max_rate_pos
Recognition Science establishes that the maximum computation rate is strictly positive by deriving it as the reciprocal of the fundamental time quantum. Researchers bounding information processing speeds in discrete-time models cite this result when confirming that the tick imposes a finite positive upper limit on operations per unit time. The proof is a direct term-mode reduction that unfolds the rate definition and invokes division positivity on the known positive tick value.
claimLet $1/τ_0$ denote the maximum computation rate, where $τ_0$ is the fundamental time quantum. Then $1/τ_0 > 0$.
background
The module IC-002 derives fundamental computation limits from Recognition Science, with the central source being temporal discreteness: the tick $τ_0$ is the minimum time quantum, so the maximum bit rate equals $1/τ_0$ in RS-native units. The definition max_computation_rate is introduced as the reciprocal of the fundamental tick, which itself equals the constant tick value of 1. Upstream results supply the positivity of this tick (tick_pos) together with the standard positivity of 1, allowing the rate to inherit strict positivity via division.
proof idea
The term proof unfolds the definition of max_computation_rate to expose the division $1$ over the fundamental tick, then applies the div_pos lemma instantiated with one_pos and tick_pos.
why it matters in Recognition Science
This theorem supplies the positivity half of the IC-002 status certificate, confirming that the tick-derived rate bound is strictly positive rather than merely non-negative. It anchors the temporal-discreteness clause in the computation-limits structure and feeds directly into the certificate string that records all IC-002 components. Within the broader framework it realizes the first of the four listed sources of computation limits by making the inverse-tick rate manifestly positive.
scope and limits
- Does not supply a numerical value for the rate in SI units.
- Does not incorporate energy or Landauer erasure costs.
- Does not address irrationality or precision limits on φ-based states.
- Does not derive an upper bound on the rate from energy-time uncertainty.
Lean usage
have h : max_computation_rate > 0 := max_rate_posformal statement (Lean)
55theorem max_rate_pos : max_computation_rate > 0 := by
proof body
Term-mode proof.
56 unfold max_computation_rate
57 apply div_pos one_pos tick_pos
58
59/-- **THEOREM IC-002.3**: Any system with n parallel operations still obeys the tick bound.
60 n operations over τ₀ time means each operation uses τ₀/n time, not less than τ₀.
61 In RS, the tick is the atomic time unit: each recognition event takes exactly 1 tick. -/