Gravity Regge Component Theorem3 Dproof Canonical Weak Field Data Row Sum

The row-sum condition

In the Regge calculus approach to general relativity, spacetime is approximated by a triangulated manifold, a space built from flat tetrahedral cells glued along their faces. The discrete Einstein equations are derived from a quadratic action, and the second derivative of that action, the Hessian, is a matrix that encodes how the action responds to small changes in the edge lengths. A central structural property of this matrix is that the entries in each row sum to zero, a consequence of the action being invariant under uniform scaling of all edge lengths. This row-sum property is what the declaration canonicalWeakFieldData_rowSum establishes, in the framework's machine-checked library of formal theorems, for a specific, canonically constructed weak-field data set.

The framework constructs this data set in two steps. First, it defines a set of dual weights, one for each unordered pair of vertices in the triangulation, where the weight is the square root of the squared length of the edge connecting them. These weights are independent geometric data, defined directly from the incidence-level edge-length chart; they are not defined by negating a Regge coefficient. Second, it feeds these weights into a Laplacian-style construction to produce the weak-field Regge data. The theorem canonicalWeakFieldData_rowSum then states that this data satisfies the SchlaefliRowSum predicate, meaning that the sum of the entries in each row of the corresponding bilinear coefficient matrix is zero.

The significance of this result is that it provides a concrete, machine-checked instance of a property that is essential for the consistency of the discrete gravity theory. It shows that the canonical weak-field data, built from the incidence geometry, automatically satisfies the row-sum condition, without any additional assumptions or adjustments. This is a necessary step toward the larger goal of proving that the genuine Regge Hessian, the true second derivative of the discrete action, also satisfies this property for arbitrary triangulations. The theorem is a component in a chain of results that aims to connect the geometric computation of dual weights to the physical Regge action.

What this theorem does not claim is that the row-sum property holds for the genuine Regge Hessian itself, the second derivative of the actual discrete Einstein-Hilbert action. It only establishes the property for a specific, canonically constructed weak-field data set, which is derived from the incidence geometry and the Laplacian construction. The theorem also does not claim that this weak-field data set is the same as the genuine Regge Hessian, only that it shares the same off-diagonal components and the row-sum property. The final target, FinalReggeComponentTarget, which would prove the property for the genuine Hessian on any triangulation, remains an open target, though the framework provides a construction that would discharge it if the canonical data were shown to be the genuine one.

theorem canonicalWeakFieldData_rowSum
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    SchlaefliRowSum (canonicalWeakFieldDataOfIncidence K hK) :=
  schlaefliRowSum_laplacianReggeData
    (Geometry.ReggeActionConcrete.canonicalDualWeight K hK)
    (Geometry.ReggeActionConcrete.canonicalDualWeight_symm K hK)

/-- Canonical weak-field Regge data induced by the incidence dual weights. -/
def canonicalWeakFieldDataOfIncidence
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    WeakFieldReggeData K.nV :=
  laplacianReggeData
    (Geometry.ReggeActionConcrete.canonicalDualWeight K hK)
    (Geometry.ReggeActionConcrete.canonicalDualWeight_symm K hK)

theorem canonicalWeakFieldData_rowSum
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    SchlaefliRowSum (canonicalWeakFieldDataOfIncidence K hK) :=
  schlaefliRowSum_laplacianReggeData
    (Geometry.ReggeActionConcrete.canonicalDualWeight K hK)
    (Geometry.ReggeActionConcrete.canonicalDualWeight_symm K hK)

What this page does not claim

The theorem does not prove that the genuine Regge Hessian satisfies the row-sum property. The theorem does not claim that the canonical weak-field data set is identical to the genuine Regge Hessian. The theorem does not establish the row-sum property for any weak-field data set other than the one constructed from the canonical incidence weights.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/ReggeComponentTheorem3DProof.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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