The new sections move into the canonical text, article/sim_numerics.md (Valeri's decision): section 7 'Where the similarity is read' with 7.1 per-place similarity, 7.2 the decision place, 7.3 the fun7 scoring artefact, 7.4 the gap decomposition; negative results 8-11 appended so items 1-7 keep their numbers; abstract paragraph; Future work rewritten (the refuted log transform removed); item 6 now says aps.13.00 is a scoring artefact, not a method failure. Translated into the .md's revised names: code 'ratio' = the tent max(0,1-|z|), code 'rational' = the ratio similarity 1/(1+|z|); the proposed method stays the .md's (tent, 3-point, projection).
Found on the way: negative item 'log change of variable' compared 3474 with 4516, the total of a DIFFERENT configuration; the honest pair is the same configuration in x, CHA 3643 -> 3474, APS 5822 -> 6940. Fixed in both texts. The gap-decomposition numbers were only traced to saved logs; they are now RECOMPUTED by verify_article.py from fresh runs of the diagnosis scripts (which return their values): 216 checks OK on Linux, 762 tests pass twice.
Open, for Valeri: the .md still says the projection's doubling is 'not a tuning constant', which simroot/pure.py (project_credit) refutes and which is why the .tex proposes credit; the .md's Tables 2-3 predate Chandrupatla's own method as a baseline (2785 APS, first place), so its 'parity' abstract is out of date; Brent 2890/4101 in the .md against 2889/4099 in the current sweep.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
@@ -10,6 +10,8 @@ Bisection places the next iterate at the midpoint because it uses no information
The derivation fixes the method completely, 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. What results is, explicitly, a Newton step with a three-point divided-difference derivative, clipped to the bracket and projected onto the bisection budget; the similarity function is scaffolding of the derivation and does not appear in the final rule. It carries a proven worst case of twice bisection plus one step and contains no constant calibrated on a problem or on the benchmark. On the 154 problems of Alefeld, Potra and Shi it needs 2806 evaluations of $f$ against 2820 for TOMS 748 — a margin of 0.5%, which is a tie — and 2890 for Brent, whom it beats on 102 problems to 40 head to head. On the 45 problems of Chandrupatla it beats Brent 30 to 4 and TOMS 748 31 to 8, and loses to ITP. The result is parity with methods built on tuned constants, obtained without any. For every data-derived scale, the ranking of the nine similarity functions matches the derived law without exception.
Reading the same similarity in one more place — which three points the local model is built on, selected by the shape of their configuration rather than by recency — lowers the Alefeld–Potra–Shi count to 2527, below every method measured here including Chandrupatla's 2785, with the guarantee intact and still no tuning constant; reading it instead on the choice of far end gives 5511 over both sets, and reading it in the decision to trust the model or halve lowers the canonical count further, to 2523. Where the similarity is read turns out to matter more than which similarity it is. The gap that remains on the wide-bracket set is taken apart on the same evaluations: a third of it is the order of the model — a convex two-point step cannot carry the negative weight of quadratic interpolation — and two thirds is not a model at all but the decision when to give the model up, which Chandrupatla takes by bisecting on up to all of his steps and which no reading measured here takes correctly for ours.
---
## 1. Introduction
@@ -100,7 +102,7 @@ $$x' \;=\; x + \operatorname{sign}(x_\mathrm{far}-x)\cdot\min\!\Bigl(\frac{|f(x)
a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. The similarity function does not appear in (4): it is scaffolding of the derivation, not machinery of the algorithm.
We state this ourselves rather than leave it to be discovered, because it locates the contribution precisely. It is not a new family of methods. It is (i) a derivation that fixes every choice — which similarity, which scale, which safeguard — before any benchmark is consulted, and then predicts the ranking of all nine similarity functions correctly; (ii) a safeguard of ITP's projection type with ITP's truncation stage, and hence all three of its constants, removed; (iii) the negative results of §7; (iv) the measurements.
We state this ourselves rather than leave it to be discovered, because it locates the contribution precisely. It is not a new family of methods. It is (i) a derivation that fixes every choice — which similarity, which scale, which safeguard — before any benchmark is consulted, and then predicts the ranking of all nine similarity functions correctly; (ii) a safeguard of ITP's projection type with ITP's truncation stage, and hence all three of its constants, removed; (iii) the negative results of §8; (iv) the measurements.
**Algorithm.** Given $[a,b]$ with $f(a)f(b)<0$ and tolerance $\varepsilon_x$:
@@ -251,7 +253,110 @@ The mechanism is step *quality*. Reconstructing each method's bracket from the s
---
## 7. Negative results
## 7. Where the similarity is read
Everything above reads the similarity in one place: the *length* of the step, $(1-S(z))$ times the distance to the far end. But a bracketing iteration takes several decisions per step, and the similarity was not consulted in any of the others: which end counts as far (decided by distance), which points the local slope is estimated from (decided by recency), and whether the step is worth taking at all (decided after the fact, by the safeguard). Each of those is a place where a similarity can be read, and the placement turns out to matter more than the form of $S$.
Three readings were added to the proposed rule of §3 (tent similarity, three-point scale), each parameter-free and each measured on the protocol of §4; every number below is re-checked against its own sweep file by `verify_article.py`, and every configuration in Table 5 keeps the $2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ guarantee (0 violations) and solves 193 of the 199 problems.
Where a safeguard other than the projection of §3.1 appears, it is one of the grid's. *safe* is the projection preceded by a step-length test — a step not at most half the step taken two evaluations earlier is replaced by a halving — with the projection still carrying the bound. *credit* replaces the scheduled allowance of (3) by one earned step by step: a surplus is banked whenever a step shrinks the bracket by more than half and drawn on when it shrinks it by less, which bounds the run by bisection plus one step. *bracket* and *both* are the replacing safeguards of §3.1, which keep a bound at the price of the order; *step* alone and *none* carry no bound.
1.**Trust before the step.** The step trusts a local slope, so the question of trust is whether that slope has settled: if two consecutive estimates have opposite signs the local model is not established, and the evaluation is spent halving instead. A sign is compared, not a magnitude.
2.**Which points the model uses.** For a linear $f$ the ratio of positions $\xi=(x-u)/(v-u)$ and the ratio of values $\varphi=(f(x)-f(u))/(f(v)-f(u))$ coincide, so $|\varphi-\xi|$ measures how unlike a linear configuration a triple is. Among the declared candidate triples the one minimising it is *selected* — an argmin over what is already in hand, with no threshold.
3.**Which end is far.** Read by dissimilarity instead of distance: the end whose $|f|$ is larger is the one less similar to the target.
**Table 5.** Reading the similarity in one more place is worth more than changing $S$. Evaluations of $f$; failures charged the full budget.
| configuration (all with the guarantee, 193/199) | APS-154 | CHA-45 | both |
| (1)+(3) far end by dissimilarity, projection | 2726 | **2785** | **5511** |
| (1)+(2)+(3), *safe* | 2884 | 2859 | 5743 |
| Chandrupatla | 2785 | 2423 | 5208 |
| TOMS 748 | 2820 | 4000 | 6820 |
| ITP, per-problem oracle | 3260 | 2611 | 5871 |
Two results here are worth stating plainly. First, the canonical set improves from 2806 to **2527**, which is below every method measured, including Chandrupatla's 2785, and it is obtained with the worst-case guarantee intact and still without a tuning constant. Second, on the two sets together the 5511 of reading (3) passes ITP's oracle (5871) but not Chandrupatla (5208): the remaining gap is entirely on the wide-bracket families, where Chandrupatla needs 2423 against our 2683 (reading (3) under *safe*).
The readings do not compose. Taken together, (2) and (3) give 2884 and 2859 — worse than either alone. Two selections that are each an improvement interfere, which is a reason to measure composition rather than assume it.
### 7.1 The similarity function belongs to the place, not to the method
Everything above varies *where* the similarity is read while keeping one function $S$ for the whole method. That is itself an assumption, and it is measurable. Proposition 1 says that the ratio similarity scaled by the far end is false position; what it does not say is *why* false position stalls. Write $|f|\sim K e^m$ near the current point, $e=|x-r|$. Then the residual ratio against the far end needs no slope at all,
and under the ratio similarity $S=1/(1+|z|)$ the step fraction is $z/(1+z)$, which is exactly inverse linear interpolation. False position therefore reads a dimensionless ratio *at order one* on an approach of order $m$: its three-century-old stall is a mismatch between the similarity function and the place it is read in. The exponent $1/m$ is not a constant — $m=1/(1-ff''/f'^2)$ comes from three evaluations already paid for, as in item 5 of §8, and is 1 whenever the power model does not apply.
Measured over the 252 cells of the grid, the correction behaves exactly as that reading predicts and not at all as a universal improvement. Paired inside each (function, safeguard) cell of the far-end reading it is cheaper in 29 of 60 cells — indistinguishable from a coin. But in the one pairing where that reading *is* exact inverse interpolation, the ratio similarity, it is cheaper under all seven safeguards:
**Table 6.** The measured order in the step place, on the far-end reading with the ratio similarity $1/(1+|z|)$. Cheaper in 7 of 7 safeguards; in 29 of 60 cells over all nine functions.
Two limits must be stated with it. The estimator is *biased low*: on a pure cube, with no noise at all, it returns 2.830 rather than 3, because a parabola through three points carries only the second difference. The bias is towards $m=1$, that is towards the uncorrected reading, which is the harmless direction, but it is a bias. And the correction does not rescue the far-end reading as a method: the best cell it reaches on the wide-bracket set is 2884, against 2683 for reading (3) above and 2423 for Chandrupatla. In the *slope* reading of §3 the same exponent is harmful, $2806\to2854$, which is the same statement read from the other side — a function that does not belong to a place does not become right by being applied there.
### 7.2 The decision place
Three places have now been measured: the step length, the far end, and the points the model is built on. The fourth is the *decision* — whether the local model deserves this evaluation or halving is the better use of it. Chandrupatla's method is strong exactly there: it refuses its model on most steps by the acceptance band $1-\sqrt{1-\xi}<\varphi<\sqrt{\xi}$, with $\xi$ and $\varphi$ the ratios of positions and of values defined above. The rule of §3 has no similarity in this place at all: a sign gate and a projection that repairs *after* the step.
Three readings of that place were measured, each on quantities already in hand and each without a constant. Two are ours: refuse when the configuration is less like a linear one than the iterate is off-centre, $|\varphi-\xi|>\min(\xi,1-\xi)$ ("defect"); and refuse when the configuration's similarity to linear falls below the iterate's similarity to the root, $S(|\varphi-\xi|)<S(z)$, with the *same* $S$ in both places ("simpair"). The third is *Chandrupatla's own band*, placed inside our step rule. It is not a proposal; it is an instrument, and it answers whether the remaining gap lives in this place.
In total, no. With Chandrupatla's band in our step, the wide-bracket set costs 3066 under the projection and 2899 under *credit*, against his 2423, while the canonical set rises from 2806 to 5432. The band is not idle — it refuses 63.6% of steps — so this is a decision that was taken and did not pay *in total*. Per family it did pay, and where: it closes almost the whole gap on the flat families and gives back more than that on the families where our step is good. §7.4 takes this apart; the short form is that the decision place does hold most of the gap, but no reading measured so far makes that decision correctly for *our* step.
Our own readings do not close it either (best 2868 on the wide-bracket set), but "defect" sets a new best on the canonical set: **2523** at 153/154 under the projection, below the 2527 of reading (2) and below every method measured, with the guarantee intact and no constant. It refuses only 7.0% of steps, the fewest of the three. Over both sets it costs 5733 against the 5511 of reading (3), because it gives back on the wide brackets (3210) what it gains on the canonical problems — so it is a second point on the same trade, not a replacement.
### 7.3 Six problems of the two sets are scored unsolved for every method
Chasing the remaining gap to Chandrupatla led to a property of the benchmarks themselves. On the five problems of the wide-bracket family `fun7`, and on the single canonical problem `aps.13.00`, the declared root is 0 while $f$ vanishes *identically* on a neighbourhood of it in floating point: $f(0\pm t)=0$ for every $t$ up to at least $3.2\times10^{-2}$, because the exponential factor underflows — at the returned points $|f(x)|$ is below the smallest normal double. Every method therefore stops after 8 to 28 evaluations at a point where $f(x)=0$ *exactly* — a genuine root — and every method is then scored as having failed, because the harness asks for $|x-\text{root}|\le10^{-6}$ while the returned point sits up to $2.8\times10^{-2}$ away.
The consequence is arithmetic. Each such problem charges the full budget of 200 evaluations to every method: about 1150 evaluations of every reported two-set total — ours, Chandrupatla's, TOMS 748's, ITP's, Brent's, bisection's — are this artefact, and the "153 of 154" and "40 of 45" that every method reports are statements about the scoring rather than about the methods. Re-scored with the criterion "a root is a root" ($|x-\text{root}|\le10^{-6}$ *or* $f(x)=0$), the same runs solve 199 of 199 for every method except Brent's 197, and the totals fall as follows.
**Table 7.** The same runs under the harness criterion and under "a root is a root". The ranking is unchanged; the magnitudes are not, and neither is the size of the remaining gap on the wide-bracket set: 253 evaluations, or 17%, not 11%.
| method | APS as scored | APS re-scored | CHA as scored | CHA re-scored |
The remaining gap looks diffuse — per family +57, +54, +46, +43, +35 and smaller — but part of it has one address. The `fun1` family is the same function with brackets of growing width, and our cost against Chandrupatla's across them is 7/8, 12/12, 19/15, 59/24, 63/44: we tie or win while the bracket is narrow and lose progressively as it widens. The endgame is not the cause — our observed order on those very problems is the higher one (3.63 against 2.02 on the $2\times10^4$-wide bracket). The cause is the rate of progress far from the root: 59 evaluations on a $2\times10^4$-wide bracket is what bisection itself needs ($\lceil\log_2(2\times10^{16})\rceil\approx54$), i.e. there we advance by about one halving per evaluation while Chandrupatla advances by about 2.3.
It is not the guarantee that caps us. With the safeguard removed entirely the same problems cost 46, 57, 61 and 68 against bisection's 57, 77, 45 and 45 — better on the wide brackets, *worse than bisection* on `fun3.1` and `fun5.1` — and still far from Chandrupatla's 24 and 44.
Nor is it that the step fails to be relative to the bracket. With the tent similarity $1-S=\min(1,|f|/(|\hat f'|D))$, so by (4) the step is exactly $\min(D,|f|/|\hat f'|)$: a *fraction* of the distance to the far end, and the similarity is that fraction. Nor does the iterate creep along one side without moving the proven bracket: measured probe by probe, 91.5% of our evaluations on the $2\times10^4$-wide bracket do shrink the bracket, slightly *more* often than Chandrupatla's 87.5%.
What differs on `fun1` is how much each probe removes. Ours shrinks the bracket by 0.93 halvings per evaluation there, Chandrupatla's by 2.26. On a wide bracket of a convex function the Newton step systematically undershoots, because a linear model of the value axis cannot see the curvature that separates the iterate from the root; a model that carries the curvature — inverse quadratic interpolation — lands far closer.
That mechanism is right for `fun1` and *wrong as a statement about the whole gap*. Replaying our own trajectories on the 33 problems where we lose, and placing both models on the *same* evaluations — one extra evaluation each, from the same state — splits the 262 evaluations of the gap in two. On `fun1` (58 of them) inverse quadratic interpolation from our own three points removes far more of the bracket than our step did: 10.37 halvings against 0.62 on the $2\times10^4$-wide bracket. There the gap is the order of the model, and the reason is structural: our step $x+(1-S)(x_\mathrm{far}-x)=S\,x+(1-S)\,x_\mathrm{far}$ is a convex combination of two points, while the quadratic interpolant's Lagrange weights at 0 always include exactly one *negative* weight when the triple brackets the root (verified on all 428 non-degenerate states; in the other five the negative weight underflowed to $-0$). A similarity in $[0,1]$ cannot produce a negative weight — Proposition 1 extended from two points to three.
But 178 of the 262 — two thirds — sit on `fun3`–`fun6`, which are pure powers of order 3, 5, 9 and 19, and there *no* interpolating model works: our step removes 0.41–0.81 halvings of the bracket per evaluation, inverse quadratic interpolation from the same points only 0.01–0.17. Chandrupatla does not win those families with his model — his traced run bisects 93, 96, 100 and 100% of his steps. We lose them because our step is slower than bisection there and our guarantee, at most twice bisection, licenses exactly that. His band inside our step confirms it family by family: under the projection it takes `fun4`, `fun5`, `fun6` from 573, 456, 456 to 296, 239, 235 — against his own 293, 228, 228 — and costs +164, +111, +170, +144 on `fun1`, `fun2`, `fun8`, `fun9`, where it refuses steps that were good.
So most of what remains is not the order of the model but the decision *when to give the model up*, and it is open because every detector measured so far confounds "the model is useless here" with something else: the band with "the configuration is not shaped for inverse quadratic interpolation", bracket progress with "the step has not moved the bracket yet" (4449 on the canonical set, §8), and the local order with nothing at all — $m\ge2$ is exactly the condition under which a tangent step loses to a halving, since it lands at $e(1-1/m)$, and it is clean (0.2% of canonical states) but blind: it reports $m\ge2$ on only 5.8% of states of families whose order is by construction 3 to 19. For the `fun1` part, three parameter-free ways round the model order have been measured and all three fail in the same way, buying the wide brackets at the price of the canonical set: a fixed fraction of the bracket (the cube-root similarity), a logarithmic change of variable, and the curvature-corrected scale of §8 item 5. The fourth way is to model the curvature directly, which is inverse quadratic interpolation — a different method rather than another reading of this one.
On the re-scored wide-bracket set of Table 7 our shape reading (2159) is *worse* than bisection (2136); only the dissimilarity reading (1740) is clearly ahead, which is the same statement seen from the other end: where the step cannot march, the reading that helps least is the one that only improves the local model.
Every statement in §§7.3–7.4 is reproduced by `diag_rescore_cha.py`, `diag_rescore_aps.py`, `diag_fun7_root.py`, `diag_cha_gap.py`, `diag_model_order.py`, `diag_model_order_checks.py`, `diag_flat_detectable.py` and `diag_band_by_family.py`, whose outputs are kept beside them.
---
## 8. Negative results
1.**The prototype-aggregation form of the method loses.** A richer construction — prototypes over *pairs* of evaluated points, softmax weights over the whole memory, similarity on both the value and the abscissa axis — leads on eight hand-picked problems but loses decisively on the real set: 0 wins, 6 ties and 147 losses against Brent. The simplest form of the idea won.
@@ -263,13 +368,21 @@ The mechanism is step *quality*. Reconstructing each method's bracket from the s
5.**The phase detector: derivation sound, estimator not.** Redoing the contraction law without the assumption $|f|\approx|f'(r)|e$ — taking $|f|\approx K e^m$ instead — does produce the march exponent on its own, $p = L/(L-\ln m)$ with $L=\ln(e/D)$, giving $p=1$ at a simple root and dropping to 0.50 and 0.23 as the approach flattens, which is the neighbourhood of the cube root that wins those families. Substituted back it collapses to the multiplier $m$, measurable as $m = 1/(1-\theta)$ with $\theta = f f''/(f')^2$ from three divided differences and no constant. It preserves the order (2.001) and improves the eight standard problems (82 against 84) — and loses on the full benchmark, 6304 against 6026. The estimator is the failure: 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 $m=34.8$, against 35.8% on the flat families. 27.9 against 35.8 is no separation. A cruder variant, correcting by the fraction of its own promise the previous step kept, fails everything: the order collapses to 0.904 and the standard problems double to 161.
6.**One problem defeats every method**, `aps.13.00`, $f(x)=x e^{-1/x^2}$, all of whose derivatives vanish at the root and which underflows to exactly zero throughout $|x|<0.086$; every method returns a machine zero at distance up to 0.086 from the root.
6.**One problem is scored unsolved for every method**, `aps.13.00`, $f(x)=x e^{-1/x^2}$, all of whose derivatives vanish at the root and which underflows to exactly zero throughout $|x|<0.086$. Every method returns a point where $f(x)=0$ exactly — a genuine root — which the harness scores as a failure because it lies more than $10^{-6}$ from the declared root. It is a property of the scoring, not a failure of the methods; §7.3 finds five more such problems in the wide-bracket set.
7.**The symmetric ratio similarity degenerates.** For $S(x,y)=\min(|x|,|y|)/\max(|x|,|y|)$ with $y$ the target 0, $S\equiv0$; the only usable reading is $1-|f|/\sigma$, which saturates and stagnates (Table 1, last row).
8.**Placement beats form, and the wrong placement is expensive.** The same "refuse before the step" idea as reading (1) of §7, with refusal decided by *promised bracket progress* instead of by the shape of the model, costs 4449 on the canonical set against 2527 — a difference of 1922 evaluations for one choice of criterion. A secant step routinely collapses the error while barely moving the proven bracket, so bracket progress rejects steps that were excellent; this is the same mechanism that makes the bracket guard of §3.1 cost 7260.
9.**Making the scale independent of the bracket does not help either.** The natural objection to the derived scale is that it defeats the very principle of a similarity step: with $\sigma=|\hat f'|D$ the normalised residual is $|f|/(|\hat f'|D)$, which on the $2\times10^4$-wide bracket of $x^3-2x-5$ at $x=10^4$ equals 0.167, so $S=0.833$ — "similar" — and the step is small although the root is some five thousand step-lengths away. The similarity is reading what fraction of the *bracket* the Newton step is, not how far the iterate is from the root. Selecting the scale between the two quantities already computed, $\sigma=\min(|\hat f'|D,\max(|f(A)|,|f(B)|))$ — no new constant, a minimum over what is in hand — does give a full-bracket step far away and Newton's step near the root, and on eight hand-picked wide problems it improves 403 evaluations to 390. Over the whole grid it is worse on both sets: 3288 on the canonical set against 2726, and 3111 on the wide-bracket set against 2683. Its apparent extra solves (194 instead of 193) are two `fun7` problems where it lands inside the flat zero region close enough to satisfy the harness — the artefact of §7.3, not a capability.
10.**Reading the similarity at the measured order does not cure false position.** Since false position's stall is a similarity read at order one (§7.1), the obvious hope is that reading it at the measured order is the parameter-free version of the classical fix. It is not. Unguarded, the correction takes the far-end reading from 127/154 and 10676 to 128/154 and 10043 — one problem and six per cent — where Illinois, at the price of one constant (halve the stale ordinate), reaches 153/154 and 7045. A sixth of the gap is not the gap. The correction is real and it is in the right place; it is simply much smaller than a constant that was chosen for exactly this failure. *Our* 10676 and the 10560 of Table 2 are not the same algorithm — our iterate starts at the midpoint of the bracket, plain false position at an end — and the two must not be differenced.
11.**A logarithmic change of variable does not buy the wide brackets.** Working in $u=\operatorname{sign}(x-c)\log(1+|x-c|/\varepsilon_x)$ makes the step multiplicative, which looked like the right diagnosis of the flat families — for the same similarity, scale and safeguard ($(1+|z|)^{-2}$, two-point, *safe*) the wide-bracket set improves from 3643 to 3474 — but the canonical set gets worse, 5822 to 6940, against the 2806 of the proposed rule. An earlier version compared the 3474 with 4516, which is the total of a different configuration. A march needs a *ratio* of distances while an identity on narrow brackets needs an *absolute* unit, and no parameter-free unit provides both. §7.4 shows the diagnosis itself was wrong: on the flat families the right move is to halve, not to march.
---
## 8. Conclusion
## 9. Conclusion
A root finder can be derived from a similarity function alone, and the derivation shows where the content of such a method actually sits. It is not in the shape of $S$ — nine shapes were tried and their ranking is dictated by a single number, the slope of $1-S$ at the origin. It is not in making the similarity change faster — that overshoots exactly as readily as flatness stagnates. It is in the *scale* against which similarity is judged: chosen, the method is brittle and merely linear; taken from the bracket, it is false position and three centuries old; read off the data, it is superlinear. And the safeguard that makes such a method safe must project the step, not replace it: replacing costs the order, projecting costs nothing on one benchmark and improves the method on the other.
@@ -279,9 +392,10 @@ This is the same movement as in parameter-free similarity control [6], where mem
### Future work
One weakness has a clear diagnosis and no fix inside this framework. On brackets many orders of magnitude wider than the distance to the root, the search must cross those decades, and an *additive* step of $|f|/|f'|$ is sized by the local slope, which knows nothing about how many decades remain. That is why the cube-root similarity, whose step is a fixed *fraction* of the bracket, beats the proposed rule 2.6-fold on those families, and why ITP keeps its lead on the Chandrupatla set.
The diagnosis of §7.4 leaves one question, and it is narrower than the one this section used to pose. An earlier version proposed a logarithmic change of variable to turn an additive step into a geometric march; that was measured and refuted (§8, item 11), and the step turned out to be a fraction of the bracket already.
What is open is a decision, not a model. On two thirds of the remaining gap the right move is to halve, and Chandrupatla halves; our guarantee permits a step that is slower than halving, and nothing we have measured tells our step when it is. The condition itself is known exactly — a tangent step on an approach of local order $m$ lands at $e(1-1/m)$ and loses to a halving when $m\ge2$ — so the constant is bisection's $\tfrac12$ and not a choice. What is missing is a reading of $m$ that sees it: the three-point estimator reports $m\ge2$ on 0.2% of canonical states, which is the right silence, and on 5.8% of states of pure powers of order 3 to 19, which is blindness. Two successive residuals of the run itself carry the order more directly than a parabola through three points — on an order-$m$ approach under tangent steps their ratio settles at $(1-1/m)^m$, while at a simple root it goes to zero — and that is the reading we would test next, under the same rule as every other here: pre-registered, with the canonical set as the price it must not charge.
The regime is therefore multiplicative versus additive movement, not one similarity function versus another — a change of the space the method works in, not of a coefficient inside it. The natural instrument is a logarithmic change of variable, $u = \operatorname{sign}(x-c)\log(1+|x-c|/s)$, in which an additive step becomes a geometric march and the exponent that the cube root fixes at $1/3$ by accident disappears as a free quantity. Two conditions decide whether that is a method or another refuted idea: the origin $c$ and unit $s$ must be read from the bracket rather than chosen, and the transform must be the identity to within rounding on narrow brackets, so that the wide-bracket families are not bought at the price of the canonical set. Every attempt reported in §7 failed on exactly that second condition.
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Claude Opus 5