Prove2Me

KN inverse check, entry (0, 0)

Proved
ExactVacuumAtlas.EinsteinMaxwell.KN.minv_0_0

by He Wang · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-vexact-vacuum-atlasgeneral-relativitynewman-penrose

KN inverse check, entry (0, 0)

Entry (0, 0) of gKN · giKN equals the identity entry where Σ, Δ, sin θ ≠ 0 (IV source lemma, inlined in IV, a separate node here).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 KN 逆矩阵检验分量（IV 源引理）。

Preamble
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_KN_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Domain
import Definitions.Def_Atlas_Kerr_InAtoms
open Real
open ExactVacuumAtlas
open scoped ContDiff
open ExactVacuumAtlas.Kerr.In (Ui Wi)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.KN
set_option maxHeartbeats 8000000
set_option maxRecDepth 100000
set_option linter.unusedVariables false
set_option linter.unusedSimpArgs false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false
variable (M a Q : ℝ)
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.KN.minv_0_0 (x : Pt 4) (hS : x 1 ^ 2 + a ^ 2 * cos (x 2) ^ 2 ≠ 0) (hD : x 1 ^ 2 - 2 * M * x 1 + a ^ 2 + Q ^ 2 ≠ 0) (hs : sin (x 2) ≠ 0) :
    (metricMatrix (gKN M a Q) x * metricMatrix (giKN M a Q) x) 0 0 = 1 := by sorry
Source
He Wang, Exact Spacetime Atlas V: From Linearized Einstein–Maxwell Equations to Kerr–Newman Radiation Systems. Local source, no public repository; commit b8cd107a558e153ed41fb4972ac5315499d14778; Atlas/EinsteinMaxwell/KN/Parts.lean#L40-L53; SHA-256 27df3365e1b568e37410fd5ea573366b73f33888fe24ccbe6f7bc357788ceccc. Registered definitions: mission/PHYSICS_BOUNDARY.md (§K–§M); literature: P. Hintz, arXiv:2609.33661 (v1); Mark–Yang–Zimmerman–Chen, arXiv:1409.5800 (v1).
Human review

  • 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).

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