Foundation Pair Kernel Bounded Coupling

Bounded coupling and its limits

In Recognition Science, a ledger (a discrete record of events) assigns a weight to every pair of entries. The question of bounded coupling asks whether that weight can be nonzero only between entries that are close together in some metric. The library's first result is a closed negative: the bare assumptions of a recognition structure and a valid atomic tick do not force this locality. There exist carriers where every pair is recognized, the weights are mean-field (all equal), and the relation is not bounded for any finite radius. This is a proved theorem, not a conjecture.

The library then generalizes the notion of finite range to an arbitrary index metric, not just the one-dimensional distance used in earlier work. This generalization is necessary because the one-dimensional export cannot express the locality of a three-dimensional lattice. The library proves a metric-parametric theorem: if a recognition relation is contained in a closed ball of radius one, and the weights are supported on that relation, then the weights are finite-range in that metric.

Under this generalized metric, the library exhibits a concrete three-dimensional inhabitant. The lattice recognition structure, where two indices recognize each other exactly when their encoded ℓ¹ distance is at most one, is bounded. The box weight, which is nonzero only for nearest neighbors on this lattice, is supported on that relation and is therefore finite-range in the encoded metric. This is a theorem about a chosen model geometry, not a force derived from the bare assumptions.

In Recognition Science, the framework's library proves that the bare assumptions do not force bounded coupling, and that a metric-bounded support relation does force finite range. The three-dimensional lattice geometry provides a concrete model where this holds. The remaining premise, named ProductionLatticeCouplingObligation, is to force or commit production pair weights onto a metric-bounded recognition relation equivalent to this lattice geometry. Until that premise is derived, the global one-dimensional finite-range statement remains a hypothesis.

The practical consequence is a clear boundary. The framework's library can certify the local behavior of a chosen three-dimensional lattice model, but it cannot yet derive that locality from more basic principles. The path to three-dimensional production physics has a theorem-grade export under the committed lattice metric, while the deeper forcing of that metric itself remains open.

/-- Bundle alias matching the named season obligation string. -/
theorem BoundedRecognitionCouplingObligation_not_forced_by_bare_RS
    (radius : ℕ) :
    ∃ n : ℕ,
      Nonempty (AtomicTick (tickCarrier n)) ∧
        ¬ BoundedRecognitionRelation (tickCarrier n).R radius ∧
          ¬ FiniteRange (meanFieldWeight n) radius := by
  rcases recognitionStructure_atomicTick_do_not_force_boundedCoupling radius with
    ⟨n, hTick, hB, _hS, hFR⟩
  exact ⟨n, hTick, hB, hFR⟩

/-- **finiteRangeOn_export_v1.** Metric-parametric form of
`finiteRange_export_v1`. -/
theorem finiteRangeOn_export_v1
    {n radius : ℕ} {R : Fin n → Fin n → Prop}
    {dist : Fin n → Fin n → ℕ} {G : WeightedLedgerGraph n}
    (hB : BoundedRecognitionRelationOn R dist radius)
    (hS : RelationSupportsWeightOn R G) :
    FiniteRangeOn G dist radius := by
  intro i j hfar
  by_contra hne
  have hR : R i j := hS i j hne
  have hle : dist i j ≤ radius := hB i j hR
  exact (not_lt_of_ge hle) hfar

theorem lattice3RecognitionRelation_bounded (L : ℕ) :
    BoundedRecognitionRelationOn
      (lattice3RecognitionRelation L) (encodedDist3 L) 1 := by
  intro i j hR
  exact hR

/-- **D=3 inhabitant of the metric export.** Under the committed lattice
geometry, nonzero box weights force `FiniteRangeOn` at radius 1. -/
theorem boxWeight_finiteRangeOn_export_v1 (L : ℕ) :
    BoundedRecognitionRelationOn
        (lattice3RecognitionRelation L) (encodedDist3 L) 1 ∧
      RelationSupportsWeightOn (lattice3RecognitionRelation L) (boxWeight L) ∧
        FiniteRangeOn (boxWeight L) (encodedDist3 L) 1 :=
  ⟨lattice3RecognitionRelation_bounded L,
    boxWeight_supported_on_lattice3 L,
    finiteRangeOn_export_v1
      (lattice3RecognitionRelation_bounded L)
      (boxWeight_supported_on_lattice3 L)⟩

What this page does not claim

The library does not prove that bounded coupling is forced by RecognitionStructure and AtomicTick alone. The three-dimensional lattice inhabitant is not derived as a forced consequence of the bare recognition assumptions. The global one-dimensional FiniteRange statement is not proved; it remains a hypothesis until the ProductionLatticeCouplingObligation is discharged.

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/PairKernelBoundedCoupling.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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