Complexified negative Bianchi (a), sources kept

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_V03C3
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Cplx
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_V03Src
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Tan
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Var
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Local
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} {U : Set (Pt 4)} {τ : CTan} {x : Pt 4}
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
variable (hτ : KNCTan M a Q U τ)
include hτ

theorem ExactVacuumAtlas.EinsteinMaxwell.NP.c_bianchi_a_neg_src (hx : x ∈ U) :
    (-1) * fdir EK 1 (cPsi3 M a Q τ) x + fdir EK 2 (cPsi4 M a Q τ) x
        + (1/2) * fdir EK 1 (cRicF M a Q τ 1 3) x - (1/2) * fdir EK 3 (cRicF M a Q τ 1 1) x
      = (2 * npGamma gK giK EK ΘK x + 4 * npMu gK giK EK ΘK x) * cPsi3 M a Q τ x
        + (npTau gK giK EK ΘK x - 4 * npBeta gK giK EK ΘK x) * cPsi4 M a Q τ x
        + (-3 * psi2 gK giK EK x + (1/2) * ricF gK giK EK 0 1 x + (1/2) * ricF gK giK EK 2 3 x)
          * cNu M a Q τ x
        + (npAlpha gK giK EK ΘK x + npBetaB gK giK EK ΘK x - (1/2) * npTauB gK giK EK ΘK x)
          * cRicF M a Q τ 1 1 x
        - (npGamma gK giK EK ΘK x + npMuB gK giK EK ΘK x) * cRicF M a Q τ 1 3 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).