Foundation Pair Kernel Newtonian3 Integrable Inv Norm Profile Comp Norm

A well-behaved integrand

The declaration integrable_invNormProfile_comp_norm is a technical lemma in the framework's machine-checked library of formal theorems. It proves that a function which is the inverse square of the norm, but only inside a ball of radius N and zero outside, is integrable over all of three-dimensional space. In plainer terms, it shows that the integral of this sharply cut-off profile is finite and well-defined.

The function in question, invNormProfile N ‖x‖, is defined as (‖x‖^2)⁻¹ when ‖x‖ < N and 0 otherwise. The theorem states that this function is integrable with respect to the standard volume measure on ℝ³. This is not a trivial statement because the inverse-square function 1/‖x‖² is not integrable over all of space; it has a logarithmic divergence at large distances. The sharp cutoff at radius N is what makes the integral converge.

This lemma is a stepping stone in a larger proof campaign. The goal is to establish the Newtonian Fourier identity, which relates the integral of a cosine over a ball to the inverse distance. This identity is a key ingredient in deriving the Newtonian potential from a Fourier-space perspective. The integrability lemma ensures that the manipulations with integrals in this proof are mathematically sound.

In Recognition Science, this work is part of a Rung 3 campaign to pin down the geometric origin of the constant 4π in three dimensions. The lemma itself is a pure mathematical result about integration in Euclidean space; it does not involve any recognition-specific concepts. It is a prerequisite for the later steps that assemble the full Newtonian potential.

What this lemma does not claim is also clear. It does not prove the Newtonian Fourier identity itself; that is a separate target. It does not make any statement about the physical interpretation of the Newtonian potential or its role in gravity. It is purely a statement about the integrability of a specific mathematical function, which is a necessary condition for the subsequent analysis to be valid.

private theorem integrable_invNormProfile_comp_norm (N : ℝ) :
    Integrable (fun x : E3 => invNormProfile N ‖x‖) := by
  rw [MeasureTheory.integrable_fun_norm_addHaar]
  simp only [finrank_euclideanSpace_fin]
  show IntegrableOn (fun y : ℝ => y ^ 2 • invNormProfile N y) (Set.Ioi 0) volume
  have hIoo : IntegrableOn (fun _ : ℝ => (1 : ℝ)) (Set.Ioo 0 N) :=
    integrableOn_const (hs := by rw [Real.volume_Ioo]; exact ENNReal.ofReal_ne_top)
  have hind : Integrable ((Set.Ioo (0 : ℝ) N).indicator fun _ => (1 : ℝ)) :=
    (integrable_indicator_iff measurableSet_Ioo).2 hIoo
  refine hind.integrableOn.congr_fun ?_ measurableSet_Ioi
  intro y hy
  rw [Set.mem_Ioi] at hy
  simp only [Set.indicator_apply, Set.mem_Ioo, smul_eq_mul, invNormProfile]
  by_cases hlt : y < N
  · rw [if_pos ⟨hy, hlt⟩, if_pos hlt,
      mul_inv_cancel₀ (pow_ne_zero 2 (ne_of_gt hy))]
  · rw [if_neg fun h => hlt h.2, if_neg hlt, mul_zero]

What this page does not claim

The Newtonian Fourier identity itself is not proved by this lemma. This lemma does not involve any recognition-specific concepts or the framework's cost function. No physical interpretation of the Newtonian potential or its role in gravity is made here.

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/Foundation/PairKernelNewtonian3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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