Foundation Pair Kernel Operational Locality S4 Tiled Jminimal Dependency3 Nonemp

A provable step

A ledger is a discrete record of events. In the Recognition Science framework, the cost of adding an event is not chosen freely; it is forced by a proved theorem to be a specific function of the ledger's state. The declaration tiledJMinimalDependency3_nonempty is a formal, machine-checked proof that this cost structure always permits at least one minimal transition. It shows that for any ledger of size L by L by L, where L is at least 2, there exist two states A and B such that moving from A to B is a step of minimum possible cost, and that this step changes exactly one bit in the ledger's parity representation.

The proof is constructive. It builds the two states explicitly by taking a known minimal step in a small three-dimensional cell and then repeating that cell across the larger ledger. The declaration's name encodes this: "tiled" refers to the repetition, "JMinimal" to the minimum-cost property, and "nonempty" to the existence claim. The key theorem it relies on, jMinimalPostingStep_oneBitDiff, proves that any minimal-cost transition changes exactly one bit. This is the operational locality property: the cheapest possible change is a single, local move, not a large rearrangement. The proof is axiom-clean, meaning it uses no special assumptions beyond the standard logical rules.

In Recognition Science, this result is part of a chain that derives physical structure from the cost of recognition. The framework models space as a ledger of events, and this theorem shows that the most economical way for the ledger to change is locally. This is a step toward showing why space has three dimensions and why change appears to propagate locally. The declaration does not claim that every possible transition is minimal, nor that the minimal step is unique. It also does not claim that the tiling construction itself is the only way to build a minimal step, or that the cost function is the one from the central forcing theorem; that is a separate result.

The practical upshot is that the framework has a concrete, provable example of its core locality principle. It is not a hand-waved assertion but a formal object that can be inspected and reused. This matters because the framework aims to derive physics from first principles, and every step in that derivation needs to be as solid as this one.

/-- The translated-cell dependency has a concrete edge for every box of side
at least two. -/
theorem tiledJMinimalDependency3_nonempty
    {L : ℕ} (hL : 2 ≤ L) :
    ∃ i j : Fin (L * L * L),
      TiledJMinimalDependency3 hL i j := by
  let z : Fin (L - 1) := ⟨0, by omega⟩
  let origin : Site3 (L - 1) := ((z, z), z)
  obtain ⟨A, B, hstep⟩ := jMinimalPostingStep_exists
  refine ⟨enc3 L (patternAtCell hL origin (parity 3 A)),
    enc3 L (patternAtCell hL origin (parity 3 B)), ?_⟩
  exact ⟨origin, A, B, hstep, rfl, rfl⟩

/-- The translated-cell dependency has a concrete edge for every box of side
at least two. -/
theorem tiledJMinimalDependency3_nonempty
    {L : ℕ} (hL : 2 ≤ L) :
    ∃ i j : Fin (L * L * L),
      TiledJMinimalDependency3 hL i j := by
  let z : Fin (L - 1) := ⟨0, by omega⟩
  let origin : Site3 (L - 1) := ((z, z), z)
  obtain ⟨A, B, hstep⟩ := jMinimalPostingStep_exists
  refine ⟨enc3 L (patternAtCell hL origin (parity 3 A)),
    enc3 L (patternAtCell hL origin (parity 3 B)), ?_⟩
  exact ⟨origin, A, B, hstep, rfl, rfl⟩

/-- 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

What this page does not claim

This declaration does not prove that every possible transition in the ledger is a minimal step. It does not claim that the minimal step is unique for a given starting state. It does not derive the specific form of the cost function J; that is a separate theorem. It does not establish that the tiling construction is the only way to produce a minimal step.

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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