Foundation Pair Kernel Locality

The locality hypothesis

In a ledger, a discrete record of events, the cost of recognition between two sites is a weight on a graph. The question of whether that weight can act between any two sites, no matter how far apart, is the question of locality. A finite-range hypothesis, one where the weight vanishes beyond a fixed distance, is a common and physically meaningful assumption. The module PairKernelLocality.lean formalizes this assumption and proves two things about it: that it is satisfiable, and that it is discriminating.

The first proof shows the hypothesis is not empty. A nearest-neighbor band graph, where each site couples only to its immediate neighbors with weight 1 and to nothing else, is an admissible ledger graph and satisfies the finite-range condition at radius 1. This graph is not trivial: it genuinely couples adjacent sites. The second proof shows the hypothesis has teeth. The mean-field graph, where every site couples to every other site with weight 1, violates the finite-range condition at every fixed radius once the carrier is large enough. This is the graph that carries a screening counterexample, an all-to-all coupling that produces a mass gap. The finite-range hypothesis rejects exactly that counterexample.

Together, these two proofs establish that the locality hypothesis is a real, load-bearing assumption. It is not derived from a more primitive principle; its provenance remains open. But it is not vacuous either. It is a named postulate that does genuine work, separating the models that are allowed from the ones that are not.

/-- 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

What this page does not claim

This module does not prove that the finite-range hypothesis excludes screening; that is a separate dispersion step. This module does not derive the finite-range hypothesis from a more primitive Recognition Science principle. This module makes no claim about the specific form of the interaction, such as 1/r versus Yukawa, from primitives.

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)

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Derived articles

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