Foundation Pair Kernel Operational Locality S4 J Minimal Posting Step One Bit Di

The one-bit posting step

In the Recognition Science framework, a ledger is a discrete record of events, and a posting is a single change to that record. The theorem jMinimalPostingStep_oneBitDiff states that if a change from one ledger state to another is both monotone (it only adds events, never removes them) and has the minimum possible cost among all such nontrivial changes, then that change is exactly a one-bit motion in the ledger's parity cube: it flips exactly one of the three binary coordinates that describe the state. The proof is a chain of two already-established results: a minimal-cost step is a posting step, and a posting step always differs by one bit.

The consequence is operational locality. The framework defines a dependency between two locations in space as existing when a minimal-cost posting at one location can affect the other. Because every such posting is a one-bit flip, the dependency relation it defines has a finite range: a posting at one site can only directly influence sites within a fixed distance, here distance 1 on a three-dimensional grid. This is proved for any grid size of at least two, and it is what makes the framework's notion of locality concrete: it is defined by realizable minimum-cost transitions, not by an arbitrary cutoff or by assumptions about how influences decay.

The theorem does not claim that every one-bit change is a minimal-cost posting, nor that the one-bit flip is the only way to achieve minimal cost in all circumstances. It also does not claim that the spatial identification used in the translated three-dimensional cell is forced by the ledger semantics alone. The pack explicitly records countermodels where balanced conservation, shift-invariant cost, and an eight-tick identity all hold while the dependency relation is all-pairs, meaning those weaker conditions do not by themselves pick out a spatial structure. The theorem's force is conditional: given the full minimal-cost posting semantics, locality follows.

/-- 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 theorem does not claim that every one-bit change is a minimal-cost posting. The theorem does not claim that the spatial identification used in the translated three-dimensional cell is forced by the ledger semantics 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/PairKernelOperationalLocalityS4.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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