IndisputableMonolith.Relativity.Geometry.LeviCivitaTheorem
The LeviCivitaTheorem module formalizes the torsion-free metric-compatible connection on 4D spacetime. Researchers deriving the Einstein field equations from lattice J-cost in the Recognition Science discrete-to-continuum bridge would cite it. The module assembles definitions of connection coefficients, the Koszul identity, and uniqueness lemmas that recover the Christoffel symbols from the metric.
claimOn a 4-dimensional spacetime manifold with metric tensor $g_{\mu\nu}$, the unique connection $\Gamma^\rho{}_{\mu\nu}$ satisfies $\nabla g = 0$ and has vanishing torsion tensor.
background
The module resides in the Relativity.Geometry section and imports tensor operations, metric structures, curvature tensors, and derivative operators. Upstream material supplies the standard basis $e_\mu$ and states that Christoffel symbols are derived from the metric. The setting is the continuum geometry required for the Recognition Science lattice-to-curvature bridge.
proof idea
This module organizes its content as a chain of definitions for connection coefficients, torsion-free and metric-compatible conditions, the Koszul identity, and lowered-connection uniqueness; the main results follow by direct algebraic substitution into the metric-compatibility equation.
why it matters in Recognition Science
The module supplies the Levi-Civita connection to the Relativity.Geometry aggregator and to DiscreteBridge. The latter maps J-cost lattice defects through the Ricci scalar to the Einstein tensor and field equations, completing the geometric step in the Recognition Science continuum limit.
scope and limits
- Does not treat connections on manifolds without a metric.
- Does not derive explicit curvature components or Bianchi identities.
- Does not address the discrete lattice approximation inside this file.
- Does not extend the uniqueness result beyond four dimensions.
used by (2)
depends on (4)
declarations in this module (17)
-
abbrev
ConnectionCoeffs -
def
IsTorsionFree -
def
cov_deriv_metric -
def
IsMetricCompatible -
theorem
levi_civita_torsion_free -
def
KoszulIdentity -
theorem
metric_compatible_of_koszul -
theorem
levi_civita_metric_compatible -
def
lowered_connection -
def
LoweredConnectionIdentity -
theorem
connection_uniqueness_lowered -
structure
FundamentalTheoremExistence -
structure
FundamentalTheoremUniqueness -
theorem
fundamental_theorem_existence -
theorem
fundamental_theorem_uniqueness -
structure
LeviCivitaCertificate -
theorem
levi_civita_certificate