Foundation Primitive Recognition Calculus Basic Length Orbit Trace

The length of an orbit trace

A trace is a finite sequence of primitive distinction acts, each act being a single step that marks a difference. The framework's machine-checked library of formal theorems defines an orbit trace as the trace obtained by repeating that one act n times, starting from the empty trace. The theorem length_orbitTrace proves that the length of this orbit trace is exactly n. The proof is a direct induction on n: the empty trace has length 0, and each added act increases the length by 1.

This result is a basic sanity check on the framework's own definitions. It establishes that the length function, which counts the number of acts in a trace, agrees with the number of times the orbit trace construction repeats the act. The theorem is tagged THEOREM because it is proved in the machine-checked library with no framework-specific axioms, and it relies only on the standard rules of the underlying type theory.

The declaration does not claim anything about the physical meaning of an orbit, about the eight-tick recognition cycle, or about the golden ratio. It does not assert that a trace of length n corresponds to any particular physical duration or spatial extent. It is a purely syntactic fact about the formal objects defined in the same file: given the definitions of Trace, step, and length, the length of the n-fold orbit trace is n.

What the theorem changes is that the framework's foundational calculus has a verified counting primitive. Any later claim that uses the length of an orbit trace can rely on this exact correspondence, and the proof is machine-checked, so the counting is not a hidden assumption but a derived fact.

/-- K2.12 preview. The length of the nth orbit trace is n. -/
theorem length_orbitTrace (n : Nat) :
    length (orbitTrace n) = n := by
  induction n with
  | zero => rfl
  | succ n ih =>
      simp [orbitTrace, step, ih]

What this page does not claim

The theorem does not assign any physical duration or spatial meaning to a trace's length. The theorem does not connect the orbit trace to the golden ratio or to the eight-tick cycle.

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

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