Foundation Simplicial Ledger Simplicial Sheaf

The Simplicial Sheaf

A sheaf is a mathematical tool for organizing local information so that it can be assembled into a consistent global object. In this framework, the ledger, a discrete record of recognition events, is modeled as a collection of tetrahedra, the simplest three-dimensional shapes, rather than as a fixed grid of cubes. The SimplicialSheaf declaration assigns to each tetrahedron a real number called a recognition potential, a value that represents the local state of the ledger at that point.

The declaration also includes a placeholder property, is_consistent, which is meant to state that these potentials agree across the shared faces of neighboring tetrahedra. This consistency condition is not yet proved; it is a scaffold for future work. What the declaration does establish is a precise way to talk about local and global cost. The local cost on a single tetrahedron is the product of the recognition cost function J and the tetrahedron's volume. The global cost is simply the sum of these local costs over all tetrahedra in the ledger, defined for finite ledgers.

In Recognition Science, the framework proves a theorem that connects these two levels. The theorem local_global_unification states that if the global J-cost is stationary, meaning its variation is zero everywhere, then every local potential must be at its stationary value, which is exactly 1. This is a formal result: it follows from the hypothesis that local stationarity everywhere implies the global condition, and it is checked by the machine-checked library of formal theorems.

The declaration also defines a recognition loop as a closed cycle of tetrahedra that induces a surjective pass through all 3-bit local pattern states. A proved theorem, eight_tick_uniqueness, states that any such loop must have length at least 8. This is the formal origin of the eight-tick cycle in the framework.

What this declaration does not claim is that the manifold covering property holds. The field is_covering is a proposition with a comment noting that its proof requires simplicial complex axioms and manifold topology, which are not yet formalized. Likewise, the consistency of potentials across boundaries is a placeholder, not a theorem. The local-global unification theorem is proved, but it depends on the hypothesis that local variation is zero everywhere; it does not prove that such a state is reached.

/-- **DEFINITION: Simplicial Sheaf**
    A sheaf assigning a recognition potential to each simplex in the ledger. -/
structure SimplicialSheaf (L : SimplicialLedger) where
  potential : Simplex3 → ℝ
  /-- The potential is consistent across simplex boundaries (placeholder). -/
  is_consistent : Prop

/-- Global J-cost summed over the ledger (for finite ledgers). -/
noncomputable def global_J_cost (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : ℝ :=
  ∑ s : L.simplices, local_J_cost s (S.potential s)

/-- **THEOREM: Local-Global Unification**
    The global J-cost is stationary if and only if every local J-cost is stationary
    within its simplicial section. -/
theorem local_global_unification (L : SimplicialLedger) (S : SimplicialSheaf L)
    [Fintype L.simplices] (h : H_LocalGlobalUnification L S)
    (h_global : ∀ s : L.simplices, local_variation s (S.potential s) = 0) :
    ∀ s : L.simplices, J_stationary (S.potential s) := h h_global

/-- **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

The manifold covering property is proved; it is a scaffold with a placeholder proposition. The consistency of potentials across simplex boundaries is established; it is a placeholder. The local-global unification theorem proves that a stationary global state is actually reached; it only proves a conditional statement.

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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