Foundation Pair Kernel Atomic Tick Countermodels Lattice3 Generator Operationall

The D=3 generator's local dependency

The declaration lattice3Generator_operationally_local is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns a specific, committed construction called the D=3 box generator, which is a site-indexed generator: a mathematical object that assigns to each site in a finite three-dimensional grid a dependency relation and a weighted graph, where a nonzero weight is only allowed on declared dependencies. The theorem states that this generator's dependency relation is bounded in the encoded D=3 site metric at radius one. In plain terms, it proves that for this particular generator, each site only depends on its immediate neighbors in the grid, not on distant sites.

The proof is direct. The generator's dependency relation is defined to be the lattice3 recognition relation, and a separate lemma establishes that this relation is bounded at radius one. The theorem then follows immediately. A further consequence, also proved, is that centers separated by more than twice the dependency radius have disjoint dependency neighborhoods: if two sites are far enough apart, they share no common dependency site. This is a clean structural fact about the geometry of the grid, not a statement about any physical response.

In Recognition Science, this result is a step toward a larger goal. The framework aims to show that three spatial dimensions are forced by the structure of recognition events. The theorem here provides an operational locality premise: it shows that a specific generator, the one committed to in the framework's Door-2 box action, satisfies a local dependency rule. This premise is target-blind, meaning it does not mention any range cutoff on a Green function or any desired long-distance law. It is a purely structural statement about the generator's dependencies.

The theorem does not claim that this generator is the only possible one, nor that the framework's physics must choose this geometry. That choice is a separate matter, a definitional commitment of the framework, not a consequence of the theorem. The theorem also does not place any range predicate on a Green response; it only concerns the generator's weights and dependencies. The all-pairs dependency, where every site depends on every other, is shown to fail the operational locality premise on any nontrivial box, which provides a countermodel to the idea that locality is automatic.

What the theorem changes is the status of the locality premise within the framework. It turns a general hypothesis into a proved property for one concrete generator. This is a necessary but not sufficient step in the framework's argument for three dimensions. The physical bridge from recognition events to the linking that would establish three-dimensional space remains an open target, not a consequence of this theorem.

/-- The D=3 generator dependency satisfies the operational premise at radius
one. This is theorem-grade for the committed box geometry, not a derivation
that production physics must choose that geometry. -/
theorem lattice3Generator_operationally_local (L : ℕ) :
    LocalOperationalDependency3 L 1
      (lattice3Generator L).dependency :=
  lattice3RecognitionRelation_bounded L

/-- **Independent consequence of operational locality.** Centers farther
apart than twice the dependency radius cannot share a dependency site. This
does not state a range property of weights and does not mention Green
response. -/
theorem separated_centers_have_disjoint_dependencies
    {L radius : ℕ}
    {dependency :
      Fin (L * L * L) → Fin (L * L * L) → Prop}
    (hlocal : LocalOperationalDependency3 L radius dependency)
    {a b : Fin (L * L * L)}
    (hsep : radius + radius < encodedDist3 L a b) :
    ¬ ∃ x, dependency a x ∧ dependency b x := by
  rintro ⟨x, hax, hbx⟩
  have hax' : encodedDist3 L a x ≤ radius := hlocal a x hax
  have hbx' : encodedDist3 L b x ≤ radius := hlocal b x hbx
  have hxb' : encodedDist3 L x b ≤ radius := by
    rw [encodedDist3_comm]
    exact hbx'
  have htri := encodedDist3_triangle L a x b
  omega

/-- 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

What this page does not claim

The theorem does not claim that the D=3 box geometry is the only possible choice for the framework's physics. The theorem does not place any range predicate on a Green response or any long-distance law. The theorem does not establish the physical recognition-to-linking bridge for three-dimensional space.

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)

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.

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