Gravity Analysis Regge Hinge4 Dstar Kernel12 Full Star Class Kernel Swap12

The swap symmetry

In four-dimensional lattice gravity, one often studies how spacetime curvature is concentrated along triangular hinges, the lower-dimensional edges where several tetrahedra meet. The Recognition Science framework formalizes this study inside a machine-checked library of formal theorems, where every statement is verified by a computer kernel. The declaration fullStarClassKernel_swap12 is a theorem about a particular classification of these hinges, and it establishes a symmetry: the classification is unchanged when the first and second coordinate directions are swapped.

To understand what is proved, one must know what the classification does. The framework divides the possible local geometric configurations around a hinge into fifteen classes, each labeled by a number. The kernel assigns a numerical weight to each class. The theorem states that if you take any class, swap the first two coordinates of its defining data, and then look up the weight of the resulting class, you get the same number as before. In symbols, for every class d, fullStarClassKernel (swap12Class d) = fullStarClassKernel d. This is a concrete, checkable identity, and the proof is a direct case check over all fifteen classes.

The theorem sits inside a larger construction that has already been verified. The same module proves that exactly four cube-simplex pairs contain the representative hinge, that all four have flat cosine zero, that the star angle sum is exactly 2π, and that the full deficit kernel takes values ±√2/2 on all fifteen classes. The swap symmetry is an additional gate: it shows the classification respects a natural coordinate exchange, a property one would expect of a well-behaved geometric invariant.

What the theorem does not claim is equally important. It does not complete the flat Hessian assembly over all hinges, which remains an open target. It does not prove that the Recognition Science action converges to the Einstein-Hilbert action, a statement named S_RS_converges_EH_4d. It does not recover the weights of the kernel by fitting them to known physics, and it does not address the complementary type (2,1) hinge, which is also open. The theorem is a single, precise piece of a larger program, not the program itself.

For a reader, the practical consequence is this: the framework now has a verified symmetry property for one family of hinges, which strengthens the case that the classification is geometrically meaningful. It also provides a template for how such symmetries can be checked mechanically, one case at a time, without relying on hand-waving or unstated assumptions.

theorem fullStarClassKernel_swap12 (d : Fin 15) :
    fullStarClassKernel (swap12Class d) = fullStarClassKernel d := by
  fin_cases d <;> rfl

theorem star_cardinality :
    (Finset.univ.filter (fun p : CubeTranslate × Fin 24 =>
      containsHinge p.1 p.2 = true)).card = 4 := by
  decide

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

What this page does not claim

The theorem does not prove convergence to the Einstein-Hilbert action. The theorem does not derive the kernel weights from first principles. The theorem does not address the complementary type (2,1) hinge.

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/ReggeHinge4DStarKernel12.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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