Prove2Me

Frame Weyl decomposition

Proved
ExactVacuumAtlas.EinsteinMaxwell.NP.NullFrameData.weylF_eq

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

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

Frame Weyl decomposition

For a null frame (no field equation): C_{IJAB} = R_{IJAB} − ½(η_{IA}R_{JB} − η_{IB}R_{JA} − η_{JA}R_{IB} + η_{JB}R_{IA}) + (R/6)(η_{IA}η_{JB} − η_{IB}η_{JA}).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 标架 Weyl 分解（无场方程）。

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_CCalc
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameConn
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameNull
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFrame
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Riemann
import Definitions.Def_Atlas_Kerr_InAtoms
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff Matrix
open Complex
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
open ExactVacuumAtlas.EinsteinMaxwell.NP.NullFrameData
variable {g gi : Metric 4} {E Θ : Fin 4 → Fin 4 → Pt 4 → ℂ} {U : Set (Pt 4)}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.NP.NullFrameData.weylF_eq (hN : NullFrameData g gi E Θ U) {x : Pt 4} (hx : x ∈ U) (I J A B : Fin 4) :
    weylF g gi E I J A B x = riemFL g gi E Θ I J A B x
      - (etaNull I A * ricF g gi E J B x - etaNull I B * ricF g gi E J A x
        - etaNull J A * ricF g gi E I B x + etaNull J B * ricF g gi E I A x) / 2
      + (ricciScalar g gi x : ℂ) * (etaNull I A * etaNull J B - etaNull I B * etaNull J A) / 6 := 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/FrameNull.lean#L353-L370; SHA-256 89b93964d08a58f81f17ec29d939043c67b8a5c7a9a375bb5c15b99bf70dff0a. 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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