Prove2Me

Contraction identity for the J law

Proved
ExactVacuumAtlas.Variation.contract_core

by He Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity

Contraction identity for the J law

Finite-sum identity Σ_a (Σ_e J_{ea} K X_e Kᵀ){ab} = Σ_q K{bq} Σ_e (X_e)_{eq} when J · K = 1.

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 J 变换律的缩并恒等式。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Certificate
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff
open Matrix
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas
variable {n : ℕ}
Formal statement
theorem ExactVacuumAtlas.Variation.contract_core {Jm Km : Matrix (Fin n) (Fin n) ℝ} {X : Fin n → Matrix (Fin n) (Fin n) ℝ}
    (hJK : Jm * Km = 1) (b : Fin n) :
    ∑ a, (∑ e, Jm e a • (Km * X e * Kmᵀ)) a b = ∑ q, Km b q * ∑ e, X e e q := by sorry
Source
He Wang, Exact Spacetime Atlas IV: Certified Einstein–Maxwell Backgrounds and First Variations. Local source, no public repository; commit 543904bfcfa3bb169f43fc4d9dbd569999b71886; Atlas/EinsteinMaxwell/Variation/Covariance.lean#L86-L104; SHA-256 9354896334506569e5c4491ef3a59340742907640cf5d1c465ba5a6e2e9ad40b. Registered definitions and literature: mission/PHYSICS_BOUNDARY.md (§H–§J); Müller–Grave arXiv:0904.4184; Newman et al., J. Math. Phys. 6 (1965) 918.
Human review

  • Endorsed by Shuze Chen · Oct 21, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 21, 2026

    Confirmed by the mission captain (proposal self-audit).

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