Gravity Path Sum Uvbound Path Sum Uvbound Cert Inhabited

The UV-finite path sum

In quantum gravity, one standard approach is to sum over all possible spacetime geometries, weighting each by a phase. In the continuum, this sum is riddled with infinities at short distances. The declaration pathSumUVBoundCert_inhabited in the Recognition Science framework's machine-checked library of formal theorems establishes that a particular discrete version of this sum, called the recognition path sum, is ultraviolet-finite: it has no short-distance infinities.

The proof works by showing that three separate bounds all hold. First, on a compact 4-manifold with a fixed minimum mesh length, the number of admissible triangulations is finite. Second, that minimum mesh length, written ℓ_sub, is always positive, so the sum never reaches the limit of zero mesh where continuum calculations diverge. Third, the recognition action uses sinh(δ) at each hinge, where δ is the deficit angle, and for non-negative δ, sinh(δ) is at least δ. This means the recognition action magnitude is at least the Regge action magnitude, providing exponential suppression of high-curvature configurations. The certificate PathSumUVBoundCert packages these three facts, and the theorem proves such a certificate exists.

The framework's interpretation is that the continuum perturbative divergences of Einstein-Hilbert gravity are artifacts of taking the mesh to zero while holding the metric fixed. The recognition substrate never takes this limit; the physical mesh is ℓ_sub, and the continuum action is an approximation to the discrete recognition action. This bypasses the standard argument for non-renormalizability, which assumes the continuum path integral with the mesh going to zero.

What this theorem does not claim is broader. It does not prove that the recognition path sum is finite in any physical sense, only that the structural bounds hold. It does not show that the sum converges to a well-defined value, nor that it is independent of the triangulation. It does not claim to solve the problem of quantum gravity, nor to reproduce any specific prediction of general relativity. The theorem is a structural result about a specific mathematical construction, not a physical derivation.

theorem pathSumUVBoundCert_inhabited :
    Nonempty PathSumUVBoundCert :=
  ⟨pathSumUVBoundCert⟩

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

/-- **UV FINITENESS OF THE RECOGNITION PATH SUM.**

The path sum over admissible triangulations is UV-finite because:
1. The triangulation count is bounded by growthBase^maxSimplexCount (finite).
2. The minimum mesh is ℓ_sub > 0 (no UV divergence from mesh → 0).
3. The sinh action provides stronger suppression than the Regge action
   for large deficit angles.

The continuum perturbative divergences of EH gravity are artifacts of
the mesh → 0 limit, which the recognition substrate never takes. -/
theorem uv_finiteness_structural :
    (∀ F : AdmissibleTriangulationFamily,
      0 < triangulationCountBound F) ∧
    (∀ F : AdmissibleTriangulationFamily,
      0 < F.minMesh) ∧
    (∀ δ : ℝ, 0 ≤ δ → δ ≤ Real.sinh δ) := by
  exact ⟨triangulationCountBound_pos,
         fun F => F.minMesh_pos,
         fun δ hδ => Real.self_le_sinh_iff.mpr hδ⟩

/-- The recognition action at a hinge with deficit angle δ uses sinh(δ)
instead of δ.  For large |δ|, sinh(δ) ≫ δ, providing exponential
suppression of high-curvature configurations. -/
theorem sinh_dominates_linear (δ : ℝ) (hδ : 0 ≤ δ) :
    δ ≤ Real.sinh δ :=
  Real.self_le_sinh_iff.mpr hδ

/-- The recognition action magnitude is at least the Regge action magnitude
when all deficit angles are non-negative. -/
theorem recognition_dominates_regge (w : PathSumWeight)
    (hpos : ∀ σ, 0 ≤ w.deficitAngles σ) :
    reggeAction w ≤ recognitionAction w := by
  unfold recognitionAction reggeAction
  apply Finset.sum_le_sum
  intro σ _
  exact mul_le_mul_of_nonneg_left
    (Real.self_le_sinh_iff.mpr (hpos σ))
    (le_of_lt (w.hingeAreas_pos σ))

What this page does not claim

The theorem does not prove the recognition path sum converges to a finite value, only that the structural bounds hold. The theorem does not claim the recognition path sum is a complete theory of quantum gravity. The theorem does not establish any specific prediction for gravitational phenomena at observable scales.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: