Gravity Analysis Regge Hinge4 Dstar Kernel22 Only Origin Corner Contains Hinge

The hinge's unique corner

In four-dimensional geometry, a hinge is the line segment where two triangular faces of a simplex meet. The declaration only_origin_corner_contains_hinge proves a precise enumeration fact: among all sixteen corners of a four-dimensional hypercube with coordinates in {-1, 0}, only the origin corner contains the specified hinge. The proof is a decidable computation over all 16 corners, so it is exhaustive and machine-checked.

The hinge in question is a type-(2,2) triangle hinge, defined by the vertex set {0, e₀+e₁, e₀+e₁+e₂+e₃}. The theorem states that if any corner contains this hinge, that corner must be the origin. This is a purely combinatorial statement about the incidence structure of the hypercube and its simplices; it does not involve any physics or dynamics.

This enumeration is one step in a larger program. The surrounding library, a machine-checked collection of formal theorems, uses this fact to build the full periodic star of the hinge: the set of all four cube-simplex pairs that contain it. The star's flat cosine multiset, angle sum, and deficit class kernel are all proved separately. The theorem itself, however, only establishes the uniqueness of the corner, not any property of the star or any physical consequence.

In Recognition Science, this kind of exact combinatorial control is a prerequisite for later analysis. The framework models spacetime as a discrete ledger of recognition events, and this hinge is a local geometric feature within that ledger. The theorem ensures that the hinge's location is unambiguous, which is necessary for any subsequent calculation of curvature or deficit angles. Without this uniqueness, the star enumeration would be ill-defined.

The declaration does not claim that this hinge is physically special, nor that the four-dimensional lattice is the actual structure of spacetime. It is a formal result about a specific geometric configuration within a mathematical model. The proof is axiom-clean, relying only on the standard axioms of the ambient type theory, and it is fully verified by the machine-checked library.

theorem only_origin_corner_contains_hinge (c : CubeCorner)
    (h : cornerContainsHinge c = true) : c = 0 := by
  revert c
  decide

/-- Absolute (2,2) hinge vertex masks. -/
def absHingeMasks : Finset ℕ := {0, 3, 15}

theorem only_origin_corner_contains_hinge (c : CubeCorner)
    (h : cornerContainsHinge c = true) : c = 0 := by
  revert c
  decide

What this page does not claim

This theorem does not prove any property of the full star, such as its angle sum or deficit kernel. This theorem does not establish any physical consequence for the Recognition Science model of spacetime. This theorem does not claim that the four-dimensional hypercube lattice is the actual structure of physical space.

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/ReggeHinge4DStarKernel22.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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