Foundation Primitive Recognition Calculus Trace Closure Canonical Succ

The successor step

A completed orbit ledger is an infinite record of positions in a counting sequence. The declaration canonical_succ proves that the canonical ledger, the one that simply lists the natural numbers in order, has the property that the position at step n+1 is the successor of the position at step n. In plainer terms, it shows that the standard counting sequence advances by exactly one each time, a fact so basic it is usually taken for granted.

The proof is a definitional equality, meaning it holds by how the objects are constructed. The canonical ledger assigns to each index n the natural number n itself. The successor of that number is n+1, which is precisely what the ledger assigns to the next index. The declaration records this identity for every natural number n, not just for a few examples. It is a theorem in the machine-checked library of formal theorems, verified by the kernel's rules of computation.

In Recognition Science, this declaration belongs to a broader construction called trace closure. A completed trace is an infinite ledger of distinction acts, and a completed orbit ledger is the natural-number side of that idea. The canonical_succ theorem is one small certificate that this infinite extension is coherent: the infinite ledger behaves exactly as the finite prefixes suggest it should. It is a consistency check on the step from finite traces to their infinite completion.

What the declaration does not claim is just as important as what it proves. It does not prove that every completed ledger advances by one; it only describes the canonical one. It does not establish that any particular physical process follows this sequence, nor that the natural numbers exist in any metaphysical sense. The theorem is about a formal object and its internal structure, not about the world. It is a piece of the framework's foundation, not a statement about empirical reality.

theorem canonical_succ (n : Nat) :
    canonical.positionAt (Nat.succ n) =
      DistinctionNat.succ (canonical.positionAt n) := by
  rfl

/-- K4.13. A completed orbit ledger is the infinite sequence of finite
δ-orbit positions. This is the natural-number side of trace closure. -/
structure CompletedOrbitLedger where
  positionAt : Nat → DistinctionNat

What this page does not claim

This declaration does not prove that every completed ledger advances by one step at a time. It does not assert that any physical process follows the canonical counting sequence. It does not establish the existence of natural numbers outside the formal system.

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/TraceClosure.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: