Geometry Freudenthal Regge Component

A concrete geometric bridge

Regge calculus is a discrete approach to general relativity where spacetime is approximated by flat triangular (or tetrahedral) pieces, and curvature is concentrated along the hinges where these pieces meet. The action is built from the areas of these hinges and the deficit angles around them. In the Recognition Science framework, a key question is whether the second-order variation of this Regge action, the part that governs small perturbations, can be written as a simple quadratic form, specifically a Dirichlet form, which is a sum over pairs of neighboring points weighted by some coefficient.

The module FreudenthalReggeComponent answers this for a concrete, finite local model. It uses a star with eight vertices, matching the vertex count of a cubic cell in a Freudenthal local chart, a standard triangulation of space. The area weights are given by the regular hinge-area formula, the area of an equilateral triangle with side length a, which is (sqrt(3)/4) * a^2. The module then constructs the second-variation coefficient matrix M_ij from these weights and proves three properties for this specific model.

First, off the diagonal, the coefficient matrix is exactly the negative of the area-weight matrix: M_ij = -A_ij for i ≠ j. Second, every row of this matrix sums to zero, a property that ensures the action is invariant under a constant shift of the potential. Third, and most importantly, the second-order Regge action reduces exactly to one half times the Dirichlet form built from the geometric area weights. This is the concrete component comparison: the discrete geometric action behaves precisely like a standard quadratic energy functional.

The module also records derivative facts: the regular triangle area has a derivative with respect to its side length, and a uniform rescaling of the tetrahedron leaves the dihedral angle constant. These are collected into a certificate structure, a single object that packages all the proven properties. This certificate is the exact interface that a future, more general computation must target.

In Recognition Science, this is a step toward connecting the framework's discrete ledger of recognition events to the continuous geometry of spacetime. The result is not a proof for arbitrary triangulations; it is the first fully concrete finite model where the Regge action and the Dirichlet form are shown to coincide, providing a template for the general case.

/-- Off diagonal, the concrete Regge coefficient matrix is the negative of the
geometric area/face-weight matrix. -/
theorem concreteM_offDiag_eq_neg_areaWeight
    (S : ConcreteReggeStar) (i j : LocalVertex) (hij : i ≠ j) :
    concreteM S i j = - areaWeight S i j := by
  unfold concreteM concreteWeakFieldReggeData
  rw [bilinearCoefficient_laplacianReggeData (areaWeight S) (areaWeight_symm S)]
  unfold laplacianCoefficient
  simp [hij]

/-- The concrete coefficient matrix has exact zero row sums. -/
theorem concreteM_rowSum_zero (S : ConcreteReggeStar) :
    ∀ i : LocalVertex, ∑ j : LocalVertex, concreteM S i j = 0 := by
  intro i
  unfold concreteM concreteWeakFieldReggeData
  simpa only [bilinearCoefficient_laplacianReggeData (areaWeight S) (areaWeight_symm S)]
    using laplacianCoefficient_row_sum (areaWeight S) i

/-- Concrete closure of the weak-field component comparison: the second-order
Regge action is exactly the geometric Dirichlet form for the concrete area
weights. -/
theorem concreteReggeSecondVariation_eq_jcostDirichlet
    (S : ConcreteReggeStar) (ε : LogPotential 8) :
    secondOrderReggeAction (concreteWeakFieldReggeData S) ε
      = (1 / 2) * dirichletForm (areaWeight S) ε := by
  simpa [concreteReggeComponentComparison] using
    componentComparison_gives_geometric_dirichlet
      (concreteWeakFieldReggeData S)
      (concreteReggeComponentComparison S)
      ε

/-- The regular hinge-area formula is derivative-ready. -/
theorem hasDerivAt_regularTriangleArea (a : ℝ) :
    HasDerivAt regularTriangleArea ((Real.sqrt 3 / 2) * a) a := by
  unfold regularTriangleArea
  have hmul : HasDerivAt (fun x : ℝ => x * x) (1 * a + a * 1) a :=
    (hasDerivAt_id a).mul (hasDerivAt_id a)
  have hsq : HasDerivAt (fun x : ℝ => x ^ 2) (2 * a) a := by
    convert hmul using 1
    · ext x
      ring
    · ring
  have h := hsq.const_mul (Real.sqrt 3 / 4)
  convert h using 1
  ring

/-- Uniform rescaling leaves a regular tetrahedral dihedral angle constant.
This records the scale-invariance fact; non-uniform edge derivatives are the
remaining hard Cayley-Menger task. -/
theorem hasDerivAt_regularDihedral_uniformScale (a : ℝ) :
    HasDerivAt (fun _s : ℝ => regularTetrahedralDihedralAngle) 0 a :=
  hasDerivAt_const a regularTetrahedralDihedralAngle

What this page does not claim

This is not a proof for arbitrary Cayley-Menger or dihedral derivative data. This does not establish the full bridge from recognition events to continuous spacetime geometry. This does not derive the fine-structure constant or any other specific coupling constant.

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)

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.

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