Geometry Freudenthal Regge Component Regular Triangle Area Nonneg

A proved inequality

The area of an equilateral triangle with side length a is given by the classical formula (√3 / 4) · a². For any real number a, squaring makes the term a² nonnegative, and √3 and 4 are both positive, so the product is always zero or greater. This is a basic fact of arithmetic, but in a machine-checked library of formal theorems, even such an obvious statement needs a proof that a computer can verify step by step.

The declaration regularTriangleArea_nonneg in the Recognition Science library is exactly that proof. It states that for every real number a, the value of the function regularTriangleArea, defined as (√3 / 4) · a², is nonnegative. The proof works by unfolding the definition and applying two standard lemmas: the product of two nonnegative numbers is nonnegative, and the square of any real number is nonnegative. The declaration is a theorem, meaning it has been checked by the framework's machine-checked library of formal theorems, with no gaps or unverified assumptions.

This small result is not an isolated curiosity. It is a building block in a larger construction: a concrete model of a Regge star, a local patch of discretized spacetime used in numerical relativity. The model assigns a nonnegative area weight to each triangular hinge, and the proof guarantees that these weights are always valid, never negative. This in turn feeds into a comparison showing that a certain second-order action equals a Dirichlet form, a key step in connecting the framework's geometric ideas to standard physics. The inequality is the first link in that chain.

What the declaration does not claim is equally important. It does not prove that the area formula is correct for arbitrary curved surfaces, nor does it handle the full Cayley-Menger determinant for a general tetrahedron. It applies only to the regular, flat triangular hinge area, not to non-uniform edge lengths or arbitrary triangulations. The proof is a single, concrete, verified fact, not a general theory of geometry.

theorem regularTriangleArea_nonneg (a : ℝ) :
    0 ≤ regularTriangleArea a := by
  unfold regularTriangleArea
  exact mul_nonneg (div_nonneg (Real.sqrt_nonneg 3) (by norm_num)) (sq_nonneg a)

theorem regularTriangleArea_nonneg (a : ℝ) :
    0 ≤ regularTriangleArea a := by
  unfold regularTriangleArea
  exact mul_nonneg (div_nonneg (Real.sqrt_nonneg 3) (by norm_num)) (sq_nonneg a)

What this page does not claim

The proof does not apply to arbitrary curved surfaces or non-regular triangles. The declaration does not establish the full Cayley-Menger determinant as a differentiable function of all edge lengths. It does not prove the dihedral-angle derivative formulas for an arbitrary Regge triangulation.

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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