Foundation Pair Kernel Locality Band Weight

The band graph

A band graph is a graph whose edges connect each site only to its immediate neighbors. In the Recognition Science framework, such a graph is a concrete example of a ledger, a discrete record of events whose entries carry weights. The declaration bandWeight defines the simplest nontrivial band graph: the weight between two sites is 1 if they are at most one index cell apart, and 0 otherwise. This is a definitional choice, not a derived result; it models, in the framework's vocabulary, a cost that depends only on nearby pairs.

The declaration's content is proved in the machine-checked library of formal theorems. One theorem, bandWeight_finiteRange, establishes that this graph satisfies the locality hypothesis FiniteRange at radius 1: no coupling extends beyond one cell. Another theorem, bandWeight_adjacent_coupled, shows that adjacent sites do couple, with weight 1, so the graph is not the trivial diagonal graph. Together these prove that the locality hypothesis is satisfiable by an admissible, nontrivial graph. The framework's library also proves the converse tooth: the mean-field graph, which couples every pair of sites uniformly, violates FiniteRange at every fixed radius once the carrier is large enough. The hypothesis is therefore discriminating: it admits the band graph and rejects the mean-field graph.

What the declaration does not claim is as important as what it proves. bandWeight does not prove that locality excludes screening; that step requires a dispersion analysis that is not formalized here. It does not derive the locality hypothesis from a more primitive Recognition Science principle; the provenance of a range cutoff remains open. And it makes no claim about 1/r versus Yukawa potentials, nor about any specific constants such as 5/8 or Z_eff. The band graph is a witness that the hypothesis is real, not a proof that the hypothesis is true.

/-- The band graph satisfies the locality hypothesis at radius `1`: nothing couples beyond
    one cell. So `FiniteRange` is not empty — a real admissible cost lives inside it. -/
theorem bandWeight_finiteRange (n : ℕ) : FiniteRange (bandWeight n) 1 := by
  intro i j hR
  simp only [bandWeight]
  exact if_neg (not_le.mpr hR)

theorem meanFieldWeight_not_finiteRange (R n : ℕ) (hn : R + 2 ≤ n) :
    ¬ FiniteRange (meanFieldWeight n) R := by
  intro hFR
  have hi : (0 : ℕ) < n := by omega
  have hj : R + 1 < n := by omega
  have hrange : R < cellDist (⟨0, hi⟩ : Fin n) (⟨R + 1, hj⟩ : Fin n) := by
    show R < Nat.dist 0 (R + 1)
    unfold Nat.dist
    omega
  have h1 := hFR ⟨0, hi⟩ ⟨R + 1, hj⟩ hrange
  simp only [meanFieldWeight] at h1
  exact one_ne_zero h1

/-- The nearest-neighbor **band** weight graph: coupling `1` between sites at most one cell
    apart, `0` otherwise. An admissible `WeightedLedgerGraph` (nonnegative, symmetric). -/
def bandWeight (n : ℕ) : WeightedLedgerGraph n where
  weight := fun i j => if cellDist i j ≤ 1 then 1 else 0
  weight_nonneg := fun i j => by
    show (0 : ℝ) ≤ if cellDist i j ≤ 1 then (1 : ℝ) else 0
    split <;> norm_num
  weight_symm := fun i j => by
    show (if cellDist i j ≤ 1 then (1 : ℝ) else 0) = if cellDist j i ≤ 1 then (1 : ℝ) else 0
    rw [cellDist_comm i j]

What this page does not claim

The band graph does not prove that locality excludes screening. The band graph does not derive the locality hypothesis from a more primitive principle. The band graph makes no claim about 1/r versus Yukawa potentials or any specific constants.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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