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

A slice integral identity

The declaration outerCubeBallSlice_r_mul_integral_eq_full_interval_ibp is a formal statement inside the Recognition Science framework's machine-checked library. It concerns a specific integral identity. The setting is a three-dimensional cube, the fundamental domain [−π, π]³, and a function defined on it. The identity relates an integral over a slice of this cube, weighted by a radial coordinate, to an integral over a full one-dimensional interval. In plain terms, the declaration establishes that a particular way of integrating a function over a three-dimensional region, when restricted to a slice and multiplied by the distance from the origin, yields the same numerical result as integrating a related function over a single line segment.

The context is the study of the lattice Green function, which is a discrete record of how influence spreads on a cubic grid. The framework models this through a Fourier representation, where the function is expressed as an integral over the cube. The specific identity in question is a technical lemma in a chain of reasoning about the behavior of this Green function. It is a step toward understanding the function's radial profile, which describes how the influence decays with distance from a source point.

What the declaration does not claim is equally important. It does not claim that the full three-dimensional Green function equals a one-dimensional integral. It does not claim that the slice integral identity holds for all functions, only for the specific integrand defined in the framework. It does not claim that the Green function itself has been fully computed or that any physical constant has been derived. The declaration is a single, precise step in a larger proof, not the final result.

The significance of this identity is that it provides a bridge between a three-dimensional integral and a one-dimensional one. This is a common and useful technique in mathematical physics, as it can simplify calculations and reveal underlying structure. The fact that this bridge is formally verified means that the framework's reasoning about the Green function can proceed with confidence, knowing that this particular transformation is sound.

/-- 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

/-- 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 full three-dimensional Green function equals a one-dimensional integral. The identity holds for all functions, only for the specific integrand defined in the framework. Any physical constant has been derived from this identity.

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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