Geometry Regge Rigorous Foundation Conformal Sq Edge At Zero

The conformal edge map

In the geometry of Regge calculus, a discrete approach to general relativity, one often studies how a tetrahedron's shape changes when its vertices are given small "potentials." The Recognition Science framework's library defines a specific conformal edge map, a rule that assigns to each pair of vertices a squared edge length: the map takes a base length ℓ₀ and vertex potentials ξ, and sets the squared length of the edge between vertices i and j to ℓ₀² times the exponential of (ξ_i + ξ_j). This is a smooth, positive formula, chosen so that when all potentials are zero, the tetrahedron is regular with all edges equal to ℓ₀.

The theorem conformalSqEdge_at_zero proves exactly that last fact: when every vertex potential is zero, the conformal edge map returns the constant squared length ℓ₀² for every edge. The proof is a direct calculation using the fact that the exponential of zero is one. This is a simple but load-bearing result: it confirms that the conformal ansatz, as defined, has the intended flat-background limit, which is the starting point for the framework's program to derive the Regge component theorem M_ij = -area(f_ij).

The declaration does not claim that this zero-point behavior holds for nonzero potentials, nor does it assert anything about the physical meaning of the potentials. It also does not, by itself, prove the Regge component theorem; that larger result depends on several external classical hypotheses, such as Schläfli's identity and the dihedral cosine formula, which the framework imports from the standard literature rather than deriving from its own axioms. The theorem is a rigorous foundation stone, not the whole building.

/-- At ξ ≡ 0, the conformal squared-edge map gives the regular constant ℓ₀². -/
theorem conformalSqEdge_at_zero (ℓ₀ : ℝ) :
    conformalSqEdge ℓ₀ (fun _ => 0) = (fun _ => ℓ₀ ^ 2) := by
  funext e
  unfold conformalSqEdge
  simp [Real.exp_zero]

/-- The conformal squared-edge map.  `ℓ₀` is the flat-background length. -/
def conformalSqEdge (ℓ₀ : ℝ) (ξ : Fin 4 → ℝ) : SqEdges :=
  fun e =>
    let v := edgeVertices e
    ℓ₀ ^ 2 * Real.exp (ξ v.1 + ξ v.2)

What this page does not claim

The theorem does not prove the full Regge component theorem M_ij = -area(f_ij). The theorem does not assert anything about the behavior of the conformal edge map for nonzero vertex potentials. The theorem does not derive the external classical hypotheses, such as Schläfli's identity, from the framework's own axioms.

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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