Foundation Pair Kernel Periodic3 Torus Laplacian Torus Spectral Response Normali

The torus inversion theorem

The declaration establishes a precise inversion property for a discrete operator on a finite three-dimensional torus. A torus here is a grid where the edges wrap around, so moving one step past the last site brings you back to the first. The operator in question, the Laplacian, measures how a value at a site differs from the average of its six immediate neighbors. The theorem states that if a source function on the torus can be written as a sum of its nonzero Fourier modes, then applying the Laplacian to a specially constructed spectral response function recovers the original source exactly.

This is the discrete, finite analogue of solving a Poisson equation, where one asks which potential produces a given source under a second-derivative operator. The construction works by expressing the source in terms of its Fourier modes, then dividing each mode by the corresponding eigenvalue of the Laplacian, and finally summing the resulting modes back together. The theorem torusLaplacian_torusSpectralResponseNormalized_eq_source_of_reconstructed verifies that this procedure is a genuine inverse: the Laplacian of the response equals the source, not merely approximately but exactly, at every site of the torus. The proof relies on the orthogonality of the Fourier modes, a standard fact that the library establishes for this finite setting.

The statement carries a hypothesis: the source must be reconstructible from its nonzero modes. This is not automatic. The zero mode, which is constant across the torus, lies in the kernel of the Laplacian, and its contribution cannot be recovered by this inversion. The theorem therefore applies to sources whose total sum over the torus is zero, a condition the library names source neutrality, meaning the total source vanishes. For such sources, the reconstruction is exact. The declaration does not assert anything about sources with a nonzero constant component, nor does it claim that the spectral response is the only function whose Laplacian yields the source.

In Recognition Science, this result is a rung in a larger chain that models physical structure from a discrete ledger of events. The framework treats space as a finite periodic lattice, and this theorem provides the analytic tool for inverting the Laplacian on that lattice, a step needed for relating sources to fields. The library proves the identity for both a raw and a normalized version of the spectral response, the latter differing by a factor of the torus volume. The theorem is a formal statement about finite sums, verified by a machine-checked library of formal theorems, and it does not by itself make any physical claim about continuous space or about the origin of the torus as a model of physical space.

/-- The normalized spectral response solves the periodic source equation under
the normalized reconstruction interface. -/
theorem torusLaplacian_torusSpectralResponseNormalized_eq_source_of_reconstructed
    {N : ℕ} [NeZero N] (rho : TorusSite3 N → ℂ)
    (hrec : torusSourceReconstructedByNonzeroModesNormalized rho)
    (p : TorusSite3 N) :
    torusLaplacian (torusSpectralResponseNormalized rho) p = rho p := by
  classical
  unfold torusSpectralResponseNormalized
  rw [torusLaplacian_smul, torusLaplacian_finset_sum]
  calc
    ((N : ℂ) ^ 3)⁻¹ *
        (∑ m ∈ torusNonzeroModes N,
          torusLaplacian
            (fun q => torusSourceTransform rho m *
              torusSpectralGreenMode m q) p) =
        ((N : ℂ) ^ 3)⁻¹ *
          (∑ m ∈ torusNonzeroModes N,
            torusSourceTransform rho m * torusFourierMode m p) := by
      congr 1
      apply Finset.sum_congr rfl
      intro m hm
      have hm0 : m ≠ 0 := (mem_torusNonzeroModes m).1 hm
      rw [torusLaplacian_smul]
      rw [torusLaplacian_torusSpectralGreenMode_eq_mode_of_ne_zero m hm0]
    _ = rho p := by
      exact (hrec p).symm

/-- The finite Fourier modes are orthogonal over the periodic sites.

The scalar is the full site count `N^3`; no normalization is hidden in the
source transform. The proof includes the `N = 1` case through the same
finite root-of-unity argument. -/
theorem torusFourierMode_orthogonality
    {N : ℕ} [NeZero N] (m n : Fin 3 → Fin N) :
    (∑ p : TorusSite3 N,
      (torusFourierMode m p)⁻¹ * torusFourierMode n p)
      = if m = n then (N : ℂ) ^ 3 else 0 := by
  classical
  have hsite_univ :
      (Finset.univ : Finset (TorusSite3 N)) =
        Fintype.piFinset
          (fun _ : Fin 3 => (Finset.univ : Finset (ZMod N))) := by
    ext p
    simp only [Finset.mem_univ, Fintype.mem_piFinset]
    simp
  simp_rw [torusFourierMode_pointwise_orthogonality_factor]
  calc
    (∑ p : TorusSite3 N,
        ∏ i : Fin 3,
          (torusRoot N ^ ((m i).val * (p i).val))⁻¹ *
            torusRoot N ^ ((n i).val * (p i).val)) =
        ∏ i : Fin 3,
          ∑ x : ZMod N,
            (torusRoot N ^ ((m i).val * x.val))⁻¹ *
              torusRoot N ^ ((n i).val * x.val) := by
          rw [hsite_univ]
          simpa [TorusSite3] using
            (Finset.sum_prod_piFinset
              (s := (Finset.univ : Finset (ZMod N)))
              (g := fun i x =>
                (torusRoot N ^ ((m i).val * x.val))⁻¹ *
                  torusRoot N ^ ((n i).val * x.val)))
    _ = if m = n then (N : ℂ) ^ 3 else 0 := by
          by_cases hmn : m = n
          · subst n
            simp [torusRoot_sum_zmod_cross]
          · have hcoord : ∃ i : Fin 3, m i ≠ n i := by
              by_contra h
              apply hmn
              funext i
              by_contra hi
              exact h ⟨i, hi⟩
            rcases hcoord with ⟨i, hi⟩
            rw [if_neg hmn]
            apply Finset.prod_eq_zero (Finset.mem_univ i)
            rw [torusRoot_sum_zmod_cross]
            simp [hi]

/-- Neutrality forces the zero source-transform coefficient to vanish. -/
theorem torusSourceTransform_zero_eq_zero_of_neutral
    {N : ℕ} [NeZero N] (rho : TorusSite3 N → ℂ)
    (hneutral : torusSourceNeutral rho) :
    torusSourceTransform rho (0 : Fin 3 → Fin N) = 0 := by
  rw [torusSourceTransform_zero_eq_totalSource]
  exact hneutral

What this page does not claim

The theorem does not apply to sources with a nonzero constant component, whose total sum over the torus is not zero. The theorem does not assert that the spectral response is the unique function whose Laplacian equals the source. The theorem makes no claim about continuous space or about the physical origin of the torus as a model of space.

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/PairKernelPeriodic3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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