Foundation Pair Kernel Locality Finite Range Is Discriminating

The locality hypothesis

In the Recognition Science framework, a ledger (a discrete record of events) is modeled as a graph whose edges carry weights, representing the strength of coupling between pairs of sites. A central question is whether the weights must be local, meaning that sites far apart do not interact directly. The declaration finiteRange_is_discriminating establishes that a specific locality hypothesis, called FiniteRange, is both satisfiable and discriminating: it is not an empty condition, and it rejects a particular all-to-all coupling that would otherwise survive other constraints.

The theorem proves two facts together. First, a nearest-neighbor band graph, where each site couples only to its immediate neighbors, satisfies FiniteRange with radius 1. This graph is admissible: its weights are nonnegative and symmetric, and it is not trivial because adjacent sites do couple. Second, the mean-field graph, where every pair of sites couples with equal weight 1, violates FiniteRange at every fixed radius once the number of sites is large enough. This mean-field graph is the carrier of a screening honest-negative, a counterexample that a difference-only cost condition could not exclude. The theorem thus shows FiniteRange does real work: it separates a plausible local model from a specific non-local one.

The declaration does not claim that FiniteRange is derived from more primitive principles. Its provenance, deriving a range cutoff from atomic-tick or recognition adjacency, remains open. It also does not prove that FiniteRange excludes screening entirely; that step requires a dispersion or Fourier analysis that is measured numerically, not proved here. The theorem only formalizes the hypothesis, proves it is non-vacuous and discriminating, and tags it as a hypothesis, not a theorem about the physical world.

/-- **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 is not derived from a more primitive principle; its provenance is open. The theorem does not prove that finite range excludes screening; that requires a separate dispersion step. The theorem does not make any claim about 1/r versus Yukawa potentials or specific constants like 5/8 or Z_eff.

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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