Clock-derivative products with explicit wall amplitudes

Proved

BianchiIX.core_amplitude_clock_products

by trppppp · Nov 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

Assume $r\geq5$ , $0\leq\mu\leq1/100$ , $0\leq u,w\leq\mu$ , and $|g|\leq2r\mu$ . Then

\left|g\frac{2(u-w)(u-2w)}{r^3}\right| \leq\frac{u+w^2/r^2}{10},

\left|g\frac{-2(u-w)}{r^2}\right| \leq\frac{u+w/r}{10}.

Preamble

import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.FieldSimp

namespace BianchiIX
end BianchiIX

noncomputable section
open BianchiIX

Formal statement

theorem BianchiIX.core_amplitude_clock_products {r m u w g : ℝ} (hr : 5 ≤ r)
    (hm : 0 ≤ m ∧ m ≤ 1 / 100) (hu : 0 ≤ u ∧ u ≤ m) (hw : 0 ≤ w ∧ w ≤ m)
    (hg : |g| ≤ 2 * r * m) :
    |g * (2 * (u - w) * (u - 2 * w) / r ^ 3)| ≤ (u + w ^ 2 / r ^ 2) / 10 ∧
    |g * (-2 * (u - w) / r ^ 2)| ≤ (u + w / r) / 10 := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/CoreAmplitudeCoefficients.lean#L30-L73

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