phase_space_finite
The theorem establishes that the phase space of the eight-tick cycle contains exactly eight discrete elements. Researchers deriving the physical Church-Turing thesis from Recognition Science ledger dynamics cite this to bound the state space of all transitions. The proof reduces directly to the definition of Phase as Fin 8.
claimThe cardinality of the eight-tick phase space satisfies $|Phase| = 8$.
background
Phase is defined as the finite set Fin 8, representing the discrete phases 0 through 7 in one eight-tick cycle. The module IC-003 derives the physical Church-Turing thesis from the discrete ledger structure: each ledger entry evolves under an eight-tick operator on this finite phase space, with transitions governed by J-cost minimization. Upstream, tick denotes the fundamental RS time quantum τ₀ = 1, while phase definitions in EightTick and related modules set the phases at kπ/4 for k = 0 to 7, enforcing periodicity with period 2π.
proof idea
The proof is a one-line wrapper that applies simp to the definition of Phase as Fin 8, whose Fintype.card is definitionally 8.
why it matters in Recognition Science
This anchors the IC-003 derivation by confirming the finite eight-phase structure required for computable ledger transitions. It feeds directly into the ic003_certificate that assembles all IC-003 components and states the status of the physical Church-Turing thesis. The result instantiates the eight-tick octave (T7) from the forcing chain, ensuring RS dynamics remain within BQP and inherit undecidability only from Turing machines.
scope and limits
- Does not prove that arbitrary continuous physical processes remain computable.
- Does not bound approximation error for real-valued ledger entries beyond the discrete phases.
- Does not address dynamics outside the eight-tick discretization.
- Does not resolve the halting problem for RS ledger transitions.
formal statement (Lean)
52theorem phase_space_finite : Fintype.card Phase = 8 := by
proof body
Term-mode proof.
53 simp [Phase]
54
55/-- **THEOREM IC-003.2**: There are finitely many functions on the 8-tick phase space.
56 |Phase → Phase| = 8^8 = 16,777,216 — a large but finite number. -/