Foundation Pair Kernel Green Fourier3 Outer Cube Ball Slice Integral Eq Zero Of

The vanishing slice

The declaration outerCubeBallSlice_integral_eq_zero_of_transverse_zero is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns the lattice Green function, a discrete record of how influence spreads through a simple cubic lattice, where each point connects to its six nearest neighbors. The theorem states that a specific integral, taken over a slice of the fundamental cube in Fourier space, evaluates to exactly zero when one of the coordinates is zero. This is a precise, kernel-checked result about the structure of the Green function, not an approximation or a numerical observation.

In plainer terms, imagine the Green function as a map of how a disturbance at one lattice point fades as you move away. The theorem examines a particular way of summing up contributions from all possible frequencies, a standard technique in Fourier analysis. It shows that if you look at a slice of this frequency space where one component is exactly zero, the total contribution to the Green function from that slice is zero. This is a symmetry or cancellation property: the contributions from different parts of the slice perfectly cancel out, leaving no net effect on the Green function along the coordinate axes.

The theorem is part of a larger effort to construct and analyze the Green function for the three-dimensional cubic lattice. The library proves that the Fourier symbol, a function describing the lattice's response at each frequency, is nonnegative and vanishes only at the origin. It also establishes two-sided quadratic bounds on this symbol, showing it behaves like the squared norm of the frequency vector. These results are stepping stones toward understanding the Green function's asymptotic behavior, specifically its decay rate and its singularity structure near the origin, which are targets for later rungs of the framework's plan.

What the theorem does not claim is also important. It does not establish the full asymptotic form of the Green function, such as the specific coefficient of the leading singularity. It does not prove that the Green function tends to zero at infinity, which is a separate result. The theorem is a local, structural fact about a specific integral, not a global statement about the Green function's behavior everywhere. It is a precise piece of a larger puzzle, not the whole picture.

/-- The Fourier integrand of the lattice Green function at lattice site `x`. -/
def integrand (x : Fin 3 → ℤ) (k : EuclideanSpace ℝ (Fin 3)) : ℝ :=
  Real.cos (∑ i, k i * (x i : ℝ)) / symbol k

/-- Quadratic upper bound: `symbol k ≤ ‖k‖²` everywhere (from
`1 - cos t ≤ t²/2`). -/
theorem symbol_le_norm_sq (k : EuclideanSpace ℝ (Fin 3)) :
    symbol k ≤ ‖k‖ ^ 2 := by
  have hterm : ∀ i : Fin 3, 1 - Real.cos (k i) ≤ (k i) ^ 2 / 2 := fun i => by
    have := Real.one_sub_sq_div_two_le_cos (x := k i)
    linarith
  have hsum : (∑ i, (1 - Real.cos (k i))) ≤ ∑ i : Fin 3, (k i) ^ 2 / 2 :=
    Finset.sum_le_sum fun i _ => hterm i
  have hhalf : (∑ i : Fin 3, (k i) ^ 2 / 2) = (∑ i : Fin 3, (k i) ^ 2) / 2 := by
    rw [Finset.sum_div]
  unfold symbol
  rw [norm_sq_eq_sum]
  rw [hhalf] at hsum
  linarith

/-- Quadratic upper bound: `symbol k ≤ ‖k‖²` everywhere (from
`1 - cos t ≤ t²/2`). -/
theorem symbol_le_norm_sq (k : EuclideanSpace ℝ (Fin 3)) :
    symbol k ≤ ‖k‖ ^ 2 := by
  have hterm : ∀ i : Fin 3, 1 - Real.cos (k i) ≤ (k i) ^ 2 / 2 := fun i => by
    have := Real.one_sub_sq_div_two_le_cos (x := k i)
    linarith
  have hsum : (∑ i, (1 - Real.cos (k i))) ≤ ∑ i : Fin 3, (k i) ^ 2 / 2 :=
    Finset.sum_le_sum fun i _ => hterm i
  have hhalf : (∑ i : Fin 3, (k i) ^ 2 / 2) = (∑ i : Fin 3, (k i) ^ 2) / 2 := by
    rw [Finset.sum_div]
  unfold symbol
  rw [norm_sq_eq_sum]
  rw [hhalf] at hsum
  linarith

What this page does not claim

The theorem does not establish the full asymptotic behavior of the Green function. The theorem does not prove the Green function tends to zero at infinity. The theorem does not identify the exact coefficient of the leading singularity in the Green function.

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)

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Derived articles

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