Geometry Regge Action Smoothness Dihedral Cos3 Sq Conformal Continuous At Zero

The smoothness guarantee

The Regge action is a way to do general relativity without a smooth spacetime, using a network of flat tetrahedra instead. The action is built from the dihedral angles along the edges where tetrahedra meet, and its behavior near flat space matters for physics. The declaration dihedralCos3Sq_conformal_continuousAt_zero proves a specific smoothness fact: the cosine-squared of a dihedral angle, when computed from a conformally rescaled set of edge lengths, is a continuous function of the vertex potentials at the point where all potentials are zero, which represents flat space.

This is not an abstract nicety. The full nonlinear Regge action is not a simple quadratic expression; it contains arccos functions that can become singular. The theorem establishes that, under a conformal rescaling that keeps tetrahedra nondegenerate, the problematic denominator in the angle formula stays away from zero. The machine-checked proof builds on earlier results showing the denominator is strictly positive for any nondegenerate tetrahedron, and that the conformal map from vertex potentials to squared edge lengths is smooth. The continuity at zero is the first step in proving the entire action is smooth at the flat configuration, a prerequisite for doing calculus, like finding equations of motion, in this discrete setting.

In Recognition Science, this is part of a larger program to show that the discrete geometry used in quantum gravity is mathematically sound. The framework's library of formal theorems provides a rigorous foundation for statements that are often taken for granted in physics. This particular declaration is a building block, not a standalone physical prediction. It does not say anything about the dynamics of gravity, the value of any coupling constant, or the existence of a particular vacuum state. It only certifies a technical condition about the smoothness of a geometric function.

What the declaration does not claim is equally important. It does not prove that the Regge action is smooth everywhere, only at the flat-space point. It does not establish that the conformal rescaling is physically preferred over other choices. And it does not address the global condition that all deficits vanish simultaneously, which is a separate assembled-triangulation condition. The theorem is a precise, local, analytic fact, and its power lies in being exactly that.

theorem dihedralCos3Sq_conformal_continuousAt_zero
    (K : Triangulation3D) (τ : Fin K.nT) (f : Fin 6) :
    ContinuousAt (fun ξ : VertexPotential K =>
      dihedralCos3Sq (conformalTetSqEdges K ξ τ) f) (zeroPotential K) := by
  have hbase := local_dihedralCos3Sq_continuousAt K τ f
  have hchart : ContinuousAt (fun ξ : VertexPotential K =>
      conformalTetSqEdges K ξ τ) (zeroPotential K) :=
    (conformalTetSqEdges_contDiff K τ 0).continuous.continuousAt
  have hbase' : ContinuousAt
      (fun x : CayleyMengerPolynomial.SqEdges => dihedralCos3Sq x f)
      (conformalTetSqEdges K (zeroPotential K) τ) := by
    simpa [conformalTetSqEdges_zero K τ] using hbase
  exact ContinuousAt.comp
    (f := fun ξ : VertexPotential K => conformalTetSqEdges K ξ τ)
    (g := fun x : CayleyMengerPolynomial.SqEdges => dihedralCos3Sq x f)
    (x := zeroPotential K)
    hbase' hchart

theorem local_dihedralDenom3_ne_zero
    (K : Triangulation3D) (τ : Fin K.nT) (f : Fin 6) :
    dihedralDenom3 ((K.tet τ).sqEdge) f ≠ 0 := by
  rw [dihedralDenom3_eq_poly]
  exact local_dihedralDenom3Poly_ne_zero K τ f

/-- Nondegenerate tetrahedra already give positive polynomial cofactor
denominators for every local dihedral angle. -/
theorem local_dihedralDenom3Poly_pos
    (K : Triangulation3D) (τ : Fin K.nT) (f : Fin 6) :
    0 < dihedralDenom3Poly ((K.tet τ).sqEdge) f :=
  dihedralDenom3Poly_pos_of_nonDegenerate (K.tet τ) f

theorem conformalTetSqEdges_contDiff
    (K : Triangulation3D) (τ : Fin K.nT) (n : ℕ∞) :
    ContDiff ℝ n (fun ξ : VertexPotential K => conformalTetSqEdges K ξ τ) := by
  rw [contDiff_pi]
  intro f
  exact conformalLocalSqEdge_contDiff K τ f n

theorem reggeAction_contDiffAt_zero_of_endpoint_free
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hEndpoint :
      ∀ τ : Fin K.nT, ∀ f : Fin 6,
        dihedralCos3Sq ((K.tet τ).sqEdge) f ≠ -1 ∧
          dihedralCos3Sq ((K.tet τ).sqEdge) f ≠ 1) :
    ContDiffAt ℝ (⊤ : ℕ∞) (reggeAction K hK) (zeroPotential K) := by
  unfold reggeAction
  simpa using
    (ContDiffAt.sum
      (s := Finset.univ)
      (f := fun e ξ =>
        hingeMeasureUnderConformal K hK ξ e * deficitAngle K ξ e)
      (x := zeroPotential K)
      (fun e _ =>
        (hingeMeasureUnderConformal_contDiffAt_zero K hK e (⊤ : ℕ∞)).mul
          (deficitAngle_contDiffAt_zero K e ⊤ hEndpoint)))

What this page does not claim

This declaration does not prove the Regge action is smooth everywhere, only at the flat-space point. It does not establish that the conformal rescaling is physically preferred over other choices. It does not address the global condition that all deficits vanish simultaneously.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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