V-03 Kerr–Newman families

-- Generated by scripts/release05/gen_platform_tree.py (ATLAS-RELEASE-05) by skeleton subtraction from Atlas/EinsteinMaxwell/NP/V03LinKN.lean
-- definition bundle: kept ExactVacuumAtlas.EinsteinMaxwell.NP.KNFam
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_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
/-!
# Atlas V-03 (block C): first-order source identities around the Kerr–Newman background

`KNFam M a Q g gi F E Θ W` packages an `NPFamily` whose `ε = 0` member is the KN background of IV / V-01 / V-02
(`gKN`, `giKN`, `faradayKN`, Kinnersley `knE`, `knTh`) with slice `{y | (0, y) ∈ W} ⊆ domReg`. No member with `ε ≠ 0`
is assumed to solve anything. For such a family with tangent `(h, f, B)`, each of V-02's eight off-shell source
identities is differentiated at `ε = 0` (genuine derivatives: the identity holds on the open slice for every `ε`
near `0`); the KN background values (V-02, all proved) then remove exactly the products with a vanishing background
factor. The frame-variation terms `Σ b_A^μ ∂_μ f₀` of the derivatives are kept in the general rule
(`hasDerivAt_fdir_param`) and drop out here only because the differentiated background functions vanish on the open
slice (`hasDerivAt_fdir_param_of_zero`, proved).

The results (`lin_*`) are linear identities among the explicit first variations (`psi*Var`, `kappaVar`, …,
`ricFVar`, `HfrVar`, `JfrVar`, `FfrFVar`) with KN background coefficients. No field equation, gauge or mode condition is
used here; the source terms (`ricFVar`, `HfrVar`, `JfrVar`) are kept. -/

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)

namespace ExactVacuumAtlas.EinsteinMaxwell.NP

attribute [-simp] Complex.ofReal_cos Complex.ofReal_sin

/-- An `NPFamily` whose `ε = 0` member is the Kerr–Newman background (class B). -/
structure KNFam (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)) : Prop where
  fam : NPFamily g gi F E Θ W
  g0 : g 0 = gKN M a Q
  gi0 : gi 0 = giKN M a Q
  F0 : F 0 = faradayKN a Q
  E0 : E 0 = knE M a Q
  Θ0 : Θ 0 = knTh M a Q
  slice : ∀ y, ((0 : ℝ), y) ∈ W → y ∈ domReg M a Q

/-! ### Lambda-form derivative combinators and an automatic builder -/

section comb
variable {f₁ f₂ : ℝ → ℂ} {d₁ d₂ : ℂ}

end comb

namespace NPFamily

variable {g gi : ℝ → Metric 4} {F : ℝ → TwoForm 4} {E Θ : ℝ → Fin 4 → Fin 4 → Pt 4 → ℂ} {W : Set (ℝ × Pt 4)}

end NPFamily

section knvals

variable {M a Q : ℝ} {x : Pt 4}

end knvals

section lin

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

end lin

end ExactVacuumAtlas.EinsteinMaxwell.NP

• 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).