Foundation Pair Kernel Relation Locality Unconstrained Relation Does Not Force F

The dead-end relation

The declaration unconstrainedRelation_does_not_force_finiteRange settles a specific provenance question in the Recognition Science framework: whether a recognition relation that relates every site to every other site can force the weight graph to be FiniteRange, meaning that only sites within a fixed index radius carry nonzero weight. The answer is no. The theorem proves, by explicit construction, that for any chosen radius there exists a finite carrier and a weight graph supported on the all-true relation that nevertheless violates the radius bound. The all-true relation is the one used by the AccountRS and tickCarrier styles, where the relation simply returns True for every pair of sites.

The construction is concrete. The theorem picks a carrier of size radius + 2, so there are sites farther apart than the chosen radius. The weight graph is the mean-field graph, which assigns a nonzero weight to every pair of sites. Because the relation is all-true, it supports every nonzero weight. But because the carrier has sites at distance radius + 1, the graph is not FiniteRange at that radius. The proof is a direct counterexample, not an indirect argument. It shows that the naive route, any recognition relation forces locality, is closed negative.

What the theorem does not claim is equally important. It does not say that locality is impossible in the framework. A separate theorem, finiteRange_export_v1, shows that if a recognition relation is metrically bounded at a fixed radius and every nonzero weight is supported on that relation, then the weight graph is FiniteRange at that radius. That is a real implication, not a rename. The framework also provides a hand-selected band relation, where sites relate only within unit distance, that inhabits this export. The surviving route to locality is therefore spatial support plus metric boundedness, not an unconstrained relation.

The open obligation is now exact. The framework must force a production recognition relation to satisfy the boundedness condition from more primitive dynamics. The theorem under discussion does not discharge that obligation. It only kills one specific route. The consequence is that any future derivation of locality must carry the boundedness premise explicitly, and the all-true relation is no longer available as a shortcut.

/-- **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 _)⟩

theorem finiteRange_export_v1
    {n radius : ℕ} {R : Fin n → Fin n → Prop} {G : WeightedLedgerGraph n}
    (hB : BoundedRecognitionRelation R radius)
    (hS : RelationSupportsWeight R G) :
    FiniteRange G radius :=
  boundedRecognitionRelation_supports_finiteRange hB hS

/-- MODEL shadow integration: the hand-selected band carrier inhabits
`finiteRange_export_v1`. This does not force the production relation. -/
theorem bandWeight_inhabits_finiteRange_export_v1 (n : ℕ) :
    BoundedRecognitionRelation (bandRelation n) 1 ∧
      RelationSupportsWeight (bandRelation n) (bandWeight n) ∧
        FiniteRange (bandWeight n) 1 :=
  ⟨bandRelation_bounded n,
    bandWeight_supported_on_bandRelation n,
    finiteRange_export_v1 (bandRelation_bounded n)
      (bandWeight_supported_on_bandRelation n)⟩

What this page does not claim

The theorem does not prove that FiniteRange is forced by Recognition Science; that remains a hypothesis. The theorem does not establish that the all-true relation is the only relation that fails to force locality. The theorem does not provide a positive construction of a production recognition relation that satisfies the boundedness condition.

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)

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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