Gravity Regge Calculus Rs Edge Length Pos

A small guarantee with a large purpose

Regge calculus, introduced by Tullio Regge in 1961, replaces smooth curved spacetime with a patchwork of flat, four-dimensional triangles called simplices. Curvature is not spread out; it is concentrated along the shared two-dimensional hinges where these blocks meet. The standard Regge action, the quantity whose variation gives the equations of motion, is a sum over all hinges of the hinge's area multiplied by its deficit angle. The deficit angle measures how much the flat blocks fail to close up around the hinge, and it is zero when the surrounding space is flat.

In the Recognition Science framework, this piecewise-flat picture is built on a discrete lattice, a regular grid of points in space and time. The edge lengths of the simplices, the distances between neighboring lattice points, are not arbitrary. They are determined by a field derived from the framework's cost function, the J-cost defect field. The specific formula for an edge length is a times the square root of g, where a is a base scale and g is a component of the metric field that encodes the geometry. The declaration rs_edge_length_pos proves a basic but essential fact: if both the base scale a and the metric component g are positive, then the resulting edge length is also positive. This is a theorem, verified by a machine-checked library of formal theorems, and it holds without exception.

This positivity guarantee is a necessary condition for the entire Regge construction to be meaningful. A triangle or a four-dimensional simplex with a negative or zero edge length is geometrically nonsensical; it cannot exist in a standard Euclidean or Lorentzian space. By ensuring all edge lengths are positive, the theorem certifies that the simplices in the framework's discrete gravity are valid geometric objects. It is a foundational piece of the larger effort to show that the framework's discrete model of gravity is well-formed and can be used for further calculations.

The theorem does not claim that the edge lengths are physically correct, that they match any particular measurement, or that they solve the Regge equations of motion. It only establishes a consistency condition: positive inputs yield positive outputs. The declaration is also silent on the specific value of the base scale a or the metric component g; it merely states a relationship between them. This is a small, precise step, but it is the kind of step that a rigorous framework must take before it can make larger claims about the nature of gravity.

theorem rs_edge_length_pos (a : ℝ) (g : ℝ) (ha : 0 < a) (hg : 0 < g) :
    0 < rs_edge_length a g :=
  mul_pos ha (Real.sqrt_pos.mpr hg)

/-- In the RS framework, the edge lengths are determined by the J-cost
    defect field. For a lattice with spacing a and defect density rho:
    L_e = a * (1 + kappa_RS * rho(x))^(1/D) approximately.

    The exact relationship is:
    L_e^2 = a^2 * g_{mu nu}(x) * dx^mu * dx^nu

    where g = eta + h is the full metric and h is determined by J-cost. -/
noncomputable def rs_edge_length (a : ℝ) (g_component : ℝ) : ℝ :=
  a * Real.sqrt g_component

/-- A 4-simplex has 5 vertices and C(5,2) = 10 edges.
    The geometry is entirely determined by the 10 edge lengths. -/
structure Simplex4D where
  edges : Fin 10 → ℝ
  all_pos : ∀ i, 0 < edges i

What this page does not claim

The theorem does not assert that the edge lengths are physically correct or match any measurement. The theorem does not solve the Regge equations of motion or determine the base scale a. The theorem does not prove that the simplices can be embedded in a higher-dimensional space without self-intersection.

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

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