Phase and wall response of the exact flow at fixed relative clock

Proved

BianchiIX.exists_actual_entrance_phase_response

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

bianchi-ix-era-statisticsdynamical-systems

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

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-2a^2+b^2+c^2+ab-2bc+ac,

Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac),

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 $(P,Q,r_1a,r_2b,r_3c)$ . 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 $E(m,v)=(m-1)^2+v^2$ .

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

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

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

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

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

The projective vector field in normalized time is

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

Write $X=\mathbb R^5$ , $Y=\mathbb R^4$ , and let $\alpha_{x,t_0}$ be the maximal solution of $\alpha'=N(\alpha)$ with initial value $x$ at $t_0$ and domain $I_{\max}(x,t_0)$ . Put

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)),

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

Define

\mathcal E_k=\{(m,u,v,w):m\geq1,\ |r(m,v)-k|\leq k/100, \ 0\leq u,w\leq1/10000,\ 0\leq v\leq1\},

and $\omega_+(m,u,v,w)=s(m,u,v,w)$ . For a normalized trajectory write

\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).

For $y\in\mathbb R^4$ , put $x_y=\Xi(y)$ ,

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 $V_+(y)$ mean that this infimum is a least element, the full interval $[0,T_+(y)]$ lies in the maximal domain with $1+(\alpha_{x_y,0}(t))_0>0$ , 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.

On the invariant plane with zero active wall, use $y=(m,u,w)\in\mathbb R^3$ , the inclusion $\iota(m,u,w)=(m,u,0,w)$ and the restriction $\pi_s(m,u,v,w)=(m,u,w)$ . Define

f(y)=\left(mu^2-(m+1)uw+w^2, \frac{1-m^2-w^2+u^2}{2}u,-(m+1+uw-u^2)w\right),

\Delta(y)=3+m^2+(u-w)^2,\qquad f_o(y)=6f(y)/\Delta(y).

Write $J=df$ , $H=d^2f$ , $J_o=df_o$ , $H_o=d^2f_o$ for their explicit Jacobian and Hessian expressions. The denominator derivatives are $d\Delta=(2m,2(u-w),2(w-u))$ and $d^2\Delta(v,z)=2v_0z_0+2(v_1-v_2)(z_1-z_2)$ . The exact original-time boundary flow and its projected variation are

\Phi(y,t)=\pi_s\Psi_0(\iota y,t),\qquad L(y,t)=\pi_s\mathcal J_{\iota y,0}(t)\iota.

For a fixed scale $K$ , set $\Phi_K(y,s)=\Phi(y,Ks)$ , $L_K(y,s)=L(y,Ks)$ and $Q_K(y,s)=d_yL_K(y,s)$ . For directions $v,z$ , define

S_K(y,s;v,z)=K\left[H_o(\Phi_K)(L_Kv,L_Kz)+J_o(\Phi_K)Q_K(v,z)\right].

All vector norms are sup norms, and operator norms are induced by them. Write $b(y)=|u|+|w|$ , $E(y)=e^{512b(y)}$ , $v_s=(v_1,v_2)$ , and

\mathcal H(y;v,z)=b(y)\|v\|\|z\|+\|v_s\|\|z\|+\|v\|\|z_s\|,

B_2(y)=192b(y)(40000E(y)^3)+10000E(y)^2.

The closed boundary region is $\mathcal S_c=\{(m,u,0,w):|m-c|\leq1/10,\ |u|,|w|\leq1/100\}$ . The boundary projection $P(m,u,w)$ is the limit of $\Phi(m,u,w;t)_0$ as $t\to\infty$ , using the total limit convention of the formal definitions.

For $y=(m,u,v,w)$ , define $p(y)=P(m,u,w)$ , $\rho(y)=\sqrt{(p(y)-1)^2+v^2}$ , $\omega(y)=m-1-u(v+w-u)$ , and $\chi(y)=(\rho(y),u,v,w)$ . Write $p_i=\partial_i p$ . The inverse differential $J_\chi(y)$ preserves the last three coordinates and has first row

\left(\frac{\rho}{(p-1)p_0},-\frac{p_1}{p_0}, -\frac{v}{(p-1)p_0},-\frac{p_3}{p_0}\right).

An admissible local inverse $\psi$ at $y$ is $C^2$ at $\chi(y)$ , maps $\chi(y)$ to $y$ , has derivative $J_\chi(y)$ there, and is a two-sided inverse to $\chi$ on neighborhoods of these points. For a scalar function $f$ , write $\bar\partial_i f=d(f\circ\psi)_{\chi(y)}e_i$ . The radial coefficients and drift are

N(y)=m-1-u(w-u)-(p-1)p_0,\qquad B(y)=\frac{N(y)}{\rho\omega},

A(y)=\frac{(p-1)((1-m)p_0+w(p_1-p_3))-v^2 -\tfrac12(p-1)vp_1}{\rho\omega},\qquad G(y)=uvA(y)+v^2B(y).

Set

S(y)=(p-1)((p-m)-(m-1)(p_0-1))+(p-1)w(p_1-p_3) -\tfrac12(p-1)vp_1,\quad H(y)=p-1-(m-1)p_0,

N_2(y)=(1-L^{-2})u^2+(-1+4L^{-2})uw-w^2/L^2,\qquad L=m+1,

R_e(y)=R_q(m,u,w),\quad N_4(y)=-R_e-(m-1)\partial_0R_e-(p-m)(p_0-1),\quad E(y)=\rho(y)-r(m,v).

Here

R_q(m,u,w)=P(m,u,w)- \left[m+\frac{mu^2}{m^2-1}-\frac{2uw}{m+1}+\frac{w^2}{2(m+1)}\right].

The logarithmic wall rates are

a_s=\frac{1-m^2-(v-w)^2+u^2}{2\omega},\qquad a_w=-1-\frac{2(1+u(v+w-u))}{\omega}.

Let $D_\chi f=df\,J_\chi$ , define

c(y)=(D_\chi G\,e_0,uD_\chi G\,e_1,wD_\chi G\,e_3),

\mathcal A_k(y)= \begin{pmatrix}uD_\chi a_s\,e_1&(w/k^2)D_\chi a_s\,e_3\\ k^2uD_\chi a_w\,e_1&wD_\chi a_w\,e_3\end{pmatrix},

and $\ell_k(y)=(\log u,k^2\log w)$ . All vector and operator norms use the sup norm. There exist $T,K,F_0,\varepsilon>0$ with the following property. Let $D(y)>0$ , $k\geq7/4$ , $s\leq t$ , and $Y(r)=\Psi_{t_0}(y,r)$ . On $[s,t]$ assume the normalized flow is in its maximal domain, $1+\alpha_{\Xi(y),t_0}(r)_0>0$ , $Y(r)\in\mathcal E_k$ , $Y(r)_2>0$ , and both stable walls are nonzero. Suppose their entry values $u_0,w_0$ are at most $\varepsilon$ . Choose $\theta=1$ , or $\theta=k^{-2}$ with $k(u_0+w_0)^2\leq1$ . For $h_2=0$ , let

V(r)=\mathcal J_{y,t_0}(r)h- \left(\frac{(\mathcal J_{y,t_0}(r)h)_2}{Y(r)_2}-\frac{h_2}{y_2}\right) \frac{F(Y(r))}{\omega(Y(r))},

X(r)=V(r)_1/Y(r)_1,\quad W(r)=V(r)_3/Y(r)_3,\quad Z(r)=d\rho_{Y(r)}V(r).

Put $v_0=Y(s)_2$ , $A_0=u_0|X(s)|+w_0|W(s)|$ ,

I_0=u_0v_0|X(s)|+ [u_0v_0w_0+\theta(v_0w_0)^2]|W(s)|,\quad B_0=KI_0/k,

\Lambda=F_0[u_0v_0+\theta(v_0w_0)^2]/k^2.

Then $0\leq\Lambda\leq1/2$ and, for every $r\in[s,t]$ ,

|Z(r)-Z(s)|\leq2B_0+2\Lambda|Z(s)|,\quad |Z(r)|\leq(1+2\Lambda)|Z(s)|+2B_0,

|X(r)-X(s)|+k^2|W(r)-W(s)| \leq T[A_0+2(\log Y(r)_2-\log v_0)(|Z(s)|+B_0)].

Preamble

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Dynamics_IntegralClock
import Definitions.Def_BianchiIX_Analysis_SignedRowGrowth
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_WallEvolution
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Algebra_ConnectionBounds
import Definitions.Def_BianchiIX_CompactPhaseExpansionData
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ProjectiveClock
import Definitions.Def_BianchiIX_Dynamics_Linearization
import Definitions.Def_BianchiIX_Analysis_ConnectionAmplitudeDifferential
import Definitions.Def_BianchiIX_Dynamics_ProjectiveOriginalTime
import Definitions.Def_BianchiIX_Dynamics_FlowComposition
import Definitions.Def_BianchiIX_Dynamics_FlowRegularity
import Definitions.Def_BianchiIX_Dynamics_FlowDifferential
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_EntrancePhaseResponseData
import Definitions.Def_BianchiIX_Dynamics_EntranceRegion
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.MeanValue
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.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Arsinh
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.SpecialFunctions.Arcosh
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
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.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.Topology.Order.Compact
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Analysis.SpecialFunctions.Pow.Real

noncomputable section
open BianchiIX
open Set MeasureTheory

Formal statement

theorem BianchiIX.exists_actual_entrance_phase_response :
    ∃ T > (0 : ℝ), ∃ K > (0 : ℝ), ∃ F > (0 : ℝ), ∃ ε > (0 : ℝ),
      ∀ (y : ProjectiveState) (t₀ s t k θ : ℝ),
      0 < projectiveStateDenom y → 7 / 4 ≤ k → s ≤ t →
      (∀ r ∈ Icc s t, r ∈ normalizedMaxDomain (fromProjectiveState y) t₀) →
      (∀ r ∈ Icc s t, 0 < 1 + normalizedMaxCurve (fromProjectiveState y) t₀ r 0) →
      let Y := projectiveOriginalFlow y t₀
      (∀ r ∈ Icc s t, entranceRegion k (Y r)) → (∀ r ∈ Icc s t, 0 < Y r 2) →
      (∀ r ∈ Icc s t, Y r 1 ≠ 0 ∧ Y r 3 ≠ 0) →
      Y s 1 ≤ ε → Y s 3 ≤ ε →
      (θ = 1 ∨ (θ = 1 / k ^ 2 ∧ k * (Y s 1 + Y s 3) ^ 2 ≤ 1)) →
      ∀ (w : ProjectiveState), w 2 = 0 →
      let v := fun r => entranceProjectedFlowVariation y t₀ r w
      let X := fun r => v r 1 / Y r 1
      let W := fun r => v r 3 / Y r 3
      let Z := fun r => fderiv ℝ entranceStableRadius (Y r) (v r)
      let A := Y s 1 * |X s| + Y s 3 * |W s|
      let I := Y s 1 * Y s 2 * |X s| +
        (Y s 1 * Y s 2 * Y s 3 + θ * (Y s 2 * Y s 3) ^ 2) * |W s|
      let B := K * I / k
      let Λ := F * (Y s 1 * Y s 2 + θ * (Y s 2 * Y s 3) ^ 2) / k ^ 2
      0 ≤ Λ ∧ Λ ≤ 1 / 2 ∧ ∀ r ∈ Icc s t,
        |Z r - Z s| ≤ 2 * B + 2 * Λ * |Z s| ∧
        |Z r| ≤ (1 + 2 * Λ) * |Z s| + 2 * B ∧
        |X r - X s| + k ^ 2 * |W r - W s| ≤
          T * (A + 2 * (Real.log (Y r 2) - Real.log (Y s 2)) * (|Z s| + B)) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/EntrancePhaseResponse.lean#L10-L138

View graph