Foundation Pair Kernel Bounded Coupling Recognition Structure Atomic Tick Do Not

Bounded coupling is not forced

The question is whether the basic ingredients of a recognition ledger force a physical constraint: that any two objects can only interact if they are close together. The framework's machine-checked library of formal theorems answers this with a definite no. Its theorem recognitionStructure_atomicTick_do_not_force_boundedCoupling proves that a valid recognition structure with an atomic tick can exist whose recognition relation is completely unrestricted, meaning every object recognizes every other object. Such a structure fails the boundedness property for any finite radius, and the associated mean-field weights also fail to have finite range.

The proof works by exhibiting a specific carrier. The carrier's recognition relation is the all-true relation: every pair of points is recognized. This relation is not bounded for any radius, because the distance between two points can be arbitrarily large. The mean-field weight graph, which assigns a nonzero weight to every pair, is then also not finite-range. The theorem states that this carrier still admits a valid atomic tick, so the bare axioms of a recognition structure plus an atomic tick are compatible with unbounded coupling. The result is a closed negative: it rules out any claim that boundedness follows from those axioms alone.

This negative result is not a dead end. The library then shows a separate, positive route to boundedness. If the recognition relation is required to be contained in a closed metric ball, and if nonzero weights are supported on that relation, then the weights are finite-range with respect to that metric. This is the metric-generalized export, and it is what allows a three-dimensional lattice geometry to work. The committed 3D lattice structure, with its nearest-neighbor recognition relation, satisfies this boundedness condition, and its box weights are finite-range in the encoded distance. This is a theorem about a specific, chosen geometry, not a consequence of the bare axioms.

What the declaration does not claim is just as important. It does not claim that bounded coupling is impossible in the framework; it only shows it is not forced by the minimal ingredients. It does not prove that the 3D lattice geometry is the only possible one, or that it is physically realized. The library names a remaining open obligation: to force or commit production weights onto a metric-bounded relation equivalent to the 3D nearest-neighbor lattice. Until that premise is derived from deeper dynamics, the global finite-range property in the original 1D form remains a hypothesis, and the 3D path rests on the committed lattice metric.

/-- **THEOREM (closed negative).** Bare `RecognitionStructure` (via
`tickCarrier`, whose `R` is all-true) plus a valid `AtomicTick` coexist with
mean-field weights that fail both `BoundedRecognitionRelation` and
`FiniteRange`. This is the Lean form of the season-1 obstruction. -/
theorem recognitionStructure_atomicTick_do_not_force_boundedCoupling
    (radius : ℕ) :
    ∃ n : ℕ,
      Nonempty (AtomicTick (tickCarrier n)) ∧
        ¬ BoundedRecognitionRelation (tickCarrier n).R radius ∧
          RelationSupportsWeight (tickCarrier n).R (meanFieldWeight n) ∧
            ¬ FiniteRange (meanFieldWeight n) radius := by
  refine ⟨radius + 2, ⟨scheduleAtomicTick (fun _ => ⟨0, by omega⟩)⟩, ?_,
    meanFieldWeight_supported_on_unconstrained (radius + 2),
    meanFieldWeight_not_finiteRange radius (radius + 2) (le_refl _)⟩
  intro hB
  have hi : (0 : ℕ) < radius + 2 := Nat.succ_pos _
  have hj : radius + 1 < radius + 2 := Nat.lt_succ_self _
  have hle :
      cellDist (⟨0, hi⟩ : Fin (radius + 2)) ⟨radius + 1, hj⟩ ≤ radius :=
    hB ⟨0, hi⟩ ⟨radius + 1, hj⟩ trivial
  have hdist :
      cellDist (⟨0, hi⟩ : Fin (radius + 2)) ⟨radius + 1, hj⟩ = radius + 1 := by
    simp only [cellDist, Nat.dist]
    omega
  omega

/-- **THEOREM (closed negative).** Bare `RecognitionStructure` (via
`tickCarrier`, whose `R` is all-true) plus a valid `AtomicTick` coexist with
mean-field weights that fail both `BoundedRecognitionRelation` and
`FiniteRange`. This is the Lean form of the season-1 obstruction. -/
theorem recognitionStructure_atomicTick_do_not_force_boundedCoupling
    (radius : ℕ) :
    ∃ n : ℕ,
      Nonempty (AtomicTick (tickCarrier n)) ∧
        ¬ BoundedRecognitionRelation (tickCarrier n).R radius ∧
          RelationSupportsWeight (tickCarrier n).R (meanFieldWeight n) ∧
            ¬ FiniteRange (meanFieldWeight n) radius := by
  refine ⟨radius + 2, ⟨scheduleAtomicTick (fun _ => ⟨0, by omega⟩)⟩, ?_,
    meanFieldWeight_supported_on_unconstrained (radius + 2),
    meanFieldWeight_not_finiteRange radius (radius + 2) (le_refl _)⟩
  intro hB
  have hi : (0 : ℕ) < radius + 2 := Nat.succ_pos _
  have hj : radius + 1 < radius + 2 := Nat.lt_succ_self _
  have hle :
      cellDist (⟨0, hi⟩ : Fin (radius + 2)) ⟨radius + 1, hj⟩ ≤ radius :=
    hB ⟨0, hi⟩ ⟨radius + 1, hj⟩ trivial
  have hdist :
      cellDist (⟨0, hi⟩ : Fin (radius + 2)) ⟨radius + 1, hj⟩ = radius + 1 := by
    simp only [cellDist, Nat.dist]
    omega
  omega

/-- **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

/-- **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

This answer does not claim that bounded coupling is impossible in the framework, only that it is not forced by the bare axioms. This answer does not claim that the three-dimensional lattice geometry is the only possible bounded geometry. This answer does not claim that global finite range in the original one-dimensional form is a theorem; it remains a hypothesis.

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)

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