Valeri's objection measured: the principle is right (a similarity step should be large far away and small near), and our sigma = |f'|D defeats it — at x=1e4 on a 2e4 bracket the normalised residual is 0.167, so S=0.833 'similar' and the step is small five thousand step-lengths from the root. Selecting sigma = min(|f'|D, max|f| at the ends) — no new constant — improves 403 to 390 on eight hand-picked wide problems but is WORSE over the grid: APS 3288 vs 2726, CHA 3111 vs 2683, so the mode is rejected by its pre-registered rule. Its extra solves are two fun7 problems landing inside the flat zero region, i.e. the scoring artefact and not a capability. Written into Negative results as the answer to the natural objection

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

PREREG_scale_selection_2026-09-21.md

@@ -0,0 +1,53 @@
+# PREREG — масштаб как ВЫБОР между двумя посчитанными (`derived3_cap`)
+
+Составлено до полного прогона. Основание — возражение Валерия 2026-09-21: «если бы шаг формировала
+похожесть, мы меняли бы шаг с очень большого на более мелкий по мере приближения к цели».
+
+## 1. Почему возражение верно и что оно вскрыло
+
+Шаг есть `(1 − S(z))·D` с `z = |f|/σ` и `σ = |slope|·D`. Проверка на `fun1.4` в точке `x = 10⁴`
+(`f ≈ 10¹²`, `f′ ≈ 3·10⁸`, ньютоновская длина ≈ 3333, `D ≈ 2·10⁴`): `z = 0.167`, то есть
+**`S = 0.833` — «похоже»** — и шаг мал, хотя до корня около 5000 его собственных длин.
+
+Причина: деление остатка на `|slope|·D` **нормирует непохожесть самой скобкой**, поэтому похожесть
+читает «какую долю скобки составляет ньютоновский шаг», а не «далеко ли я». Принцип «крупный шаг
+далеко, мелкий близко» формулой не реализован — он требует `σ`, не зависящего от `D`.
+
+## 2. Что проверяется
+
+`σ = min(|slope|·D, max(|F_A|, |F_B|))` — **выбор между двумя уже посчитанными** масштабами:
+значением на концах скобки и наклонным. Ни новой константы, ни порога: `min` по имеющемуся.
+Ожидаемое поведение: далеко на широкой скобке минимум даёт значение на конце ⇒ `z ≈ 1` ⇒ шаг во всю
+скобку; у корня минимум даёт наклонный ⇒ ньютоновское поведение и прежний порядок.
+
+**Режим:** `scale="derived3_cap"`; прежние шкалы и поведение по умолчанию не тронуты.
+
+## 3. Точка отсчёта и цели
+
+Сравнение — против той же конфигурации без новой шкалы (`slope`-гейт + `dissimilar`):
+APS **2726**, CHA **2785** (по-наборно 2683), два набора **5511**, 193/199, `over2x = 0`.
+Лучшее на APS у нас — **2527** (`shape`-чтение).
+
+| величина | цель |
+|---|---|
+| APS-154 | **≤ 2726** (не хуже точки отсчёта); идеально ≤ 2527 |
+| CHA-45 | **< 2683** (улучшение против нашего лучшего) |
+| два набора | **< 5511** |
+| решено | ≥ 193/199 |
+| гарантия | `over2x = 0` |
+
+## 4. Предсказания и фальсификаторы
+
+| # | Предсказание | ФАЛЬСИФИКАТОР → вывод |
+|---|---|---|
+| **K1** | CHA улучшается (<2683) | не улучшается → выбор масштаба не бьёт в измеренную причину (недолёт линейной модели) |
+| **K2** | APS не ухудшается (≤2726) | ухудшается → тот же трейд «широкие за счёт узких», четвёртый раз; режим отвергается |
+| **K3** | гарантия цела | нарушена → немедленное закрытие |
+| **K4** | порядок сходимости у корня не падает: наблюдаемый порядок на узких задачах не ниже, чем у `derived3` | падает → выбор `min` мешает концовке, и это записывается |
+
+🔴 **Оговорка о смещении:** предварительная проба на 8 **широких** задачах (403 → 390, 4 лучше / 3
+хуже / 1 вровень) выбрана мной под механизм и **свидетельством не является**. Решает полный прогон по
+всей сетке, где широкие задачи составляют меньшинство.
+
+**Правило остановки:** K2 или K3 ложны ⇒ режим не принимается. K1 ложна ⇒ измеренная причина
+(недолёт линейной модели) выбором масштаба не устраняется, и это пишется прямо.

article/sim_numerics.tex

@@ -677,6 +677,22 @@ 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 cost $7260$.
+\item \textbf{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 $\sigma=|f'|D$ the normalised residual is
+$|f|/(|f'|D)$, 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 \emph{bracket} the Newton step is, not how far the
+iterate is from the root. Selecting the scale between the two quantities already
+computed, $\sigma=\min(|f'|D,\max(|f(A)|,|f(B)|))$ --- no new constant, a
+\texttt{min} 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
+\texttt{fun7} problems where it lands inside the flat zero region close enough to
+satisfy the harness --- the artefact of \S\ref{sec:fun7}, not a capability.
 \item \textbf{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 is the right diagnosis of the flat families --- CHA

results_sweep_cap.txt

@@ -0,0 +1,64 @@
+sweep: 63 configurations x 199 problems (FULL)
+  10/63 done, 1s elapsed
+  20/63 done, 1s elapsed
+  30/63 done, 1s elapsed
+  40/63 done, 1s elapsed
+  50/63 done, 1s elapsed
+  60/63 done, 1s elapsed
+  63/63 done, 2s elapsed
+wrote sweep_cap.json  (2s total)
+====================================================================================
+APS: 154 problems   (top 20 of 63)
+====================================================================================
+configuration                              solved   total fev  over 2x
+------------------------------------------------------------------------------------
+pure+dissim+slope.exp_abs.derived3_cap.none   153/154        3288        0
+pure+dissim+slope.exp_abs.derived3_cap.project   153/154        3288        0
+pure+dissim+slope.exp_abs.derived3_cap.step   153/154        3385        0
+pure+dissim+slope.exp_abs.derived3_cap.safe   153/154        3385        0
+pure+dissim+slope.rational.derived3_cap.step   153/154        3460        0
+pure+dissim+slope.rational.derived3_cap.safe   153/154        3460        0
+pure+dissim+slope.rational.derived3_cap.none   153/154        3497        0
+pure+dissim+slope.rational.derived3_cap.project   153/154        3497        0
+pure+dissim+slope.exp_abs.derived3_cap.bracket   153/154        3537        0
+pure+dissim+slope.exp_abs.derived3_cap.both   153/154        3619        0
+pure+dissim+slope.exp_abs.derived3_cap.credit   153/154        4125        0
+pure+dissim+slope.ratio.derived3_cap.bracket   153/154        4333        0
+pure+dissim+slope.ratio.derived3_cap.both   153/154        4903        0
+pure+dissim+slope.rational.derived3_cap.credit   153/154        5001        0
+pure+dissim+slope.ratio.derived3_cap.step   153/154        5044        0
+pure+dissim+slope.ratio.derived3_cap.safe   153/154        5044        0
+pure+dissim+slope.sqrt.derived3_cap.step   153/154        6184        0
+pure+dissim+slope.sqrt.derived3_cap.safe   153/154        6184        0
+pure+dissim+slope.sqrt.derived3_cap.both   153/154        6216        0
+pure+dissim+slope.sqrt.derived3_cap.bracket   153/154        6580        0
+
+====================================================================================
+CHA: 45 problems   (top 20 of 63)
+====================================================================================
+configuration                              solved   total fev  over 2x
+------------------------------------------------------------------------------------
+pure+dissim+slope.sqrt.derived3_cap.none    42/45         2969        0
+pure+dissim+slope.sqrt.derived3_cap.project    41/45         3111        0
+pure+dissim+slope.exp_abs.derived3_cap.step    40/45         2658        0
+pure+dissim+slope.exp_abs.derived3_cap.safe    40/45         2658        0
+pure+dissim+slope.exp_abs.derived3_cap.both    40/45         2689        0
+pure+dissim+slope.rational.derived3_cap.step    40/45         2790        0
+pure+dissim+slope.rational.derived3_cap.safe    40/45         2790        0
+pure+dissim+slope.exp_abs.derived3_cap.bracket    40/45         2796        0
+pure+dissim+slope.exp_abs.derived3_cap.none    40/45         2808        0
+pure+dissim+slope.exp_abs.derived3_cap.project    40/45         2808        0
+pure+dissim+slope.ratio.derived3_cap.both    40/45         2906        0
+pure+dissim+slope.ratio.derived3_cap.step    40/45         2959        0
+pure+dissim+slope.ratio.derived3_cap.safe    40/45         2959        0
+pure+dissim+slope.rational.derived3_cap.none    40/45         2970        0
+pure+dissim+slope.rational.derived3_cap.project    40/45         2970        0
+pure+dissim+slope.ratio.derived3_cap.bracket    40/45         2978        0
+pure+dissim+slope.rational.derived3_cap.credit    40/45         2982        0
+pure+dissim+slope.exp_abs.derived3_cap.credit    40/45         2991        0
+pure+dissim+slope.cbrt.derived3_cap.both    40/45         3041        0
+pure+dissim+slope.sqrt.derived3_cap.step    40/45         3066        0
+
+====================================================================================
+THE STAGE-0C LAW ACROSS THE GRID
+====================================================================================

simroot/pure.py

@@ -505,7 +505,7 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
     if similarity not in PURE_SIMS:
         raise ValueError(f"unknown similarity {similarity!r}")
     if scale not in ("derived", "derived3", "derived_rho", "derived3_rho",
-                     "derived3_m", "start", "bracket"):
+                     "derived3_m", "derived3_cap", "start", "bracket"):
         raise ValueError(f"unknown scale {scale!r}")
     if space not in SPACES:
         raise ValueError(f"unknown space {space!r}")
@@ -515,11 +515,12 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
         raise ValueError(f"unknown points rule {points!r}")
     if direction not in DIRECTIONS:
         raise ValueError(f"unknown direction {direction!r}")
-    if points == "shape" and scale not in ("derived3", "derived3_rho", "derived3_m"):
+    if points == "shape" and scale not in ("derived3", "derived3_rho", "derived3_m",
+                                          "derived3_cap"):
         # the triple only exists for the three-point scale; selecting one under a
         # two-point scale would change nothing and hide that
         raise ValueError("points='shape' requires a three-point scale (derived3*)")
-    if prestep == "slope" and scale not in ("derived", "derived3"):
+    if prestep == "slope" and scale not in ("derived", "derived3", "derived3_cap"):
         # there is no slope to judge under a chosen or bracket scale; judging something
         # else would hide that, so the call fails instead
         raise ValueError("prestep='slope' requires scale in {'derived', 'derived3'}")
@@ -603,6 +604,19 @@ def pure_root(f, a, b, xtol, maxfev, similarity="ratio", scale="derived",
                     else:
                         slope = _slope3(x, fx, x_prev, f_prev, x_prev2, f_prev2)
                     sc = abs(slope) * abs(far - x)
+                elif scale == "derived3_cap":
+                    # Valeri, 2026-09-21: "if the similarity formed the step, the step would be
+                    # large far from the target and small near it".  It does not, and the reason
+                    # is the normalisation: with sigma = |slope| * D the similarity reads "what
+                    # fraction of the BRACKET the Newton step is", not "how far am I".  On a
+                    # 2e4-wide bracket of x^3-2x-5 at x = 1e4 that fraction is 0.167, so S =
+                    # 0.833 - "similar" - and the step is small exactly where it should be large.
+                    # The scale is therefore SELECTED between the two already-computed ones:
+                    # far away the value scale of the bracket ends is the smaller and the step
+                    # opens up; near the root the slope scale is the smaller and Newton (and the
+                    # order) returns.  A min over what is in hand - no constant, no threshold.
+                    slope = _slope3(x, fx, x_prev, f_prev, x_prev2, f_prev2)
+                    sc = min(abs(slope) * abs(far - x), max(abs(FA), abs(FB)))
                     if scale == "derived3_m":
                         # REFUTED as an improvement - kept for reproducibility.
                         # The DERIVATION holds: dropping the linearity

sweep_full.py

@@ -198,6 +198,15 @@ def build_grid_log(smoke):
             for sim in sims for sc in ("derived", "derived3") for gu in guards]
 
 
+def build_grid_cap(smoke):
+    """Scale as a SELECTION between the two computed ones (PREREG 2026-09-21): identical to
+    the slope+dissimilar configuration except scale = derived3_cap, so one thing changes."""
+    sims = ["ratio", "sqrt"] if smoke else sorted(PURE_SIMS)
+    guards = ("credit", "safe") if smoke else GUARD_MODES
+    return [("pure", sim, "derived3_cap", gu, "x", "slope", "recent", "dissimilar")
+            for sim in sims for gu in guards]
+
+
 def build_grid_combo(smoke):
     """A+B together (PREREG 2026-09-21 §2: licensed only because each mode alone improved
     APS — A 2527 and B 2726 against the published 2806)."""
@@ -264,10 +273,12 @@ def main():
     logmode = "--log" in sys.argv          # the u-space grid, written to its OWN file
     gatemode = "--gate" in sys.argv        # the pre-step gate grid, likewise
     slopemode = "--slope" in sys.argv      # the slope-stability gate grid
+    capmode = "--cap" in sys.argv          # scale selected between the two computed ones
     combomode = "--combo" in sys.argv      # modes A and B together
     pointsmode = "--points" in sys.argv    # mode A: triple by shape
     dirmode = "--direction" in sys.argv    # mode B: far end by dissimilarity
-    specs = (build_grid_combo(smoke) if combomode else
+    specs = (build_grid_cap(smoke) if capmode else
+             build_grid_combo(smoke) if combomode else
              build_grid_points(smoke) if pointsmode else
              build_grid_direction(smoke) if dirmode else
              build_grid_slope(smoke) if slopemode else
@@ -288,8 +299,9 @@ def main():
 
     # the published sweep_full.json is what verify_article.py checks the paper
     # against, so the log grid never writes into it
-    if combomode or pointsmode or dirmode:
+    if capmode or combomode or pointsmode or dirmode:
-        base = ("sweep_combo" if combomode else
+        base = ("sweep_cap" if capmode else
+                "sweep_combo" if combomode else
                 "sweep_points" if pointsmode else "sweep_direction")
         out = f"{base}_smoke.json" if smoke else f"{base}.json"
         with open(out, "w", encoding="utf8") as fh: