Step C

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII16R.stepC

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

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

Step C. From the real realPC residual (incl. the J-term): b′ → b₁ and f₁ − 2b₁ = h. The constraint is an exact quadratic u q² + β q + x·ρ = 0 in q = x b′/b (e3_core), with ρ → c = 2h − 2f₁ + 2b₁. q = −1 is excluded near 0; by continuity q stays on one side of −1 (K4); q < −1 would make x·b decreasing, contradicting x·b → 0⁺; on q > −1 the quadratic forces q/x → −c/2, and the value bound identifies −c/2 = b₁.

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_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
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild
open Filter Set Topology
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Compact
open Filter Set Topology ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Compact

Formal statement

theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII16R.stepC {h Q J : ℝ} {F b A : ℝ → ℝ} {f₁ a₁ b₁ C_F C_b C_A δ : ℝ} (hδ : 0 < δ)
    (hu : ∀ x ∈ Ioo (0:ℝ) δ, 1 / 2 ≤ 1 - h * x)
    (hF0 : ∀ x ∈ Ioo (0:ℝ) δ, 0 < F x) (hb0 : ∀ x ∈ Ioo (0:ℝ) δ, 0 < b x)
    (hvalF : ∀ x ∈ Ioo (0:ℝ) δ, |F x - (1 + f₁ * x)| ≤ C_F * x ^ 2)
    (hvalb : ∀ x ∈ Ioo (0:ℝ) δ, |b x - (1 + b₁ * x)| ≤ C_b * x ^ 2)
    (hvalA : ∀ x ∈ Ioo (0:ℝ) δ, |A x - a₁ * x| ≤ C_A * x ^ 2)
    (hb1d : ∀ x ∈ Ioo (0:ℝ) δ, HasDerivAt b (deriv b x) x)
    (hb2d : ∀ x ∈ Ioo (0:ℝ) δ, HasDerivAt (deriv b) (deriv (deriv b) x) x)
    (hresC : ∀ x ∈ Ioo (0:ℝ) δ, realPC h Q J F b A x = 0)
    (hAlim : Tendsto (deriv A) (𝓝[>] (0:ℝ)) (𝓝 a₁))
    (hFlim : Tendsto (deriv F) (𝓝[>] (0:ℝ)) (𝓝 f₁)) :
    Tendsto (deriv b) (𝓝[>] (0:ℝ)) (𝓝 b₁) ∧ f₁ - 2 * b₁ = h := 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/VII16RFirstOrder.lean#L478-L682; SHA-256 c37ebd91fc832f0ca92cc961a78b3d0265b3e23c4aed113f53001c44db1c66e9. 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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