Geometry Freudenthal Regge Component Has Deriv At Regular Dihedral Uniform Scale

A geometric stability fact

A dihedral angle is the angle between two faces of a solid shape, like the angle at which two faces of a pyramid meet. For a regular tetrahedron, the shape made of four identical equilateral triangles, this angle is always the same, about 70.53 degrees, no matter how large or small the tetrahedron is. The theorem in question, hasDerivAt_regularDihedral_uniformScale, states this fact in the precise language of calculus: if you scale the entire tetrahedron uniformly, the derivative of that angle with respect to the scale factor is exactly zero. In plain terms, the angle does not change when you blow the shape up or shrink it down.

This result is proved in a machine-checked library of formal theorems, meaning a computer has verified every step of the logical argument. It is part of a larger project within the Recognition Science framework, which models physical structure from a discrete record of events called a ledger. The theorem itself is a small, concrete fact about geometry, not a claim about the whole framework. It is one component in a certificate that verifies other properties of a local geometric model, such as how area weights behave and how a certain action reduces to a Dirichlet form.

The theorem does not claim anything about non-uniform scaling, where different edges of the tetrahedron are stretched by different amounts. That case, involving full Cayley-Menger determinants and dihedral-angle derivatives for arbitrary edge lengths, remains an open and harder task. The theorem also does not say anything about the physical meaning of the scaling or connect directly to the framework's larger claims about dimensions or constants. It is a precise, narrow statement: under uniform scaling, the regular tetrahedral dihedral angle is constant.

/-- 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

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

What this page does not claim

The theorem does not handle non-uniform edge scaling or arbitrary Cayley-Menger data. The theorem does not derive the dihedral angle value itself, only its invariance under uniform scaling. The theorem does not make any claim about the physical interpretation of the scaling or the framework's larger constants.

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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