barrierPeriod_pos
The theorem establishes positivity of the consciousness barrier period in RS-native units. Researchers bounding holographic maintenance costs for conscious boundaries cite it to guarantee non-negative demand over the integration window. The proof is a one-line wrapper that unfolds the definition to 360 times the unit tick and applies numerical normalization.
claim$0 < 360 · τ_0$ where $τ_0$ is the fundamental recognition time quantum and the factor 360 is the least common multiple of the eight-tick octave and 45.
background
In the Consciousness Bandwidth module the barrier period is the minimum integration window for consciousness, defined as 360 ticks so that zero accumulated cost forces zero defect at every step. The upstream definition of tick supplies the RS-native time quantum with $τ_0$ as its abbreviation; barrierTicks is the natural number 360 and barrierPeriod multiplies that integer by $τ_0$ to obtain a real number. The module documentation states that this window equals 45 octaves and supplies the holographic budget comparison $N_{events} = L^2 / (4ℓ_P^2 · ln(φ))$ against integrated J-cost demand.
proof idea
The term proof unfolds barrierPeriod, $τ_0$, tick and barrierTicks, reducing the goal to the concrete inequality 0 < 360, then invokes norm_num to discharge it by arithmetic simplification.
why it matters in Recognition Science
The result is invoked directly by maintenanceDemand_nonneg and maintenanceDemand_zero_iff to obtain non-negativity and the zero-iff condition at unit scale; it therefore closes the positivity scaffolding required for the holographic constraint theorems higher_Z_more_demand and bandwidth_constrains_extent. It instantiates the eight-tick octave structure (T7) inside the 360-tick barrier that spans 45 octaves, supplying the time base for the demand-versus-budget comparison in the module.
scope and limits
- Does not derive the numerical value 360 from a deeper forcing-chain step.
- Does not constrain the spatial extent L or the critical scale L_crit.
- Does not incorporate Z-complexity or J-cost dependence.
- Does not address units outside RS-native time.
formal statement (Lean)
72theorem barrierPeriod_pos : 0 < barrierPeriod := by
proof body
Term-mode proof.
73 unfold barrierPeriod τ₀ tick barrierTicks
74 norm_num
75