Foundation Pair Kernel Atomic Tick Countermodels

What the countermodels show

In Recognition Science, an AtomicTick is the framework's basic clock: a discrete record of events in which exactly one site posts at each tick. The framework's machine-checked library of formal theorems, in the module PairKernelAtomicTickCountermodels, proves a striking fact about this clock. The current AtomicTick interface is too weak to distinguish a bounded local recognition relation from an unbounded all-pairs relation. Both admit the same posting schedule, because AtomicTick only says that one site posts at each tick; it has no site-indexed dependency semantics.

The library makes this precise with two explicit countermodels. The first, localAtomicTick, places the clock on a recognition structure where each site connects only to neighbors within a fixed radius. The second, globalAtomicTick, places the same clock on a structure where every site connects to every other site. The theorem atomicTick_admits_local_and_global_dependency_models proves that for any radius, both models are valid AtomicTicks with the same constant posting schedule. The clock cannot tell the two worlds apart.

The module then introduces a new premise, LocalOperationalDependency3, to break the tie. This premise says that a site-indexed generator dependency is bounded in the encoded D=3 site metric: if site i depends on site j, then the distance between them is at most some radius. The premise is target-blind: it mentions neither range cutoffs on weights nor any Green response. Its independent consequence, proved as separated_centers_have_disjoint_dependencies, is that centers separated by more than twice the dependency radius cannot share a dependency site.

With this premise in hand, the library constructs the D=3 box generator, lattice3Generator, and proves it satisfies the operational premise at radius one. The theorem lattice3Generator_finiteRangeOn then shows that the generator weights have bounded support in the encoded metric. The all-pairs dependency, by contrast, fails the premise on any nontrivial box, as allPairsDependency_not_local3 proves. The module thus establishes that a target-blind locality premise on generator dependencies, not on Green responses, is what selects local generator weights.

The consequence for the framework is concrete: the AtomicTick clock needs a dependency field to carry physical locality. The countermodels show that without it, the clock is compatible with both local and global physics. With the new premise, the D=3 box generator becomes provably local, and the all-pairs decoy is excluded. This is a step toward making the framework's spatial dimension claim depend on operational premises rather than on the clock alone.

/-- **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

/-- Every schedule is an `AtomicTick`: the current interface has no dependency
or update-support field. -/
def atomicTickOfSchedule (M : RecognitionStructure) (s : ℕ → M.U) :
    AtomicTick M where
  postedAt := fun t u => u = s t
  unique_post := by
    intro t
    refine ⟨s t, rfl, ?_⟩
    intro u hu
    exact hu

/-- **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

/-- 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

/-- 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 module does not prove that production physics must choose the D=3 box geometry. The module does not place any range predicate on a Green response. The module does not derive the three spatial dimensions from AtomicTick alone.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: