Induction step

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII17.Std.step

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

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

Induction step Inv k → Inv (k+1): Maxwell S (primitive gain), T and D (Euler gain), then F, b, Z via F′ = FZ/u, b′ = b·rootE, Z′ = zrhsE.

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_CompactifiedBridge
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_CompactifiedResiduals
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_FiniteJetObstruction
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_FiniteTaylorJet
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_JetRecursion
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_RadialBVP
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_RadialBVPEquivalence
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_ScalarCertificate
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_SmoothBranch
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_VII13FiniteBranchSpec
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_VII16AsymptoticBridge
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_VII17Algebra
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_VII17Analysis
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_VII17C6Bridge
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII17
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII17.Std
open Filter Set Topology ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Compact
open scoped ContDiff
variable {h Q J f₁ b₁ a₁ : ℝ} {F b A : ℝ → ℝ} (S : Std h Q J f₁ b₁ a₁ F b A)
include S

Formal statement

theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII17.Std.step (k : ℕ) (H : Std.Inv (h := h) (f₁ := f₁) (a₁ := a₁) (F := F) (b := b) (A := A) k) :
    Std.Inv (h := h) (f₁ := f₁) (a₁ := a₁) (F := F) (b := b) (A := A) (k + 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/VII17C6Bridge.lean#L370-L416; SHA-256 4db645ee1ed6a785adc2eecde4f7cdbe7ab576f706b3c7cd3dc8d264520bebdb. 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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