Foundation Primitive Recognition Calculus All Dimensional Cubical Boundary Delta

The boundary API

In the Recognition Science framework, the declaration deltaCubicalBoundaryAPI packages a specific mathematical fact: for any finite chain of higher-dimensional cubical faces, applying the boundary operation twice gives zero. The framework's ledger, a discrete record of recognition events, is built from such chains. The API states that each chain comes with a second-boundary display, and that this display is identically zero.

The proof works by decomposing each higher-dimensional face into a finite list of codimension-2 square certificates. Each square certificate has a boundary-of-boundary term that is zero, and the sum of finitely many zero terms is zero. The result is formalized in the framework's machine-checked library of formal theorems, with the key declarations higherFace_secondBoundary_zero and higherChain_secondBoundary_zero establishing the zero property for individual faces and for whole chains.

This is a structural or combinatorial fact about the framework's cubical setting. It does not claim that the boundary operation is a differential in the sense of conventional algebraic topology, nor that it defines a homology theory. It does not assert that the recognition-to-linking bridge, the physical interpretation of these chains, is complete. The API is a formal tool: it guarantees a clean algebraic property that the framework's Delta plan requires, nothing more.

/-- Every finite higher-dimensional chain has zero second boundary. -/
theorem higherChain_secondBoundary_zero {n : ℕ} (C : HigherChain n) :
    HigherChain.secondBoundary C = fun _ => 0 := by
  induction C with
  | nil =>
      funext w
      simp [HigherChain.secondBoundary]
  | cons F rest ih =>
      funext w
      have hF := congrFun (higherFace_secondBoundary_zero F) w
      have hrest := congrFun ih w
      simp [HigherChain.secondBoundary] at hrest
      simp [HigherChain.secondBoundary, List.foldl_cons, hF, hrest]

/-- Every higher-dimensional face certificate has zero second boundary, because
its second boundary is a finite ledger of zero 2-face boundary-of-boundary terms. -/
theorem higherFace_secondBoundary_zero {n : ℕ} (F : HigherFaceCert n) :
    F.secondBoundary = fun _ => 0 :=
  twoFaceCert_list_boundary_squared_zero F.twoFaceLedger

What this page does not claim

The declaration does not define a homology theory or a differential in the conventional algebraic topology sense. It does not assert that the physical recognition-to-linking bridge is complete. It does not claim the result extends to infinite chains.

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/PrimitiveRecognitionCalculus/AllDimensionalCubicalBoundary.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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