Foundation Pair Kernel Green Fourier3 Integrable On Outer Post Ibp Amplitude Com

A well-defined integral

The declaration integrableOn_outer_post_ibp_amplitude_complex establishes that a specific mathematical expression, an amplitude arising in the framework's analysis of a three-dimensional lattice, has a well-defined integral. In plain terms, it proves that the sum of infinitely many contributions, each representing a possible interaction across the lattice, does not blow up to infinity or oscillate without settling on a value. The integral exists, which is the necessary precondition for any further calculation to be meaningful.

This is a statement about a particular integrand, a function built from the framework's lattice Green function (a measure of influence spreading through a discrete grid of points) and a complex exponential factor. The proof shows this product is integrable over the relevant domain, meaning its total contribution is finite. This is not a statement about the value of the integral itself, only about its existence. It is a technical but essential step, akin to proving that a series converges before you can talk about its sum.

In the broader context of the framework's development, this declaration is part of a larger effort to construct and analyze a specific physical quantity. It is a foundational piece, ensuring that later steps, which may involve further manipulation or approximation, are built on solid ground. The declaration itself does not claim to have computed the integral's value, nor does it assert any physical interpretation of the result. It simply certifies that the mathematical object is well-behaved enough to be part of the ongoing analysis.

theorem integrableOn_remInvPartial0_cube :
    IntegrableOn remInvPartial0 cube :=
  integrableOn_remInvPartial0

What this page does not claim

This answer does not claim the declaration provides the numerical value of the integral. This answer does not claim the declaration assigns a physical meaning to the amplitude. This answer does not claim the declaration proves the integral is finite over all of space, only over its specified domain.

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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