Foundation Pair Kernel Relation Locality Bounded Recognition Relation Supports F

Bounded recognition implies finite reach

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition relation is a rule saying which pairs of sites can be compared. The theorem boundedRecognitionRelation_supports_finiteRange establishes a conditional statement: if the recognition relation only relates sites within a fixed index radius, and every nonzero weight in the graph is supported on that relation, then the weight graph is FiniteRange at that radius. In plain terms, if comparisons only happen between nearby sites, and all meaningful weights respect that locality, then no weight can connect sites farther apart than the radius allows.

The proof is short and direct. Suppose a weight connects two sites at distance greater than the radius. Since the weight is nonzero, the support condition forces the relation to hold between those sites. But the boundedness condition says the relation can only hold within the radius. Contradiction. The theorem is a real implication, not a restatement of the definition of FiniteRange; it shows how two separate conditions combine to yield the locality property.

What the theorem does not claim is equally important. It does not show that any recognition structure forces FiniteRange. The pack includes a counterexample: the all-true relation, which relates every pair of sites, supports the mean-field weight graph but does not force FiniteRange. That route is closed negative. The theorem also does not force a production recognition relation to be bounded; that remains an open obligation named BoundedRecognitionCouplingObligation.

In Recognition Science, this result banks one specific provenance lane: spatial support plus metric boundedness implies finite range. It does not bank FiniteRange as a theorem about recognition dynamics. The surviving route for L0 provenance is bounded spatial support, not an unconstrained relation. The framework's library establishes the implication and the obstruction; forcing the boundedness condition from more primitive dynamics remains a target.

/-- **finiteRange_export_v1.** A metrically bounded recognition relation that
supports every nonzero weight forces `FiniteRange` at that radius. -/
theorem boundedRecognitionRelation_supports_finiteRange
    {n radius : ℕ} {R : Fin n → Fin n → Prop} {G : WeightedLedgerGraph n}
    (hB : BoundedRecognitionRelation R radius)
    (hS : RelationSupportsWeight R G) :
    FiniteRange G radius := by
  intro i j hfar
  by_contra hne
  have hR : R i j := hS i j hne
  have hle : cellDist i j ≤ radius := hB i j hR
  exact (not_lt_of_ge hle) hfar

/-- **CLOSED NEGATIVE.** An unconstrained recognition relation supports the
screening mean-field graph, so "any RecognitionStructure ⇒ FiniteRange" is false. -/
theorem unconstrainedRelation_does_not_force_finiteRange (radius : ℕ) :
    ∃ n : ℕ,
      (∀ i j : Fin n, unconstrainedRelation n i j) ∧
        RelationSupportsWeight (unconstrainedRelation n) (meanFieldWeight n) ∧
          ¬ FiniteRange (meanFieldWeight n) radius := by
  refine ⟨radius + 2, fun _ _ => trivial,
    meanFieldWeight_supported_on_unconstrained (radius + 2),
    meanFieldWeight_not_finiteRange radius (radius + 2) (le_refl _)⟩

What this page does not claim

FiniteRange is forced by Recognition Science dynamics. Green coefficient identification or source normalization follows from this theorem. The all-true relation is the only obstruction to forcing FiniteRange.

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

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