sim-numerics/NEXT_PAPER_logarithmic.md

Next paper: the logarithmic change of variable for wide brackets

Not part of the current paper. Recorded here so the reasoning that leads to it is not reconstructed from scratch.

Why this is a separate paper and not a fix

The current paper's method is additive: every step moves the iterate by an absolute amount, |f|/|f'|. On the Chandrupatla families fun3.*, fun4.*, fun5.* the bracket is up to 10^12 wide while the root sits at |x| ≤ 3, so the search must cross twelve orders of magnitude. An additive step cannot do that efficiently: it is sized by the local slope, which says nothing about how many decades remain.

That is exactly what stage 7 measured. The cube-root similarity beats the proposed rule on all five members of fun4 — 32/32/40/47/59 against 89/93/113/125/153, a factor of 2.6 — and it does so because its step is (|f|/σ)^(1/3)·D rather than (|f|/σ)·D, i.e. a fixed fraction of the bracket rather than an absolute distance. It marches geometrically.

So the regime is not "a different similarity function". It is multiplicative versus additive movement, and that is a change of the space the method works in, not of a coefficient inside it.

What was already tried and must not be repeated

All of this is measured, in the repository, and pinned by tests.

1. Use cbrt where it wins. A switch chosen per benchmark set is worse than not switching: 6905 evaluations against 6026. cbrt wins the fun families and loses the rest of Chandrupatla (4099 against 3220 over that set). The regime is narrower than the set.
2. Detect the regime from the value ratio. rho = 1 − |f_n|/|f_{n−1}| destroys the order (2.000 → 0.904) and doubles the cost on the standard problems (84 → 161). Near the root both values are negligible and their ratio is noise.
3. Detect it from the local order of f. Dropping the linearity assumption from the contraction law does produce the march exponent, p = L/(L − ln m) with L = ln(e/D), and it collapses to the multiplier m = 1/(1 − θ), θ = f·f''/(f')². The derivation is sound and the order survives (2.001), but the estimator is not sharp enough: on Alefeld–Potra–Shi, where almost every root is simple and the truth is m = 1, it reports m > 1 on 27.9% of steps with excursions to m = 34.8; on the flat families it fires on 35.8%. 27.9 against 35.8 is not a separation, and the false positives cost 273 evaluations.
4. Separate the regimes by anything the method can see before it commits. Bracket width does not do it: the winners overlap completely, the proposed scale winning on widths from 10^0 to 10^12.

The proposal

Work in u = sign(x − c)·log(1 + |x − c|/s) rather than in x, with the origin c and the unit s read from the bracket rather than chosen. An additive step in u is a multiplicative step in x, so the same similarity rule, the same derived scale and the same projection safeguard become a geometric march without any new coefficient — and the march exponent, which cbrt fixed at 1/3 by accident, disappears as a free quantity.

Two things must be established before this is a paper:

• The transform must be parameter-free. c and s have to come from the bracket ends and the sign change, not from a choice. If they cannot, the line closes the same way the phase detector did.
• It must not cost anything on ordinary problems. The transform has to be the identity, to within rounding, when the bracket is narrow — otherwise it buys the wide-bracket families at the price of the canonical set, which is exactly the trade every attempt so far has failed.

What would settle it

The same protocol as the current paper: the 154 Alefeld–Potra–Shi problems and the 45 Chandrupatla problems, evaluations of f as the cost, failures charged the full budget, the worst-case bound checked on every solved run, and the configuration fixed by the derivation before the ranking is consulted.

Target to beat: 6026 evaluations over the two sets, 193 of 199 solved, with the 2⌈log₂(w/xtol)⌉ + 3 guarantee intact. On the Chandrupatla set alone the target is ITP's 2857, which the current method does not reach (3220).