MVT witness

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.VII16R.mvt_bound

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

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

MVT witness: a value bound |g x − c₁x| ≤ Cx² gives, below each x, a point where the derivative is bounded by a constant (no derivative hypothesis beyond differentiability).

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.mvt_bound {g g' : ℝ → ℝ} {c₁ C δ : ℝ}
    (hd : ∀ x ∈ Ioo (0:ℝ) δ, HasDerivAt g (g' x) x)
    (hb : ∀ x ∈ Ioo (0:ℝ) δ, |g x - c₁ * x| ≤ C * x ^ 2) :
    ∀ x ∈ Ioo (0:ℝ) δ, ∃ ξ ∈ Ioo (0:ℝ) x, |g' ξ| ≤ |c₁| + 3 * |C| * δ := 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#L207-L241; 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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