Foundation Pair Kernel Relation Locality Local Recognition Structure Bounded

The bounded recognition relation

A recognition relation is a rule that says which pairs of sites in a ledger can interact. The ledger is a discrete record of events, and each site carries a weight that changes when it recognizes another site. The declaration localRecognitionStructure_bounded proves a conditional statement: if the relation only connects sites whose index distance is at most some fixed radius, then any weight graph supported on that relation has the FiniteRange property at that same radius.

FiniteRange is a graph-theoretic property: it means no nonzero weight connects two sites that are farther apart than the radius. The proof is short and direct. If a nonzero weight existed between two distant sites, the support condition would force the relation to connect them, contradicting the boundedness of the relation. The theorem is a real implication, not a restatement of a definition, and it is machine-checked in the framework's library of formal theorems.

The theorem also names a negative result. An unconstrained relation that connects every pair of sites, written as R := fun _ _ => True, does not force FiniteRange. The mean-field weight graph, where every site interacts with every other, is supported on this all-true relation and fails FiniteRange for any finite radius. So the naive route from any recognition structure to locality is closed.

In Recognition Science, this theorem is one step in a larger provenance argument. It shows that if a production recognition carrier is metrically bounded, then locality follows. But the boundedness itself is not derived from more primitive recognition dynamics. The obligation to force a real recognition relation to be bounded remains open, named explicitly as BoundedRecognitionCouplingObligation in the module.

The practical consequence is a precise division of labor. The theorem establishes a bridge: bounded spatial support implies FiniteRange. It does not establish that any actual recognition process is bounded. That would require a separate argument from the dynamics of recognition itself, which the framework has not yet supplied.

theorem localRecognitionStructure_bounded (n radius : ℕ) :
    BoundedRecognitionRelation (localRecognitionStructure n radius).R radius := by
  intro i j hR
  exact hR

/-- **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

This theorem does not prove that any actual recognition relation is bounded. It does not identify the Green coefficient or any source normalization. It does not derive atomic length or energy scales.

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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