Prove2Me

The exact strong-wall derivative has the ordinary amplitude scale

Proved
BianchiIX.connectionLog_actual_strong_variation_bounds

by trppppp · Nov 5, 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.

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).

For the positive reflected phase variable mm, put

n(a,b)=a2+b2,j(a,b)=1n(a,b)(n(a,b)+a),n(a,b)=\sqrt{a^2+b^2},\quad j(a,b)=\frac1{n(a,b)(n(a,b)+a)}, ϕ=n(m+1,v)+n(m−1,w)−m,h=m+1n(m+1,v)+m−1n(m−1,w)−1,\phi=n(m+1,v)+n(m-1,w)-m,\quad h=\frac{m+1}{n(m+1,v)}+\frac{m-1}{n(m-1,w)}-1, Q=mh+v2n(m+1,v)+w2n(m−1,w),k=j(m+1,v)−j(m−1,w).Q=mh+\frac{v^2}{n(m+1,v)}+\frac{w^2}{n(m-1,w)},\quad k=j(m+1,v)-j(m-1,w).

Write κ=u(v+w−u)\kappa=u(v+w-u), s=m−1−κs=m-1-\kappa, b=(1−m2−(v−w)2+u2)/2b=(1-m^2-(v-w)^2+u^2)/2, and

Mv=−v2n(m+1,v)h,Mw=−w2n(m−1,w)h.M_v=-\frac{v^2}{n(m+1,v)h},\qquad M_w=-\frac{w^2}{n(m-1,w)h}.

The four coordinate differential operators, in the order (ϕ,log⁡u,log⁡v,log⁡w)(\phi,\log u,\log v,\log w), are

L0=h−1∂m,L1=u∂u,L2=Mv∂m+v∂v,L3=Mw∂m+w∂w.L_0=h^{-1}\partial_m,\quad L_1=u\partial_u,\quad L_2=M_v\partial_m+v\partial_v,\quad L_3=M_w\partial_m+w\partial_w.

Subscripts ii below denote the rational expressions obtained by these rules, with total real-field division. On the regular positive-wall region they are the corresponding coordinate partial derivatives. Set

N=h(v−w)+(v+w−u)Q,R=h(v−w)+(v+w−u)(Q−s),N=h(v-w)+(v+w-u)Q,\quad R=h(v-w)+(v+w-u)(Q-s), A=−N/s,B=k/s,G=uA+(vw)2B,α=b/s,β=−1−2(1+κ)/s.A=-N/s,\quad B=k/s,\quad G=uA+(vw)^2B,\quad \alpha=b/s,\quad \beta=-1-2(1+\kappa)/s.

The coefficient matrix CC has rows (Gi)i(G_i)_i, (αi)i(\alpha_i)_i and (βi)i(\beta_i)_i.

For a projective state yy, evaluate the reflected formulas at (m,u,v,w)=(−y0,y1,y2,y3)(m,u,v,w)=(-y_0,y_1,y_2,y_3). Define

χ(y)=(ϕ,log⁡y1,log⁡y2,log⁡y3),\chi(y)=(\phi,\log y_1,\log y_2,\log y_3), Tyξ=(−hξ0+y2ξ2n(−y0+1,y2)+y3ξ3n(−y0−1,y3),ξ1y1,ξ2y2,ξ3y3).\mathcal T_y\xi=\left(-h\xi_0+ \frac{y_2\xi_2}{n(-y_0+1,y_2)}+ \frac{y_3\xi_3}{n(-y_0-1,y_3)}, \frac{\xi_1}{y_1},\frac{\xi_2}{y_2},\frac{\xi_3}{y_3}\right).

Let Ayd=(∑iC0idi,∑iC1idi,∑iC2idi,0)\mathcal A_y d=(\sum_i C_{0i}d_i,\sum_i C_{1i}d_i, \sum_i C_{2i}d_i,0) and Wμd=(d1,μ2d2)W_\mu d=(d_1,\mu^2d_2), with the sup norm. The original-time clock rate is Ω(y)=6ω−(y)/D(y)\Omega(y)=6\omega_-(y)/D(y). For Y=Ψt0(y,t)Y=\Psi_{t_0}(y,t) and ξ=dyΨt0(y,t)v\xi=d_y\Psi_{t_0}(y,t)v, set

Py,t0,tv=ξ−(ξ3/Y3−v3/y3)H(Y),Dy,t0,tv=TY(Py,t0,tv).\mathcal P_{y,t_0,t}v=\xi-(\xi_3/Y_3-v_3/y_3)H(Y),\qquad \mathcal D_{y,t_0,t}v=\mathcal T_Y(\mathcal P_{y,t_0,t}v).

Thus D\mathcal D is the logarithmic variation at fixed relative growing-wall clock.

For entry amplitudes u,pu,p, define the feedback budget

Λμ(u,p)=64(2uμ3+p24μ3)+8192(4uμ2+p216μ3).\Lambda_\mu(u,p)=64\left(\frac{2u}{\mu^3}+\frac{p^2}{4\mu^3}\right) +8192\left(\frac{4u}{\mu^2}+\frac{p^2}{16\mu^3}\right).

Let a≤ba\leq b, μ≥4\mu\geq4, D(y)>0D(y)>0, and q(a)=t0q(a)=t_0. For every z∈[a,b]z\in[a,b], suppose q(z)∈Imax⁡(Ξ(y),t0)q(z)\in I_{\max}(\Xi(y),t_0), 1+(αΞ(y),t0(q(z)))0>01+(\alpha_{\Xi(y),t_0}(q(z)))_0>0, and Y(z)=Ψt0(y,q(z))∈CμY(z)=\Psi_{t_0}(y,q(z))\in\mathcal C_\mu with all three walls positive. Assume q′(z)=1/Ω(Y(z))q'(z)=1/\Omega(Y(z)), y1≤y2y_1\leq y_2, y1≤1/(10000μ)y_1\leq1/(10000\mu) and y3≤1/10000y_3\leq1/10000. Write Dt=Dy,t0,q(t)\mathcal D_t=\mathcal D_{y,t_0,q(t)} and τ=t−a\tau=t-a. For eu=(0,1,0,0)e_u=(0,1,0,0) and every t∈[a,b]t\in[a,b],

∣(Dteu)0∣≤70/μ,|(\mathcal D_te_u)_0|\leq70/\mu, ∣(Dteu)1∣≤1/y1+256+8960τ/μ,∣(Dteu)2∣≤(256+8960τ/μ)/μ2.|(\mathcal D_te_u)_1|\leq1/y_1+256+8960\tau/\mu,\qquad |(\mathcal D_te_u)_2|\leq(256+8960\tau/\mu)/\mu^2.
Preamble
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Algebra_ConnectionBounds
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Analysis_MatchingPhaseWeighted
import Definitions.Def_BianchiIX_Dynamics_Linearization
import Definitions.Def_BianchiIX_Analysis_ConnectionAmplitudeDifferential
import Definitions.Def_BianchiIX_Analysis_MatchingKernelDifferential
import Definitions.Def_BianchiIX_Analysis_MatchingKernelVariation
import Definitions.Def_BianchiIX_Analysis_ConnectionPhaseKernel
import Definitions.Def_BianchiIX_Dynamics_ProjectiveOriginalTime
import Definitions.Def_BianchiIX_Analysis_MatchingPhaseWeightDifferential
import Definitions.Def_BianchiIX_Dynamics_FlowDifferential
import Definitions.Def_BianchiIX_Analysis_MatchingPhaseWallDifferential
import Definitions.Def_BianchiIX_Analysis_ConnectionPhaseNumerator
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Analysis_ConnectionPhaseLinearization
import Definitions.Def_BianchiIX_Dynamics_ConnectionClock
import Definitions.Def_BianchiIX_Dynamics_ConnectionClockDifferential
import Definitions.Def_BianchiIX_Dynamics_ConnectionFlowVariation
import Definitions.Def_BianchiIX_Analysis_ConnectionCoefficientMatrix
import Definitions.Def_BianchiIX_Analysis_ConnectionLogCoordinates
import Definitions.Def_BianchiIX_Analysis_ConnectionLogLinearization
import Definitions.Def_BianchiIX_Analysis_ConnectionWeightedVariation
import Definitions.Def_BianchiIX_Dynamics_ConnectionLogFlowVariation
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
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Module
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Topology.Order.ProjIcc
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
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

noncomputable section
open BianchiIX
open Set
Formal statement
theorem BianchiIX.connectionLog_actual_strong_variation_bounds {y : ProjectiveState} {q : ℝ → ℝ}
    {t₀ μ a b : ℝ} (hab : a ≤ b) (hμ : 4 ≤ μ) (hden : 0 < projectiveStateDenom y)
    (hdom : ∀ z ∈ Icc a b, q z ∈ normalizedMaxDomain (fromProjectiveState y) t₀)
    (hchart : ∀ z ∈ Icc a b, 0 < 1 + normalizedMaxCurve (fromProjectiveState y) t₀ (q z) 0)
    (hm : ∀ z ∈ Icc a b, |-projectiveOriginalFlow y t₀ (q z) 0 - μ| ≤ 1 / 2)
    (hu : ∀ z ∈ Icc a b, 0 < projectiveOriginalFlow y t₀ (q z) 1 ∧
      projectiveOriginalFlow y t₀ (q z) 1 ≤ 1 / 10000)
    (hv : ∀ z ∈ Icc a b, 0 < projectiveOriginalFlow y t₀ (q z) 2 ∧
      projectiveOriginalFlow y t₀ (q z) 2 ≤ 1)
    (hw : ∀ z ∈ Icc a b, 0 < projectiveOriginalFlow y t₀ (q z) 3 ∧
      projectiveOriginalFlow y t₀ (q z) 3 ≤ 1)
    (hentry : y 1 ≤ y 2) (hsmall : y 1 ≤ (1 / 10000) / μ) (hwall : y 3 ≤ 1 / 10000)
    (hq : ∀ z ∈ Icc a b,
      HasDerivAt q (1 / connectionOriginalClockRate (projectiveOriginalFlow y t₀ (q z))) z)
    (hqa : q a = t₀) :
    ∀ t ∈ Icc a b,
      |connectionLogFlowVariation y t₀ (q t) ![0, 1, 0, 0] 0| ≤ 70 / μ ∧
      |connectionLogFlowVariation y t₀ (q t) ![0, 1, 0, 0] 1| ≤ 1 / y 1 + 256 + 8960 * (t - a) / μ ∧
      |connectionLogFlowVariation y t₀ (q t) ![0, 1, 0, 0] 2| ≤ (256 + 8960 * (t - a) / μ) / μ ^ 2 := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ConnectionActualStrongFeedback.lean#L10-L67

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