Foundation Pair Kernel Atomic Tick Countermodels Lattice3 Generator Separated De

Separated centers

The declaration lattice3Generator_separated_dependencies_disjoint establishes a precise separation property for a three-dimensional lattice model in the Recognition Science framework. It proves that if two recognition centers are farther apart than twice a fixed dependency radius, then no third site can be a dependency of both. In plain terms: when the distance between two centers exceeds the reach of their local interactions, their neighborhoods of influence cannot overlap. The theorem is machine-checked, meaning it has been verified by a computer from the framework's axioms.

The proof rests on two ingredients. First, the lattice3Generator is defined as a site-indexed generator whose dependency relation is bounded: it only connects sites within a radius of one in the encoded three-dimensional metric. Second, a general lemma, separated_centers_have_disjoint_dependencies, shows that any dependency relation satisfying this boundedness condition yields the disjointness property. The specific theorem applies this lemma to the lattice generator, using the fact that its dependency relation is operationally local at radius one.

What the declaration does not claim is equally important. It does not assert that the lattice geometry is the one physics must choose; the framework's library states this is a model choice, not a derivation. It does not place any range condition on a Green response or inverse operator. It only concerns the generator's declared dependency relation, not the behavior of solutions to any field equation. The theorem is a structural fact about the model, not a statement about which model is physically correct.

The practical consequence is that in this three-dimensional lattice model, locality at the level of generator dependencies implies a clean separation property: distant centers cannot influence each other through a shared intermediate site. This is a necessary ingredient for any later claim that the model exhibits finite-range propagation of recognition effects. The theorem itself does not prove that propagation, but it rules out a certain kind of nonlocal interference at the level of dependencies.

/-- Separated D=3 generator centers have disjoint dependency neighborhoods. -/
theorem lattice3Generator_separated_dependencies_disjoint
    (L : ℕ) {a b : Fin (L * L * L)}
    (hsep : 2 < encodedDist3 L a b) :
    ¬ ∃ x,
      (lattice3Generator L).dependency a x ∧
      (lattice3Generator L).dependency b x :=
  separated_centers_have_disjoint_dependencies
    (lattice3Generator_operationally_local L) hsep

/-- Separated D=3 generator centers have disjoint dependency neighborhoods. -/
theorem lattice3Generator_separated_dependencies_disjoint
    (L : ℕ) {a b : Fin (L * L * L)}
    (hsep : 2 < encodedDist3 L a b) :
    ¬ ∃ x,
      (lattice3Generator L).dependency a x ∧
      (lattice3Generator L).dependency b x :=
  separated_centers_have_disjoint_dependencies
    (lattice3Generator_operationally_local L) hsep

/-- The committed Door-2 box action packaged as a site-indexed generator. -/
def lattice3Generator (L : ℕ) :
    SiteIndexedGenerator (L * L * L) where
  dependency := lattice3RecognitionRelation L
  graph := boxWeight L
  weight_uses_dependency := boxWeight_supported_on_lattice3 L

What this page does not claim

The theorem does not prove that the three-dimensional lattice is the physically correct geometry. It does not place any range condition on a Green response or inverse operator. It does not show that recognition effects propagate only finitely; it only concerns declared dependencies.

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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