Root (actual null Ricci sum of squares)

Proved

ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.NullEnergy.vii09_actual_null_ricci_sos

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

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

Root (actual null Ricci sum of squares). At an actual domain point with f, m, N > 0, alphaSq > 0 and the actual Einstein–Maxwell equations, every actually null k satisfies Ric(k,k) = c₀L² + c₂ξ₂² + c₄ξ₄² with c₀ = 8; mu > 0; Ric(k,k) ≥ mu·misalign ≥ 0; and Ric(k,k) = 0 iff the three residuals vanish iff misalign = 0. The bound is pointwise in the fixed chart and numerical units; misalign is not an invariant or a physical observable.

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_MaxwellFrame
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_NullRicciEnergy
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_RicciNullDirections
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_ScalarCertificate
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_KerrSchild_SmoothBranch
open scoped Matrix
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.Radial
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.NullDirections
open ExactVacuumAtlas.EinsteinMaxwell.D (einsteinMaxwellD)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild
open ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.NullEnergy
variable {I : Set ℝ} {f m N w p a : ℝ → ℝ}

Formal statement

theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.KerrSchild.NullEnergy.vii09_actual_null_ricci_sos (hP : PaperData I f m N w p a) {x : Pt 5}
    (hx : x ∈ esDomain I) (hf : 0 < f (x 1)) (hm : 0 < m (x 1)) (hN : 0 < N (x 1))
    (ha2 : 0 < alphaSq f m N w p a (x 1))
    (hEin : ∀ i j, einsteinMaxwellD (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w)
      (esInverse f (Dfun f m N) (Afun f m) (Hfun m N) w) (esFaraday p a) i j x = 0)
    {k : Fin 5 → ℝ} (hnull : contract2 (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w) x k = 0) :
    contract2 (ricciOf (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w)
        (esInverse f (Dfun f m N) (Afun f m) (Hfun m N) w)) x k = nullSOS f m N w p a x k ∧
      coeff0 f m N (x 1) = 8 ∧ 0 < muCoeff f m N w p a x ∧
      muCoeff f m N w p a x * misalign f m N w p a x k ≤
        contract2 (ricciOf (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w)
          (esInverse f (Dfun f m N) (Afun f m) (Hfun m N) w)) x k ∧
      0 ≤ muCoeff f m N w p a x * misalign f m N w p a x k ∧
      (contract2 (ricciOf (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w)
          (esInverse f (Dfun f m N) (Afun f m) (Hfun m N) w)) x k = 0 ↔
        resL f m N w p a x k = 0 ∧ frameInv w x k 2 = 0 ∧ frameInv w x k 4 = 0) ∧
      (contract2 (ricciOf (esMetric f (Dfun f m N) (Afun f m) (Hfun m N) w)
          (esInverse f (Dfun f m N) (Afun f m) (Hfun m N) w)) x k = 0 ↔
        misalign f m N w p a x k = 0) := 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/NullRicciEnergy.lean#L208-L248; SHA-256 70e4af6f35074a925369db1c80c437687dda58ca8dd01131091fded7e18d928f. 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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