The entrance strong wall decays faster than the active wall grows

Proved

BianchiIX.entrance_strong_clock_rate

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

bianchi-ix-era-statisticsdynamical-systems

For $y=(m,u,v,w)\in\mathbb R^4$ , define

K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u),

F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u,

F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad F=(F_m,F_u,F_v,F_w),

and $E(m,v)=(m-1)^2+v^2$ .

Put $c=u(v+w-u)$ , $s(m,u,v,w)=m-1-c$ , $b(m,u,v,w)=(1-m^2-(v-w)^2+u^2)/2$ , $D(m,u,v,w)=3+m^2+K(u,v,w)$ , and $r(m,v)=\sqrt{(m-1)^2+v^2}$ .

Assume $k\geq7/4$ , $4k/5\leq m-1\leq101k/100$ , $0\leq u,w\leq1/10000$ , and $0\leq v\leq1$ . Then $b(m,u,v,w)/s(m,u,v,w)\leq-(1+k/3)$ .

Preamble

import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Algebra_ConnectionBounds
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.Calculus.Deriv.Inv
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt

noncomputable section
open BianchiIX

Formal statement

theorem BianchiIX.entrance_strong_clock_rate {k m u v w : ℝ} (hk : 7 / 4 ≤ k)
    (hm : 4 / 5 * k ≤ m - 1 ∧ m - 1 ≤ 101 / 100 * k)
    (hu : 0 ≤ u ∧ u ≤ 1 / 10000) (hv : 0 ≤ v ∧ v ≤ 1)
    (hw : 0 ≤ w ∧ w ≤ 1 / 10000) :
    connectionStrongRate m u v w / connectionSpeed m u v w ≤ -(1 + k / 3) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/EntranceBounds.lean#L39-L58

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