Gravity Path Sum Uvbound Sinh Over Linear Monotone Statement

The suppression lemma

The declaration establishes a simple inequality: for any non-negative real number δ, δ ≤ sinh(δ). The hyperbolic sine function sinh(δ) is defined as (e^δ - e^-δ)/2. For small positive δ, sinh(δ) is only slightly larger than δ, but as δ grows, sinh(δ) grows much faster, since e^δ dwarfs e^-δ. The inequality is a formal theorem in the framework's machine-checked library of formal theorems, proved from the standard properties of the exponential function. Its name, sinh_over_linear_monotone_statement, refers to the ratio sinh(δ)/δ, which increases monotonically; the theorem itself states the weaker but sufficient fact that this ratio never drops below 1.

The inequality matters because of how the framework models spacetime. In this model, the fabric of space is a discrete record of events, a ledger, not a continuous sheet. The framework's path sum for gravity is a sum over triangulations, or piecewise-linear approximations, of a compact four-dimensional manifold. Each triangulation has a minimum mesh size, a smallest length scale, denoted ℓ_sub. The action, or the quantity whose exponential gives the path sum weight, is built from the deficit angles at each hinge of the triangulation. The deficit angle measures how much curvature is concentrated at that hinge. The recognition action uses sinh(δ) at each hinge, while the older Regge action uses just δ. The theorem recognition_dominates_regge, which follows directly from the inequality, shows that the recognition action is always at least as large as the Regge action when all deficit angles are non-negative.

This dominance is the core of the UV finiteness claim. In the standard continuum path integral for Einstein-Hilbert gravity, one sums over all possible metrics, and the mesh size is taken to zero. This limit produces the famous perturbative non-renormalizability of gravity, with divergences appearing at each loop order. The framework's path sum never takes this limit. The mesh is fixed at ℓ_sub, a positive number. The number of distinct triangulations is finite, bounded by an exponential of the maximum simplex count. The sinh action then provides an additional suppression: configurations with large deficit angles, meaning high local curvature, are exponentially more suppressed than they would be in the Regge sum. The theorem uv_finiteness_structural packages these three facts, the finite count, the positive mesh, and the sinh inequality, into a single statement.

In Recognition Science, the framework models the continuum Einstein-Hilbert action as an effective approximation to the discrete recognition action, valid only at scales much larger than ℓ_sub. The perturbative divergences of standard gravity are therefore not a property of nature but an artifact of a limit the framework never takes. This is a structural theorem, not a numerical prediction. It does not, for instance, compute a specific value for ℓ_sub or for the cosmological constant. The theorem establishes the logical consistency of the framework's approach to quantum gravity, showing that its central object, the path sum, is finite. It does not claim that this path sum is the correct theory of quantum gravity, nor that it reproduces any specific observable.

/-- The suppression ratio sinh(δ)/δ grows monotonically for δ > 0:
larger deficit angles are more suppressed relative to the Regge action.
(Monotonicity follows from d/dδ[sinh(δ)/δ] = (δcosh(δ) - sinh(δ))/δ² ≥ 0,
which holds because tanh(δ) ≤ δ for δ ≥ 0.  Statement only; proof deferred
to hard PDE content.) -/
theorem sinh_over_linear_monotone_statement :
    ∀ δ : ℝ, 0 ≤ δ → δ ≤ Real.sinh δ :=
  fun δ hδ => 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))

/-- **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⟩

What this page does not claim

The theorem does not prove that the recognition path sum is the correct theory of quantum gravity. The theorem does not compute the value of the substrate length ℓ_sub or any other physical constant. The theorem does not address the infrared behavior of the path sum or the cosmological constant problem.

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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