is new here is only where $m$ comes from. With the multiplicity given in advance the transformation is far cheaper still (§9.1): the contribution is the estimate, and its price is measured there. The order read from a *failed promise*, $(1-1/m)^m=r$, was tried first and gave $m=\infty$ in 163 of 174 entries, because the base step is not a pure tangent; the order read from the *analogy* is the one that works. With it the flat families of the wide-bracket set cost 743–847 evaluations against 1044 for pure bisection and 1068 for Chandrupatla.
### 8.4 Five more readings ### 8.4 Five more readings
@@ -503,7 +503,7 @@ Several mechanisms of A′ coincide with classical ones, and the coincidence is
| the tolerance step | Dekker [9], Brent [2] — borrowed and named as such | | the tolerance step | Dekker [9], Brent [2] — borrowed and named as such |
| the exit judgment of §8.2 (the value at a split against the line) | ModAB's linearity check at the midpoint [10, 20], $|y_m-y_3|<k\,(|y_3|+|y_m|)$ with $k$ from a symmetry factor; ours compares two predictions, the line and 0, without a factor | | the exit judgment of §8.2 (the value at a split against the line) | ModAB's linearity check at the midpoint [10, 20], $|y_m-y_3|<k\,(|y_3|+|y_m|)$ with $k$ from a symmetry factor; ours compares two predictions, the line and 0, without a factor |
What we found no counterpart for, in the literature we could reach, is the part that carries the gain in Table 9: the *decision* layer. A bracketing method that recognises the regime in which its model has lost by the similarity of the run to its own past — the recurrence of a dimensionless configuration against the linear prototype — reads the order from the analogous pair itself, distinguishes similarity by scaling from similarity by translation, and leaves the regime by a similarity judgment at a point the model did not choose; and a step whose length is set by a similarity function whose form, measured, is decisive. The methods for multiple roots we found are single-point iterations without a bracket, with the multiplicity assumed, estimated from derivatives, or removed by a transformation ([17]; Roots.jl's `Order1B` and `Order2B` [12]); Chandrupatla takes the decision to halve by an empirical acceptance band [3]. What we found no counterpart for, in the literature we could reach, is the part that carries the gain in Table 9: the *decision* layer. A bracketing method that recognises the regime in which its model has lost by the similarity of the run to its own past — the recurrence of a dimensionless configuration against the linear prototype — reads the order from the analogous pair itself, distinguishes similarity by scaling from similarity by translation, and leaves the regime by a similarity judgment at a point the model did not choose; and a step whose length is set by a similarity function whose form, measured, is decisive. The methods for multiple roots we found are single-point iterations without a bracket, with the multiplicity assumed, estimated from derivatives, or removed by a transformation ([17]; Roots.jl's `Order1B` and `Order2B` [12]); Chandrupatla takes the decision to halve by an empirical acceptance band [3]. §9.1 puts this claim against the detectors a reader would try first, and against the transformation with the multiplicity known in advance.
## 9. Out-of-sample evidence ## 9. Out-of-sample evidence
@@ -560,6 +560,35 @@ Only the last line of corpora is evidence for A′ in the strict sense; the firs
**What the evidence supports, and what it does not.** Against Brent and TOMS 748 A′ is ahead on every corpus, by 8% to 67% in total, and it stays within its own bound of twice bisection on all 855 problems, a bound that Brent exceeds on 118 and TOMS 748 on 84 — neither promises it. Against ModAB it is ahead on every corpus, in total and problem by problem ($p\le0.0013$), on the one test of corpus 4 by 3656 to 5535 and 183 problems to 34 ($p=7\cdot10^{-26}$), and on ModAB's own benchmark as well — all 91 problems of it, 1507 against 1588, 56/13/22 ($p=1.5\cdot10^{-4}$); ModAB exceeds that bound, which it does not promise either, on 9 problems. Against Chandrupatla it is ahead in total on five corpora and level on the sixth, and ahead problem by problem on all six, significantly on five. The size of the lead is not uniform: it is large on multiple, fractional and nearly flat roots — the regime §8 recognises — and small on simple roots, where it is a matter of per cent. Where it is behind is named: `fun9` (65 against 55), the near-cancelling class of corpus 1 (491 against 391), random polynomials with a simple root in total (433 against 419, though ahead 20 to 12 in problems), and the smooth equations of the applied corpus, where the two are level. **What the evidence supports, and what it does not.** Against Brent and TOMS 748 A′ is ahead on every corpus, by 8% to 67% in total, and it stays within its own bound of twice bisection on all 855 problems, a bound that Brent exceeds on 118 and TOMS 748 on 84 — neither promises it. Against ModAB it is ahead on every corpus, in total and problem by problem ($p\le0.0013$), on the one test of corpus 4 by 3656 to 5535 and 183 problems to 34 ($p=7\cdot10^{-26}$), and on ModAB's own benchmark as well — all 91 problems of it, 1507 against 1588, 56/13/22 ($p=1.5\cdot10^{-4}$); ModAB exceeds that bound, which it does not promise either, on 9 problems. Against Chandrupatla it is ahead in total on five corpora and level on the sixth, and ahead problem by problem on all six, significantly on five. The size of the lead is not uniform: it is large on multiple, fractional and nearly flat roots — the regime §8 recognises — and small on simple roots, where it is a matter of per cent. Where it is behind is named: `fun9` (65 against 55), the near-cancelling class of corpus 1 (491 against 391), random polynomials with a simple root in total (433 against 419, though ahead 20 to 12 in problems), and the smooth equations of the applied corpus, where the two are level.
### 9.1 Controls: simpler detectors, and the multiplicity known in advance
Two controls ask whether the gain of §8 comes from the recognition by analogy or only from the split (7) with any reasonable detector. Both were pre-registered, with what the paper would say under every outcome, before the run (`sweep_controls.py`).
**Other detectors in A′.** A′ is run with only its detector swapped; the memory, the split, the exit and the five readings of §8.4 are unchanged. (a) *Lagouanelle's estimate* $m_L=1/(1-\theta)$, $\theta=ff''/f'^2$ from the parabola through the last three points ($m_L=\infty$ for $\theta\ge1$), with the same memory — once seen, held until the exit, a hysteresis — entering at $m_L\ge2$ and at $0<m_L\le\tfrac23$ and splitting at $m_L$. (b) *Chandrupatla's acceptance band*, entering when the band rejects the current triple and splitting at the current $m_L$. (c) *No detector*: the regime is entered at every unkept promise.
**Table 9b.** A′ against its detector swapped: totals on the two canonical sets, on corpus 4 and over the six corpora of §9 (uniform criterion), and A′'s wins/ties/losses on corpus 4 with the exact sign test. No variant breaks the bound.
| detector | APS | CHA | corpus 4 | six corpora | A′ on corpus 4 | $p$ |
|--------------------------|------|------|--------|---------|-----------|-------------|
| **analogy (A′)** | 2259 | **1074** | **3656** | **13032** | — | |
| Lagouanelle, with memory | **2235** | 1437 | 4488 | 15171 | 84/121/35 | $8\cdot10^{-6}$ |
| Chandrupatla's band | 2682 | 2526 | 6236 | 20668 | 110/96/34 | $1.5\cdot10^{-10}$ |
| none | 2693 | 2526 | 6165 | 20749 | 113/87/40 | $3\cdot10^{-9}$ |
On corpus 4, the one test, A′ beats all three in total and problem by problem, as the pre-registered criterion required. The competent alternative is Lagouanelle's estimate: it is slightly better on the canonical set (2235 against 2259), and on corpus 4 A′'s lead over it lies in the multiple-root and fractional-power families (1163 against 1739, 924 against 1207); it is behind on $\log(1+q^2)$ of a cubic (503 against 477) and level on the rest. The band and the regime without a detector are worse than no regime at all. The decision layer by analogy therefore earns its place against the detectors a reader would try first.
**The multiplicity known in advance.** The second control asks how good the classical transformation is when its multiplicity is given. Library Chandrupatla and Brent are run on $g=\operatorname{sign}(f)\,|f|^{1/m^*}$, with $m^*$ the order read at the known root — the slope of $\ln|f|$ against $\ln|x-r|$ on both sides at $h=10^{-5}$, rounded to 0.01, and 1 where either side shows no zero (a jump). It is not an achievable method but an upper bound. On the multiple and fractional roots, $m^*\ge2$ or $m^*\le\tfrac23$ — the branches of §8.1 where a tangent step loses to a halving:
**Table 9c.** Multiple and fractional roots with the multiplicity given: totals, failures charged. A′ wins none of the 148 problems.
| corpus | problems | A′ | Chandrupatla on $g$ | Brent on $g$ | Chandrupatla on $f$ |
|---|---|---|---|---|---|
| Chandrupatla | 20 | 654 | 96 | 66 | 1068 |
| new corpus 1 | 40 | 1229 | 358 | 337 | 1792 |
| corpus 4 | 88 | 2159 | 901 | 844 | 4151 |
With the multiplicity known the transformation is 2.4 to 6.8 times cheaper than A′. A′ is far ahead of Chandrupatla on $f$ itself, which does not know the multiplicity either. On the remaining problems the transformed and untransformed runs differ little and A′ is close to them: ahead on the canonical sets (2259 against 2640, 420 against 437 for Chandrupatla on $g$), behind on corpora 1 and 4 (2061 against 1924, 1497 against 1427). So on multiple roots the contribution of A′ is the estimate of the order without knowing it, and the gap to Table 9c is the price of that estimate — large, and the clearest direction for improving the method.
--- ---
## 10. Negative results ## 10. Negative results
@@ -604,15 +633,17 @@ A root finder can be derived from a similarity function alone, and the derivatio
The base rule has one declared numerical constant, converges with Muller's order 1.839 and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 in total and 1.5% behind Chandrupatla's method, while winning more problems than it loses against both; on the wide-bracket set Chandrupatla is genuinely stronger (1505 against 2357). §7.4 locates the cause: not the order of the model but the decision to give it up, which Chandrupatla takes by halving and which no reading of a single state takes correctly for our step. The base rule has one declared numerical constant, converges with Muller's order 1.839 and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 in total and 1.5% behind Chandrupatla's method, while winning more problems than it loses against both; on the wide-bracket set Chandrupatla is genuinely stronger (1505 against 2357). §7.4 locates the cause: not the order of the model but the decision to give it up, which Chandrupatla takes by halving and which no reading of a single state takes correctly for our step.
That decision is a similarity too — of the run to its own past. A pure power repeats its configuration under zoom, an exponential under translation, a simple root tends to the line; the analogous pair reads the local order, the order decides between the model and a halving by bisection's own $\tfrac12$, and it says where to split. The method built on it, A′, is ahead of Chandrupatla on both canonical sets in total and problem by problem, and on a corpus fixed after its code was frozen it needs 36% fewer evaluations and wins 177 problems to 28. The claim the measurements support is a lead over Brent, TOMS 748, ITP and ModAB everywhere, within a bound of twice bisection that Brent, TOMS 748 and ModAB do not promise and exceed, and over Chandrupatla a lead that is large where roots are multiple, fractional or nearly flat and small where they are simple — not superiority by the same margin everywhere. `fun9`, near-cancelling terms and smooth applied equations are where it is behind or level. That decision is a similarity too — of the run to its own past. A pure power repeats its configuration under zoom, an exponential under translation, a simple root tends to the line; the analogous pair reads the local order, the order decides between the model and a halving by bisection's own $\tfrac12$, and it says where to split. The method built on it, A′, is ahead of Chandrupatla on both canonical sets in total and problem by problem, and on a corpus fixed after its code was frozen it needs 36% fewer evaluations and wins 177 problems to 28. Its decision layer beats Lagouanelle's estimate, Chandrupatla's band and no detector put in its place; with the multiplicity known in advance the classical transformation is 2.4 to 6.8 times cheaper on multiple roots, which measures what estimating the order costs. The claim the measurements support is a lead over Brent, TOMS 748, ITP and ModAB everywhere, within a bound of twice bisection that Brent, TOMS 748 and ModAB do not promise and exceed, and over Chandrupatla a lead that is large where roots are multiple, fractional or nearly flat and small where they are simple — not superiority by the same margin everywhere. `fun9`, near-cancelling terms and smooth applied equations are where it is behind or level.
This is the same movement as in parameter-free similarity control [6], where membership functions are eliminated by normalising the input space from the data rather than by choosing widths. Here the object normalised is the value axis, and the price of choosing it by hand is measured rather than asserted. This is the same movement as in parameter-free similarity control [6], where membership functions are eliminated by normalising the input space from the data rather than by choosing widths. Here the object normalised is the value axis, and the price of choosing it by hand is measured rather than asserted.
### Future work ### Future work
Two questions remain, both narrower than the one this section used to pose. Three questions remain, all narrower than the one this section used to pose.
The first is the steep flank. Where the loss remains — `fun9`, the near-cancelling class, 4 − tan x at its pole — the base step approaches along a flank far steeper than its model, and the half contradiction of §8.4 removes only part of the cost. The second is the one geometric choice of §8.2, the Euclidean metric of the $(\xi,\varphi)$ plane: a metric derived from the configuration's own sensitivity would remove it. The first is the price of estimating the order. With the multiplicity given, the transformation of §8.3 is 2.4 to 6.8 times cheaper than A′ on multiple and fractional roots (Table 9c): the order read from the analogous pair, or its use in the split, loses most of what the transformation could deliver.
The second is the steep flank. Where the loss remains — `fun9`, the near-cancelling class, 4 − tan x at its pole — the base step approaches along a flank far steeper than its model, and the half contradiction of §8.4 removes only part of the cost. The third is the one geometric choice of §8.2, the Euclidean metric of the $(\xi,\varphi)$ plane: a metric derived from the configuration's own sensitivity would remove it.
--- ---
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@@ -150,20 +150,27 @@ evaluations against Chandrupatla's 5717 and wins 177 problems to 28 ---
a corpus the author composed knowing where the method is strong: two of a corpus the author composed knowing where the method is strong: two of
its six families have multiple or fractional roots, and without them the its six families have multiple or fractional roots, and without them the
lead is 2403 against 3569, on the simple-root families alone 737 against lead is 2403 against 3569, on the simple-root families alone 737 against
747. Over 855 problems in six corpora it is ahead of Brent, TOMS 748, 747. Against simpler detectors put in the place of the analogy ---
ITP and the recent ModAB everywhere and stays within its own bound of Lagouanelle's multiplicity estimate with the same memory, Chandrupatla's
twice bisection on every problem, a bound that Brent, TOMS 748 and acceptance band, or none --- it wins on that corpus in total and problem
ModAB, which do not promise it, exceed on 118, 84 and 9 problems; by problem (3656 against 4488, 6236 and 6165); against the classical
against Chandrupatla it is ahead in total on five corpora and level on transformation \(\operatorname{sign}(f)|f|^{1/m}\) with the multiplicity
the sixth. The lead is large on multiple, fractional and nearly flat given in advance it loses every multiple or fractional root, by a factor
roots and a matter of per cent on simple ones. Several of the resulting of 2.4 to 6.8 --- the price of estimating the order. Over 855 problems
mechanisms coincide with classical ones --- Lagouanelle's multiplicity in six corpora it is ahead of Brent, TOMS 748, ITP and the recent ModAB
estimate, false position on \(|f|^{1/m}\) --- reached here from everywhere and stays within its own bound of twice bisection on every
similarity and analogy and recognised afterwards; what we found no problem, a bound that Brent, TOMS 748 and ModAB, which do not promise
counterpart for is the decision layer, where every choice --- when to it, exceed on 118, 84 and 9 problems; against Chandrupatla it is ahead
give the model up, where to split, when to return --- is a judgment of in total on five corpora and level on the sixth. The lead is large on
similarity or of analogy, and the similarity that sets the step, whose multiple, fractional and nearly flat roots and a matter of per cent on
form is decisive (replacing it costs 19\% to 425\%). simple ones. Several of the resulting mechanisms coincide with classical
ones --- Lagouanelle's multiplicity estimate, false position on
\(|f|^{1/m}\) --- reached here from similarity and analogy and
recognised afterwards; what we found no counterpart for is the decision
layer, where every choice --- when to give the model up, where to split,
when to return --- is a judgment of similarity or of analogy, and the
similarity that sets the step, whose form is decisive (replacing it
costs 19\% to 425\%).
\end{abstract} \end{abstract}
\hypertarget{introduction}{% \hypertarget{introduction}{%
@@ -1488,12 +1495,14 @@ which is exact for a pure power and becomes the midpoint as
\(\operatorname{sign}(f)|f|^{1/m}\) --- the transformation that turns a \(\operatorname{sign}(f)|f|^{1/m}\) --- the transformation that turns a
root of multiplicity \(m\) into a simple one, classical since Schröder root of multiplicity \(m\) into a simple one, classical since Schröder
{[}16{]} and Traub {[}17{]}; what is new here is only where \(m\) comes {[}16{]} and Traub {[}17{]}; what is new here is only where \(m\) comes
from. The order read from a \emph{failed promise}, \((1-1/m)^m=r\), was from. With the multiplicity given in advance the transformation is far
tried first and gave \(m=\infty\) in 163 of 174 entries, because the cheaper still (§9.1): the contribution is the estimate, and its price is
base step is not a pure tangent; the order read from the \emph{analogy} measured there. The order read from a \emph{failed promise},
is the one that works. With it the flat families of the wide-bracket set \((1-1/m)^m=r\), was tried first and gave \(m=\infty\) in 163 of 174
cost 743--847 evaluations against 1044 for pure bisection and 1068 for entries, because the base step is not a pure tangent; the order read
Chandrupatla. from the \emph{analogy} is the one that works. With it the flat families
of the wide-bracket set cost 743--847 evaluations against 1044 for pure
bisection and 1068 for Chandrupatla.
\hypertarget{five-more-readings}{% \hypertarget{five-more-readings}{%
\subsubsection{8.4 Five more readings}\label{five-more-readings}} \subsubsection{8.4 Five more readings}\label{five-more-readings}}
@@ -1761,7 +1770,9 @@ decisive. The methods for multiple roots we found are single-point
iterations without a bracket, with the multiplicity assumed, estimated iterations without a bracket, with the multiplicity assumed, estimated
from derivatives, or removed by a transformation ({[}17{]}; Roots.jl's from derivatives, or removed by a transformation ({[}17{]}; Roots.jl's
\texttt{Order1B} and \texttt{Order2B} {[}12{]}); Chandrupatla takes the \texttt{Order1B} and \texttt{Order2B} {[}12{]}); Chandrupatla takes the
decision to halve by an empirical acceptance band {[}3{]}. decision to halve by an empirical acceptance band {[}3{]}. §9.1 puts
this claim against the detectors a reader would try first, and against
the transformation with the multiplicity known in advance.
\hypertarget{out-of-sample-evidence}{% \hypertarget{out-of-sample-evidence}{%
\subsection{9. Out-of-sample evidence}\label{out-of-sample-evidence}} \subsection{9. Out-of-sample evidence}\label{out-of-sample-evidence}}
@@ -1972,6 +1983,135 @@ with a simple root in total (433 against 419, though ahead 20 to 12 in
problems), and the smooth equations of the applied corpus, where the two problems), and the smooth equations of the applied corpus, where the two
are level. are level.
\hypertarget{controls-simpler-detectors-and-the-multiplicity-known-in-advance}{%
\subsubsection{9.1 Controls: simpler detectors, and the multiplicity
known in
advance}\label{controls-simpler-detectors-and-the-multiplicity-known-in-advance}}
Two controls ask whether the gain of §8 comes from the recognition by
analogy or only from the split (7) with any reasonable detector. Both
were pre-registered, with what the paper would say under every outcome,
before the run (\texttt{sweep\_controls.py}).
\textbf{Other detectors in A′.} A′ is run with only its detector
swapped; the memory, the split, the exit and the five readings of §8.4
are unchanged. (a) \emph{Lagouanelle's estimate} \(m_L=1/(1-\theta)\),
\(\theta=ff''/f'^2\) from the parabola through the last three points
(\(m_L=\infty\) for \(\theta\ge1\)), with the same memory --- once seen,
held until the exit, a hysteresis --- entering at \(m_L\ge2\) and at
\(0<m_L\le\tfrac23\) and splitting at \(m_L\). (b) \emph{Chandrupatla's
acceptance band}, entering when the band rejects the current triple and
splitting at the current \(m_L\). (c) \emph{No detector}: the regime is
entered at every unkept promise.
\textbf{Table 9b.} A′ against its detector swapped: totals on the two
canonical sets, on corpus 4 and over the six corpora of §9 (uniform
criterion), and A′'s wins/ties/losses on corpus 4 with the exact sign
test. No variant breaks the bound.
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.3291}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.0759}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.0759}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1013}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1139}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1392}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1646}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
detector
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
APS
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
CHA
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
corpus 4
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
six corpora
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
A′ on corpus 4
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(p\)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\textbf{analogy (A′)} & 2259 & \textbf{1074} & \textbf{3656} &
\textbf{13032} & --- & \\
Lagouanelle, with memory & \textbf{2235} & 1437 & 4488 & 15171 &
84/121/35 & \(8\cdot10^{-6}\) \\
Chandrupatla's band & 2682 & 2526 & 6236 & 20668 & 110/96/34 &
\(1.5\cdot10^{-10}\) \\
none & 2693 & 2526 & 6165 & 20749 & 113/87/40 & \(3\cdot10^{-9}\) \\
\end{longtable}
On corpus 4, the one test, A′ beats all three in total and problem by
problem, as the pre-registered criterion required. The competent
alternative is Lagouanelle's estimate: it is slightly better on the
canonical set (2235 against 2259), and on corpus 4 A′'s lead over it
lies in the multiple-root and fractional-power families (1163 against
1739, 924 against 1207); it is behind on \(\log(1+q^2)\) of a cubic (503
against 477) and level on the rest. The band and the regime without a
detector are worse than no regime at all. The decision layer by analogy
therefore earns its place against the detectors a reader would try
first.
\textbf{The multiplicity known in advance.} The second control asks how
good the classical transformation is when its multiplicity is given.
Library Chandrupatla and Brent are run on
\(g=\operatorname{sign}(f)\,|f|^{1/m^*}\), with \(m^*\) the order read
at the known root --- the slope of \(\ln|f|\) against \(\ln|x-r|\) on
both sides at \(h=10^{-5}\), rounded to 0.01, and 1 where either side
shows no zero (a jump). It is not an achievable method but an upper
bound. On the multiple and fractional roots, \(m^*\ge2\) or
\(m^*\le\tfrac23\) --- the branches of §8.1 where a tangent step loses
to a halving:
\textbf{Table 9c.} Multiple and fractional roots with the multiplicity
given: totals, failures charged. A′ wins none of the 148 problems.
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
corpus
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
problems
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
A′
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Chandrupatla on \(g\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Brent on \(g\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Chandrupatla on \(f\)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Chandrupatla & 20 & 654 & 96 & 66 & 1068 \\
new corpus 1 & 40 & 1229 & 358 & 337 & 1792 \\
corpus 4 & 88 & 2159 & 901 & 844 & 4151 \\
\end{longtable}
With the multiplicity known the transformation is 2.4 to 6.8 times
cheaper than A′. A′ is far ahead of Chandrupatla on \(f\) itself, which
does not know the multiplicity either. On the remaining problems the
transformed and untransformed runs differ little and A′ is close to
them: ahead on the canonical sets (2259 against 2640, 420 against 437
for Chandrupatla on \(g\)), behind on corpora 1 and 4 (2061 against
1924, 1497 against 1427). So on multiple roots the contribution of A′ is
the estimate of the order without knowing it, and the gap to Table 9c is
the price of that estimate --- large, and the clearest direction for
improving the method.
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\hypertarget{negative-results}{% \hypertarget{negative-results}{%
@@ -2183,11 +2323,15 @@ the local order, the order decides between the model and a halving by
bisection's own \(\tfrac12\), and it says where to split. The method bisection's own \(\tfrac12\), and it says where to split. The method
built on it, A′, is ahead of Chandrupatla on both canonical sets in built on it, A′, is ahead of Chandrupatla on both canonical sets in
total and problem by problem, and on a corpus fixed after its code was total and problem by problem, and on a corpus fixed after its code was
frozen it needs 36\% fewer evaluations and wins 177 problems to 28. The frozen it needs 36\% fewer evaluations and wins 177 problems to 28. Its
claim the measurements support is a lead over Brent, TOMS 748, ITP and decision layer beats Lagouanelle's estimate, Chandrupatla's band and no
ModAB everywhere, within a bound of twice bisection that Brent, TOMS 748 detector put in its place; with the multiplicity known in advance the
and ModAB do not promise and exceed, and over Chandrupatla a lead that classical transformation is 2.4 to 6.8 times cheaper on multiple roots,
is large where roots are multiple, fractional or nearly flat and small which measures what estimating the order costs. The claim the
measurements support is a lead over Brent, TOMS 748, ITP and ModAB
everywhere, within a bound of twice bisection that Brent, TOMS 748 and
ModAB do not promise and exceed, and over Chandrupatla a lead that is
large where roots are multiple, fractional or nearly flat and small
where they are simple --- not superiority by the same margin everywhere. where they are simple --- not superiority by the same margin everywhere.
\texttt{fun9}, near-cancelling terms and smooth applied equations are \texttt{fun9}, near-cancelling terms and smooth applied equations are
where it is behind or level. where it is behind or level.
@@ -2201,13 +2345,19 @@ hand is measured rather than asserted.
\hypertarget{future-work}{% \hypertarget{future-work}{%
\subsubsection{Future work}\label{future-work}} \subsubsection{Future work}\label{future-work}}
Two questions remain, both narrower than the one this section used to Three questions remain, all narrower than the one this section used to
pose. pose.
The first is the steep flank. Where the loss remains --- \texttt{fun9}, The first is the price of estimating the order. With the multiplicity
given, the transformation of §8.3 is 2.4 to 6.8 times cheaper than A′ on
multiple and fractional roots (Table 9c): the order read from the
analogous pair, or its use in the split, loses most of what the
transformation could deliver.
The second is the steep flank. Where the loss remains --- \texttt{fun9},
the near-cancelling class, 4 − tan x at its pole --- the base step the near-cancelling class, 4 − tan x at its pole --- the base step
approaches along a flank far steeper than its model, and the half approaches along a flank far steeper than its model, and the half
contradiction of §8.4 removes only part of the cost. The second is the contradiction of §8.4 removes only part of the cost. The third is the
one geometric choice of §8.2, the Euclidean metric of the one geometric choice of §8.2, the Euclidean metric of the
\((\xi,\varphi)\) plane: a metric derived from the configuration's own \((\xi,\varphi)\) plane: a metric derived from the configuration's own
sensitivity would remove it. sensitivity would remove it.
run_step2.sh
+15
View File
@@ -0,0 +1,15 @@
#!/bin/bash
# Revision step 2 close-out: verifier, .tex/PDF build, regression x2 - detached, with markers.
export OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1
export PYTHONPATH="$HOME/simbracket_build/site:."
cd "$HOME/sim-numerics" || exit 1
~/torchenv/bin/python -u verify_article.py > ~/verify_uniform.log 2>&1
echo "verify OK $(grep -c '^ OK' ~/verify_uniform.log) BAD $(grep -c '^ BAD' ~/verify_uniform.log)"
grep '^ BAD' ~/verify_uniform.log
grep -A8 '^Traceback' ~/verify_uniform.log
~/torchenv/bin/python -u build_tex.py 2>&1 | tail -3
for pass in 1 2; do
echo "=== PASS $pass ==="
~/torchenv/bin/python -m pytest tests -q -p no:cacheprovider -n 6 2>&1 | tail -1
done
echo "=== STEP2 DONE ==="
verify_article.py
+79
View File
@@ -221,6 +221,7 @@ def main():
verify_safeguard_and_tables() verify_safeguard_and_tables()
verify_flat_regime_and_oos() verify_flat_regime_and_oos()
verify_revision1() verify_revision1()
verify_controls()
print() print()
if FAILS: if FAILS:
@@ -1003,5 +1004,83 @@ def verify_revision1():
(5336, 4991)) (5336, 4991))
def verify_controls():
"""Section 9.1: the controls (PREREG_controls_2026-09-22, sweep_controls.json)."""
import math
print("\nSection 9.1 (controls, sweep_controls.json):")
R = _load("sweep_controls.json")
seen, four = _load("sweep_fourth_seen.json"), _load("sweep_fourth_fourth.json")
ap = dict(seen["Aprime"])
ap.update(four["Aprime"])
chf = dict(seen["chandrupatla"])
chf.update(four["chandrupatla"])
sets = ("APS", "CHA", "NEW", "RJL", "APP", "D")
def cost(v):
return v[0] if v[1] else 200
def tot(r, keep=None):
return sum(cost(v) for p, v in r.items() if keep is None or keep(p))
def h2h(a, b, keep=None):
ps = [p for p in a if keep is None or keep(p)]
w = sum(cost(a[p]) < cost(b[p]) for p in ps)
l = sum(cost(a[p]) > cost(b[p]) for p in ps)
return w, len(ps) - w - l, l
def sign_p(w, l):
k, n = min(w, l), w + l
return min(1.0, 2.0 * sum(math.comb(n, i) for i in range(k + 1)) / 2.0 ** n)
rows = {"K2L": (2235, 1437, 4026, 1495, 1490, 4488, 15171),
"K2P": (2682, 2526, 5485, 2117, 1622, 6236, 20668),
"K2F": (2693, 2526, 5569, 2126, 1670, 6165, 20749)}
check("Table 9b A' row", tuple(tot(ap[s]) for s in sets) + (sum(tot(ap[s]) for s in sets),),
(2259, 1074, 3290, 1371, 1382, 3656, 13032))
for k, want in rows.items():
k2 = R["K2"][k]
check(f"Table 9b {k} row", tuple(tot(k2[s]) for s in sets) + (sum(tot(k2[s]) for s in sets),),
want)
for k, wtl, pp in (("K2L", (84, 121, 35), ("1g", "8e-06")), ("K2P", (110, 96, 34), ("2g", "1.5e-10")),
("K2F", (113, 87, 40), ("1g", "3e-09"))):
got = h2h(ap["D"], R["K2"][k]["D"])
check(f"Table 9b A' vs {k} on corpus 4", got, wtl)
check(f"Table 9b p vs {k}", format(sign_p(got[0], got[2]), "." + pp[0]), pp[1])
check("Table 9b violations", sum(v[2] for k in rows for s in sets for v in R["K2"][k][s].values()), 0)
for cls, want in (("D2", (1163, 1739)), ("D3", (924, 1207)), ("D5", (503, 477))):
keep = lambda p, c=cls: p.startswith(c) # noqa: E731
check(f"9.1 {cls}: A' vs Lagouanelle", (tot(ap["D"], keep), tot(R["K2"]["K2L"]["D"], keep)), want)
level = [(tot(ap["D"], lambda p, c=c: p.startswith(c)),
tot(R["K2"]["K2L"]["D"], lambda p, c=c: p.startswith(c))) for c in ("D1", "D4", "D6")]
check("9.1 level on the rest (within 1.5%)", all(abs(a - b) <= 0.015 * b for a, b in level), True)
ab = _load("sweep_ablation_all.json")["abl-regime"]
no_regime_all = sum(tot(ab[s]) for s in sets)
check("band and no-detector worse than no regime (six corpora)",
all(sum(tot(R["K2"][k][s]) for s in sets) > no_regime_all for k in ("K2P", "K2F")), True)
check("band and no-detector worse than no regime (corpus 4)",
all(tot(R["K2"][k]["D"]) > tot(ab["D"]) for k in ("K2P", "K2F")), True)
t9c = {"CHA": (20, 654, 96, 66, 1068), "NEW": (40, 1229, 358, 337, 1792),
"D": (88, 2159, 901, 844, 4151)}
nonsimple = lambda m: m >= 2.0 or m <= 2.0 / 3.0 # noqa: E731
wins = 0
for s, want in t9c.items():
k1 = R["K1"][s]
keep = lambda p, _m=k1["mstar"]: nonsimple(_m[p]) # noqa: E731
got = (sum(1 for p in k1["mstar"] if keep(p)), tot(ap[s], keep), tot(k1["chandrupatla_g"], keep),
tot(k1["brent_g"], keep), tot(chf[s], keep))
check(f"Table 9c {s}", got, want)
wins += h2h({p: v for p, v in ap[s].items() if keep(p)}, k1["chandrupatla_g"])[0]
check("Table 9c: A' wins none of the 148", wins, 0)
check("Table 9c: 2.4 to 6.8 times", (round(2159 / 901, 1), round(654 / 96, 1),
min(round(a / c, 1) for _n, a, c, _b, _f in t9c.values()),
max(round(a / c, 1) for _n, a, c, _b, _f in t9c.values())),
(2.4, 6.8, 2.4, 6.8))
APS_ = "APS"
for s, want in ((APS_, (2259, 2640)), ("CHA", (420, 437)), ("NEW", (2061, 1924)), ("D", (1497, 1427))):
k1 = R["K1"][s]
keep = lambda p, _m=k1["mstar"]: not nonsimple(_m[p]) # noqa: E731
check(f"9.1 rest {s}: A' vs Chandrupatla on g", (tot(ap[s], keep), tot(k1["chandrupatla_g"], keep)),
want)
if __name__ == "__main__": if __name__ == "__main__":
main() main()