Key actual identity

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geometry.KSData.ksC_M

by He Wang · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-viiexact-vacuum-atlasfive-dimensional-gravitygeneral-relativitykerr-schild

Key actual identity: M^a_p = Σ_q C^a_{pq} k^q = ½[k(v) k^a ℓ_p + v A^a ℓ_p + v k^a a_p].

Scope. Supporting result in the formal proof chain of Atlas VII Unit A: the restricted geodesic Kerr–Schild obstruction on the actual asymptotic branch of the five-dimensional, $\Lambda = 0$ , source-free Einstein–Maxwell radial system with two equal-magnitude angular momenta. It asserts exactly the Lean statement of this node for the data stated there; it does not claim the existence of a solution, a horizon or a black hole.

Preamble

import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_D_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Cert
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Data
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Reduced
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Tables
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_CoordinateAcceleration
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_KSConnection
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_KSMetric
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_MaxwellFrame
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_PaperGauge
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_RadialBVP
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_RadialBVPEquivalence
import Definitions.Def_Atlas_Transport_Connection
open scoped ContDiff Topology Matrix
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.NullDirections (contract2)
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geodesic
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geometry
variable {b bi : Metric 5} {v : Pt 5 → ℝ} {k : Pt 5 → Fin 5 → ℝ} {U : Set (Pt 5)}
variable (hD : KSData b bi v k U)
include hD

Formal statement

theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geometry.KSData.ksC_M {x : Pt 5} (hx : x ∈ U) (a p : Fin 5) :
    ∑ q, ksC b bi v k a p q x * k x q =
      (dprof v k x * k x a * lowerK b k x p + v x * accel b bi k x a * lowerK b k x p
        + v x * k x a * accL b bi k x p) / 2 := by sorry

Source

He Wang, Exact Spacetime Atlas VII, Unit A (restricted Kerr–Schild obstruction on the actual asymptotic branch, VII17). Local source, no public repository; commit f64ba9c823ec; Atlas/EinsteinMaxwell/ES5/KerrSchild/KSConnection.lean#L354-L378; SHA-256 960e18dab71d5cb1d6ea0bc530be65229e8dc8727bb3912bd4c64612ec1c6996. Registered definitions: mission/PHYSICS_BOUNDARY.md (Atlas VII sections P–XVII). Physical Kerr–Schild obstruction: Fröb–Khavkine–Málek–Pravda, Eur. Phys. J. C 82 (2022) 215, arXiv:2112.13266 (Eqs. 23–25, 33, 59–60).

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