Foundation Pair Kernel Local Generator Global Green

Local action, global response

In Recognition Science, a ledger is a discrete record of events, and the framework asks how a local rule, one that only touches nearby entries, can shape the entire record. This result answers that question for a three-dimensional periodic grid, a torus. It separates two objects that are easy to conflate: the generator, a local rule that assigns weights to the six nearest neighbors of each site, and the Green response, the global inverse that solves the source equation across all sites at once.

The local generator is proved to have finite range: it only connects sites within distance one in the encoded three-dimensional metric. The global response, by contrast, is defined as a sum over every nonzero Fourier mode, which is a way of saying it has no locality restriction at all. The proof then shows the two fit together: applying the canonical torus Laplacian to the global response reproduces the original source. The source is a dipole, a pair of opposite charges at two sites, and an explicit scale factor multiplies it. The proof shows that scaling the dipole does not change its neutrality, meaning the total source remains zero.

The theorem localGenerator_globalGreen_consumer states the conjunction plainly: the generator's graph is finite-range, and the Laplacian of the Green response equals the dipole source. A second consumer theorem extends this to any weighted graph whose nonzero weights satisfy the minimal-J dependency condition, which is a structural constraint from the framework's cost theory. Both theorems are machine-checked, with no unproved assumptions and no new axioms in the formal development.

In Recognition Science, the significance is that locality and globality are not in tension. A rule that only sees its immediate neighbors can still generate a response that is genuinely global, because the inverse operation, summing over all modes, is what spreads the influence. The result deliberately leaves one seam open: it proves the canonical torus Laplacian has this global response, but it does not yet prove that the arbitrary weighted graph G is identical to that canonical operator. That identification remains an open target.

For a reader outside the framework, the portable lesson is about inverse problems. A local differential operator can have a global inverse, and the two are different kinds of objects. The result makes that distinction precise and checkable, which is what a formal library is for: not to replace intuition, but to make it exact.

/-- **Compiling downstream consumer.** The D=3 action generator is local while
its normalized spectral inverse solves the globally represented source
equation. Source scale is explicit and no response-range conclusion is made. -/
theorem localGenerator_globalGreen_consumer
    {N : ℕ} [NeZero N]
    (sourceScale : ℂ) (a b p : TorusSite3 N) :
    FiniteRangeOn
        (lattice3Generator N).graph (encodedDist3 N) 1 ∧
      torusLaplacian
          (scaledDipoleGreenResponse sourceScale a b) p =
        scaledTorusDipole sourceScale a b p := by
  refine ⟨lattice3Generator_finiteRangeOn N, ?_⟩
  unfold scaledDipoleGreenResponse
  apply
    torusLaplacian_torusSpectralResponseNormalized_eq_source_of_reconstructed
  exact
    torusSourceReconstructedByNonzeroModesNormalized_of_neutral _
      (scaledTorusDipole_neutral sourceScale a b)

/-- 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 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

What this page does not claim

The result does not prove that the arbitrary weighted graph G equals the canonical torus Laplacian. The result does not assign a finite-range predicate to the Green response field. The result does not derive the value of the source scale from the framework.

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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