Prove2Me

Integrated radial column estimate for the exact logarithmic exit flow

Proved
BianchiIX.exitClock_radial_column_bound

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

bianchi-ix-era-statisticsdynamical-systems

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.

Write r(m,v)=(m−1)2+v2r(m,v)=\sqrt{(m-1)^2+v^2} and define

g(R,v)=R2−v2,d(R,u,v,w)=g(R,v)+u(v+w−u),g(R,v)=\sqrt{R^2-v^2},\qquad d(R,u,v,w)=g(R,v)+u(v+w-u), fR=−u(v+w−u)R−g(R,v)(u−w)2/Rd,f_R=\frac{-u(v+w-u)R-g(R,v)(u-w)^2/R}{d}, as=2g(R,v)−R2+2vw−w2+u22d,aw=g(R,v)−2−u(v+w−u)d.a_s=\frac{2g(R,v)-R^2+2vw-w^2+u^2}{2d},\qquad a_w=\frac{g(R,v)-2-u(v+w-u)}d.

The exit clock rate is ω(y)=1−m+u(v+w−u)\omega(y)=1-m+u(v+w-u), and its field is Flog⁡(y)=F(y)/ω(y)F_{\log}(y)=F(y)/\omega(y). Division and square roots have their total real-valued Lean meanings.

For Y=(R,u,w)Y=(R,u,w) define Ev(Y)=(fR,asu,aww)E_v(Y)=(f_R,a_su,a_ww), evaluated at (R,u,v,w)(R,u,v,w). The coordinate maps are Q(m,u,v,w)=(r(m,v),u,w)Q(m,u,v,w)=(r(m,v),u,w) and Qv−1(R,u,w)=(1−g(R,v),u,v,w)Q_v^{-1}(R,u,w)=(1-g(R,v),u,v,w). For a curve Y(s)Y(s) set

Mi(Y,t)=exp⁡∫0tai(Y0(s),Y1(s),e−s,Y2(s)) ds(i=s,w).M_i(Y,t)=\exp\int_0^t a_i(Y_0(s),Y_1(s),e^{-s},Y_2(s))\,ds \quad(i=s,w).

Let Rα,b,A,D\mathcal R_{\alpha,b,A,D} consist of the states satisfying

7/4≤R≤17/4,∣u∣,∣w∣≤1/100,∣u∣+∣w∣≤Ab,∣R−(2+2α)∣≤Db.7/4\leq R\leq17/4,\quad |u|,|w|\leq1/100,\quad |u|+|w|\leq Ab,\quad |R-(2+2\alpha)|\leq Db.

Put

B(α,b,A,D,T)=[10(α+Db)+100(Ab)2]T+100Ab+5,B(\alpha,b,A,D,T)=[10(\alpha+Db)+100(Ab)^2]T+100Ab+5, L(η,α,J)=log⁡(1/η)+Jlog⁡(1/α).L(\eta,\alpha,J)=\log(1/\eta)+J\log(1/\alpha).

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.

For Y=(k,u,w)∈R3Y=(k,u,w)\in\mathbb R^3, define

e−(Y)=(1−k2−1,u,1,w),xY=Ξ(e−(Y)),βY(t)=Ψ0(e−(Y),t),g(x)=(Π(x))v.e_-(Y)=(1-\sqrt{k^2-1},u,1,w),\quad x_Y=\Xi(e_-(Y)),\quad \beta_Y(t)=\Psi_0(e_-(Y),t),\quad g(x)=(\Pi(x))_v.

For any real height η\eta, put

Hη,Y={t>0:t∈Imax⁡(xY,0), g(αxY,0(t))=η},H_{\eta,Y}=\{t>0:t\in I_{\max}(x_Y,0),\ g(\alpha_{x_Y,0}(t))=\eta\}, Tη(Y)=inf⁡Hη,Y,Pη(Y)=βY(Tη(Y)).T_\eta(Y)=\inf H_{\eta,Y},\qquad P_\eta(Y)=\beta_Y(T_\eta(Y)).

The infimum uses the total real-order convention. Let TailValid⁡η(Y)\operatorname{TailValid}_\eta(Y) mean that Tη(Y)T_\eta(Y) is the least element of Hη,YH_{\eta,Y}, and, writing T=Tη(Y)T=T_\eta(Y), that

t∈Imax⁡(xY,0),1+(αxY,0(t))0>0(0≤t≤T),t\in I_{\max}(x_Y,0),\quad 1+(\alpha_{x_Y,0}(t))_0>0 \quad(0\leq t\leq T), g(αxY,0(t))>η(0≤t<T),g(\alpha_{x_Y,0}(t))>\eta\quad(0\leq t<T), dgαxY,0(t)N(αxY,0(t))<0,∣(Π(αxY,0(t)))w∣<1/2(0≤t≤T).dg_{\alpha_{x_Y,0}(t)}N(\alpha_{x_Y,0}(t))<0,\quad |(\Pi(\alpha_{x_Y,0}(t)))_w|<1/2\quad(0\leq t\leq T).

All vector and operator norms use the finite-product supremum norms.

For y∈R4y\in\mathbb R^4, put xy=Ξ(y)x_y=\Xi(y),

T+(y)=inf⁡{t>0:t∈Imax⁡(xy,0), (Ψ0(y,t))v=1},P+(y)=Ψ0(y,T+(y)).T_+(y)=\inf\{t>0:t\in I_{\max}(x_y,0),\ (\Psi_0(y,t))_v=1\}, \qquad P_+(y)=\Psi_0(y,T_+(y)).

Let V+(y)V_+(y) mean that this infimum is a least element, the full interval [0,T+(y)][0,T_+(y)] lies in the maximal domain with 1+(αxy,0(t))0>01+(\alpha_{x_y,0}(t))_0>0, the active wall is below one before the endpoint, and its normalized-time derivative is strictly positive throughout the interval. The infimum has its total real-order meaning.

For y=(m,u,v,w)y=(m,u,v,w) define the cyclic chart by

H(y)=(m+1)2+K(u,v,w),A(y)=(1−4(m+1)/H,4v/H,4w/H,4u/H).H(y)=(m+1)^2+K(u,v,w),\qquad \mathcal A(y)=(1-4(m+1)/H,4v/H,4w/H,4u/H).

Its normalized symmetry is

A(p,s,a,b,c)=((−p+3s)/2,(−3p−s)/2,b,c,a).A(p,s,a,b,c)=((-p+\sqrt3s)/2,(-\sqrt3p-s)/2,b,c,a).

Put zh(Y)=A(Ph(Y))z_h(Y)=\mathcal A(P_h(Y)) and

T(h,Y)=Th(Y)+T+(zh(Y)),P(h,Y)=P+(zh(Y)),xYc=AΞ(e−(Y)).\mathcal T(h,Y)=T_h(Y)+T_+(z_h(Y)),\qquad \mathcal P(h,Y)=P_+(z_h(Y)),\qquad x_Y^c=A\Xi(e_-(Y)).

Write r(m,v)=(m−1)2+v2r(m,v)=\sqrt{(m-1)^2+v^2} and g∗(x)=(Π(x))vg_*(x)=(\Pi(x))_v.

Let β(Y,t)=Ψ0(e−(Y),t)\beta(Y,t)=\Psi_0(e_-(Y),t) and c(Y,t)=−log⁡β(Y,t)2c(Y,t)=-\log\beta(Y,t)_2. Call (Y,t)(Y,t) regular when Y0>1Y_0>1, D(e−(Y))>0D(e_-(Y))>0, the normalized flow exists at tt with positive chart clock, and β2>0\beta_2>0, β0<1\beta_0<1, ω(β)>0\omega(\beta)>0. For an interval II, let τI(Y,s)\tau_I(Y,s) be a solution in II of c(Y,t)=sc(Y,t)=s, selected by choice, or zero if there is none. Define

XI(Y,s)=Q(β(Y,τI(Y,s))),ℓi(Y,s)=∫0sai(XI(Y,r)0,XI(Y,r)1,e−r,XI(Y,r)2) dr.X_I(Y,s)=Q(\beta(Y,\tau_I(Y,s))),\qquad \ell_i(Y,s)=\int_0^s a_i(X_I(Y,r)_0,X_I(Y,r)_1,e^{-r},X_I(Y,r)_2)\,dr.

Here i=0,1i=0,1 denotes as,awa_s,a_w respectively. Write

V(α,b,A,D,T)=2B(α,b,A,D,T)+2048(1+AbT).V(\alpha,b,A,D,T)=2B(\alpha,b,A,D,T)+2048(1+AbT).

The tail box is 7/4≤R≤17/47/4\leq R\leq17/4, ∣u∣,∣w∣≤1/100|u|,|w|\leq1/100, 0≤v≤10\leq v\leq1. Write GR=∇R,u,wfRG_R=\nabla_{R,u,w}f_R, Gs=∇R,u,wasG_s=\nabla_{R,u,w}a_s, Gw=∇R,u,wawG_w=\nabla_{R,u,w}a_w for their explicit algebraic gradients there. Let II be open and convex, UU open with Y∈UY\in U, and all of U×IU\times I regular. Suppose T≥0T\geq0, [0,T]⊂I[0,T]\subset I, 0≤s≤c(Y,T)0\leq s\leq c(Y,T), K≥1K\geq1, ∣Y1∣+∣Y2∣≤b|Y_1|+|Y_2|\leq b. For every r∈[0,s]r\in[0,s], assume XI(Y,r)X_I(Y,r) is in the tail box at v=e−rv=e^{-r}, ∥dYXI(Y,r)∥≤K\|d_YX_I(Y,r)\|\leq K, eℓi(Y,r)≤Ke^{\ell_i(Y,r)}\leq K, and ∥dYℓi(Y,r)∥≤Kr\|d_Y\ell_i(Y,r)\|\leq Kr for i=0,1i=0,1. For every direction vv with v1=0v_1=0,

∣(dYXI(Y,s)v)0−v0∣≤512K3[(2∣Y1∣+b2s(1+s))∥v∥+b∣v2∣s].|(d_YX_I(Y,s)v)_0-v_0| \leq512K^3[(2|Y_1|+b^2s(1+s))\|v\|+b|v_2|s].
Preamble
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_TerminalConeData
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_TerminalConnectionData
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ProjectiveOriginalTime
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_CoreTail
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Tactic.Module
import Mathlib.Analysis.Calculus.Deriv.Pow
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
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.LinearCombination
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
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Topology.Order.ProjIcc
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Topology.Order.Compact
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Tactic.Positivity
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
import Mathlib.Topology.Instances.Matrix
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp

noncomputable section
open BianchiIX
open Set Filter MeasureTheory
open scoped Topology
Formal statement
theorem BianchiIX.exitClock_radial_column_bound {I : Set ℝ} (hIo : IsOpen I) (hI : Convex ℝ I)
    {U : Set ExitTailState} (hUo : IsOpen U) {Y : ExitTailState} (hY : Y ∈ U)
    (H : ∀ Z ∈ U, ∀ r ∈ I, exitEntryRegular Z r) {T s K b : ℝ} (hT : 0 ≤ T)
    (hcover : Icc 0 T ⊆ I) (hs : s ∈ Icc 0 (exitEntryClock (Y, T))) (hK : 1 ≤ K)
    (hb : |Y 1| + |Y 2| ≤ b)
    (hreg : ∀ r ∈ Icc 0 s, (7 / 4 ≤ exitClockFlow I (Y, r) 0 ∧ exitClockFlow I (Y, r) 0 ≤ 17 / 4) ∧
      |exitClockFlow I (Y, r) 1| ≤ 1 / 100 ∧ |exitClockFlow I (Y, r) 2| ≤ 1 / 100)
    (hbound : ∀ r ∈ Icc 0 s, ‖fderiv ℝ (fun Z => exitClockFlow I (Z, r)) Y‖ ≤ K)
    (hmult : ∀ r ∈ Icc 0 s, ∀ i : Fin 2, Real.exp (exitClockLogMultiplier I i Y r) ≤ K ∧
      ‖fderiv ℝ (fun Z => exitClockLogMultiplier I i Z r) Y‖ ≤ K * r)
    (v : ExitTailState) (hv : v 1 = 0) :
    |fderiv ℝ (fun Z => exitClockFlow I (Z, s)) Y v 0 - v 0| ≤
      512 * K ^ 3 * ((2 * |Y 1| + b ^ 2 * s * (1 + s)) * ‖v‖ + b * |v 2| * s) := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ExitClockRadialColumn.lean#L11-L84

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