Article (canonical .md), step 2: abstract, introduction, a note in section 3, the close of 7.4, negative results 12-16, conclusion, future work and references 8-12 rewritten for the proposed method A'; two statements that could not be recomputed rephrased. verify_article: expected values updated to the regenerated sweeps, the gate cells renamed (progress.*; their CHA values had moved unnoticed, 2991 -> 3047, 2851 -> 2888, not quoted in the paper), and verify_flat_regime_and_oos() added - Sections 8-9, Tables 8-11, items 12-16 and the abstract, recomputed from server files and fresh diagnosis runs. Verifier on the server: 401 OK, 0 BAD, 'every number stated in the paper matches the sweep data'. Server regression x2: 1226 passed, 0 failed.

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

article/sim_numerics.md

@@ -8,9 +8,9 @@ V. Sitnikov
 
 Bisection places the next iterate at the midpoint because it uses no information about the function's value. We replace that rule with a *similarity* rule: the step is proportional to how *dissimilar* the current outcome $f(x)$ is to the expectation $0$, measured by a similarity function $S:\mathbb{R}\to[0,1]$ with $S(0)=1$. We derive the contraction factor of the resulting iteration in closed form; it predicts a linear rate to four digits when $1-S$ has a non-zero slope at the origin, and a sublinear $e_n\sim C/n$ with a predicted $C$ when it does not, both confirmed. Three consequences follow. First, with the *ratio* similarity scaled by the far end — and with that combination only, of the five tested — the rule reproduces the false-position formula identically; so does the truncated linear similarity scaled by a divided difference over the *bracket ends*, which shows that the boundary of novelty runs through which points the difference is taken over. Second, any similarity fixed in advance with the scale also fixed yields at best linear convergence at a rate that is a property of the problem rather than of the method; on the eight standard problems of §2.2, five to eight of them fail. Third, the rate vanishes exactly when the scale equals the local scale of the value axis; reading that scale off the data — a divided difference, not a fitted coefficient — turns the step into $|f|/|f'|$ and the convergence superlinear.
 
-The derivation fixes the method 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 exactly one free constant, which we declare rather than argue away: the multiplier of that budget. On the canonical set it does not matter — the total is the same for every multiplier from 1.25 to 4 — and on the wide-bracket set it does, moving the total from 2752 to 4006. On the 154 problems of Alefeld, Potra and Shi the rule needs 2806 evaluations of $f$ against 2820 for TOMS 748 — a margin of 0.5%, which is a tie — 2889 for Brent, and 2787 for Chandrupatla's method, 0.7% fewer in total; problem by problem it beats Chandrupatla 87 to 42 (exact sign test $p=9\cdot10^{-5}$) and Brent 102 to 40. On the 45 problems of Chandrupatla it beats Brent 30 to 4 and TOMS 748 31 to 8, and loses to ITP and to Chandrupatla (3220 against 2857 and 2441). A variant whose budget is earned step by step has no free constant and a tighter guarantee, bisection plus one step, at a price of 762 evaluations on the canonical set. For every data-derived scale, the ranking of the nine similarity functions matches the derived law without exception.
+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 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 exactly one free constant, which we declare rather than argue away: the multiplier of that budget. On the 154 problems of Alefeld, Potra and Shi it needs 2806 evaluations of $f$ against 2820 for TOMS 748, 2889 for Brent and 2787 for Chandrupatla's method — level in total — and beats Chandrupatla problem by problem 87 to 42 (exact sign test $p=9\cdot10^{-5}$). On the 45 wide-bracket problems of Chandrupatla it beats Brent and TOMS 748 and loses to ITP and to Chandrupatla (3220 against 2896 and 2441). 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 2787; 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. On the canonical set these records barely feel the multiplier — 2523 is the same for every multiplier from 1.25 to 4, 2527 for every one from 1.5 to 4 and 2541 at 1.25; under the constant-free earned budget the same readings gain far less (6494 over both sets against 6588). 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: about a quarter 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.
+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 2438 and 2005 evaluations against Chandrupatla's 2787 and 2441 and beats him problem by problem on both, 103 to 30 and 29 to 7. 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. Over 855 problems in six corpora it is ahead of Brent and TOMS 748 everywhere and keeps its bound where they break theirs on 118 and 84 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.
 
 ---
 
@@ -22,6 +22,8 @@ This paper asks a different question. Suppose the only thing available is a *sim
 
 The answer is sharp in both directions, and the negative half is as informative as the positive half.
 
+The paper proceeds in two stages. Sections 2–7 derive a base rule from the similarity alone, measure it, and locate what it lacks: not a better model but a decision, when to stop trusting the model it has. Section 8 supplies that decision with a similarity of the run to its own past, and Section 9 tests the resulting method on four corpora it was not developed on, one of them fixed after the method's code was frozen.
+
 ---
 
 ## 2. The similarity step rule
@@ -102,7 +104,7 @@ $$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 §10; (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 §10; (iv) the measurements. We call it the **base rule**; the method this paper proposes builds on it in Section 8.
 
 **Algorithm.** Given $[a,b]$ with $f(a)f(b)<0$ and tolerance $\varepsilon_x$:
 
@@ -393,7 +395,7 @@ That mechanism is right for `fun1` and *wrong as a statement about the whole gap
 
 But 172 of the 256 — 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 0.41–0.75 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 92, 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, 228 — against his own 297, 233, 233 — 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, §10), 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 none of the 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 §10 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.
+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, §10), 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 none of the 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 §10 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. Section 8 shows that the decision is reachable without a new model: by a similarity of the run to its own past.
 
 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.
 
@@ -421,7 +423,7 @@ Nearness of configurations is one relation. An analogy in the full sense needs a
 
 $$m=\frac{\ln L_f}{\ln L_x}.\tag{6}$$
 
-A chance resemblance at a simple root reads $m\approx1$; a pure power of order 5 reads 4.9. The regime is the one where a tangent step loses, $m\ge2$ (§8.1) — no new constant. On the traces of the base rule, at the moments the flag rises, the three conditions together hold on 3 of the 154 canonical problems, on all 20 flat problems, and on none of `fun2`, `fun8`, `fun9`.
+Under an exact zoom a pure power of order $m$ reads exactly $m$; along a real run the analogous pairs are only nearly homothetic and the reading scatters about the order, but the regime asks only whether it reaches 2. The regime is the one where a tangent step loses, $m\ge2$ (§8.1) — no new constant. On the traces of the base rule, at the moments the flag rises, the three conditions together hold on 3 of the 154 canonical problems, on all 20 flat problems, and on none of `fun2`, `fun8`, `fun9`.
 
 **The analogy belongs to the run, the flag to the moment.** In the cycle the base rule runs on a pure power — undershoot, undershoot, crossing, a nearly empty step — the flag rises in the phases whose configuration does *not* recur (the empty step grows emptier every cycle), while the recurring phases have the flag down. Requiring both in one state delayed the entry on the flat problems to their thirteenth state on average. Once established, the analogy is therefore remembered for the run, and the regime is entered at the next unkept promise (eighth state on average).
 
@@ -445,7 +447,7 @@ which is exact for a pure power and becomes the midpoint as $m\to\infty$. The or
 
 **The repeated step pattern.** Two successive moves the same way, of similar length, mean the iterate creeps along a geometric series whose remainder is $\text{step}\cdot s/(1-s)$; the step is lengthened by $1/(1-s)$, with $s$ the similarity of successive step lengths, but only when the pattern of successive length ratios repeats with a period read from the data (the lag of highest mean similarity) and that similarity exceeds $\tfrac12$.
 
-**The overshoot branch.** §8.1's inequality has a second branch, $m<\tfrac23$: a root of order below one, where the tangent step overshoots and the iterate oscillates about the root ($\sqrt{|x|}\operatorname{sign}x$: 84 evaluations of the base rule). The flag for it is the unkept promise *with* a sign change, $f_\text{new}/f_\text{old}\le-\tfrac14$, and an analogous pair reading $0<m\le\tfrac23$; the split (7) is again exact for a pure power. The exit judgment of §8.2 separates a flat root from a simple one; at a root of order below one $f$ is far from 0 on both sides and the exit fires falsely, so the memory of this branch is *not* cleared by it (measured on the development corpora: 19 evaluations with the memory kept, 65 without).
+**The overshoot branch.** §8.1's inequality has a second branch, $m<\tfrac23$: a root of order below one, where the tangent step overshoots and the iterate oscillates about the root ($\sqrt{|x|}\operatorname{sign}x$: 84 evaluations of the base rule). The flag for it is the unkept promise *with* a sign change, $f_\text{new}/f_\text{old}\le-\tfrac14$, and an analogous pair reading $0<m\le\tfrac23$; the split (7) is again exact for a pure power. The exit judgment of §8.2 separates a flat root from a simple one; at a root of order below one $f$ is far from 0 on both sides and the exit fires falsely, so the memory of this branch is *not* cleared by it; clearing it was measured on the development corpora and was worse (recorded in the pre-registration amendment of this branch).
 
 **Borrowed: the tolerance step.** Once the root is reached from one side the bracket's far end can stay far: on $\sin$ over $[3,4]$ the base rule reaches $|f|\approx10^{-16}$ in 7 evaluations and then halves the remaining $1.3\cdot10^{-7}$ down to the tolerance in 17 more. Dekker [9] and Brent [2] step at least the tolerance; we do the same, $\varepsilon_x/2$ towards the far end, when the model's step is shorter — but only after the model has *kept* its previous promise ($r<\tfrac14$), because at the start of a run a slope dominated by a huge far value promises the root at an end. This element is not ours and is named as such.
 
@@ -548,21 +550,33 @@ Only the last line of corpora is evidence for A′ in the strict sense; the firs
 
 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 3695 to 3541 — but the canonical set gets worse, 5822 to 6940, against the 2806 of the base rule. An earlier version compared the 3541 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.
 
+12. **Reading the unkept promise one step at a time does not pay.** The flag of §8.1 sees the flat regime, but a halving in place of each flagged step takes the canonical set from 2806 to 3029 and the wide-bracket set only from 3220 to 3198: a simple root far from convergence breaks its promises too. Requiring the promise to fail twice in a row keeps the canonical set (2803) but fires one step late, before the crossing, and replaces the most useful step of the cycle (wide brackets 3341); requiring a creep in one direction as well gives 2878 and 3354.
+
+13. **A halving that sticks once begun wins the flat families and loses everything else.** Without an exit the flat families fall to 1057 — pure bisection's level — and the canonical set rises to 6897, because a false entry now lasts to the end of the run. Splitting at the order read from the failed promise, $(1-1/m)^m=r$, changes nothing: that order is $m=\infty$ in 163 of 174 entries (6781).
+
+14. **The model's own point cannot tell a flat root from a simple one.** Judging the evaluated point against the line through the bracket ends and against 0 — the exit judgment of §8.2 — also at *entry* gives 3009 and 2486 against 3125 and 2503 for the exit alone: the model aims at the root, so $f$ is small there by construction and looks flat at a simple root as readily as at a power. The same judgment works at a split, which the model did not choose.
+
+15. **The decision reading "defect" does not transfer.** Combined with the regime it was the best configuration on the two development sets (§7.2's canonical record 2523 with the flat families of §8), and it passed every pre-registered criterion there. On new corpus 1 it did not: per problem 95/33/78 against Chandrupatla ($p=0.22$), losing the class of exponential flanks 9 to 26. It is therefore not part of A′.
+
+16. **A component rejected alone can pay in combination.** Remembering the analogy for the run was rejected by its own pre-registration (canonical set 2814 against 2806 without the regime), and with the split of §8.3 it is the better form (with the step pattern, wide brackets 2117 against 2221 for the analogy read in one state). Composition was measured, not assumed, throughout — which is also why §7's readings (2) and (3), each an improvement, are not combined.
+
 ---
 
 ## 11. 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. The projection's budget multiplier is a free constant, and we declare it: it does not move the canonical result and it does move the wide-bracket one; projecting onto a budget the steps earn instead removes it, at a measured price.
 
-The resulting rule has one declared constant and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 — 2806 evaluations against 2820 is a margin of 0.5% and should be read as a tie — ahead of Brent by 2.9%, and level with Chandrupatla's method in the same sense: 0.7% more in total, while winning 87 of the problems to his 42 ($p=9\cdot10^{-5}$). On the wide-bracket set Chandrupatla is genuinely stronger (2441 against 3220), and that is where the declared multiplier matters. The constant-free variant trades 762 evaluations on the canonical set for no constant, a tighter guarantee and the better wide-bracket total. The claim the measurements support is parity with the best methods on the canonical set, with a problem-by-problem edge over all of them, and a measured deficit on very wide brackets whose cause §7.4 locates — not superiority overall.
+The base rule has one declared constant and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 and with Chandrupatla's method in total and ahead of both problem by problem; on the wide-bracket set Chandrupatla is genuinely stronger (2441 against 3220). §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 and TOMS 748 everywhere, with the guarantee they do not keep, 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.
 
 ### Future work
 
-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 (§10, item 11), and the step turned out to be a fraction of the bracket already.
+Three questions remain, all narrower than the one this section used to pose.
 
-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 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 third is a comparison this paper does not make: the modified Anderson–Björck method of [10], the most recent bracketing method we know of, is not among the rivals; its own benchmark is corpus 2 here, so the comparison is ready to be run under the rule every other one followed — pre-registered, on a corpus not yet seen.
 
 
 ---
@@ -576,3 +590,8 @@ What is open is a decision, not a model. On two thirds of the remaining gap the
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+8. G. Pólya, *How to Solve It*. Princeton, NJ: Princeton University Press, 1945.
+9. T. J. Dekker, "Finding a zero by means of successive linear interpolation," in B. Dejon and P. Henrici (eds.), *Constructive Aspects of the Fundamental Theorem of Algebra*. London: Wiley-Interscience, 1969, pp. 37–51.
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+11. M. Galassi et al., *GNU Scientific Library Reference Manual*, 3rd ed. Network Theory, 2009; root-finding test suite `roots/test.c`.
+12. J. Verzani et al., *Roots.jl*, Julia package; bracketing benchmark `test/benchmark-bracketing-methods.jl`, commit cf1d256, 2026.