Prove2Me

The strong to active wall ratio decreases in logarithmic connection time

Proved
BianchiIX.connectionLog_ratio_antitone

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

For real variables p,s,a,b,cp,s,a,b,c, set

K=a2+b2+c2−2(ab+bc+ca),S=p2+s2,C=S+K.K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K.

The shear components and wall rates are

P=Kp−2a2+b2+c2+ab−2bc+ac,P=Kp-2a^2+b^2+c^2+ab-2bc+ac, Q=Ks+3(−b2+c2+ab−ac),Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac), r1=−S+2p,r2=−S−p+3s,r3=−S−p−3s.r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s.

These functions define the normalized vacuum Bianchi IX vector field (P,Q,r1a,r2b,r3c)(P,Q,r_1a,r_2b,r_3c). For y=(m,u,v,w)∈R4y=(m,u,v,w)\in\mathbb R^4, define

K(u,v,w)=u2+v2+w2−2(uv+vw+wu),κ=u(v+w−u),K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u), Fm=−v2+w2+u(v−w)−mκ,Fu=12(1−m2−(v−w)2+u2)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, Fv=(m−1−κ)v,Fw=(−m−1−κ)w,F=(Fm,Fu,Fv,Fw),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+v2E(m,v)=(m-1)^2+v^2.

For x=(p,s,a,b,c)x=(p,s,a,b,c), put

ψ(p,z)=3 z1+p,Π(x)=(ψ(p,s),ψ(p,a),ψ(p,b),ψ(p,c)).\psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad \Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)).

For y=(m,u,v,w)y=(m,u,v,w), set D(y)=3+m2+K(u,v,w)D(y)=3+m^2+K(u,v,w) and

ΞA(y)=(A−1,Am3,Au3,Av3,Aw3),Ξ(y)=Ξ6/D(y)(y).\Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3}, \frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad \Xi(y)=\Xi_{6/D(y)}(y).

Write N(x)=(P,Q,r1a,r2b,r3c)N(x)=(P,Q,r_1a,r_2b,r_3c) for the normalized vector field and C(x)=C(p,s,a,b,c)\mathcal C(x)=C(p,s,a,b,c) for its constraint.

The normalized and projective state spaces are R5\mathbb R^5 and R4\mathbb R^4 with their finite-product norms. The maps Π,Ξ,D,F\Pi,\Xi,D,F act on these state spaces.

The projective vector field in normalized time is

G(y)=6D(y)F(y).G(y)=\frac6{D(y)}F(y).

Write X=R5X=\mathbb R^5, Y=R4Y=\mathbb R^4, and let αx,t0\alpha_{x,t_0} be the maximal solution of α′=N(α)\alpha'=N(\alpha) with initial value xx at t0t_0 and domain Imax⁡(x,t0)I_{\max}(x,t_0). Put

Jx,t0(t)=dxαx,t0(t),Ψt0(y,t)=Π(αΞ(y),t0(t)),J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)), Jy,t0(t)=dΠαΞ(y),t0(t)∘JΞ(y),t0(t)∘dΞy.\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.

Put c=u(v+w−u)c=u(v+w-u), s(m,u,v,w)=m−1−cs(m,u,v,w)=m-1-c, b(m,u,v,w)=(1−m2−(v−w)2+u2)/2b(m,u,v,w)=(1-m^2-(v-w)^2+u^2)/2, D(m,u,v,w)=3+m2+K(u,v,w)D(m,u,v,w)=3+m^2+K(u,v,w), and r(m,v)=(m−1)2+v2r(m,v)=\sqrt{(m-1)^2+v^2}.

Define

Cμ={(m,u,v,w):∣−m−μ∣≤1/2, 0≤u≤1/10000, 0≤v,w≤1},\mathcal C_\mu=\{(m,u,v,w):|-m-\mu|\leq1/2, \ 0\leq u\leq1/10000,\ 0\leq v,w\leq1\}, ω−(y)=s(−ym,yu,yv,yw),H(y)=F(y)/ω−(y).\omega_-(y)=s(-y_m,y_u,y_v,y_w),\qquad H(y)=F(y)/\omega_-(y).

For a normalized trajectory write

β(t)=Π(αx,t0(t)),A(t)=1+(αx,t0(t))0,I=Imax⁡(x,t0).\beta(t)=\Pi(\alpha_{x,t_0}(t)),\quad A(t)=1+(\alpha_{x,t_0}(t))_0, \quad I=I_{\max}(x,t_0).

Let JJ be convex and μ≥4\mu\geq4. Suppose Y′=H(Y)Y'=H(Y), Y∈CμY\in\mathcal C_\mu and Yv>0Y_v>0 on JJ. Then Yu/YvY_u/Y_v is nonincreasing on JJ.

Preamble
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Algebra_ConnectionBounds
import Definitions.Def_BianchiIX_Dynamics_ConnectionClock
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
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.SpecialFunctions.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial

noncomputable section
open BianchiIX
open Set MeasureTheory
Formal statement
theorem BianchiIX.connectionLog_ratio_antitone {I : Set ℝ} (hI : Convex ℝ I)
    {Y : ℝ → ProjectiveState} {μ : ℝ} (hμ : 4 ≤ μ)
    (hY : ∀ z ∈ I, HasDerivAt Y (connectionLogField (Y z)) z)
    (hm : ∀ z ∈ I, |-Y z 0 - μ| ≤ 1 / 2)
    (hu : ∀ z ∈ I, 0 ≤ Y z 1 ∧ Y z 1 ≤ 1 / 10000)
    (hv : ∀ z ∈ I, 0 < Y z 2 ∧ Y z 2 ≤ 1)
    (hw : ∀ z ∈ I, 0 ≤ Y z 3 ∧ Y z 3 ≤ 1) :
    AntitoneOn (fun z => Y z 1 / Y z 2) I := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ConnectionPrefix.lean#L34-L58

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