Linearised NP Bianchi (b) at KN, 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_V03Family
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03LinKN
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

theorem ExactVacuumAtlas.EinsteinMaxwell.NP.lin_bianchi_b (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 1 (psi0Var gK giK h EK B) x - fdir EK 2 (psi1Var gK giK h EK B) x
        + (1/2) * fdir EK 0 (ricFVar gK giK h EK B 2 2) x - (1/2) * fdir EK 2 (ricFVar gK giK h EK B 0 2) x
      = (-2 * npBeta gK giK EK ΘK x - 4 * npTau gK giK EK ΘK x) * psi1Var gK giK h EK B x
        + (4 * npGamma gK giK EK ΘK x - npMu gK giK EK ΘK x) * psi0Var gK giK h EK B 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)
          * sigmaVar gK h EK B x
        + (npPiB gK giK EK ΘK x - npBeta gK giK EK ΘK x) * ricFVar gK giK h EK B 0 2 x
        + (1/2) * npRhoB gK giK EK ΘK x * ricFVar gK giK h EK B 2 2 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).