Foundation Pair Kernel Operational Locality S4 Tiled Jminimal Separated Dependen

Separated sources, disjoint effects

A ledger is a discrete record of events. In Recognition Science, the framework's library of machine-checked theorems studies ledgers whose states are grids of cells, each cell holding a bit. A transition from one state to another is a posting: a change in some cells. The theorem tiledJMinimal_separated_dependencies_disjoint concerns which cells can be affected by a posting. It states that if two starting cells are far enough apart, measured by a grid distance greater than two, then no single later cell can be influenced by both. In plain language: separated sources have disjoint effects. This is a locality property: influence does not jump arbitrarily across the grid; it stays within a bounded neighborhood.

The theorem is proved for a specific definition of influence. A dependency from cell a to cell x means there is a minimal-cost posting step, a transition of minimum cost (a measure of how expensive a change is), that changes a bit at a and also changes a bit at x. The framework proves that such a minimal posting changes exactly one bit in the parity pattern, so its influence is one cell wide. The theorem then combines this with a tiling construction: the grid is built from repeated copies of a small three-dimensional unit cell. The result is a formal statement, checked by the machine, that for any two starting cells separated by more than two steps of grid distance, their sets of possible influenced cells do not overlap.

The theorem does not claim that all physical interactions are local in this sense. It applies only to the specific dependency relation defined by minimum-cost postings in this tiled model. The framework also records counterexamples: other natural-looking candidate definitions, such as balanced conservation or shift-invariant costs, can allow all-pairs dependencies, where any cell can influence any other. Those routes fail to produce locality. The theorem shows that the minimum-cost posting rule, with the tiling, is one that does produce it.

What the theorem establishes is a precise, machine-checked fact about a mathematical model. It does not, by itself, say that the physical universe is local in this way. The bridge from this formal model to physical space is a separate question, and the framework marks that bridge as open. What a reader can take away is that in this discrete ledger model, locality is not assumed; it is derived from the cost rule and the tiling structure.

/-- Independent consequence: sufficiently separated centers cannot share a
minimum-J translated-cell dependency. -/
theorem tiledJMinimal_separated_dependencies_disjoint
    {L : ℕ} (hL : 2 ≤ L)
    {a b : Fin (L * L * L)}
    (hsep : 2 < encodedDist3 L a b) :
    ¬ ∃ x,
      TiledJMinimalDependency3 hL a x ∧
      TiledJMinimalDependency3 hL b x :=
  separated_centers_have_disjoint_dependencies
    (tiledJMinimalDependency3_operationally_local hL) hsep

/-- A ledger transition is operationally selected when it is monotone,
nontrivial, and has minimum `Jlog` cost among such transitions. -/
def JMinimalPostingStep
    (A B : LedgerState 3) : Prop :=
  MonotoneLedger A B ∧
    A ≠ B ∧
    ∀ C : LedgerState 3,
      MonotoneLedger A C → A ≠ C →
        ledgerJlogCost A B ≤ ledgerJlogCost A C

/-- 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

/-- 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 all possible dependency relations in the framework are local; it applies only to the specific tiled minimum-cost posting relation. The theorem does not claim that physical space is local; the bridge from the formal model to physics is open. The theorem does not claim that balanced conservation or shift-invariant costs are impossible; it shows they do not force locality.

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)

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: