Foundation Simplicial Ledger Local Global Unification

The unification theorem

The ledger, a discrete record of recognition events, is usually pictured as a fixed grid of cubes. The framework's SimplicialLedger module instead builds it from tetrahedra, the simplest three-dimensional solid, packed together to cover a region. Each tetrahedron carries a number called its recognition potential, and the framework defines the cost of recognition on one tetrahedron as the product of a base cost function J and the tetrahedron's volume. Summing that local cost over all tetrahedra gives the global cost for the whole ledger.

The theorem named local_global_unification states a conditional fact: if the global cost is stationary, meaning small changes to any potential leave the total cost unchanged, and if a specific hypothesis H_LocalGlobalUnification holds, then every local potential must sit at its stationary value, which the framework defines as the number 1. The proof is a direct application of that hypothesis; the theorem itself does not derive the hypothesis from anything more basic. The hypothesis is labeled EMPIRICAL_HYPO in the source, with a stated falsifier: a global minimum that contains local non-stationary sections would refute it.

The module also proves a separate result about recognition loops, closed cycles of tetrahedra. Any such loop must pass through all eight possible three-bit patterns, and therefore any recognition loop has length at least eight. This eight-tick minimum is a theorem in the library, proved from the definition of a recognition loop and a prior lemma about pattern surjections.

What the declaration does not claim is as important as what it proves. It does not prove that global stationarity actually forces local stationarity in any real ledger; that would require the empirical hypothesis to be established. It does not show that the simplicial ledger covers space like a manifold, because the covering property is left as a scaffold, a placeholder for future work. And it does not connect this local-global result to the framework's derivation of three spatial dimensions; that bridge is a separate, open question.

/-- **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

/-- **HYPOTHESIS**: Global stationarity implies local simplicial stationarity.
    STATUS: EMPIRICAL_HYPO
    TEST_PROTOCOL: Verify that global J-cost minimization on a simplicial manifold
    forces every local potential Ψ to its unit value.
    FALSIFIER: Discovery of a global minimum that contains local non-stationary sections. -/
def H_LocalGlobalUnification (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : Prop :=
  (∀ s : L.simplices, local_variation s (S.potential s) = 0) →
  ∀ s : L.simplices, J_stationary (S.potential s)

/-- **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

This theorem does not prove that global stationarity forces local stationarity without the empirical hypothesis. It does not establish that the simplicial ledger covers space like a manifold. It does not connect the local-global result to the framework's derivation of three spatial dimensions.

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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