Gravity Path Sum Uvbound Triangulation Count Bound Pos

A positive count bound

The declaration triangulationCountBound_pos is a small but load-bearing result inside the Recognition Science framework's treatment of gravity. The framework models spacetime as a discrete record of events, and its path sum, the analogue of a sum over all possible histories, runs over triangulations: ways of chopping a curved four-dimensional space into flat tetrahedral pieces. The declaration proves that the bound on how many such triangulations exist is always a positive real number, never zero, for any admissible family of triangulations.

The bound itself is an exponential expression, growthBase raised to the power of maxSimplexCount. The growthBase is a positive real number, and maxSimplexCount is a positive natural number. The theorem triangulationCountBound_pos simply assembles these two positivity facts: a positive number raised to a positive power is positive. That is the entire content of the declaration, and it is proved in the machine-checked library of formal theorems with no unproved assumptions.

Why does this matter? The path sum in this framework is a sum over admissible triangulations, each weighted by an action built from the hyperbolic sine of deficit angles, the angles by which a triangulated space fails to be flat. For the sum to be a well-defined object, the number of triangulations must be finite and the bound on that number must not collapse to zero. The declaration guarantees the bound is nonzero, so the counting side of the path sum has a solid foundation.

In Recognition Science, this positivity result is one ingredient in a larger claim: that the path sum is ultraviolet-finite, meaning it does not blow up at short distances. The framework argues that the continuum divergences of ordinary Einstein-Hilbert gravity are artifacts of taking the mesh size to zero, a limit the discrete substrate never takes. The declaration triangulationCountBound_pos does not itself prove ultraviolet finiteness. It proves only the positivity of the count bound, a necessary but far from sufficient condition.

The declaration also does not claim that the bound is tight or that it gives a realistic estimate of the number of triangulations. It says nothing about the physical correctness of the framework's model of gravity, and it does not assert that the path sum converges. It is a narrow, exact statement about a positive real number, and its value lies in being one verified brick in a larger formal edifice.

/-- The triangulation count bound is positive. -/
theorem triangulationCountBound_pos (F : AdmissibleTriangulationFamily) :
    0 < triangulationCountBound F :=
  pow_pos F.growthBase_pos _

/-- The triangulation count bound is positive. -/
theorem triangulationCountBound_pos (F : AdmissibleTriangulationFamily) :
    0 < triangulationCountBound F :=
  pow_pos F.growthBase_pos _

What this page does not claim

The declaration does not prove that the recognition path sum is ultraviolet-finite. The declaration does not claim the triangulation count bound is tight or physically realistic. The declaration does not assert anything about the empirical correctness of the framework's model of gravity.

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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