Foundation Simplicial Ledger H Local Global Unification

Local and global cost

A ledger, a discrete record of events, can be built from tiny tetrahedral blocks called 3-simplices. The Recognition Science library defines a simplicial complex, a shape assembled by gluing these blocks face to face, as the ledger's topology. Each block carries a number, its recognition potential, and the library defines a local cost for each block and a global cost for the whole assembly. The local cost is the framework's J-cost multiplied by the block's volume; the global cost is the sum of all local costs.

The declaration H_LocalGlobalUnification states a one-way implication: if every block's local cost is stationary, meaning its variation with respect to the potential is zero, then every block's potential equals its unit value. The library also proves a theorem, local_global_unification, that takes this declaration as a premise and derives the conclusion. In plainer terms, the theorem says that when a certain condition holds, a system at rest at the local level is also at rest at the global level. The declaration itself is a definition, a formal statement of intent, not a proved fact.

The library explicitly marks the stronger claim as a hypothesis. The docstring for H_LocalGlobalUnification labels it EMPIRICAL_HYPO and names a falsifier: a global minimum that contains local non-stationary sections. This means the framework does not claim to have proved that global stationarity forces local stationarity. It only defines the implication and proves a conditional theorem that relies on it. The manifold covering property, the condition that the blocks form a whole without gaps, is also listed as a scaffold, a placeholder for future work, not a completed proof.

This distinction matters for anyone reading the framework's claims. The library contains a proved theorem, but the theorem's power depends on an unproved hypothesis. The honest summary is that the framework has formalized a proposed link between local and global behavior, and has proved what follows if that link holds. It has not proved that the link itself is true in the physical world the framework aims to describe.

/-- **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: 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)

What this page does not claim

The framework does not claim that global stationarity forces local stationarity; that is an explicit hypothesis with a named falsifier. The framework does not claim that the manifold covering property is proved; it is marked as a scaffold. The framework does not claim that the local-global unification applies to any physical system; the theorem is conditional on the hypothesis.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: