Foundation Pair Kernel Operational Locality S4 Conserved Elementary Posting Can

The nonlocal posting

In the Recognition Science framework, a ledger is a discrete record of events, and a posting is a single change to that record. The declaration conserved_elementaryPosting_can_be_nonlocal proves a precise fact: there exist two positions in a small three-dimensional cell such that a single posting connects them, and that posting conserves the total balance of the ledger. The distance between the two positions is exactly two units, and the sum of all changes across the cell is zero. This is a theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The theorem is a counterexample to a tempting guess. One might think that a minimal-cost posting, the cheapest possible change to a ledger, would always connect nearby entries. The framework's own results show that a minimal posting changes exactly one bit of the ledger's parity pattern, a fact that sounds local. But this declaration shows that the one-bit change can land at a position two units away from its origin. The conservation law, the balance-preserving property, does not force the posting to be local. The two properties are independent in this sense.

The framework uses this result to sharpen its own definitions. It constructs a dependency relation from these minimal postings, and proves that this relation is operationally local: it is defined by realizable transitions, not by a range cutoff or a generator support. The theorem also records exact countermodels for weaker candidate routes. Balanced conservation, shift-invariant cost, an eight-tick identity, and a valid atomic tick can all coexist with an all-pairs dependency. Those weaker conditions do not choose the spatial identification that the translated-cell semantics uses.

What the declaration does not claim is just as important. It does not say that all conserved postings are nonlocal, nor that nonlocality is the typical case. It exhibits one example. It does not derive a physical speed or a causal structure; it operates entirely within the abstract ledger model. And it does not prove that the framework's spatial identification is forced. The theorem is a building block for the framework's account of locality, not a claim about the physical world on its own.

/-- Conservation permits an elementary posting whose endpoints are farther
than one D=3 lattice step. -/
theorem conserved_elementaryPosting_can_be_nonlocal :
    ∃ a b : Fin (2 * 2 * 2),
      encodedDist3 2 a b = 2 ∧
        (∑ i : Fin (2 * 2 * 2),
          divF (elementaryPosting a b) i) = 0 := by
  let o : Site3 2 :=
    ((⟨0, by decide⟩, ⟨0, by decide⟩), ⟨0, by decide⟩)
  let d : Site3 2 :=
    ((⟨1, by decide⟩, ⟨1, by decide⟩), ⟨0, by decide⟩)
  refine ⟨enc3 2 o, enc3 2 d, ?_, ?_⟩
  · simp only [encodedDist3, Equiv.symm_apply_apply, dist3, o, d]
    decide
  · exact elementaryPosting_sum_div_zero (enc3 2 o) (enc3 2 d)

/-- The committed J-cost theorem turns a minimum-J transition into exactly
one bit of parity motion. -/
theorem jMinimalPostingStep_oneBitDiff
    {A B : LedgerState 3}
    (h : JMinimalPostingStep A B) :
    OneBitDiff (parity 3 A) (parity 3 B) := by
  have hpost : PostingStep A B :=
    minJlogCost_monotoneStep_implies_postingStep
      h.1 h.2.1 h.2.2
  exact postingStep_oneBitDiff hpost

/-- **S4 locality theorem.** The operational dependency induced by minimum-J
posting in translated D=3 cells satisfies `LocalOperationalDependency3`
at radius one. The theorem is exact on the account-axis MODEL identification;
production selection of that identification remains the physical hard fork. -/
theorem tiledJMinimalDependency3_operationally_local
    {L : ℕ} (hL : 2 ≤ L) :
    LocalOperationalDependency3 L 1 (TiledJMinimalDependency3 hL) := by
  intro i j hij
  rcases hij with ⟨origin, A, B, hstep, rfl, rfl⟩
  have hdist :
      dist3
        (patternAtCell hL origin (parity 3 A))
        (patternAtCell hL origin (parity 3 B)) = 1 := by
    rw [patternAtCell_dist3]
    exact oneBitDiff_patternDist3
      (jMinimalPostingStep_oneBitDiff hstep)
  simpa only [encodedDist3, Equiv.symm_apply_apply] using hdist.le

/-- Atomicity, balanced conservation, shift-invariant J-cost, period eight,
D=3, and a nontrivial distinction all coexist with an all-pairs dependency.
Thus none of those scalar or temporal declarations chooses spatial locality. -/
theorem committed_candidate_routes_admit_allPairs :
    Nonempty (AtomicTick (tickCarrier 8)) ∧
      Conserves (globalBalancedLedger 8) ∧
      ShiftInvariant (meanFieldLedgerCost 8) ∧
      EightTickFromDimension 3 = eight_tick ∧
      (∃ a b : Fin (2 * 2 * 2), a ≠ b) ∧
      ¬ LocalOperationalDependency3 2 1
        (allPairsDependency :
          Fin (2 * 2 * 2) → Fin (2 * 2 * 2) → Prop) := by
  refine ⟨⟨globalAtomicTick (fun _ => 0)⟩,
    globalBalancedLedger_conserves 8,
    meanFieldLedgerCost_shift_invariant 8,
    rfl, ?_, allPairsDependency_not_local3 2 (by omega)⟩
  exact ⟨0, 1, by decide⟩

What this page does not claim

The declaration does not claim that all conserved postings are nonlocal. It does not claim to derive a physical speed or causal structure from the ledger model. It does not claim that the framework's spatial identification is forced by the theorem.

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/PairKernelOperationalLocalityS4.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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