Prove2Me

The full moving reference flight remains in the flow domain and chart

Proved
BianchiIX.eventually_referenceCore_interval_regular

by trppppp · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

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.

For real k,tk,t, define the boundary pulse

βk(t)=(mk(t),0,vk(t),0)=(1−ktanh⁡(kt),0,kcosh⁡(kt),0).\beta_k(t)=(m_k(t),0,v_k(t),0) =\left(1-k\tanh(kt),0,\frac{k}{\cosh(kt)},0\right).

Set

hk=arcosh⁡kk,Tk=(k2+4)arcosh⁡k3k,dk=k2−1,h_k=\frac{\operatorname{arcosh}k}{k},\qquad T_k=\frac{(k^2+4)\operatorname{arcosh}k}{3k},\qquad d_k=\sqrt{k^2-1}, Hk(t)=k2+46t−13log⁡cosh⁡(kt).H_k(t)=\frac{k^2+4}{6}t-\frac13\log\cosh(kt).

Use the total real conventions x/0=0x/0=0, x=max⁡(x,0)\sqrt{x}=\sqrt{\max(x,0)}, log⁡0=0\log0=0, and log⁡x=log⁡∣x∣\log x=\log|x| for x<0x<0, with arcosh⁡x=log⁡(x+x2−1)\operatorname{arcosh}x=\log(x+\sqrt{x^2-1}). Write αx,0\alpha_{x,0} for the maximal normalized solution starting at xx at time zero, with domain Imax⁡(x,0)I_{\max}(x,0).

The positive transverse modes and crossing multipliers are

Mk,r(t)=e−rtcosh⁡(kt),μu(k)=e−karcosh⁡k,μw(k)=e−4arcosh⁡k/k.M_{k,r}(t)=e^{-rt}\cosh(kt),\qquad \mu_u(k)=e^{-k\operatorname{arcosh}k},\qquad \mu_w(k)=e^{-4\operatorname{arcosh}k/k}.

Define the variational velocity with tangent forcing by

Vk(t,ξ,a)=dFβk(t)ξ+aF(βk(t)).V_k(t,\xi,a)=dF_{\beta_k(t)}\xi+aF(\beta_k(t)).

For ξ=(ξm,ξu,ξv,ξw)\xi=(\xi_m,\xi_u,\xi_v,\xi_w), set

Jk(t,ξ)=2(mk(t)−1)ξm+2vk(t)ξv−4vk(t)ξu,J_k(t,\xi)=2(m_k(t)-1)\xi_m+2v_k(t)\xi_v-4v_k(t)\xi_u, Pkξ=(ξm−ξv/dk,ξu,0,ξw).P_k\xi=(\xi_m-\xi_v/d_k,\xi_u,0,\xi_w).

Write G(y)=6F(y)/D(y)G(y)=6F(y)/D(y) for the projective vector field in normalized time.

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.

Define xk−=Ξ(βk(−hk))x_k^-=\Xi(\beta_k(-h_k)), xk+=Ξ(βk(hk))x_k^+=\Xi(\beta_k(h_k)), and the active-wall observable g(x)=(Π(x))2g(x)=(\Pi(x))_2, with zero-based indices. For x∗=αx,t0(t)x_*=\alpha_{x,t_0}(t), set

ρx,t0,t=dgx∗(N(x∗)),Lx,t0,t=−ρx,t0,t−1dgx∗∘Jx,t0(t),\rho_{x,t_0,t}=dg_{x_*}(N(x_*)),\qquad \mathscr L_{x,t_0,t}=-\rho_{x,t_0,t}^{-1}dg_{x_*}\circ J_{x,t_0}(t), Hx,t0,t(v)=Jx,t0(t)v+Lx,t0,t(v)N(x∗).\mathscr H_{x,t_0,t}(v)=J_{x,t_0}(t)v+ \mathscr L_{x,t_0,t}(v)N(x_*).

Here 0−1=00^{-1}=0 in the total definitions.

Define the clock based at entry and the actual projective variation by

ck(t)=Hk(t)−Hk(−hk),Wk(w,t)=Jβk(−hk),0(ck(t))w.c_k(t)=H_k(t)-H_k(-h_k),\qquad W_k(w,t)=\mathcal J_{\beta_k(-h_k),0}(c_k(t))w.

Let Z=R3Z=\mathbb R^3 with its finite-product norm and define

e(z0,z1,z2)=(z0,z1,1,z2),q(m,u,v,w)=(m,u,w),χ=Ξ∘e,e(z_0,z_1,z_2)=(z_0,z_1,1,z_2),\qquad q(m,u,v,w)=(m,u,w),\qquad \chi=\Xi\circ e, zk−=(1+dk,0,0),zk+=(1−dk,0,0).z_k^-=(1+d_k,0,0),\qquad z_k^+=(1-d_k,0,0).

For k>1k>1, let

Ak(v0,v1,v2)=(−v0+2dk(1−μu(k))v1,μu(k)v1,μw(k)v2).A_k(v_0,v_1,v_2)= \left(-v_0+\frac2{d_k}(1-\mu_u(k))v_1,\mu_u(k)v_1,\mu_w(k)v_2\right).

Suppose k>1k>1 and τ:Z→R\tau:Z\to\mathbb R is continuous at zk−z_k^- with τ(zk−)=Tk\tau(z_k^-)=T_k. Then for all zz sufficiently near zk−z_k^- and every t∈[0,τ(z)]t\in[0,\tau(z)],

t∈Imax⁡(χ(z),0),1+(αχ(z),0(t))0>0.t\in I_{\max}(\chi(z),0),\qquad 1+(\alpha_{\chi(z),0}(t))_0>0.
Preamble
import Definitions.Def_BianchiIX_Dynamics_IntegralClock
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Dynamics_ReferencePulse
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ReferenceReturn
import Definitions.Def_BianchiIX_Dynamics_FlowRegularity
import Definitions.Def_BianchiIX_Dynamics_ReferenceEvents
import Definitions.Def_BianchiIX_Dynamics_ReferenceActualVariation
import Definitions.Def_BianchiIX_Dynamics_ReferenceSectionMap
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.Linarith
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Ring
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.Calculus.Deriv.Inv
import Mathlib.Tactic.FieldSimp
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.Calculus.ImplicitContDiff

noncomputable section
open BianchiIX
open Set Filter
open scoped Topology
Formal statement
theorem BianchiIX.eventually_referenceCore_interval_regular {k : ℝ} (hk : 1 < k)
    {tau : CoreCoordinates → ℝ}
    (hτ : tau (referenceCoreInput k) = referenceNormalizedTime k)
    (hcτ : ContinuousAt tau (referenceCoreInput k)) :
    ∀ᶠ z in 𝓝 (referenceCoreInput k), ∀ t ∈ Icc 0 (tau z),
      t ∈ normalizedMaxDomain (coreSectionChart z) 0 ∧
        0 < 1 + normalizedMaxCurve (coreSectionChart z) 0 t 0 := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ReferenceFirstReturn.lean#L119-L139

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