If v = t K_λ with t, λ ≠ 0 (actual principal vector), then f v⁰/e = t, v¹ ≠ 0 and the actual lowered components 0, 3, 4 are -t p', -t u a', -t (1-u) a'

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geodesic.multiple_principal_facts

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

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

If v = t K_λ with t, λ ≠ 0 (actual principal vector), then f v⁰/e = t, v¹ ≠ 0 and the actual lowered components 0, 3, 4 are -t p', -t u a', -t (1-u) a'.

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_ChargedTail
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_CompactifiedBridge
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_CompactifiedResiduals
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_CoordinateAcceleration
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_FiniteJetObstruction
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_FiniteTaylorJet
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_GeodesicCongruenceObstruction
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_JetRecursion
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_MaxwellFrame
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_OneSidedExtraction
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_PaperGauge
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_PolynomialJet
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_PrincipalGeodesic
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_RadialBVP
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_RadialBVPEquivalence
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_RicciNullDirections
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_ScalarCertificate
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_SmoothBranch
open scoped ContDiff Topology Matrix
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Radial
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Compact
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.SmoothExt
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.NullDirections
open ExactVacuumAtlas.EinsteinMaxwell.D (einsteinMaxwellD EMCertificateD)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geodesic
variable {I : Set ℝ} {f m N w p a : ℝ → ℝ}

Formal statement

theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geodesic.multiple_principal_facts {y : Pt 5} (hf : f (y 1) ≠ 0) (hm : m (y 1) ≠ 0)
    (hN : N (y 1) ≠ 0) (he : eFun w p a (y 1) ≠ 0) {t lam : ℝ} (ht : t ≠ 0) (hl : lam ≠ 0)
    {v : Fin 5 → ℝ} (hv : v = t • principalCoord f m N w p a lam y) :
    f (y 1) * v 0 / eFun w p a (y 1) = t ∧ v 1 ≠ 0 ∧
      (metricMatrix (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w) y *ᵥ v) 0 =
        -(t * deriv p (y 1)) ∧
      (metricMatrix (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w) y *ᵥ v) 3 =
        -(t * (y 2 * deriv a (y 1))) ∧
      (metricMatrix (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w) y *ᵥ v) 4 =
        -(t * ((1 - y 2) * deriv a (y 1))) := 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/GeodesicCongruenceObstruction.lean#L57-L80; SHA-256 40c3447b42aa46d0a5a728dada1ccd4d602bdb3f202b0c97df6a35ba79c094e7. 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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