IsSubcritical
Systems whose demanded recognition rate stays strictly below the bandwidth limit set by their boundary area satisfy the subcritical condition. Control theorists modeling recognition-based physics would cite this predicate when proving stability bounds for semantic condensation. The definition reduces directly to an inequality against the bandwidth function derived from holographic bounds and the eight-tick cadence.
claimA recognition system with boundary area $A$ and demand rate $D$ is subcritical precisely when $D < R_{max}(A)$, where $R_{max}(A)$ is the maximum number of recognition events per unit time permitted by the holographic bound on area $A$.
background
Recognition bandwidth of a region with boundary area $A$ is the maximum number of recognition events per unit time that the holographic bound permits, given each event costs $k_R = ln phi$ bits and processing is limited to one cycle per eight fundamental time units: $R_{max}(A) = A / (4 ell_P^2 cdot k_R cdot 8 tau_0)$. The module sketches a control theorem for the operating regime of recognition systems. The load ratio rho = R_dem / R_max must satisfy rho_min < rho < 1 for healthy operation, with stability judged on the 360-tick supervisory horizon forced by lcm(8,45).
proof idea
Direct definition: the predicate holds exactly when the demanded rate is strictly below the bandwidth computed from the area's holographic limit and the native eight-tick processing cadence.
why it matters in Recognition Science
This predicate supplies the upper stability bound in the critical recognition loading control theorem. It is invoked by the certificates criticalRecognitionLoading_certificate and forcedCriticalRecognitionLoading_certificate to assert that a controller state remains subcritical while satisfying attention and zeta bounds. The construction ties directly to the eight-tick octave through the bandwidth function, closing one structural lemma in the unification of recognition dynamics with physical constants.
scope and limits
- Does not define the minimum load ratio for the critical band.
- Does not incorporate the 360-tick supervisory horizon.
- Does not address attention or zeta bounds on the controller state.
- Does not provide numerical evaluation of bandwidth in SI units.
formal statement (Lean)
72def IsSubcritical (area demand : ℝ) : Prop :=
proof body
Definition body.
73 demand < bandwidth area
74
75/-- Overloaded systems hit or exceed the bandwidth ceiling. -/