eightTickCadence_eq
The equality fixes the eight-tick cadence at exactly 8 in units where the fundamental time quantum equals 1. Researchers deriving recognition bandwidth from holographic bounds would cite this to anchor the cycle time in the rate formula. The proof is a term-mode reduction that unfolds the cadence definition and applies ring normalization.
claimWith the fundamental time quantum set to unity, the eight-tick cadence equals 8, so that one processing cycle occupies exactly eight base intervals.
background
Recognition Science defines the eight-tick cadence as the period for one full cycle of the recognition operator, expressed as 8 times the fundamental time quantum. The module on recognition bandwidth combines this with the holographic bound on information capacity and the per-bit recognition cost to obtain a maximum event rate. Upstream results supply the base time quantum as the constant 1 and its abbreviation, together with the structure of ledger constants from the law of existence.
proof idea
The term proof unfolds the definition of the eight-tick cadence together with the abbreviations for the time quantum and tick, then invokes the ring tactic to reduce the resulting arithmetic expression to 8.
why it matters in Recognition Science
This pins the numerical factor in the recognition bandwidth formula that unifies holographic capacity, per-bit cost, and the eight-tick processing rate, realizing the T7 octave landmark of the forcing chain. The module positions the result as one of the core connections among previously separate elements of Recognition Science. No open questions remain as the equality is fully discharged.
scope and limits
- Does not derive the choice of eight ticks from prior axioms.
- Does not evaluate bandwidth for concrete boundary geometries.
- Does not incorporate the phi-ladder mass formula or alpha band.
- Does not address consciousness boundary costs listed in the module.
formal statement (Lean)
54theorem eightTickCadence_eq : eightTickCadence = 8 := by
proof body
Term-mode proof.
55 unfold eightTickCadence τ₀ tick
56 ring
57
58/-! ## §2. Recognition Bandwidth -/
59
60/-- **DEFINITION**: Recognition bandwidth of a region with boundary area A.
61
62 The maximum number of recognition events per unit time that the holographic
63 bound permits within the region, given that each event costs k_R = ln(φ) bits
64 and the 8-tick cadence limits processing to one cycle per 8τ₀.
65
66 R_max(A) = A / (4ℓ_P² · k_R · 8τ₀)
67
68 Units: events per unit time.
69
70 This combines three previously disconnected elements:
71 - Holographic capacity: A/(4ℓ_P²) [from Quantum.HolographicBound]
72 - Per-bit cost: k_R = ln(φ) [from Constants.BoltzmannConstant]
73 - Processing rate: 8τ₀ per cycle [from Foundation.EightTick] -/