Foundation Pair Kernel Operational Locality S4 Committed Candidate Routes Admit

What the route admits

The declaration committed_candidate_routes_admit_allPairs is a result in the Recognition Science framework's machine-checked library of formal theorems. It concerns the framework's central picture of reality as a ledger, a discrete record of events, where each event is a posting that changes the state of the record. The theorem states that a set of five weaker conditions can all hold at once while still permitting a dependency relation in which every cell is connected to every other cell. In plain terms, it shows that those conditions do not, by themselves, force a notion of 'nearby' in space.

The five conditions are: a valid atomic tick, a ledger that conserves some quantity, a cost function that is shift-invariant, an eight-tick identity derived from dimension three, and a nontrivial distinction between two states. The theorem's conclusion is that all five can coexist with an all-pairs dependency relation, which fails the framework's operational locality property. That property, defined in the same module, requires that a dependency only exists between cells that are within a bounded distance of each other. The all-pairs relation connects every cell to every other, so it violates that bound.

The result is a negative one: it rules out five candidate routes for deriving spatial locality from the ledger semantics. Each route looked like a plausible way to get a local structure, but the theorem shows they are all too weak. The framework's own path to locality uses a different, stronger condition: a transition of minimum cost, called a J-minimal posting step. The library proves that such a step changes exactly one bit in the ledger's parity pattern, and that the resulting dependency relation is operationally local. That positive result stands apart from the countermodels in this declaration.

What the declaration does not claim is just as important. It does not say that the five conditions are inconsistent, or that they lead to contradiction. It does not say that the all-pairs dependency is the only possible outcome, only that it is one possible outcome. And it does not say that the weaker routes are useless; it says they do not, on their own, choose the spatial identification that the translated-cell semantics uses. The theorem is a boundary marker: it shows where the easy paths stop and why the framework's actual construction must go further.

/-- 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⟩

/-- 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

What this page does not claim

The five weaker conditions are inconsistent with each other. The all-pairs dependency is the only possible outcome of those conditions. The weaker routes are useless for deriving other properties.

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

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