Prove2Me

Variation of a frame derivative of a family vanishing on the KN slice

Proved
ExactVacuumAtlas.EinsteinMaxwell.NP.KNFam.hasDerivAt_fdir_zero

by He Wang · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-vexact-vacuum-atlasgeneral-relativitynewman-penrose

Variation of a frame derivative of a family vanishing on the KN slice

For a KNFam and a jointly C^∞ Φ with Φ₀ ≡ 0 on domReg: δ(D_A Φ)(x) = D_A ψ (x) at Kerr–Newman.

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 KN 切片上为零的族：标架导数的变分只剩 D_A ψ。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Domain
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Bianchi
import Definitions.Def_Atlas_EinsteinMaxwell_NP_CCalc
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameConn
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameNull
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNComm
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFaraday
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFrame
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNSpin
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNWeyl
import Definitions.Def_Atlas_EinsteinMaxwell_NP_ModeOperators
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPBianchi
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPMaxwell
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPRicci
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPScalars
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Riemann
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Family
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03LinKN
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Realize
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Var
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Defs
import Definitions.Def_Atlas_Kerr_InAtoms
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
import Definitions.Def_Hintz2026_Core
import Definitions.Def_Hintz2026_Domain
import Definitions.Def_Hintz2026_System1
import Definitions.Def_Hintz2026_System4
open Complex Filter Topology
open scoped ComplexConjugate Matrix ContDiff
open ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.Variation
open ExactVacuumAtlas.EinsteinMaxwell.KN (Sig Del gKN giKN domReg domKN faradayKN)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
variable {M a Q : ℝ} {g gi : ℝ → Metric 4} {F : ℝ → TwoForm 4} {E Θ : ℝ → Fin 4 → Fin 4 → Pt 4 → ℂ}
  {W : Set (ℝ × Pt 4)} {h : Metric 4} {f : TwoForm 4} {B : Fin 4 → Fin 4 → Pt 4 → ℂ}
local notation "gK" => gKN M a Q
local notation "giK" => giKN M a Q
local notation "EK" => knE M a Q
local notation "ΘK" => knTh M a Q
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.NP.KNFam.hasDerivAt_fdir_zero (hK : KNFam M a Q g gi F E Θ W) (ht : NPTangent g F E W h f B) {x : Pt 4}
    (hx0 : ((0 : ℝ), x) ∈ W) (A : Fin 4) {Φ : ℝ → Pt 4 → ℂ} {ψ : Pt 4 → ℂ}
    (hΦ : ContDiffOn ℝ ∞ (fun p : ℝ × Pt 4 => Φ p.1 p.2) W)
    (hψ : ∀ y, ((0 : ℝ), y) ∈ W → HasDerivAt (fun ε => Φ ε y) (ψ y) 0)
    (h0 : ∀ y ∈ domReg M a Q, Φ 0 y = 0) :
    HasDerivAt (fun ε => fdir (E ε) A (Φ ε) x) (fdir EK A ψ x) 0 := by sorry
Source
He Wang, Exact Spacetime Atlas V: From Linearized Einstein–Maxwell Equations to Kerr–Newman Radiation Systems. Local source, no public repository; commit b8cd107a558e153ed41fb4972ac5315499d14778; Atlas/EinsteinMaxwell/NP/V03LinKN.lean#L126-L136; SHA-256 9c8ef1119adf30602b657e281877a44735f55b9af3e5d202336d742bdd8d3197. Registered definitions: mission/PHYSICS_BOUNDARY.md (§K–§M); literature: P. Hintz, arXiv:2609.33661 (v1); Mark–Yang–Zimmerman–Chen, arXiv:1409.5800 (v1).
Human review

  • Endorsed by Shuze Chen · Oct 24, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 24, 2026

    Confirmed by the mission captain (proposal self-audit).

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