Foundation Pair Kernel Locality Finite Range

A locality rule for the ledger

In Recognition Science, the ledger (a discrete record of events) is modeled as a graph whose sites carry weights. The declaration FiniteRange states a simple locality rule: the weight between two sites is zero whenever their index distance exceeds a fixed bound R. In plain terms, it says that no coupling, no influence, reaches beyond a certain range. This is a postulate, a named hypothesis, not something the framework derives from more primitive principles. Its provenance, the question of where a range cutoff would come from, remains open.

The declaration earns its place by doing real work. A mean-field graph, one where every site couples to every other with equal weight, violates FiniteRange at every fixed radius once the carrier is large enough. That graph is exactly the carrier of a screening honest-negative, a counterexample that survives other assumptions. FiniteRange rejects it. At the same time, the hypothesis is not empty: a nearest-neighbor band graph, where coupling exists only between adjacent sites, satisfies FiniteRange at radius one. So the postulate is satisfiable, non-trivial, and discriminating. It rules out the all-to-all coupling that would otherwise produce a mass gap away from zero momentum.

What FiniteRange does not claim is as important as what it does. It does not prove that locality excludes screening; that step, the dispersion or Fourier analysis, is numerical and measured in a separate harness. It does not claim to be forced by the framework; its derivation from atomic-tick or recognition adjacency is a target, not a result. And it says nothing about specific force laws, such as an inverse-square versus a Yukawa form, or about constants like 5/8 or 27/16. The declaration is a load-bearing hypothesis, cleanly scoped, that rejects one specific counterexample and is satisfiable by a real cost.

/-- **L0 (finite-range / locality hypothesis).** The weight between two sites more than `R`
    index-cells apart is zero: no coupling beyond a fixed range. This is the separate
    postulate `L1` / the bridge do not supply; it is stated as a named hypothesis, NOT
    derived from anything more primitive in the Lean surface today (its provenance is OPEN,
    expected-closure via the atomic-tick / recognition adjacency being nearest-neighbor). -/
def FiniteRange {n : ℕ} (G : WeightedLedgerGraph n) (R : ℕ) : Prop :=
  ∀ i j : Fin n, R < cellDist i j → G.weight i j = 0

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 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)

/-- **L0 status bundle.** `FiniteRange` is (a) satisfiable by an admissible non-trivial
    graph (the band graph, radius `1`), and (b) violated by the mean-field graph that
    carries the built screening honest-negative (at every fixed radius, for large enough
    carriers). A hypothesis with both properties is neither vacuous nor trivially true: it
    does real work. It remains HYPOTHESIS-tier — its RS provenance (a forced range cutoff)
    is OPEN. -/
theorem finiteRange_is_discriminating :
    (∀ n, FiniteRange (bandWeight n) 1) ∧
      (∀ R n, R + 2 ≤ n → ¬ FiniteRange (meanFieldWeight n) R) :=
  ⟨bandWeight_finiteRange, meanFieldWeight_not_finiteRange⟩

What this page does not claim

FiniteRange does not prove that locality excludes screening. FiniteRange is not derived from more primitive Recognition Science principles. FiniteRange says nothing about inverse-square versus Yukawa force laws.

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.

Derived articles

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