Prove2Me

First variation of a frame derivative

Proved
ExactVacuumAtlas.EinsteinMaxwell.NP.hasDerivAt_fdir_param

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

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

First variation of a frame derivative

For a jointly C^∞ family Φ and frame family E with ε-derivatives ψ and B: δ(D_A Φ)(x) = Σ_μ (B_A^μ ∂_μ Φ₀(x) + e_A^μ ∂_μ ψ(x)); the frame-variation term is kept.

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 标架导数的一阶变分（保留标架变分项）。

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_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
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
variable {n : ℕ}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.NP.hasDerivAt_fdir_param {E : ℝ → Fin n → Fin n → Pt n → ℂ} {B : Fin n → Fin n → Pt n → ℂ}
    {Φ : ℝ → Pt n → ℂ} {ψ : Pt n → ℂ} {W : Set (ℝ × Pt n)} {A : Fin n} (hW : IsOpen W)
    (hΦ : ContDiffOn ℝ ∞ (fun p : ℝ × Pt n => Φ p.1 p.2) W) {x : Pt n} (hx : ((0 : ℝ), x) ∈ W)
    (hB : ∀ μ, HasDerivAt (fun ε => E ε A μ x) (B A μ x) 0)
    (hψ : ∀ y, ((0 : ℝ), y) ∈ W → HasDerivAt (fun ε => Φ ε y) (ψ y) 0) :
    HasDerivAt (fun ε => fdir (E ε) A (Φ ε) x)
      (∑ μ, (B A μ x * cpdC μ (Φ 0) x + E 0 A μ x * cpdC μ 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/V03Joint.lean#L147-L157; SHA-256 9464f44ddafc9a1d1a85a42c012db1d6667cf2a8c76b4b99a4a754b380af6ae3. 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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