Gravity Analysis Freudenthal Stencil Preflight Canonical Periodic No Self Loop E

A clean mesh, edge by edge

In the study of discrete gravity, a triangulation is a mesh of points connected by edges, and a self-loop is an edge that starts and ends at the same point. Such a loop is a defect: it makes the geometry ambiguous and the energy formulas ill-defined. The declaration canonicalPeriodic_noSelfLoopEdges establishes that a specific, standard triangulation, the canonical periodic Freudenthal triangulation at side length N, has no self-loops, provided N is greater than 2. This is a proved theorem in the framework's machine-checked library of formal theorems, not an assumption or a numerical check.

The proof works by showing that every edge in the triangulation connects two distinct vertices. The argument uses the periodic boundary conditions, which wrap the mesh around like a torus, and the fact that the displacement between the endpoints of any edge is never zero. The condition N > 2 is essential: for smaller meshes, the wrapping can identify distinct points, creating a self-loop. This theorem is a foundational step, a preflight check that the mesh is well-behaved before any energy or continuum limit is computed.

In Recognition Science, this result is part of a larger campaign to derive the continuum limit of a discrete gravity action. The framework models spacetime as a discrete ledger of events, and here it proves that a particular discrete geometry is free of a basic pathology. The theorem does not claim that all triangulations are free of self-loops, nor does it say anything about the physical content of the continuum limit. It is a necessary condition, not a sufficient one, for the validity of the subsequent stencil analysis.

The practical consequence is that the energy calculations on this triangulation are well-defined. The proof removes a potential source of error, allowing the framework to proceed with confidence to the next stage, where it derives the exact stencil identity and the moment tensor. This no-self-loop theorem is a small but load-bearing piece of the larger argument that the discrete theory has a sensible continuum limit.

/-- For `N > 2` the canonical periodic triangulation has no self-loop edges:
the side-length assumption rules out `x + d = x` wraparound. -/
theorem canonicalPeriodic_noSelfLoopEdges (hN : 2 < N) :
    NoSelfLoopEdges (canonicalPeriodicTriangulation N N N) := by
  intro e h
  rw [canonical_edgeVerts_eq] at h
  exact PeriodicEdge.endpoints_ne hN hN hN (edgeFinEquiv N N N e)
    ((vertexFinEquiv N N N).symm.injective h)

What this page does not claim

The theorem does not apply to triangulations with side length N equal to 2 or less. The theorem does not prove that all periodic triangulations are free of self-loops. The theorem does not establish the physical validity of the continuum limit, only the well-definedness of the discrete energy.

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

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