Foundation Pair Kernel Atomic Tick Countermodels All Pairs Dependency Not Local3

The locality exclusion

In the Recognition Science framework, a recognition event is a discrete record that one site in a system acknowledges another. The framework's ledger, a discrete record of such events, must specify which sites can depend on which others when updating. One extreme possibility is an all-pairs dependency, where every site depends on every other site simultaneously. The theorem allPairsDependency_not_local3 establishes that this all-pairs relation cannot satisfy a bounded locality premise on a three-dimensional box of side length at least 2.

The locality premise in question, LocalOperationalDependency3, states that if one site depends on another, the encoded three-dimensional distance between them is at most some fixed radius. For radius one, this means a site may only depend on its immediate neighbors. The theorem proves that the all-pairs dependency, where every site depends on every other, violates this premise. The proof is constructive: on a box with side length at least 2, one can find two sites whose encoded distance exceeds the radius, yet the all-pairs relation declares a dependency between them.

This exclusion is a target-blind result. The premise LocalOperationalDependency3 mentions neither a Green function nor any response range; it only constrains the dependency relation itself. The theorem therefore provides a clean countermodel: it shows that the all-pairs dependency is not operationally local in the encoded D=3 metric, without invoking any physical law or coupling constant. The framework's machine-checked library of formal theorems records this as a proved statement, with the exclusion witness being the all-pairs dependency itself.

The consequence is that the framework's atomic tick interface, which only requires one site to post at each tick, does not by itself select between local and global dependency models. Both a bounded local relation and an unbounded all-pairs relation admit valid tick schedules. The locality premise is the additional structure that rules out the all-pairs case. This theorem does not claim that the all-pairs dependency is physically impossible, only that it fails the specific bounded-locality premise at radius one on a nontrivial three-dimensional box.

/-- On any nontrivial D=3 box, the all-pairs dependency is not radius-one
operational locality. This is the exclusion witness for the new premise. -/
theorem allPairsDependency_not_local3 (L : ℕ) (hL : 2 ≤ L) :
    ¬ LocalOperationalDependency3 L 1
      (allPairsDependency :
        Fin (L * L * L) → Fin (L * L * L) → Prop) := by
  intro hlocal
  let o : Site3 L :=
    ((⟨0, by omega⟩, ⟨0, by omega⟩), ⟨0, by omega⟩)
  let e : Site3 L :=
    ((⟨1, by omega⟩, ⟨1, by omega⟩), ⟨0, by omega⟩)
  have hle : encodedDist3 L (enc3 L o) (enc3 L e) ≤ 1 :=
    hlocal (enc3 L o) (enc3 L e) trivial
  have hdist : encodedDist3 L (enc3 L o) (enc3 L e) = 2 := by
    simp only [encodedDist3, Equiv.symm_apply_apply, dist3, o, e]
    unfold Nat.dist
    omega
  omega

/-- **Operational locality premise (HYPOTHESIS in production).** A generator
dependency is bounded in the encoded D=3 site metric. The statement is
target-blind: it does not mention `FiniteRangeOn`, a Green function, a source,
or a desired long-distance law. -/
def LocalOperationalDependency3
    (L radius : ℕ)
    (dependency :
      Fin (L * L * L) → Fin (L * L * L) → Prop) : Prop :=
  ∀ i j, dependency i j → encodedDist3 L i j ≤ radius

/-- On any nontrivial D=3 box, the all-pairs dependency is not radius-one
operational locality. This is the exclusion witness for the new premise. -/
theorem allPairsDependency_not_local3 (L : ℕ) (hL : 2 ≤ L) :
    ¬ LocalOperationalDependency3 L 1
      (allPairsDependency :
        Fin (L * L * L) → Fin (L * L * L) → Prop) := by
  intro hlocal
  let o : Site3 L :=
    ((⟨0, by omega⟩, ⟨0, by omega⟩), ⟨0, by omega⟩)
  let e : Site3 L :=
    ((⟨1, by omega⟩, ⟨1, by omega⟩), ⟨0, by omega⟩)
  have hle : encodedDist3 L (enc3 L o) (enc3 L e) ≤ 1 :=
    hlocal (enc3 L o) (enc3 L e) trivial
  have hdist : encodedDist3 L (enc3 L o) (enc3 L e) = 2 := by
    simp only [encodedDist3, Equiv.symm_apply_apply, dist3, o, e]
    unfold Nat.dist
    omega
  omega

/-- **Explicit local/global countermodels.** At every proposed radius, the
same constant posting schedule is valid for both a bounded local relation and
an unbounded all-pairs relation. Current `AtomicTick` therefore cannot select
generator locality. -/
theorem atomicTick_admits_local_and_global_dependency_models (radius : ℕ) :
    ∃ n : ℕ,
      Nonempty (AtomicTick (localRecognitionStructure n radius)) ∧
        BoundedRecognitionRelation
          (localRecognitionStructure n radius).R radius ∧
        Nonempty (AtomicTick (tickCarrier n)) ∧
        ¬ BoundedRecognitionRelation (tickCarrier n).R radius := by
  let n := radius + 2
  let s : ℕ → Fin n := fun _ => ⟨0, by simp [n]⟩
  refine ⟨n, ⟨localAtomicTick s⟩,
    localRecognitionStructure_bounded n radius, ⟨globalAtomicTick s⟩, ?_⟩
  intro hB
  have hi : (0 : ℕ) < n := by simp [n]
  have hj : radius + 1 < n := by simp [n]
  have hle :
      cellDist (⟨0, hi⟩ : Fin n) ⟨radius + 1, hj⟩ ≤ radius :=
    hB ⟨0, hi⟩ ⟨radius + 1, hj⟩ trivial
  have hdist :
      cellDist (⟨0, hi⟩ : Fin n) ⟨radius + 1, hj⟩ = radius + 1 := by
    simp only [cellDist, Nat.dist]
    omega
  omega

What this page does not claim

The all-pairs dependency is physically impossible, only that it fails the bounded-locality premise at radius one. The theorem establishes that any locality premise is necessary for the framework, only that this specific premise excludes the all-pairs case. The theorem derives a physical law or coupling constant from the locality exclusion.

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

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