Foundation Pair Kernel Green Fourier3 Outer Post Ibp Derivative Integral Tendsto

The vanishing boundary term

The declaration outer_post_ibp_derivative_integral_tendsto_zero is a machine-checked lemma in the framework's library of formal theorems. It concerns the lattice Green function, which in this context is a discrete analog of the fundamental solution to Laplace's equation on a three-dimensional cubic lattice. The lemma establishes that a specific boundary integral, arising from an integration by parts procedure, tends to zero as a parameter grows without bound. This is a technical but necessary step in proving that the Green function decays like 1/r along a lattice axis, a classical result for such discrete operators.

The proof relies on the Fourier representation of the lattice Green function. The function is defined as an integral over a cube in frequency space, with an integrand involving a cosine divided by the Fourier symbol of the lattice Laplacian. The symbol, symbol k = 2 * Σᵢ (1 - cos kᵢ), is the discrete analog of |k|². The lemma shows that the contribution from the boundary of this cube vanishes in the limit, allowing the main integral to be evaluated. This is a standard technique in harmonic analysis, here formalized in the framework's library.

The lemma does not claim that the Green function itself equals 1/(4πr) exactly. That asymptotic identification remains a later target in the framework's development. The lemma also does not claim that the boundary integral is zero for any finite parameter; it only establishes the limit as the parameter tends to infinity. In addition, the lemma does not assert any physical law or property of the recognition lattice beyond the mathematical statement about the integral. It is a foundational result, not a phenomenological one.

theorem integrableOn_remInvPartial0_cube :
    IntegrableOn remInvPartial0 cube :=
  integrableOn_remInvPartial0

What this page does not claim

The lemma does not prove the full 1/(4πr) asymptotic for the Green function. The lemma does not assert any physical law about the recognition lattice. The lemma does not show the boundary integral is zero for any finite parameter.

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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