Geometry Regge Rigorous Foundation Cm3 Conformal Cont Diff

A Smoothness Result for Tetrahedral Volumes

The volume of a tetrahedron is not a simple function of its edge lengths. In 3D, the squared volume is given by the Cayley-Menger polynomial, a degree-3 expression in the six squared edge lengths. The declaration cm3_conformal_contDiff establishes that this volume-squared function remains smooth when the edge lengths are varied according to a specific, exponential rule tied to vertex potentials.

This exponential rule, called the conformal edge ansatz, sets the length of the edge between vertices i and j to L₀·exp((ξᵢ+ξⱼ)/2), where ξ is a set of four real numbers assigned to the tetrahedron's vertices. This is a common parameterization in geometry, as it preserves angles locally. The theorem proves that the composition of the Cayley-Menger polynomial with this conformal map is infinitely differentiable (ContDiff ℝ n for any n). The proof is a direct composition of two already-established smoothness results: the polynomial is smooth, and the exponential map is smooth.

This result is a foundational building block, not a finished physical theory. It does not, by itself, prove the Regge component theorem, which relates the Hessian of the Regge action to the areas of triangles in a 3D lattice. That theorem requires additional, named classical hypotheses: Schläfli's identity, the dihedral cosine formula via Cayley-Menger cofactors, and the smoothness of dihedral angles on the realisability cone. These are imported from the standard Regge calculus literature, not derived from within the Recognition Science framework.

In Recognition Science, this machine-checked result is a rigorously verified first step. It guarantees that the volume-squared function behaves well under the conformal ansatz, providing a solid foundation for future work on the Regge component theorem. The framework's library of formal theorems now contains a clean, verified proof of this smoothness property, ready to be used in the larger program.

theorem cm3_conformal_contDiff (ℓ₀ : ℝ) (n : ℕ∞) :
    ContDiff ℝ n (fun ξ : Fin 4 → ℝ => cm3 (conformalSqEdge ℓ₀ ξ)) := by
  exact (cm3_contDiff n).comp (conformalSqEdge_contDiff ℓ₀ n)

theorem cm3_conformal_contDiff (ℓ₀ : ℝ) (n : ℕ∞) :
    ContDiff ℝ n (fun ξ : Fin 4 → ℝ => cm3 (conformalSqEdge ℓ₀ ξ)) := by
  exact (cm3_contDiff n).comp (conformalSqEdge_contDiff ℓ₀ n)

What this page does not claim

The full Regge component theorem M_ij = -area(f_ij) is not proved by this declaration. The classical hypotheses (Schläfli identity, dihedral cosine formula) are not derived within the framework. The result does not imply that the Regge action itself is smooth under the conformal ansatz.

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/ReggeRigorousFoundation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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