Foundation Pair Kernel Green Fourier3 Outer Cube Ball Slice R Mul Integral Eq Tw

A slice integral identity

The declaration outerCubeBallSlice_r_mul_integral_eq_two_interval_ibp is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns the lattice Green function, which describes how influence spreads on an infinite cubic lattice where each point connects to its six nearest neighbors. The theorem proves that a certain integral over a slice of the cube, weighted by the radial distance, equals another integral over a two-dimensional interval. In plain terms, it shows that two different mathematical procedures for computing a quantity along a radial direction produce identical results.

This identity is part of a larger effort to understand the lattice Green function's behavior. The function is defined through a Fourier representation, which expresses it as an integral over a cube in frequency space. The theorem in question helps establish that this representation is consistent when one restricts attention to a slice of the cube, a technical step needed for later analysis. The proof is fully formalized and checked by the Lean kernel, meaning it is a verified result within the framework's axiomatic system.

In Recognition Science, this Green function is not just a mathematical curiosity. It models how a discrete ledger of recognition events propagates through a lattice, and its long-range behavior is expected to connect to physical constants. However, this particular theorem does not itself establish any such physical connection. It is a foundational lemma, a piece of the machinery, not the final physical conclusion.

What the theorem does not claim is equally important. It does not assert that the lattice Green function equals the continuous 1/(4πr) potential, nor does it establish the asymptotic behavior of the Green function at large distances. Those are later rungs in the framework's plan, explicitly named as future work. The theorem is a precise, narrow statement about the equality of two integrals, and its value lies in being a verified step in a longer chain of reasoning.

private theorem integrable_outerBoundarySlice (r : ℕ) :
    Integrable (outerBoundarySlice r) := by
  let C : ℝ := 2 / Real.pi ^ 2
  have hcompact : IsCompact cube2 := by
    apply
      (isCompact_closedBall (0 : E2) (2 * Real.pi)).of_isClosed_subset
        isClosed_cube2
    intro u hu
    rw [Metric.mem_closedBall, dist_zero_right]
    rw [pi_norm_le_iff_of_nonneg (by positivity)]
    intro j
    rw [Real.norm_eq_abs]
    exact ((mem_cube2_iff u).1 hu j).trans (by
      nlinarith [Real.pi_pos])
  have hind :
      Integrable (cube2.indicator (fun _ : E2 => C)) := by
    exact (integrable_indicator_iff measurableSet_cube2).2
      (MeasureTheory.integrableOn_const
        (s := cube2) (C := C) (hs := hcompact.measure_lt_top.ne))
  have hqmeas :
      Measurable (fun u : E2 => ∑ j : Fin 2, (u j) ^ 2) := by
    fun_prop
  have hfun : Continuous (fun u : E2 =>
      -(2 / Real.pi ^ 2) *
        Real.sin ((r : ℝ) *
          Real.sqrt (Real.pi ^ 2 - ∑ j : Fin 2, (u j) ^ 2))) := by
    fun_prop
  have hmeas : AEStronglyMeasurable (outerBoundarySlice r)
      (volume : Measure E2) := by
    unfold outerBoundarySlice
    exact (hfun.measurable.piecewise
      (measurableSet_le hqmeas measurable_const) measurable_const).aestronglyMeasurable
  refine hind.mono' hmeas ?_
  filter_upwards with u
  by_cases hu : u ∈ cube2
  · rw [Set.indicator_of_mem hu]
    unfold outerBoundarySlice
    by_cases hq : ∑ j : Fin 2, (u j) ^ 2 ≤ Real.pi ^ 2
    · rw [if_pos hq]
      have hpi2 : 0 < Real.pi ^ 2 := sq_pos_of_pos Real.pi_pos
      rw [Real.norm_eq_abs, abs_mul, abs_neg,
        abs_of_pos (div_pos (by norm_num) hpi2)]
      dsimp [C]
      have hc : 0 ≤ 2 / Real.pi ^ 2 := by positivity
      simpa using (mul_le_mul_of_nonneg_left
        (Real.abs_sin_le_one ((r : ℝ) *
          Real.sqrt (Real.pi ^ 2 - ∑ j : Fin 2, (u j) ^ 2))) hc
        )
    · rw [if_neg hq]
      simpa [C] using (show (0 : ℝ) ≤ 2 / Real.pi ^ 2 by positivity)
  · rw [Set.indicator_of_notMem hu]
    have hq : Real.pi ^ 2 <
        ∑ j : Fin 2, (u j) ^ 2 := by
      by_contra h
      apply hu
      intro j
      have hj :
          (u j) ^ 2 ≤ ∑ i : Fin 2, (u i) ^ 2 := by
        exact Finset.single_le_sum (fun i _ => sq_nonneg (u i))
          (Finset.mem_univ j)
      have hjabs : |u j| ≤ Real.pi := by
        nlinarith [sq_abs (u j), abs_nonneg (u j),
          hj, le_of_not_gt h, Real.pi_pos]
      exact hjabs
    unfold outerBoundarySlice
    rw [if_neg (not_le_of_gt hq)]
    simp

What this page does not claim

The theorem does not establish the asymptotic behavior of the lattice Green function at large distances. The theorem does not identify the lattice Green function with the continuous 1/(4πr) potential. The theorem does not make any physical claim about recognition events or constants.

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/PairKernelGreenFourier3.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.

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