Geometry Regge Action First Variation

The Regge action and its critical point

The Regge action is a way to do general relativity without a smooth metric, by triangulating spacetime into flat tetrahedra and summing a deficit angle times an edge length over every edge. Its first variation is the discrete analogue of the Einstein field equations: setting it to zero picks out the allowed geometries. The framework's machine-checked library of formal theorems establishes that the flat conformal potential, where all edge lengths are equal, is a critical point of this action.

The proof rests on two classical ideas. First, Schläfli's differential identity, which relates small changes in a tetrahedron's dihedral angles to changes in its edge lengths, causes the variation from each tetrahedron's interior to cancel locally. Second, at the flat configuration every deficit angle is zero, so the remaining boundary terms vanish. The library records the exact analytic statement: the Fréchet derivative of the Regge action at the zero potential equals zero, written as `fderiv ℝ (reggeAction K hK) (zeroPotential K) = 0`.

In Recognition Science, this result is a structural checkpoint, not a dynamical law. The framework models a discrete record of events, its ledger, and derives physical constants from a forced cost function. The Regge action's critical point at flatness is a consistency condition the framework's geometry must satisfy, and the library proves it without invoking any additional physical postulate.

The library's proof is organized around a directional derivative: it shows that along any line through the flat potential, the action's derivative vanishes. This is packaged as `ReggeActionDirectionalCriticalAtZero`, and a theorem converts it into the full Fréchet derivative statement. The key local cancellation is `local_conformal_schlaefli_cancellation`, which states that the sum over a tetrahedron's six edges of the edge length times the angle derivative is zero.

What this establishes in plain language is that the flat geometry is a stationary point of the discrete gravitational action. It is a necessary condition for the flat configuration to be a solution of the discrete Einstein equations, and it is the first step toward understanding the action's second variation, which would determine stability. The module does not yet expand the full derivative calculation from the closed-form local Schläfli identities; that remains a named input, recorded as `ReggeActionFirstVariationInput`.

/-- Phase-C first-variation theorem, conditional on the named analytic
first-variation input. -/
theorem reggeAction_firstVariation_zero
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_first : ReggeActionFirstVariationInput K hK h_flat) :
    fderiv ℝ (reggeAction K hK) (zeroPotential K) = 0 :=
  h_first.firstVariation_zero

theorem reggeActionCriticalAtZero_of_directional
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hdir : ReggeActionDirectionalCriticalAtZero K hK) :
    ReggeActionCriticalAtZero K hK := by
  unfold ReggeActionCriticalAtZero
  ext η
  exact hdir η

/-- Local Schläfli cancellation for the conformal length direction on a single
tetrahedron.  This is pure finite-sum algebra plus the already proved local
Schläfli identity. -/
theorem local_conformal_schlaefli_cancellation
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (η : VertexPotential K) (τ : Fin K.nT) :
    (∑ f : Fin 6,
      Real.sqrt ((K.tet τ).sqEdge f) *
        localAngleLengthChainDeriv K hK η τ f) = 0 := by
  unfold localAngleLengthChainDeriv
  calc
    (∑ f : Fin 6,
      Real.sqrt ((K.tet τ).sqEdge f) *
        (∑ k : Fin 6,
          ((triangulationSchlaefliData_of_incidence K hK).tetData τ).dihedralDeriv f k *
            localEdgeLengthDirectionalDeriv K η τ k))
        =
      ∑ k : Fin 6,
        localEdgeLengthDirectionalDeriv K η τ k *
          (∑ f : Fin 6,
            Real.sqrt ((K.tet τ).sqEdge f) *
              ((triangulationSchlaefliData_of_incidence K hK).tetData τ).dihedralDeriv f k) := by
          simp_rw [Finset.mul_sum]
          rw [Finset.sum_comm]
          refine Finset.sum_congr rfl ?_
          intro k _
          refine Finset.sum_congr rfl ?_
          intro f _
          ring
    _ = ∑ k : Fin 6, localEdgeLengthDirectionalDeriv K η τ k * 0 := by
          refine Finset.sum_congr rfl ?_
          intro k _
          rw [((triangulationSchlaefliData_of_incidence K hK).tetData τ).schlaefli k]
    _ = 0 := by simp

What this page does not claim

The full derivative calculation from the closed-form local Schläfli identities is not expanded in this module. The flat configuration is not claimed to be a global minimum of the Regge action. This result does not derive the Einstein field equations from the framework's cost function.

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

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