Geometry Freudenthal Regge Component Regular Tetrahedral Dihedral Angle Eq

The tetrahedral angle

A regular tetrahedron is the four-sided solid made of four equilateral triangles, the simplest of the Platonic solids. The dihedral angle is the angle between two faces measured along their shared edge. For a regular tetrahedron that angle is the same at every edge, and its cosine is exactly 1/3, which makes the angle about 70.53 degrees. This is a classical fact of solid geometry, known since the ancient Greeks studied the five regular solids.

The machine-checked library of formal theorems contains a declaration, regularTetrahedralDihedralAngle_eq, that proves this value. The proof is a definitional equality: the framework defines the regular tetrahedral dihedral angle as the angle whose cosine is 1/3, so the statement reduces to the arccosine function by rewriting. It is a theorem in the sense that the library verifies it, but it is a trivial one, a record of a definition rather than a deep derivation.

In Recognition Science, this angle appears inside a larger construction called a Regge component, a discrete model of curved space made of flat triangular pieces. The library proves that for a regular tetrahedral piece, the angle stays constant under uniform rescaling of the edge lengths. It also proves that the second-order action of this discrete geometry equals a Dirichlet form with geometric weights, a step toward connecting the framework's cost function to classical gravity.

What the declaration does not claim is more important than what it proves. It does not derive the dihedral angle from any deeper principle; it simply states the known value. It does not handle arbitrary tetrahedra or general Regge triangulations, where the dihedral angle depends on all edge lengths in a complicated way. The full Cayley-Menger determinant machinery, which would compute angles for any edge lengths, is not part of this component. The library's own documentation says this is the first concrete finite model, not the general solution.

The practical upshot is a checked foundation stone. Any later computation that needs the regular tetrahedral dihedral angle can cite this theorem and know the value is correct to the last digit. The hard work, the general derivative formulas for arbitrary triangulations, remains open, and the library says so explicitly.

theorem regularTetrahedralDihedralAngle_eq :
    regularTetrahedralDihedralAngle = Real.arccos (1 / 3) := rfl

/-- 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

/-- 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

What this page does not claim

The declaration does not derive the dihedral angle from any deeper principle; it simply states the known value. It does not handle arbitrary tetrahedra or general Regge triangulations. It does not provide the full Cayley-Menger determinant machinery.

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.

Derived articles

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