eightTickCadence_pos
plain-language theorem explainer
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
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.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.