Geometry Deficit Linearization Linear Regge Vanishes

The vanishing first-order action

The Regge action is a discrete version of the Einstein-Hilbert action of general relativity, built from a simplicial complex (a space made of flat triangles, tetrahedra, or higher-dimensional simplices glued together). Its basic variables are the edge lengths, and its curvature is concentrated along hinges, the (n-2)-dimensional faces where simplices meet. The deficit angle at a hinge measures how much the total angle around it deviates from the flat value of 2π; a nonzero deficit signals curvature there. The action sums the area of each hinge times its deficit angle.

Around a flat background, where all deficit angles are zero, a small perturbation η of each edge length changes the deficit angle at each hinge. To first order, that change is a linear combination of the perturbations, with coefficients given by partial derivatives of the deficit angle with respect to edge lengths, evaluated at the flat background. This is the Piran-Williams (1986) linearization of Regge calculus, and the existence of these coefficients is classical. The key consequence, proved in the framework's machine-checked library of formal theorems, is that the first-order part of the Regge action vanishes identically: the sum over hinges of hinge area times the linearized deficit equals zero, for any perturbation, whenever the background is flat and Schläfli's identity holds for the linearization coefficients. Schläfli's identity is a classical relation among the derivatives of dihedral angles with respect to edge lengths, and here it makes the linear term disappear edge by edge.

The vanishing has a direct structural meaning: the leading non-trivial Regge action is quadratic in the perturbation, not linear. This matches the form of a Dirichlet energy, a quadratic functional of a field, and is exactly the content the framework's Phase C5 needs for its cost-function program. The theorem is stated for a finite simplicial complex with finitely many hinges and edges, and it assumes the flat-sum condition on every hinge and positive flat edge lengths.

In Recognition Science, the framework models the cost of recognition events as a forced quantity, and this theorem is part of the chain that connects that cost to geometry. The framework's library proves the vanishing result with zero sorry and zero new axioms, meaning the proof is fully checked and relies only on the ambient type theory's standard postulates. The result does not, however, establish that the Regge action itself is quadratic, nor that the linearization coefficients for a general simplicial complex are the same as those for the cubic lattice, where the coefficients are integer-linear in the perturbations. It also does not prove that the Regge action equals the J-cost Dirichlet energy; it shows only that the first-order term vanishes, leaving the quadratic term as the leading contribution.

/-- The linear (first-order) part of the Regge action vanishes under
    Schläfli's identity. -/
theorem linear_regge_vanishes {nH nE : ℕ}
    (W : WellShapedData nH nE) (η : EdgePerturbation nE) :
    (∑ h : Fin nH, (W.complex.hinges h).area *
      linearizedDeficit W.coeffs η h) = 0 := by
  unfold linearizedDeficit
  -- Rewrite the sum: move the minus sign out, then swap summation order.
  have h_swap :
      (∑ h : Fin nH, (W.complex.hinges h).area *
        -(∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
      = - ∑ e : Fin nE,
          η.eta e * (∑ h : Fin nH, (W.complex.hinges h).area * W.coeffs.dThetadL h e) := by
    rw [show (∑ h : Fin nH, (W.complex.hinges h).area *
              -(∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
            = -(∑ h : Fin nH, (W.complex.hinges h).area *
                 (∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
         from by
           rw [← Finset.sum_neg_distrib]
           apply Finset.sum_congr rfl
           intro h _; ring]
    rw [show (∑ h : Fin nH, (W.complex.hinges h).area *
              (∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
            = ∑ h : Fin nH, ∑ e : Fin nE,
               (W.complex.hinges h).area * W.coeffs.dThetadL h e * η.eta e
         from by
           apply Finset.sum_congr rfl
           intro h _
           rw [Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro e _; ring]
    rw [Finset.sum_comm]
    congr 1
    apply Finset.sum_congr rfl
    intro e _
    rw [← Finset.sum_mul]
    ring
  rw [h_swap]
  -- Now apply Schläfli's identity per edge.
  have h_each : ∀ e : Fin nE,
      η.eta e * (∑ h : Fin nH, (W.complex.hinges h).area * W.coeffs.dThetadL h e) = 0 := by
    intro e
    rw [W.schlaefli e, mul_zero]
  rw [Finset.sum_eq_zero (fun e _ => h_each e), neg_zero]

/-- The linear (first-order) part of the Regge action vanishes under
    Schläfli's identity. -/
theorem linear_regge_vanishes {nH nE : ℕ}
    (W : WellShapedData nH nE) (η : EdgePerturbation nE) :
    (∑ h : Fin nH, (W.complex.hinges h).area *
      linearizedDeficit W.coeffs η h) = 0 := by
  unfold linearizedDeficit
  -- Rewrite the sum: move the minus sign out, then swap summation order.
  have h_swap :
      (∑ h : Fin nH, (W.complex.hinges h).area *
        -(∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
      = - ∑ e : Fin nE,
          η.eta e * (∑ h : Fin nH, (W.complex.hinges h).area * W.coeffs.dThetadL h e) := by
    rw [show (∑ h : Fin nH, (W.complex.hinges h).area *
              -(∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
            = -(∑ h : Fin nH, (W.complex.hinges h).area *
                 (∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
         from by
           rw [← Finset.sum_neg_distrib]
           apply Finset.sum_congr rfl
           intro h _; ring]
    rw [show (∑ h : Fin nH, (W.complex.hinges h).area *
              (∑ e : Fin nE, W.coeffs.dThetadL h e * η.eta e))
            = ∑ h : Fin nH, ∑ e : Fin nE,
               (W.complex.hinges h).area * W.coeffs.dThetadL h e * η.eta e
         from by
           apply Finset.sum_congr rfl
           intro h _
           rw [Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro e _; ring]
    rw [Finset.sum_comm]
    congr 1
    apply Finset.sum_congr rfl
    intro e _
    rw [← Finset.sum_mul]
    ring
  rw [h_swap]
  -- Now apply Schläfli's identity per edge.
  have h_each : ∀ e : Fin nE,
      η.eta e * (∑ h : Fin nH, (W.complex.hinges h).area * W.coeffs.dThetadL h e) = 0 := by
    intro e
    rw [W.schlaefli e, mul_zero]
  rw [Finset.sum_eq_zero (fun e _ => h_each e), neg_zero]

/-- The certificate is inhabited by the proved `linear_regge_vanishes`. -/
theorem deficitLinearizationCert : DeficitLinearizationCert where
  linear_vanishes := fun W η => linear_regge_vanishes W η

What this page does not claim

The Regge action itself is quadratic in the perturbation; only the first-order term vanishes. The linearization coefficients for a general simplicial complex are integer-linear in the perturbations, as they are for the cubic lattice. The Regge action equals the J-cost Dirichlet energy; the theorem shows only that the first-order term vanishes.

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