Foundation Pair Kernel Local Generator Global Green Scaled Dipole Green Response

The global response

In the Recognition Science framework, a ledger (a discrete record of events on a periodic three-dimensional grid) can carry a dipole source: a positive unit at one site and a negative unit at another. The declaration scaledDipoleGreenResponse_eq_global_mode_sum states that the response to such a source, computed by the framework's canonical inverse operator, is exactly equal to a specific sum over every nonzero Fourier mode of the grid. The equation is scaledDipoleGreenResponse sourceScale a b p = (N^3)^(-1) * ∑ m ∈ torusNonzeroModes N, torusSourceTransform (scaledTorusDipole sourceScale a b) m * torusSpectralGreenMode m p. In plain terms: the global answer is the sum of its spectral parts, with no approximation and no omitted modes.

This is a Green's function statement (the response to a point source) in the framework's native setting. The theorem is proved in the machine-checked library of formal theorems, meaning the equality holds by construction, not by numerical experiment. The source scale (a complex number multiplying the dipole) is an explicit input throughout; the neutrality of the dipole, that its total source is zero, is a separate proved theorem. The response field itself is deliberately not assigned a finite-range predicate, meaning the framework makes no claim that the response is confined to a neighborhood of the source.

In Recognition Science, the broader context is a pairing: a local generator (the six-neighbor action weights) and a global response (the spectral inverse). The theorem scaledDipoleGreenResponse_eq_global_mode_sum is the seam that connects the two, showing that the global response is exactly the full normalized nonzero-mode inverse. This makes the type separation inspectable downstream: a local rule can produce a global answer without the answer being local itself.

What the declaration does not claim is as important as what it proves. It does not claim that the response is finite-range, nor that the local generator and the canonical Laplacian are the same operator. That identification, between the arbitrary graph G and the canonical local operator, remains an open problem in the framework. The theorem also does not assign physical meaning to the source scale; it is a parameter, not a derived constant.

/-- The consumer exposes the global response as the full normalized
nonzero-mode inverse, making the local-generator/global-response type
separation inspectable downstream. -/
theorem scaledDipoleGreenResponse_eq_global_mode_sum
    {N : ℕ} [NeZero N]
    (sourceScale : ℂ) (a b p : TorusSite3 N) :
    scaledDipoleGreenResponse sourceScale a b p =
      ((N : ℂ) ^ 3)⁻¹ *
        (∑ m ∈ torusNonzeroModes N,
          torusSourceTransform (scaledTorusDipole sourceScale a b) m *
            torusSpectralGreenMode m p) := by
  rfl

/-- 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

/-- Global response to the scaled dipole. The normalized spectral inverse
sums all nonzero Fourier modes; it is deliberately not assigned a
`FiniteRangeOn` predicate. -/
def scaledDipoleGreenResponse {N : ℕ} [NeZero N]
    (sourceScale : ℂ) (a b : TorusSite3 N) :
    TorusSite3 N → ℂ :=
  torusSpectralResponseNormalized (scaledTorusDipole sourceScale a b)

What this page does not claim

The theorem does not claim that the response is finite-range or confined to a neighborhood of the source. The theorem does not claim that the local generator and the canonical Laplacian are the same operator. The theorem does not assign physical meaning to the source scale 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/PairKernelLocalGeneratorGlobalGreen.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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