Gravity Regge Calculus Cube Dihedral Is Right Angle

The cube's right angle

Regge calculus, introduced by Tullio Regge in 1961, replaces smooth spacetime with a patchwork of flat blocks, called a simplicial complex. Curvature is not spread through the blocks but concentrated along their edges and corners, the hinges where blocks meet. The key quantity is the deficit angle: the amount by which the angles around a hinge fall short of or exceed a full turn. Where blocks fit together perfectly, the deficit is zero and spacetime is flat; where they do not, the deficit measures the curvature concentrated there.

In the Recognition Science (RS) framework, this patchwork is built on a lattice of cubes, a four-dimensional grid of space and time. The framework's machine-checked library of formal results contains a small but precise result about these cubes: the dihedral angle of a cube, the angle between two faces that share an edge, is a right angle, exactly pi over 2 radians, which is 90 degrees. The declaration cube_dihedral_is_right_angle states this fact as a theorem, and the proof is immediate from the definition: the angle is defined as pi over 2, and the theorem confirms that definition is correct.

This right angle is not an accident. A companion theorem in the same library, cubic_lattice_flat, shows that when four cubes meet around an edge, their four right angles sum to a full turn: 4 times pi over 2 equals 2 pi. The deficit angle is therefore zero, and the cubic lattice is flat. This is the discrete analogue of a familiar fact: a flat sheet of paper can be tiled with squares, but a sphere cannot be tiled with squares without wrinkling or cutting. The cube's right angle is what makes the RS lattice a flat background on which gravity can be built as a small perturbation.

In Recognition Science, the framework models this flat lattice as the starting point for its discrete gravity programme. The library proves that the Regge action, the sum over hinges of area times deficit angle, vanishes for flat configurations, and it derives a coupling constant from the golden ratio. But the right-angle theorem itself is a statement about the geometry of a cube, not a claim about physics. It does not assert that physical space is a cube lattice, nor that gravity is actually described by Regge calculus. It establishes a consistency condition: if the framework uses this lattice, the lattice is flat, and the machinery of Regge calculus applies cleanly.

What the theorem does not claim is just as important. It does not prove that the cube lattice is the only possible lattice, nor that the Regge action is the correct action for quantum gravity. It does not derive the value of any physical constant, and it does not say anything about the dynamics of the lattice, only its geometry. The right angle is a foundation stone, not a building. It tells a reader that the framework's chosen starting point is geometrically coherent, and that the flat background it assumes is not a hidden source of curvature.

/-- The dihedral angle of a cube is pi/2 (90 degrees). -/
theorem cube_dihedral_is_right_angle :
    cube_dihedral_angle = Real.pi / 2 := rfl

/-- 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

/-- 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]

/-- For a CUBE (all edges = a, all right angles), the dihedral angle is pi/2. -/
noncomputable def cube_dihedral_angle : ℝ := Real.pi / 2

What this page does not claim

The theorem does not claim that physical space is a cube lattice. It does not assert that Regge calculus is the correct theory of quantum gravity. It does not derive any physical constant or describe the dynamics of the lattice.

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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