V-03 zero complex tangent

-- Generated by scripts/release05/gen_platform_tree.py (ATLAS-RELEASE-05) by skeleton subtraction from Atlas/EinsteinMaxwell/NP/V03Main.lean
-- definition bundle: kept ExactVacuumAtlas.EinsteinMaxwell.NP.CTan.zero
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_V03Mode
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Realize
import Definitions.Def_Atlas_EinsteinMaxwell_NP_V03Res
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_Chiral
import Definitions.Def_Hintz2026_Core
import Definitions.Def_Hintz2026_Domain
import Definitions.Def_Hintz2026_System1
import Definitions.Def_Hintz2026_System4
/-!
# Atlas V-03 (block D, part 3): composition with III's main theorems; a non-empty instance

* `pos_sys1` / `neg_sys1`: under the hypotheses of the V-03 contract only — Kerr–Newman background parameters
  (`p : Hintz2026.SubextremalKN`, domain `domKN`), a complex tangent whose real parts are compatible real tangents
  (`KNCTan`), IV's linearised Einstein–Maxwell system complexified (`CLinEM`), the explicit complex phantom gauge
  (`δΦ₀^ℂ = 0` positive / `δΦ₂^ℂ = 0` negative) and the single-mode condition (`PosMode` / `NegMode`) — III's
  system (1) holds on `U p`.
* `pos_sys7` / `neg_sys7`: III's `main_pos_on_U` / `main_neg_on_U` (imported unchanged; `PosSmooth`/`NegSmooth`
  derived from the mode condition) then give III's system (7) for III's chiral variables `yPos`/`yNeg`.
* `atlasV03_main`: both branches together (both gauges, as in the contract's joint statement).
* `zero_instance`: the hypothesis bundle is non-empty (zero tangent, zero amplitudes, every `ω`, `m`). This does
  not replace the general theorems.

No gauge reachability, inverse reconstruction, spectral/stability statement, global flow or nonlinear
integrability is claimed. -/

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

section main

variable (p : Hintz2026.SubextremalKN) {τ : CTan} {om mm : ℝ}

end main

/-! ### A non-empty instance of the hypothesis bundle (zero tangent) -/

section zero

/-- The zero complex tangent. -/
def CTan.zero : CTan := ⟨0, 0, 0, 0, 0, 0⟩

variable {M a Q : ℝ}

variable {U : Set (Pt 4)} (hU : IsOpen U) (hUD : U ⊆ domReg M a Q)
include hU hUD

end zero

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).