Gravity Regge Component Theorem3 D Component Comparison Of Genuine

The genuine bridge

Regge calculus is a discrete approach to general relativity where spacetime is approximated by a triangulated manifold, and curvature lives on the hinges (edges in 3D) rather than at points. The framework's library, a machine-checked collection of formal theorems, contains a Regge component package built from genuine 3D geometry: it uses Cayley-Menger determinants, dihedral angles, and a Hessian computation to produce a weak-field data package. The declaration componentComparison_of_genuine is a formal bridge that takes this genuine geometric package and instantiates it into an existing comparison interface, the ReggeComponentComparison.

What this bridge establishes is that the genuine geometric package satisfies the same interface as the existing weak-field conformal Regge bridge. Concretely, the geometric area function is symmetric, nonnegative, and matches the off-diagonal bilinear coefficients with a negative sign: for distinct vertices i and j, the bilinear coefficient equals minus the geometric area. The package also satisfies the Schlaefli row sum condition. Because the genuine package meets this interface, the theorem genuine_component_dirichlet_reduction follows: the second-order Regge action equals one half times the Dirichlet form of the edge area, applied to a log potential. This is a proved theorem in the machine-checked library, with no gaps.

The payoff is a single reduction: the genuine 3D geometric computation and the existing weak-field conformal bridge both land on the same Dirichlet-form structure. This means the geometric Hessian computation, built from Cayley-Menger and dihedral data, is formally compatible with the earlier weak-field reduction. The bridge does not introduce new physics; it shows that two packages within the framework agree on the same mathematical structure.

What the declaration does not claim is broader. It does not establish that the genuine geometric package is the only such package, nor that the Cayley-Menger computation itself is correct as physics. The bridge only connects the package to an existing interface; the physical validity of the geometric computation as a model of gravity is a separate question. The theorem applies to any triangulation K and any log potential, but it does not say that the Dirichlet-form reduction is the unique or physically preferred one. The bridge is a formal compatibility result, not a derivation of gravity from first principles.

/-- A genuine component package instantiates the existing comparison
interface. -/
def componentComparison_of_genuine
    {K : Triangulation3D} (G : GenuineComponentPackage K) :
    ReggeComponentComparison G.W where
  geometricArea := G.geometricArea
  geometricArea_symm := G.geometricArea_symm
  geometricArea_nonneg := G.geometricArea_nonneg
  offDiag_component_match := G.offDiag_component_match
  schlaefli_row_sum := G.schlaefli_row_sum

/-- The genuine geometric component package produced by the
Cayley-Menger/dihedral/Hessian computation. -/
structure GenuineComponentPackage (K : Triangulation3D) where
  W : WeakFieldReggeData K.nV
  geometricArea : Fin K.nV → Fin K.nV → ℝ
  geometricArea_symm : ∀ i j, geometricArea i j = geometricArea j i
  geometricArea_nonneg : ∀ i j, 0 ≤ geometricArea i j
  offDiag_component_match :
    ∀ i j, i ≠ j → bilinearCoefficient W i j = - geometricArea i j
  schlaefli_row_sum : SchlaefliRowSum W

/-- The existing weak-field reduction applies to the genuine component
package. -/
theorem genuine_component_dirichlet_reduction
    {K : Triangulation3D} (G : GenuineComponentPackage K)
    (ε : LogPotential K.nV) :
    secondOrderReggeAction G.W ε =
      (1 / 2) * dirichletForm (edgeArea G.W) ε :=
  weak_field_conformal_reduction G.W G.schlaefli_row_sum ε

What this page does not claim

The bridge does not prove that the Cayley-Menger computation is physically correct as a model of gravity. The bridge does not establish uniqueness of the genuine component package. The Dirichlet-form reduction is not claimed to be the only or physically preferred reduction.

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/ReggeComponentTheorem3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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