Geometry Freudenthal Regge Component Has Deriv At Regular Triangle Area

The derivative of a regular triangle's area

The area of a regular (equilateral) triangle with side length a is given by the classical formula (√3/4) a². This is one of the oldest results in geometry, known since antiquity. The declaration hasDerivAt_regularTriangleArea in the framework's machine-checked library of formal theorems establishes a calculus fact about this formula: its derivative with respect to the side length a is (√3/2) a. In plain words, if you increase the side length by a tiny amount, the area increases at a rate proportional to that side length, with the constant √3/2.

This derivative is what you would compute by applying the power rule to the area formula: d/da [(√3/4) a²] = (√3/4) · 2a = (√3/2) a. The declaration does not introduce a new geometric law. It verifies, in a formal system, that the familiar differentiation rule holds for this specific function. The derivation is a direct computation from the definition of the area function and the standard derivative of a square, so the result is fully grounded in the machine-checked library's axioms.

In the Recognition Science framework, this derivative is a small but necessary piece of a larger structure. The framework models physical geometry through a discrete ledger of recognition events, and it uses Regge calculus, a discrete approximation to general relativity, to study how geometry changes. The area derivative appears when the framework computes how the second-order action, the discrete analogue of the Einstein-Hilbert action, responds to small variations in edge lengths. The declaration is a certificate that this particular component is derivative-ready, meaning the framework can use it in variational calculations without adding new geometric assumptions.

The declaration's scope is deliberately narrow. It establishes the derivative for the regular triangle area formula only. It does not establish derivative formulas for arbitrary Cayley-Menger determinants, which describe volumes of general simplices, nor for dihedral angles in a general Regge triangulation. Those remain open targets in the library. The declaration also does not claim that the regular triangle area formula is the only possible area function, or that it is forced by the framework's principles. It simply states a calculus fact about a specific, classical formula.

What this means for the framework is concrete: the first fully concrete finite model of a Regge component now has a verified derivative. This is the exact interface that a future, more general derivative computation must target. The declaration closes a small gap in the library, and it does so in a way that any reader familiar with basic calculus can check by hand.

/-- 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

/-- 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

What this page does not claim

The declaration does not establish that the regular triangle area formula is the only possible area function. The declaration does not introduce a new geometric law or derive the area formula from the framework's principles. The declaration does not cover non-uniform edge derivatives or arbitrary Cayley-Menger data.

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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