Gravity Analysis Regge Hinge4 Dstar Kernel

The flat star kernel

In four-dimensional geometry, a hinge is a two-dimensional triangle where several four-dimensional simplexes meet, just as an edge in three dimensions is where several tetrahedra meet. Regge calculus, a standard method for studying gravity, approximates curved spacetime by gluing together many flat simplexes and measuring how the angles around each hinge fail to sum to the flat value of 2π. That failure, the deficit angle, encodes curvature.

The module ReggeHinge4DStarKernel examines one specific hinge in a periodic lattice, the seed triangle with vertices at 0, e₀, and e₀+e₁. The lattice is built from cubes, each split into 24 simplexes. The module proves that exactly six cube-simplex pairs contain this hinge, a fact established by exhaustive enumeration. For each of these six simplexes, the module computes the cosine of the dihedral angle at the hinge from the simplex's own Gram vector, the matrix of inner products of its edge vectors.

Four of the six simplexes have a flat cosine of 1/√2, corresponding to an angle of π/4, and two have a flat cosine of 0, an angle of π/2. The sum of these angles is exactly 2π, the flat-space value. This is the flatness gate: the seed hinge, in the absence of curvature, has no deficit. The module then assembles a full-star deficit class kernel, a function assigning a numerical weight to each of the 15 lattice classes, with values (-1,-1,+1,-1,+1,+1,-1) on the seven classes that appear.

In Recognition Science, this kernel is a machine-checked increment in a campaign to derive gravity from the framework's core principles. The module proves four gates: the kernel is nonvacuous, it is symmetric under swapping two coordinates, it is a decoy against uniform scaling, and it is stationary under homothety, meaning the directional derivative along a uniform scaling direction is zero. Each theorem is checked by the machine-checked library of formal theorems, with no unproved assumptions beyond the standard three axioms of the underlying type theory.

What this establishes in plain language is that the seed hinge, surrounded by its full periodic star of six simplexes, is exactly flat before any curvature is introduced. The deficit kernel built from it is a concrete, verified object that later modules can use to assemble the full Hessian of the Einstein-Hilbert action. The module does not complete that assembly, and it does not prove that the framework's action converges to the Einstein-Hilbert action in four dimensions. Those remain targets.

theorem star_cardinality :
    (Finset.univ.filter (fun p : CubeTranslate × Fin 24 =>
      containsHinge p.1 p.2 = true)).card = 6 := by
  decide

def starFlatCosines : Fin 6 → ℝ
  | ⟨0, _⟩ | ⟨1, _⟩ => 1 / Real.sqrt 2
  | ⟨2, _⟩ | ⟨3, _⟩ => 0
  | ⟨4, _⟩ | ⟨5, _⟩ => 1 / Real.sqrt 2

theorem star_flat_angle_sum_two_pi : starFlatAngleSum = 2 * Real.pi := by
  simp only [starFlatAngleSum, flatAngleSeedOpp_eq, flatAngleOrth_eq]
  ring

theorem fullStar_homothety_stationary :
    fullStarDirectional (fun d => (classWeightNat d : ℝ)) = 0 := by
  simp only [fullStarDirectional]
  rw [sum15_support _ (fun d h2 h3 h6 h7 h10 h11 h14 => by
    rw [fullStarClassKernel_zero_off d h2 h3 h6 h7 h10 h11 h14, mul_zero])]
  have w2 : classWeightNat 2 = 2 := by decide
  have w3 : classWeightNat 3 = 1 := by decide
  have w6 : classWeightNat 6 = 3 := by decide
  have w7 : classWeightNat 7 = 1 := by decide
  have w10 : classWeightNat 10 = 3 := by decide
  have w11 : classWeightNat 11 = 2 := by decide
  have w14 : classWeightNat 14 = 4 := by decide
  simp [fullStarClassKernel, w2, w3, w6, w7, w10, w11, w14]
  norm_num

What this page does not claim

The module does not complete the flat Hessian assembly over all hinges. The module does not prove S_RS_converges_EH_4d. The module does not reverse-engineer weights from the Einstein-Hilbert action.

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/Analysis/ReggeHinge4DStarKernel.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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