Foundation Pair Kernel Tick Locality Atomic Tick Does Not Force Finite Range

The scope of a tick

In the Recognition Science framework, a ledger is a discrete record of events, and an atomic tick is a single moment in that record when exactly one site posts an entry. The framework's library, a machine-checked collection of formal theorems, proves a negative result about this primitive: the fact that exactly one site posts each tick does not force any limit on which sites can couple to each other. The theorem, named atomicTick_does_not_force_finiteRange, closes a proposed route to deriving a spatial locality cutoff from the temporal schedule alone.

The reason is structural. An atomic tick is a temporal schedule: it says only that one site posts per tick, and every possible posting sequence satisfies that condition. The pair weight graph, by contrast, describes spatial coupling: which pairs of sites carry a nonzero interaction strength. These two structures are independent. The theorem demonstrates this with a decoy: for any radius R, a carrier of size R + 2 admits a valid atomic tick while its all-to-all mean-field weight graph fails to be finite-range at that radius. The all-to-all graph couples every site to every other site, so no fixed radius cutoff exists once the carrier is large enough.

A second witness sharpens the point. A valid atomic tick schedule can post site 0 at tick 0 and site n − 1 at every later tick, so the first two consecutive posts are index-maximally separated, a distance of n − 1 apart. Even if one tried to define coupling from posting adjacency, the induced range would live in tick-order, not in the index metric. Mapping it to the index distance would require a separate commitment that the schedule is index-local, which atomic-tick does not supply.

The theorem does not say locality is impossible in the framework. It says the locality cutoff is not carried by the temporal atomic-tick primitive; it must come from the spatial recognition-relation geometry. The framework's recognition structure carries a relation R that defines adjacency, and a local coupling graph called bandWeight already satisfies finite-range with radius 1. The open work is deriving that the recognition relation itself is finite-neighborhood, which remains a target, not an impossibility.

/-- **Null test / decoy.** For every radius `R`, a carrier of size `R + 2` carries a valid
    `AtomicTick` instance while its all-to-all mean-field weight graph fails `FiniteRange R`.
    Atomic-tick validity does not make the weight graph finite-range. -/
theorem atomicTick_does_not_force_finiteRange (R : ℕ) :
    ∃ n : ℕ, Nonempty (AtomicTick (tickCarrier n)) ∧ ¬ FiniteRange (meanFieldWeight n) R := by
  refine ⟨R + 2, ⟨scheduleAtomicTick (fun _ => ⟨0, by omega⟩)⟩, ?_⟩
  exact meanFieldWeight_not_finiteRange R (R + 2) (le_refl _)

/-- **Null test / decoy.** For every radius `R`, a carrier of size `R + 2` carries a valid
    `AtomicTick` instance while its all-to-all mean-field weight graph fails `FiniteRange R`.
    Atomic-tick validity does not make the weight graph finite-range. -/
theorem atomicTick_does_not_force_finiteRange (R : ℕ) :
    ∃ n : ℕ, Nonempty (AtomicTick (tickCarrier n)) ∧ ¬ FiniteRange (meanFieldWeight n) R := by
  refine ⟨R + 2, ⟨scheduleAtomicTick (fun _ => ⟨0, by omega⟩)⟩, ?_⟩
  exact meanFieldWeight_not_finiteRange R (R + 2) (le_refl _)

/-- The first two (tick-adjacent) posts of `maxSeparatedSchedule` are index-maximally separated:
    `cellDist (s 0) (s 1) = n − 1`. Tick-adjacency is not index-adjacency. -/
theorem maxSeparatedSchedule_consecutive_maximally_separated (n : ℕ) (hn : 2 ≤ n) :
    cellDist (maxSeparatedSchedule n hn 0) (maxSeparatedSchedule n hn 1) = n - 1 := by
  have h0 := maxSeparatedSchedule_val_zero n hn
  have h1 := maxSeparatedSchedule_val_one n hn
  unfold cellDist Nat.dist
  rw [h0, h1]
  omega

/-- **L0 provenance verdict (bundle).** At carrier size `n = R + 2` all three facts hold at once:
    (1) a valid `AtomicTick` instance exists, (2) the all-to-all mean-field weight graph fails
    `FiniteRange R` (the null test), and (3) a valid `AtomicTick` schedule posts index-maximally
    separated sites at consecutive ticks (`cellDist (s 0) (s 1) = n − 1 = R + 1`). Together:
    atomic-tick posting adjacency does not force `FiniteRange`; the route to a THEOREM-grade `L0`
    via the tick schedule is CLOSED NEGATIVE. `L0` stays HYPOTHESIS, its provenance the spatial
    coupling geometry (`M.R` / lattice adjacency), not the temporal tick primitive. -/
theorem atomicTick_finiteRange_provenance_closed_negative (R : ℕ) :
    Nonempty (AtomicTick (tickCarrier (R + 2))) ∧
      ¬ FiniteRange (meanFieldWeight (R + 2)) R ∧
      cellDist (maxSeparatedSchedule (R + 2) (by omega) 0)
          (maxSeparatedSchedule (R + 2) (by omega) 1) = R + 1 := by
  refine ⟨⟨scheduleAtomicTick (maxSeparatedSchedule (R + 2) (by omega))⟩,
    meanFieldWeight_not_finiteRange R (R + 2) (le_refl _), ?_⟩
  have h := maxSeparatedSchedule_consecutive_maximally_separated (R + 2) (by omega)
  simpa using h

What this page does not claim

This does not claim that FiniteRange is false, since bandWeight satisfies it. This does not claim that atomic-tick validity is impossible or self-contradictory. This does not claim that the framework admits screening, since FiniteRange still excludes the mean-field route.

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

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