Geometry Freudenthal Regge Component Freudenthal Regge Component Cert

A concrete certificate

In the calculus of variations, the second variation of an action functional determines whether a stationary point is a minimum, a maximum, or a saddle. For Regge calculus, a discrete approximation to general relativity where spacetime is built from flat simplices, the second variation of the action is a matrix whose entries depend on edge lengths and dihedral angles. The declaration freudenthalReggeComponentCert is a machine-checked certificate that, for one concrete eight-vertex model, this matrix has a precise geometric form: its off-diagonal entries are the negative of the hinge areas, its rows sum to zero, and the second-order action equals a Dirichlet form, a standard quadratic energy functional.

The model is a local star with eight vertices, matching the vertex count of a cubic cell or a Freudenthal local chart. The certificate packages five theorems. First, the area of a regular triangle, (sqrt 3 / 4) a^2, has the expected derivative with respect to edge length. Second, the dihedral angle of a regular tetrahedron, arccos(1/3), is invariant under uniform rescaling, so its derivative is zero. Third and fourth, the coefficient matrix M_ij satisfies M_ij = -A_ij for i ≠ j and has zero row sums, where A_ij is the hinge area weight. Fifth, the second-order Regge action equals (1/2) times the Dirichlet form with those weights. These are not assumptions; each is a proved theorem in the framework's machine-checked library of formal theorems.

The certificate is deliberately narrow. It does not prove these properties for arbitrary Cayley-Menger determinants or for dihedral-angle derivatives of a general Regge triangulation. That remains an open target. The certificate's value is as a concrete, fully checked instance: it pins down the exact interface that a future, more general derivative computation must satisfy. It is a proof of concept for the weak-field bridge, not a proof of the bridge itself. The distinction matters: a certificate for one model is a foundation stone, not a cathedral.

What the certificate changes is the status of the component comparison. Before it, the comparison between the second-order Regge action and the Dirichlet form was a statement about the abstract construction. Now, for the eight-vertex regular flat-sector model, it is a closed, verified fact. A reader can take the certificate, apply it to any concrete star with the given structure, and know that the action reduces exactly to the geometric Dirichlet form, with no hidden assumptions. That is a small but solid step: it makes the bridge between Regge data and the Dirichlet energy a checked reality for one concrete case, and it names precisely what a general proof would need to cover.

theorem freudenthalReggeComponentCert : FreudenthalReggeComponentCert where
  area_derivative := hasDerivAt_regularTriangleArea
  dihedral_uniform_scale_derivative := hasDerivAt_regularDihedral_uniformScale
  off_diag := concreteM_offDiag_eq_neg_areaWeight
  row_sum := concreteM_rowSum_zero
  dirichlet := concreteReggeSecondVariation_eq_jcostDirichlet

structure FreudenthalReggeComponentCert where
  area_derivative : ∀ a : ℝ,
    HasDerivAt regularTriangleArea ((Real.sqrt 3 / 2) * a) a
  dihedral_uniform_scale_derivative : ∀ a : ℝ,
    HasDerivAt (fun _s : ℝ => regularTetrahedralDihedralAngle) 0 a
  off_diag : ∀ (S : ConcreteReggeStar) (i j : LocalVertex),
    i ≠ j → concreteM S i j = - areaWeight S i j
  row_sum : ∀ S : ConcreteReggeStar,
    ∀ i : LocalVertex, ∑ j : LocalVertex, concreteM S i j = 0
  dirichlet : ∀ (S : ConcreteReggeStar) (ε : LogPotential 8),
    secondOrderReggeAction (concreteWeakFieldReggeData S) ε
      = (1 / 2) * dirichletForm (areaWeight S) ε

What this page does not claim

The certificate does not prove the component comparison for arbitrary Cayley-Menger or dihedral derivative data. The certificate does not establish that the second-order Regge action equals the Dirichlet form for any triangulation beyond the eight-vertex model. The certificate does not claim that the dihedral angle is invariant under non-uniform rescaling.

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

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