Foundation Simplicial Ledger Simplicial Ledger

The simplicial ledger

A ledger is a discrete record of events. The Recognition Science framework's ledger is such a record, and its usual picture is a fixed cubic lattice of cells. The declaration SimplicialLedger replaces that picture with a more flexible one: a collection of tetrahedra, called 3-simplices, that fit together to cover a volume. Each tetrahedron is a tiny atom of space, and the whole collection is the ledger. The declaration defines this object formally, along with a way to assign a number, a recognition potential, to each tetrahedron.

On that structure the framework defines a cost. The local cost on one tetrahedron is the framework's standard cost function J multiplied by the tetrahedron's volume. The global cost is the sum over all tetrahedra, for a finite ledger. The declaration then states a theorem: if every local cost is stationary, meaning its numerical variation is zero, then every local potential equals 1. This is a formal statement, but it is a conditional one. The hypothesis that global stationarity implies local stationarity is marked as an empirical hypothesis, not a proved theorem.

The declaration also defines a recognition loop, a closed cycle of tetrahedra. It proves that any such loop must have length at least 8. The proof relies on a surjective mapping onto the 8 possible local pattern states of 3 bits. This is the eight-tick cycle, the framework's minimal self-consistent loop. The theorem is proved from the definition of a recognition loop and a prior lemma about the minimum length of a surjective sequence onto 8 states.

In Recognition Science, this declaration is a step toward a coordinate-free description of space. The framework models the ledger as a simplicial 3-complex rather than a fixed lattice, and it provides a sheaf representation to unify local and global cost variations. The payoff is a picture of space built from recognition events themselves, not from a pre-existing grid. The declaration does not claim that this picture is physically real; it claims that the formal object is defined and that one theorem about its loops holds.

/-- **DEFINITION: Simplicial Ledger**
    A collection of 3-simplices that form a manifold covering. -/
structure SimplicialLedger where
  simplices : Set Simplex3
  /-- The simplices form a non-empty set (non-vacuity). -/
  non_empty : simplices.Nonempty
  /-- SCAFFOLD: Manifold covering property.
    Proof requires simplicial complex axioms and manifold topology.
    See: LaTeX Manuscript, Chapter "Gravity as Recognition", Section "Simplicial Ledger". -/
  is_covering : 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 not proved; it is a declared placeholder. The hypothesis that global stationarity implies local stationarity is not a theorem; it is marked as an empirical hypothesis. The declaration does not claim that the simplicial ledger is the physically real structure of space.

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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