eightTickCadence_pos
The theorem establishes strict positivity of the eight-tick cadence, the minimum duration of one full recognition cycle in RS-native units. Researchers deriving holographic bandwidth limits or black-hole entropy relations cite it to guarantee positive denominators in rate formulas. The proof is a one-line wrapper that unfolds the definition and reduces the inequality via numerical normalization.
claim$0 < 8 τ_0$, where $τ_0$ is the fundamental time quantum.
background
The RecognitionBandwidth module defines recognition bandwidth as the maximum rate of ledger events inside a holographically bounded region, given by $R_max = A / (4 ℓ_P² · ln(φ) · 8 τ_0)$. The eight-tick cadence supplies the time denominator and equals eight times the fundamental tick $τ_0$. The tick itself is the basic RS time quantum, normalized to 1 in native units.
proof idea
The proof is a one-line wrapper that unfolds eightTickCadence in terms of τ₀ and tick, then applies norm_num to confirm 0 < 8 * 1.
why it matters in Recognition Science
This positivity result is invoked by entropy_is_bandwidth_capacity (which equates black-hole entropy to bandwidth capacity times cycle cost) and by horizonBandwidth_pos. It closes the positivity requirement for the eight-tick octave (T7) inside the forcing chain, ensuring the recognition-rate ceiling remains well-defined. The result touches the open question of how the cadence couples to the ILG parameters and consciousness-boundary cost listed in the module header.
scope and limits
- Does not compute a numerical value for the cadence beyond its definition as eight ticks.
- Does not establish positivity of k_R or planckArea without separate lemmas.
- Does not address time-dependent or spatially varying bandwidth.
Lean usage
have h : 0 < k_R * eightTickCadence := mul_pos k_R_pos eightTickCadence_posformal statement (Lean)
50theorem eightTickCadence_pos : 0 < eightTickCadence := by
proof body
Term-mode proof.
51 unfold eightTickCadence τ₀ tick
52 norm_num
53