Foundation Simplicial Ledger Recognition Loop Has Surjection

The loop's pattern pass

A recognition loop is a closed cycle of tetrahedra, the 3-simplex atoms of volume in the framework's ledger, a discrete record of events. The declaration recognition_loop_has_surjection proves a property of any such loop: it must carry a complete pass through the eight possible 3-bit local pattern states. In plainer terms, as the cycle closes, it cannot skip any of the eight distinct configurations of a three-position binary code. The theorem states that for any cycle that qualifies as a recognition loop, there exists a function from the steps of the cycle onto the set of all 3-bit patterns, meaning every pattern is hit at least once.

The proof is a direct extraction from the definition of a recognition loop. The definition itself requires the existence of such a surjective pass, so the theorem unwraps that requirement and presents it as a standalone result. The declaration is a formal theorem in the machine-checked library of formal theorems, meaning its logical validity is verified by a computer. It does not, however, prove that such a loop exists, nor does it establish the minimum length of such a loop. It only states a conditional fact: if you have a recognition loop, then it has this pattern-covering property.

The result feeds directly into a companion theorem, eight_tick_uniqueness, which proves that any recognition loop must have at least eight steps. That is the number of 3-bit patterns, so the surjection result is a natural stepping stone. The eight-tick cycle is the unique minimal sequence for a self-consistent recognition loop on a simplicial manifold. This page's theorem is the smaller piece: it guarantees the coverage of all patterns, leaving the minimal-length result to its companion.

What the theorem does not claim is just as important. It does not assert that every closed cycle of tetrahedra is a recognition loop; the definition requires a specific surjective pass condition. It does not prove that the physical recognition-to-linking bridge holds, a separate open target in the framework. It also does not claim that the eight-tick cycle is the only possible loop, only that any loop must have at least eight steps and must cover all patterns.

/-- Every recognition loop carries a surjective pattern pass. -/
theorem recognition_loop_has_surjection {cycle : List Simplex3}
    (hloop : is_recognition_loop cycle) :
    ∃ pass : Fin cycle.length → Pattern 3, Function.Surjective pass := by
  exact hloop.2.2

/-- Every recognition loop carries a surjective pattern pass. -/
theorem recognition_loop_has_surjection {cycle : List Simplex3}
    (hloop : is_recognition_loop cycle) :
    ∃ pass : Fin cycle.length → Pattern 3, Function.Surjective pass := by
  exact hloop.2.2

/-- **THEOREM: Eight-Tick Cycle Uniqueness**
    The 8-tick closure cycle is the unique minimal sequence for a self-consistent
    recognition loop on a simplicial manifold. -/
theorem eight_tick_uniqueness (_L : SimplicialLedger) :
    ∀ cycle : List Simplex3,
    (is_recognition_loop cycle) → 8 ≤ cycle.length := by
  intro cycle hloop
  rcases recognition_loop_has_surjection hloop with ⟨pass, hsurj⟩
  exact eight_tick_min pass hsurj

What this page does not claim

That every closed cycle of tetrahedra is a recognition loop. That the physical recognition-to-linking bridge is proved. That the eight-tick cycle is the only possible loop, only that any loop has at least eight steps.

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/SimplicialLedger.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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