Main root (VII-08 contracted Ricci identity)

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geometry.vii08_ks_ricci_contraction

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

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

Main root (VII-08 contracted Ricci identity). Same chart, raw KSData on open U (non-constant, possibly non-diagonal background; no flatness, signature, v ≠ 0 or k ≠ 0): Ric_g(k,k) = Ric_b(k,k) − v · b(A,A) with A = accel b bi k, all actual registered objects.

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_KSRicci
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)}

Formal statement

theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Geometry.vii08_ks_ricci_contraction (hD : KSData b bi v k U) {x : Pt 5} (hx : x ∈ U) :
    contract2 (ricciOf (ksMetric b v k) (ksInverse bi v k)) x (k x) =
      contract2 (ricciOf b bi) x (k x) - v x * contract2 b x (accel b bi k x) := 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/KSRicci.lean#L315-L333; SHA-256 aafee68ecbc6cbecc405990bce4ac1e2aab6836af1a98ba510524447fce6f4ea. 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).

View graph