Foundation Pair Kernel Local Generator Global Green Scaled Torus Dipole Neutral

A neutral source

A dipole is the simplest arrangement of two opposite charges: one positive, one negative, close together. On a three-dimensional grid with periodic boundaries, a torus, the declaration scaledTorusDipole_neutral establishes a basic fact about such a pair. It proves that the total charge of a dipole, defined as the sum of its values over every grid point, is exactly zero. The proof is carried out in the framework's machine-checked library of formal theorems, and it holds for any complex scaling factor applied to the source.

The theorem's statement is precise. A dipole source is defined by choosing two distinct grid sites, a and b, and assigning a value of +1 at a, -1 at b, and 0 everywhere else. Multiplying this entire pattern by a complex number sourceScale produces a scaled dipole. The theorem scaledTorusDipole_neutral states that for any such scale, the sum of the scaled values over the whole torus remains zero. This is the neutrality property: the positive and negative contributions always cancel exactly.

The result is a lemma within a larger construction. The same module defines a local generator, a set of weights that act only between neighboring sites, and a global response, a function defined by summing over all nonzero Fourier modes on the torus. The neutrality theorem is a precondition for connecting these two objects: it ensures that the source is compatible with the global spectral inverse, which is only defined for sources with zero total charge. The theorem itself, however, makes no claim about the response field, its range, or its physical interpretation.

What the declaration does not establish is equally important. It does not prove that the response to the dipole is local, nor does it assign any finite-range property to the response field. The theorem only concerns the source, not the solution to the equation it generates. The connection between the local generator and the global response is the subject of a separate theorem, and the identification of the arbitrary graph with the canonical local operator remains an open problem.

/-- The explicit source scale does not change dipole neutrality. -/
theorem scaledTorusDipole_neutral {N : ℕ} [NeZero N]
    (sourceScale : ℂ) (a b : TorusSite3 N) :
    torusSourceNeutral (scaledTorusDipole sourceScale a b) := by
  classical
  unfold torusSourceNeutral torusTotalSource scaledTorusDipole
  rw [← Finset.mul_sum]
  simp

/-- The explicit source scale does not change dipole neutrality. -/
theorem scaledTorusDipole_neutral {N : ℕ} [NeZero N]
    (sourceScale : ℂ) (a b : TorusSite3 N) :
    torusSourceNeutral (scaledTorusDipole sourceScale a b) := by
  classical
  unfold torusSourceNeutral torusTotalSource scaledTorusDipole
  rw [← Finset.mul_sum]
  simp

What this page does not claim

No claim is made about the range or locality of the response field. The theorem does not establish that the Green response solves the source equation for the arbitrary graph G.

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/PairKernelLocalGeneratorGlobalGreen.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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