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

The Green function's boundary identity

The declaration outer_cube_ball_r_mul_integral_eq_boundary_add_derivative is a theorem in the framework's machine-checked library of formal theorems. It concerns the Green function, a standard tool in physics and mathematics that measures how a disturbance at one point in a lattice spreads to another point. The theorem establishes an exact identity for the Green function of the three-dimensional cubic lattice with six nearest-neighbor connections.

In plain terms, the identity relates the Green function evaluated at a point on the coordinate axis to two other quantities. The first is an integral over the boundary of a ball of radius r, and the second is a derivative of the Green function. The theorem states that multiplying the Green function by r and integrating over this boundary equals the boundary term plus the derivative term. This is a structural decomposition, not a numerical value: it shows how the Green function's behavior on a sphere relates to its behavior at a single point and its rate of change.

The theorem is part of a larger effort within the framework to establish that the lattice Green function decays like 1/r, where r is the distance from the origin. The identity provides a precise relationship that can be used to analyze this decay. The framework's library has already proved that the Fourier symbol of the lattice Laplacian, which appears in the Green function's integral representation, is bounded between two quadratic forms and approaches the squared norm near the origin. These results are stepping stones toward the 1/r asymptotic, but that final result is not yet in the library.

The theorem does not claim that the Green function itself equals 1/(4πr), the known asymptotic form for the three-dimensional Laplacian. It does not establish the decay rate of the Green function. It does not identify the Green function with the finite-box Green object from another module. These are later rungs on the framework's ladder of results. The identity is one exact, machine-checked step, not the final destination.

/-- The infinite-lattice Green function of the D=3 cubic recognition lattice
(Laplacian-inverse normalization), as its Fourier integral over the
fundamental cube. -/
def Gint (x : Fin 3 → ℤ) : ℝ :=
  (1 / (2 * Real.pi) ^ 3) * ∫ k in cube, integrand x k

/-- The infinite-lattice Green function of the D=3 cubic recognition lattice
(Laplacian-inverse normalization), as its Fourier integral over the
fundamental cube. -/
def Gint (x : Fin 3 → ℤ) : ℝ :=
  (1 / (2 * Real.pi) ^ 3) * ∫ k in cube, integrand x k

What this page does not claim

The theorem does not state that the Green function equals 1/(4πr). The theorem does not establish the decay rate of the Green function. The theorem does not identify this Green function with the finite-box Green object.

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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