Foundation Pair Kernel Newtonian3 Newtonian Fourier Target Of Aligned Axis

The aligned-axis theorem

The declaration newtonianFourierTarget_of_aligned_axis is a theorem in the framework's machine-checked library of formal theorems. It states that a certain Fourier integral over three-dimensional space, truncated to a ball of radius N, converges to the Newtonian potential 1/(4π|x|) as N grows without bound. The name encodes the proof strategy: rotation invariance lets the argument reduce every nonzero separation vector to its norm along the first coordinate axis, so the full three-dimensional target follows from a one-dimensional scalar limit.

The theorem is the culmination of a module that deliberately imports no lattice or recognition structure. Its job is to pin the geometric origin of the constant 4π in three dimensions, independently of the recognition carrier. The module proves that three times the volume of the unit ball in ℝ³ equals 4π, equivalently that 4π is the surface area of the unit 2-sphere. It also proves the one-dimensional sinc integral ∫ cos(ρt) dt from -1 to 1 equals 2 sinc ρ, and the Dirichlet leaf ∫₀ᴺ sin(t)/t dt → π/2 with an explicit 2/R remainder bound.

What the theorem does not claim is just as important. The full Newtonian Fourier identity, written as lim_N (2π)⁻³ ∫_{‖k‖

In plain terms, the theorem shows that the inverse-square law's Fourier representation is not an independent physical assumption but a consequence of three-dimensional geometry plus a sharp-cutoff limiting procedure. The framework uses this as a stepping stone toward deriving Newtonian gravity from recognition costs, but the theorem itself is a pure mathematical statement about integrals and spheres. It says nothing about masses, forces, or the recognition ledger; those connections belong to later rungs of the campaign.

theorem newtonian_fourier_target : NewtonianFourierTarget := by
  intro x hx
  have ha : 0 < ‖x‖ := norm_pos_iff.2 hx
  have hscale :
      Filter.Tendsto (fun N : ℝ => ‖x‖ * N) Filter.atTop Filter.atTop :=
    Filter.tendsto_atTop.2 fun b => by
      filter_upwards [Filter.eventually_ge_atTop (b / ‖x‖)] with N hN
      have h := (div_le_iff₀ ha).1 hN
      simpa [mul_comm] using h
  have hdir :=
    dirichlet_sine_integral_tendsto.comp hscale
  have haxis :
      Filter.Tendsto
        (fun N : ℝ =>
          ∫ k in Metric.ball (0 : E3) N,
            Real.cos (‖x‖ * k 0) / ‖k‖ ^ 2)
        Filter.atTop
        (nhds ((4 * Real.pi / ‖x‖) * (Real.pi / 2))) := by
    apply (hdir.const_mul (4 * Real.pi / ‖x‖)).congr'
    filter_upwards [Filter.eventually_gt_atTop (0 : ℝ)] with N hN
    exact (axis_ball_integral_formula ‖x‖ N ha hN).symm
  have horiginal :
      Filter.Tendsto
        (fun N : ℝ =>
          ∫ k in Metric.ball (0 : E3) N,
            Real.cos (∑ i, k i * x i) / ‖k‖ ^ 2)
        Filter.atTop
        (nhds ((4 * Real.pi / ‖x‖) * (Real.pi / 2))) := by
    apply haxis.congr'
    exact Filter.Eventually.of_forall fun N =>
      (newtonian_integral_rotate_to_axis x N).symm
  have hnormalized :=
    horiginal.const_mul (1 / (2 * Real.pi) ^ 3)
  convert hnormalized using 1
  field_simp [Real.pi_ne_zero, ha.ne']
  ring_nf

private noncomputable def axisIsometry (x : E3) : E3 ≃ₗᵢ[ℝ] E3 :=
  ((ℝ ∙ (x - ‖x‖ • axisVector))ᗮ).reflection

/-- The ambient three-dimensional Euclidean carrier for the Newtonian angular
leaf. -/
abbrev E3 : Type := EuclideanSpace ℝ (Fin 3)

What this page does not claim

The full Newtonian Fourier identity is not proved in this module; it remains a stated target. The theorem says nothing about masses, forces, or the recognition ledger; those connections belong to later rungs. The module imports no lattice or recognition structure, so the 4π constant is purely geometric, not recognition-derived.

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)

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

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