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

A slice of the Green function

The declaration outerCubeBallSlice_r_mul_integral_eq_boundary_add_post_ibp_ae is a theorem in the framework's machine-checked library of formal theorems. It concerns the lattice Green function, which measures the expected number of visits by a random walker on the infinite cubic lattice to a given site, starting from the origin. The theorem states an identity for a slice of this function: when you multiply the slice by the radial distance r and integrate, the result equals a boundary term plus another integral. This is a technical but necessary step in a longer proof chain.

The identity is an equality that holds almost everywhere, meaning it can fail on a set of measure zero, which is standard in Lebesgue integration theory. It is derived using integration by parts, a classical technique that relates the integral of a derivative to the function's values at the boundary. The theorem is part of Rung 0 of a larger campaign to show that the Green function decays like 1/(4πr) at large distances, a result that would match the known Coulomb-like behavior in three dimensions. The declaration itself does not prove that decay; it only establishes this intermediate identity.

The theorem does not claim that the Green function equals 1/(4πr). It does not claim that the boundary term vanishes. It does not claim that the integral is finite or that the identity holds everywhere. It is a precise, narrow statement about a specific integral relation, and its proof is checked by the Lean kernel, meaning it is axiomatically sound. The broader goal of identifying the full asymptotic behavior remains an open target in the framework's development plan.

/-- 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

What this page does not claim

The theorem does not prove the 1/(4πr) decay of the Green function. The theorem does not claim the boundary term vanishes. The theorem does not assert the identity holds at every point, only almost everywhere.

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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