Gravity Path Sum Uvbound Triangulation Count Bound Ne Zero

The positive count bound

The declaration triangulationCountBound_ne_zero is a small but load-bearing result in the framework's treatment of quantum gravity. It states that for any admissible family of triangulations, the bound on the number of distinct triangulations is a positive real number, never zero. The bound is computed as growthBase raised to the power maxSimplexCount, where growthBase is a positive growth rate and maxSimplexCount is a positive natural number. Since a positive base raised to any power is positive, the bound is strictly greater than zero. The theorem is proved in the machine-checked library of formal theorems with no axioms beyond the standard three (propext, Classical.choice, Quot.sound).

The result matters because it underpins the UV finiteness argument for the recognition path sum. The path sum is a sum over admissible triangulations of a compact 4-manifold, with mesh bounded below by a substrate length ℓ_sub. The count bound being positive guarantees that the sum has at least one term, and combined with the finiteness of the count bound itself, ensures the sum does not diverge from having infinitely many configurations. The framework models gravity as a discrete sum over triangulations rather than a continuum path integral, and this positivity is a necessary condition for the sum to be well-defined.

The theorem does not claim that the actual number of triangulations is positive, only that the bound on that number is positive. It does not establish UV finiteness by itself; that requires the full conjunction of the positive count bound, positive mesh, and sinh suppression of high-curvature configurations. The declaration is a structural guarantee about the formalism, not a physical prediction about gravity. It also does not claim that the bound is tight or that the growth rate is known; growthBase is a parameter of the admissible family, not a derived constant.

/-- The triangulation count bound is a concrete positive real number. -/
theorem triangulationCountBound_ne_zero (F : AdmissibleTriangulationFamily) :
    triangulationCountBound F ≠ 0 :=
  ne_of_gt (triangulationCountBound_pos F)

/-- The triangulation count bound is a concrete positive real number. -/
theorem triangulationCountBound_ne_zero (F : AdmissibleTriangulationFamily) :
    triangulationCountBound F ≠ 0 :=
  ne_of_gt (triangulationCountBound_pos F)

/-- An admissible triangulation family for the recognition path sum.
Members have mesh bounded below by ℓ_sub and simplex count bounded
above by the volume constraint. -/
structure AdmissibleTriangulationFamily where
  /-- Maximum simplex count in any admissible triangulation. -/
  maxSimplexCount : ℕ
  maxSimplexCount_pos : 0 < maxSimplexCount
  /-- Growth rate of the triangulation count: the number of
  combinatorially distinct triangulations with at most N simplices
  is bounded by growthBase^N. -/
  growthBase : ℝ
  growthBase_pos : 0 < growthBase
  /-- The minimum mesh length, equal to ℓ_sub. -/
  minMesh : ℝ
  minMesh_pos : 0 < minMesh

What this page does not claim

This theorem does not claim that the actual number of triangulations is positive, only that the bound on that number is positive. This theorem does not by itself establish UV finiteness of the recognition path sum. This theorem does not claim that growthBase or maxSimplexCount are derived from physical principles rather than chosen parameters.

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

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