The decomposed connection phase rate and kernel derivatives

Definition

BianchiIX_Analysis_ConnectionPhaseKernel

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

bianchi-ix-era-statisticsdynamical-systems

For the positive reflected phase variable $m$ , put

n(a,b)=\sqrt{a^2+b^2},\quad j(a,b)=\frac1{n(a,b)(n(a,b)+a)},

\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+\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 $\kappa=u(v+w-u)$ , $s=m-1-\kappa$ , $b=(1-m^2-(v-w)^2+u^2)/2$ , and

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 $(\phi,\log u,\log v,\log w)$ , are

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 $i$ 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,\quad R=h(v-w)+(v+w-u)(Q-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 $C$ has rows $(G_i)_i$ , $(\alpha_i)_i$ and $(\beta_i)_i$ .

The phase rate is $G=uA+(vw)^2B$ . Its kernel part uses $s_i=L_i s$ and $B_i=k_i/s-ks_i/s^2$ .

Definition code

import Definitions.Def_BianchiIX_Algebra_ConnectionBounds
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Analysis_MatchingPhaseWeighted
import Definitions.Def_BianchiIX_Analysis_MatchingKernelVariation
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.Sqrt
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.ExpDeriv

noncomputable section

namespace BianchiIX

open Filter
open scoped Topology

def connectionPhaseA (m u v w : ℝ) : ℝ :=
  -(matchingPhaseSlope m v w * (v - w) + (v + w - u) * matchingPhaseWeight m v w) /
    connectionSpeed m u v w

def connectionPhaseB (m u v w : ℝ) : ℝ := matchingPhaseKernel m v w / connectionSpeed m u v w

def connectionPhaseG (m u v w : ℝ) : ℝ := matchingPhaseRate m u v w / connectionSpeed m u v w



def connectionSpeedPartials (m u v w : ℝ) : Fin 4 → ℝ :=
  ![1 / matchingPhaseSlope m v w, -u * (v + w - 2 * u),
    matchingPhaseVVariation m v w - u * v, matchingPhaseWVariation m v w - u * w]

def connectionPhaseBPartials (m u v w : ℝ) (i : Fin 4) : ℝ :=
  matchingPhaseKernelPartials m v w i / connectionSpeed m u v w -
    matchingPhaseKernel m v w * connectionSpeedPartials m u v w i / connectionSpeed m u v w ^ 2







end BianchiIX

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Analysis/ConnectionPhaseKernel.lean

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