Foundation Pair Kernel Locality Mean Field Weight Not Finite Range

A locality test that bites

In physics, a locality assumption says that distant objects do not directly influence each other: a particle at one site couples only to neighbors within a fixed range, not to every other particle in the system. The Recognition Science framework formalizes this idea on a ledger, a discrete record of events, by assigning a weight to every pair of sites. The declaration meanFieldWeight_not_finiteRange proves a negative result: the specific all-to-all coupling pattern called the mean-field graph, where every site connects to every other site with weight 1, violates the locality condition at every fixed radius once the system has enough sites.

The proof is direct. For any chosen range R, once the number of sites n is at least R+2, the theorem finds two sites, 0 and R+1, that are more than R cells apart yet still carry a nonzero coupling weight of 1. This single counterexample shows the mean-field graph is not finite-range. The same result carries over to the full cost function built on that graph, meanFieldLedgerCost_not_finiteRange, which is the object that actually appears in the framework's screening analysis. The theorem is not vacuous: a separate proof shows a nearest-neighbor band graph, where only adjacent sites couple, does satisfy the locality condition at radius 1, and it is a valid, nontrivial graph because adjacent sites do couple.

In Recognition Science, this pair of results does real work. The framework's earlier analysis had shown that a shift-invariant cost, one depending only on differences between site indices, could not exclude a screened, Yukawa-like interaction. The mean-field graph is exactly such a shift-invariant, all-to-all coupled cost, and it produces a mass gap away from zero momentum. The locality hypothesis, called FiniteRange, is the separate postulate that rejects this screening carrier. The theorem proves the hypothesis is discriminating: it is satisfiable by the band graph and violated by the mean-field graph. This makes the locality assumption load-bearing rather than empty, a genuine constraint that rules out a specific counterexample.

The declaration does not prove that locality excludes screening. That step, a dispersion or Fourier analysis, is numerical and measured in a separate harness, not formalized here. It also does not derive the locality hypothesis from any more primitive principle; its provenance, a forced range cutoff from atomic-tick or recognition adjacency, remains an open target. The theorem only establishes the hypothesis is non-vacuous and discriminating, a real assumption that does work, not a null hypothesis.

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 same teeth, stated on the built honest-negative cost `meanFieldLedgerCost`: its
    weight graph (`meanFieldWeight`) is not finite-range. This is the direct link — L0 is
    exactly the hypothesis that rejects the screening carrier that survives L1. -/
theorem meanFieldLedgerCost_not_finiteRange (R n : ℕ) (hn : R + 2 ≤ n) :
    ¬ FiniteRange (meanFieldLedgerCost n).G R :=
  meanFieldWeight_not_finiteRange R n hn

/-- 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)

/-- Sanity: the band graph does couple adjacent sites (weight `1` on the nearest-neighbor
    pair `0,1`), so it is not the trivial diagonal graph — the witness carries real content. -/
theorem bandWeight_adjacent_coupled (n : ℕ) (hn : 2 ≤ n) :
    (bandWeight n).weight ⟨0, by omega⟩ ⟨1, by omega⟩ = 1 := by
  have hd : cellDist (⟨0, by omega⟩ : Fin n) (⟨1, by omega⟩ : Fin n) ≤ 1 := by
    show Nat.dist 0 1 ≤ 1
    unfold Nat.dist
    omega
  simp only [bandWeight]
  exact if_pos hd

What this page does not claim

The declaration does not prove that locality excludes screening; that is a numerical dispersion step. It does not derive the locality hypothesis from any more primitive principle. It makes no claim about 1/r versus Yukawa potentials 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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