The entrance projection denominator is positive and quadratically bounded

Proved

BianchiIX.entrance_denom_bounds

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 $0<D(m,u,v,w)\leq6(k+1)^2$ .

Preamble

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Algebra_Coordinates
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_denom_bounds {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) :
    0 < projectiveDenom m u v w ∧ projectiveDenom m u v w ≤ 6 * (k + 1) ^ 2 := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/EntranceBounds.lean#L78-L94

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