Gravity Analysis Regge Hinge4 Dstar Kernel13

The hinge kernel

In four-dimensional spacetime, a hinge is a two-dimensional plane where several triangular building blocks meet, like the crease where pages of a book come together. Regge calculus, a standard approach to quantum gravity, approximates curved spacetime by gluing together flat four-dimensional blocks along these hinges. The curvature then lives at the hinges, measured by how much the angles around them fail to sum to the full 2π that flat space requires. This deficit angle is the basic currency of the theory.

The module ReggeHinge4DStarKernel13 examines one specific type of hinge, labeled type (1,3), and proves in a machine-checked library of formal theorems that its local geometry is exactly flat. The proof works by enumerating all six triangular blocks that contain the hinge, showing each has a flat cosine of 1/2, and then verifying that their angles sum to precisely 2π. The kernel then computes the derivative of this angle sum with respect to all ten edge lengths, yielding a list of values that describe how the geometry responds to small changes. These values form the kernel, a mathematical object that can be used to build larger structures.

In Recognition Science, this kernel is a step toward a larger goal: deriving the dynamics of spacetime from a discrete ledger of recognition events. The framework models physical space as a lattice of points, and the kernel provides a verified piece of the Hessian, the matrix of second derivatives that governs how the geometry bends. The module proves that this particular hinge is a flat, regular piece of the lattice, with a homothety stationarity property: scaling all edge lengths uniformly leaves the kernel unchanged. This is a consistency check that the discrete geometry behaves like its continuous counterpart.

The work is explicitly scoped. It does not complete the full Hessian assembly over all hinges, and it does not prove that the entire framework converges to the Einstein-Hilbert action of general relativity. The complementary type (3,1) hinge is related by symmetry, but transporting this kernel to that case remains an open problem. What the module does establish is a clean, verified fact about the geometry of a single hinge, a necessary brick in a much larger construction.

def starFlatCosines : Fin 6 → ℝ := fun _ => (1 / 2 : ℝ)

theorem star_flat_angle_sum_two_pi : starFlatAngleSum = 2 * Real.pi := by
  simp only [starFlatAngleSum, flatAngleT13_eq]
  ring

theorem only_origin_contains_hinge :
    (Finset.univ.filter (fun c : CubeOffset =>
      cubeContainsHinge c = true)) = {originOffset} := by
  decide

theorem fullStar_homothety_stationary :
    fullStarDirectional (fun d => (classWeightNat d : ℝ)) = 0 := by
  simp only [fullStarDirectional]
  rw [sum15_support _ (fun d h1 h3 h5 h7 h9 h11 h13 => by
    rw [fullStarClassKernel_zero_off d h1 h3 h5 h7 h9 h11 h13, mul_zero])]
  have w1 : classWeightNat 1 = 1 := by decide
  have w3 : classWeightNat 3 = 1 := by decide
  have w5 : classWeightNat 5 = 2 := by decide
  have w7 : classWeightNat 7 = 1 := by decide
  have w9 : classWeightNat 9 = 2 := by decide
  have w11 : classWeightNat 11 = 2 := by decide
  have w13 : classWeightNat 13 = 3 := by decide
  simp [fullStarClassKernel, w1, w3, w5, w7, w9, w11, w13]
  ring

What this page does not claim

This module does not prove convergence to the Einstein-Hilbert action. This module does not complete the full Hessian assembly. This module does not reverse-engineer weights from general relativity.

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/ReggeHinge4DStarKernel13.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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