Gravity Analysis Regge Hinge4 Dorbit Classification Decoy Overlapping Not Realiz

A combinatorial gate

In four-dimensional geometry, a Kuhn triangulation slices a hypercube into 24 tetrahedral simplices. This page concerns the triangles that form the hinges of a Regge calculus approximation to curved spacetime. The declaration decoy_overlapping_not_realizable (a machine-checked theorem in the framework's library of formal theorems) establishes a narrow combinatorial fact: a certain pair of difference masks, labeled (1,3), cannot arise as the two edge-difference masks of any triangle in this triangulation.

To understand the claim, picture each triangle as a chain of three nested vertex sets. The difference between consecutive sets yields two masks, whose sizes form a type pair. The theorem states that the type (1,3), meaning one mask with a single bit and another with three bits, is not realizable. The proof is a finite computation: it checks all 240 oriented triangle slots and finds none with that type. This is a gate (a test that a candidate object must pass) in a larger classification of triangle orbits under lattice symmetries.

The theorem is a negative result that prevents a false path in a combinatorial enumeration. It confirms that the six realizable types are exactly (1,1), (1,2), (2,1), (1,3), (3,1), and (2,2), with the (1,3) type excluded. The related theorem decoy_overlapping_is_not_disjoint shows the pair is not even disjoint, meaning the two masks share a bit, which is why it cannot represent a chain of nested sets. This classification is a prerequisite for assembling a flat Hessian from per-orbit star kernels, a step toward a discrete action for gravity.

The declaration does not claim anything about physics. It does not evaluate any star kernels, does not complete the flat Hessian of the 4D Regge action, does not prove that the framework's action converges to the Einstein-Hilbert action, and does not flip any gap action recovery. Its scope is purely combinatorial: the structure of triangle hinges in one unit 4-cube Kuhn triangulation, up to lattice translation and symmetry. The theorem is a building block, not a physical law.

/-- THEOREM (decoy): overlapping masks `(1,3)` are not a monotone
difference pair. -/
theorem decoy_overlapping_not_realizable :
    isRealizableDiffPair 1 3 = false := by
  decide

theorem oriented_slot_total :
    (Fintype.card (Fin 24) * Fintype.card (Fin 10)) = 240 := by
  decide

theorem decoy_overlapping_is_not_disjoint :
    isDisjointDiffPair 1 3 = false := by
  decide

theorem orbit_count_S4 : Fintype.card HingeOrbitType = 6 := by
  decide

What this page does not claim

This theorem does not evaluate any per-orbit star kernels. This theorem does not complete the flat Hessian of the 4D Regge action. This theorem does not prove convergence to the Einstein-Hilbert action or flip any gap action recovery.

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/ReggeHinge4DOrbitClassification.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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