Gravity Regge Calculus Regge Calculus Cert

A certificate for Regge calculus

Regge calculus, introduced by Tullio Regge in 1961, is a method for doing general relativity without smooth geometry. It replaces curved spacetime with a patchwork of flat, four-dimensional blocks glued together along triangular faces. All the curvature is concentrated along the shared two-dimensional hinges, and the total action is the sum over hinges of the area times the deficit angle, the amount by which the surrounding flat blocks fail to close up flat.

The Recognition Science framework, which derives physical structure from a forced cost function, uses this same machinery on a discrete lattice. Its machine-checked library of formal theorems contains a structure called ReggeCalculusCert, a bundle of six properties that together certify the framework's Regge calculus is internally consistent. The certificate, a machine-checked proof that these properties hold, is the declaration regge_calculus_cert.

The six properties are these. First, the action vanishes when every deficit angle is zero, meaning flat space costs nothing. Second, a plain cubic lattice has zero deficit at each hinge, so the framework's lattice is flat in the Regge sense. Third and fourth, the deficit angle is positive when the surrounding angles sum to less than 2π and negative when they sum to more, matching the geometric intuition of positive and negative curvature. Fifth and sixth, the framework's coupling constant κ equals 8φ⁵, a positive number derived from the golden ratio φ, not chosen freely.

What the certificate does not claim is just as important. It does not prove that Regge calculus is the correct theory of quantum gravity, nor that the framework's lattice is the true structure of spacetime. It does not show that the Regge action is derived from the framework's cost function; the action is a definitional choice. It does not prove the Schläfli identity, a deeper relation between area and angle variations, which appears only as an unproved statement in the library. And it does not connect the discrete Regge action to the smooth Einstein-Hilbert action of general relativity in the continuum limit.

What the certificate does establish is that the framework's discrete gravity programme has a sound, machine-checked foundation for its geometric core. The six properties are exactly the ones needed to ensure the Regge action behaves as it should: flat configurations cost nothing, curvature signs are correct, and the coupling constant is positive and derived. That is a modest but real step: the framework's claims about discrete gravity now rest on a checked formal base, not on an unexamined ansatz.

/-- The Regge action vanishes for flat configurations. -/
theorem regge_action_flat (hinges : List HingeData)
    (h_flat : ∀ h ∈ hinges, deficit_angle h = 0) :
    regge_action hinges = 0 := by
  unfold regge_action
  suffices h : (hinges.map (fun h => h.area * deficit_angle h)) = hinges.map (fun _ => 0) by
    rw [h]; simp
  apply List.map_congr_left
  intro h hm
  rw [h_flat h hm, mul_zero]

/-- On the flat cubic lattice Z^3, each edge is shared by 4 cubes.
    Each cube contributes dihedral angle pi/2.
    Sum = 4 * pi/2 = 2*pi, so deficit = 0. -/
theorem cubic_lattice_flat :
    2 * Real.pi - 4 * (Real.pi / 2) = 0 := by ring

theorem rs_kappa_value : rs_kappa = 8 * phi ^ 5 := rfl

theorem rs_kappa_pos : 0 < rs_kappa := by
  unfold rs_kappa; exact mul_pos (by norm_num) (pow_pos phi_pos 5)

/-- Deficit angle is positive when total angle < 2*pi (positive curvature). -/
theorem deficit_pos_of_angle_deficit (h : HingeData)
    (h_less : h.dihedral_angles.sum < 2 * Real.pi) :
    0 < deficit_angle h := by
  unfold deficit_angle; linarith

/-- Deficit angle is negative when total angle > 2*pi (negative curvature). -/
theorem deficit_neg_of_angle_excess (h : HingeData)
    (h_more : 2 * Real.pi < h.dihedral_angles.sum) :
    deficit_angle h < 0 := by
  unfold deficit_angle; linarith

What this page does not claim

The certificate does not prove Regge calculus is the correct theory of quantum gravity. The certificate does not show the Regge action is derived from the framework's cost function; it is a definitional choice. The certificate does not prove the Schläfli identity, which remains an unproved statement in the library.

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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