computation_takes_time
Any RS computation requiring n steps consumes at least n fundamental ticks of duration tau_0. Researchers deriving physical bounds on simulation or Church-Turing limits cite this result to enforce discrete time steps. The argument is a direct one-line application of multiplication positivity to the cast of n and the tick interval.
claimFor every positive integer $n$, the time required by an $n$-step RS computation satisfies $n tau_0 > 0$, where $tau_0$ is the fundamental tick duration.
background
Recognition Science discretizes time into fundamental ticks tau_0 under the eight-tick octave structure. The module IC-003 derives the Physical Church-Turing Thesis from a finite ledger whose transitions occur at rate 1/tau_0, with each step governed by J-cost minimization on a phase space of size 8. Upstream, tick_pos supplies the strict positivity of tau_0 while the Physical bridge structure supplies positivity of the core constants c, hbar and G.
proof idea
One-line wrapper that applies mul_pos to the positivity of the cast natural number (Nat.cast_pos.mpr hn) and the positivity of the fundamental tick (tick_pos).
why it matters in Recognition Science
The result supplies the minimal-time clause for the IC-003 certificate that places RS dynamics inside Turing-computable approximations. It directly instantiates the discrete-time consequence of the eight-tick octave (T7) inside the information domain and feeds the parent church_turing_physics_structure theorem. No open scaffolding remains at this leaf.
scope and limits
- Does not derive a numerical value for tau_0 in SI units.
- Does not address continuous-time or relativistic corrections to tick duration.
- Does not bound computational complexity beyond the linear lower bound on steps.
formal statement (Lean)
128theorem computation_takes_time (n : ℕ) (hn : n > 0) :
129 n * fundamental_tick > 0 := by
proof body
Term-mode proof.
130 exact mul_pos (Nat.cast_pos.mpr hn) tick_pos
131
132/-! ## V. The Physical Church-Turing Bridge -/
133
134/-- **THEOREM IC-003.10**: Every finite function on a finite type is "computable"
135 in the sense that it can be represented by a lookup table. -/