Linearised negative Maxwell combination at KN

import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Domain
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Bianchi
import Definitions.Def_Atlas_EinsteinMaxwell_NP_CCalc
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameConn
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameNull
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNComm
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFaraday
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFrame
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNSpin
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNWeyl
import Definitions.Def_Atlas_EinsteinMaxwell_NP_ModeOperators
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPBianchi
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPMaxwell
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPRicci
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPScalars
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Riemann
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Family
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03LinKN
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03LinMx
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Realize
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Var
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Defs
import Definitions.Def_Atlas_Kerr_InAtoms
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
import Definitions.Def_Hintz2026_Core
import Definitions.Def_Hintz2026_Domain
import Definitions.Def_Hintz2026_System1
import Definitions.Def_Hintz2026_System4
open Complex Filter Topology
open scoped ComplexConjugate Matrix ContDiff
open ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.Variation
open ExactVacuumAtlas.EinsteinMaxwell.KN (Sig Del gKN giKN domReg domKN faradayKN)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
variable {M a Q : ℝ} {g gi : ℝ → Metric 4} {F : ℝ → TwoForm 4} {E Θ : ℝ → Fin 4 → Fin 4 → Pt 4 → ℂ}
  {W : Set (ℝ × Pt 4)} {h : Metric 4} {f : TwoForm 4} {B : Fin 4 → Fin 4 → Pt 4 → ℂ}
local notation "gK" => gKN M a Q
local notation "giK" => giKN M a Q
local notation "EK" => knE M a Q
local notation "ΘK" => knTh M a Q
local notation "FK" => faradayKN a Q

theorem ExactVacuumAtlas.EinsteinMaxwell.NP.lin_maxwell_combo_neg (hK : KNFam M a Q g gi F E Θ W) (ht : NPTangent g F E W h f B) {x : Pt 4}
    (hx0 : ((0 : ℝ), x) ∈ W) :
    fdir EK 3 (srcS4 FK f gK giK h EK ΘK B) x
        + (2 * npPi gK giK EK ΘK x + npAlpha gK giK EK ΘK x + npBetaB gK giK EK ΘK x - npTauB gK giK EK ΘK x)
          * srcS4 FK f gK giK h EK ΘK B x
      - (fdir EK 1 (srcS3 FK f gK giK h EK ΘK B) x
        + (2 * npMu gK giK EK ΘK x + npMuB gK giK EK ΘK x + npGamma gK giK EK ΘK x - npGammaB gK giK EK ΘK x)
          * srcS3 FK f gK giK h EK ΘK B x)
      = 2 * phi1 EK x FK * (2 * psi3Var gK giK h EK B x + fdir EK 0 (fun y => omLVar gK h EK B 1 1 3 y) x
            - fdir EK 2 (fun y => omLVar gK h EK B 1 3 3 y) x
            + (npPiB gK giK EK ΘK x + npTau gK giK EK ΘK x + 3 * npBeta gK giK EK ΘK x - npAlphaB gK giK EK ΘK x)
              * lambdaVar gK h EK B x
            - (3 * npEps gK giK EK ΘK x + npEpsB gK giK EK ΘK x + npRho gK giK EK ΘK x - npRhoB gK giK EK ΘK x)
              * nuVar gK h EK B x)
        - nuVar gK h EK B x * fdir EK 0 (fun z => phi1 EK z FK) x
        + lambdaVar gK h EK B x * fdir EK 2 (fun z => phi1 EK z FK) x := by sorry

• Endorsed by Shuze Chen · Oct 24, 2026

  Confirmed by the moderator at approval.

• Endorsed by He Wang · Oct 24, 2026

  Confirmed by the mission captain (proposal self-audit).