Geometry Regge Action Smoothness Tet Dihedral Angle Under Conformal Cont Diff At

Smoothness at the flat potential

In the geometry of a tetrahedron, the dihedral angle between two faces is a basic measure of shape. The declaration tetDihedralAngleUnderConformal_contDiffAt_zero is a theorem in the framework's machine-checked library of formal theorems. It states that this angle, viewed as a function of a conformal deformation of the tetrahedron, is infinitely differentiable at the point where the deformation potential is zero. In plainer terms, as you smoothly reshape the tetrahedron through its flat, undeformed state, the dihedral angle changes without any kinks, jumps, or sudden breaks; its rate of change is itself smooth to all orders.

The theorem is not unconditional. It requires that the cosine of the dihedral angle, squared, is not equal to 1 or -1 at the flat state. This condition keeps the angle away from 0 and 180 degrees, where the tetrahedron degenerates and the angle is no longer well-defined. The proof builds on earlier results showing that the squared edge lengths and the denominator in the angle formula are smooth under the same conformal map, and that the denominator is strictly positive for a non-degenerate tetrahedron. The theorem is a local statement: it guarantees smoothness at the single point of the flat potential, not across the entire space of all possible deformations.

This result is a component of a larger effort to show that the full, nonlinear Regge action, a discretized model of gravity built from tetrahedral deficits, is smooth at the flat configuration. The theorem provides the analytic input needed for that broader claim. It does not, by itself, prove that the action is smooth; it only establishes the smoothness of one ingredient, the dihedral angle of a single tetrahedron under a conformal change. The global condition that all deficits vanish at the flat potential is a separate, assembled-triangulation property that does not follow from the local non-degeneracy of each tetrahedron.

theorem tetDihedralAngleUnderConformal_contDiffAt_zero
    (K : Triangulation3D) (τ : Fin K.nT) (f : Fin 6) (n : ℕ∞)
    (hEndpoint :
      dihedralCos3Sq ((K.tet τ).sqEdge) f ≠ -1 ∧
        dihedralCos3Sq ((K.tet τ).sqEdge) f ≠ 1) :
    ContDiffAt ℝ n (fun ξ : VertexPotential K =>
      tetDihedralAngleUnderConformal K ξ τ f) (zeroPotential K) := by
  unfold tetDihedralAngleUnderConformal DihedralDerivatives.dihedralAngle3Sq
  have hcos := dihedralCos3Sq_conformal_contDiffAt_zero K τ f n
  have hm :
      dihedralCos3Sq (conformalTetSqEdges K (zeroPotential K) τ) f ≠ -1 := by
    simpa [conformalTetSqEdges_zero K τ] using hEndpoint.1
  have hp :
      dihedralCos3Sq (conformalTetSqEdges K (zeroPotential K) τ) f ≠ 1 := by
    simpa [conformalTetSqEdges_zero K τ] using hEndpoint.2
  have hacos : ContDiffAt ℝ n Real.arccos
      (dihedralCos3Sq (conformalTetSqEdges K (zeroPotential K) τ) f) :=
    Real.contDiffAt_arccos hm hp
  simpa [Function.comp_def] using
    (ContDiffAt.comp (x := zeroPotential K) hacos hcos)

theorem tetDihedralAngleUnderConformal_contDiffAt_zero
    (K : Triangulation3D) (τ : Fin K.nT) (f : Fin 6) (n : ℕ∞)
    (hEndpoint :
      dihedralCos3Sq ((K.tet τ).sqEdge) f ≠ -1 ∧
        dihedralCos3Sq ((K.tet τ).sqEdge) f ≠ 1) :
    ContDiffAt ℝ n (fun ξ : VertexPotential K =>
      tetDihedralAngleUnderConformal K ξ τ f) (zeroPotential K) := by
  unfold tetDihedralAngleUnderConformal DihedralDerivatives.dihedralAngle3Sq
  have hcos := dihedralCos3Sq_conformal_contDiffAt_zero K τ f n
  have hm :
      dihedralCos3Sq (conformalTetSqEdges K (zeroPotential K) τ) f ≠ -1 := by
    simpa [conformalTetSqEdges_zero K τ] using hEndpoint.1
  have hp :
      dihedralCos3Sq (conformalTetSqEdges K (zeroPotential K) τ) f ≠ 1 := by
    simpa [conformalTetSqEdges_zero K τ] using hEndpoint.2
  have hacos : ContDiffAt ℝ n Real.arccos
      (dihedralCos3Sq (conformalTetSqEdges K (zeroPotential K) τ) f) :=
    Real.contDiffAt_arccos hm hp
  simpa [Function.comp_def] using
    (ContDiffAt.comp (x := zeroPotential K) hacos hcos)

What this page does not claim

The theorem does not prove that the full Regge action is smooth; it only proves the smoothness of a single dihedral angle. The theorem does not establish that all deficit angles vanish at the flat potential, which is a separate global condition. The theorem does not describe the behavior of the dihedral angle away from the flat potential.

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)

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