Foundation Substrate Axioms Compatibility Trivial

The Gray cycle guarantee

A Gray code is a sequence of binary strings in which consecutive entries differ by exactly one bit. The declaration compatibility_trivial proves that for every dimension D, a bijective closed walk exists on the D-dimensional cube graph, visiting each vertex exactly once and returning to the start, with each step changing one bit. This is the standard binary reflected Gray cycle, and the proof is constructive: the witness is the usual recursive construction.

The theorem belongs to the framework's substrate axioms, a set of structural inputs for its dimension route. The statement is a pure combinatorial fact about cube graphs, and the proof is a direct construction. It does not depend on any smooth topology, homology, or manifold theory. The declaration is a theorem in the machine-checked library, meaning the construction and its verification are formalized and checked by a computer.

In the framework, this compatibility claim says that the realized recognition cycle, an eight-tick process, can be embedded into the cube graph in any dimension. The theorem itself establishes the combinatorial possibility, not the physical realization. The framework models the recognition cycle as a Gray cycle, and this theorem shows that such a cycle exists for every dimension, not just for the physically realized dimension three.

What the declaration does not claim is equally important. It does not assert that any smooth manifold substrate exists, that any cube-graph embedding into a manifold is tame, or that any retraction back to the cube graph is available. Those clauses were formerly typed as trivial truths and have been deleted from the library because no honest predicates for them exist in the current vocabulary. The theorem is a combinatorial existence result, not a geometric or topological one.

/-- Every dimension admits a BRGC Gray-cycle witness. -/
theorem compatibility_trivial (D : Dimension) :
    CompatibilityWithRealizedCycle D where
  witness_is_closed_walk := by
    classical
    refine ⟨brgcPath D, ?_, ?_⟩
    · have h_inj : Function.Injective (brgcPath D) := brgcPath_injective D
      have h_card : Fintype.card (Fin (2 ^ D)) = Fintype.card (Pattern D) := by
        simp
      exact (Fintype.bijective_iff_injective_and_card (brgcPath D)).2 ⟨h_inj, h_card⟩
    · intro i
      cases D with
      | zero => exact Or.inl rfl
      | succ d =>
        exact Or.inr (brgc_oneBit_step (d := d + 1) (Nat.succ_pos d) i)

What this page does not claim

The theorem does not claim that any smooth manifold substrate exists in any dimension. It does not claim that any cube-graph embedding into a manifold is tame or that a retraction back to the cube graph exists. It does not claim that the recognition cycle is physically realized in any dimension; it only establishes a combinatorial possibility.

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

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Derived articles

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