Gravity Analysis Regge Hinge4 Dstar Kernel13 Only Origin Contains Hinge

The unique hinge location

In a four-dimensional lattice built from simplices, a hinge is a shared edge where several triangular faces meet, like the spine of a book where pages converge. The declaration only_origin_contains_hinge proves a precise fact about a specific type of hinge in a periodic lattice: among all possible cube-shaped regions that could contain this hinge, only the one centered at the origin does. The proof is a decidable search: it checks every candidate cube in a neighborhood and finds exactly one match, the origin.

This result is part of a larger effort to understand how gravity might emerge from a discrete, lattice-based model of spacetime. The hinge in question is a type-(1,3) triangle hinge, defined by a specific set of vertex masks. The theorem establishes that this hinge has a unique location, a fact that simplifies the analysis of the lattice's geometry. It is a kernel-checked theorem in the framework's machine-checked library of formal theorems, meaning it is proved with no gaps in its logical derivation.

What the declaration does not claim is equally important. It does not prove that this hinge is the only one in the entire lattice; it only proves that among the cube translates in a range from -1 to 1 in each dimension, the origin is the sole container. It does not establish that the full assembly of flat hinges over the entire lattice is complete, nor does it prove that the framework's action converges to the Einstein-Hilbert action of general relativity. It also does not reverse-engineer any weights from Einstein-Hilbert gravity. The theorem is a narrow, foundational step, not a complete theory of quantum gravity.

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

/-- Local masks of the type-`(1,3)` hinge in the origin cube. -/
def localHingeMasks : Finset ℕ := {0, 1, 15}

theorem star_cube_cardinality :
    (Finset.univ.filter (fun c : CubeOffset =>
      cubeContainsHinge c = true)).card = 1 := by
  decide

What this page does not claim

This theorem does not claim that the hinge is the only one in the entire infinite lattice. This theorem does not claim that the full flat Hessian assembly over all hinges is complete. This theorem does not claim that the framework's action converges to 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/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.

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