numPhases
Recognition Science fixes the phase space cardinality at eight to match the eight-tick evolution period. Physicists deriving the Physical Church-Turing Thesis from discrete ledger dynamics cite this constant to bound the state space per tick. The definition is a direct constant assignment with no further computation required.
claimLet $N$ denote the number of distinct phases in one 8-tick cycle. Then $N=8$.
background
The module derives the Physical Church-Turing Thesis from the discrete ledger structure of Recognition Science, where each ledger entry evolves under an 8-tick operator on a finite phase space. Upstream, tick is the fundamental time quantum with value 1 (RS-native units), and phase from the EightTick module returns the eight angles $kπ/4$ for $k=0,…,7$. The local setting requires that every physical process maps finite states to finite states at rate $1/τ_0$, with no access to unbounded resources per step.
proof idea
Direct definition that assigns the constant 8, matching the cardinality of the phase set defined in the EightTick module.
why it matters in Recognition Science
This supplies the finite cardinality used by the downstream theorem phase_space_bounded, which states that no RS process can access more than eight phases in one tick and thereby prevents hypercomputation. It realizes the eight-tick octave (T7) of the forcing chain and supports the finite-memory argument in the module's derivation of the Physical Church-Turing Thesis.
scope and limits
- Does not derive the value 8 from the J-cost functional or Recognition Composition Law.
- Does not prove that eight phases suffice for simulating arbitrary physical systems.
- Does not address phase spaces outside the discrete 8-tick ledger model.
Lean usage
theorem phase_space_bounded : numPhases ≤ 8 := by unfold numPhases; norm_numformal statement (Lean)
49def numPhases : ℕ := 8
proof body
Definition body.
50
51/-- **THEOREM IC-003.1**: The 8-tick phase space has exactly 8 elements. -/