loadPenalty
loadPenalty defines the non-negative cost incurred when a recognition system's load ratio falls below rhoMin or exceeds 1. Control theorists working on recognition bandwidth and semantic condensation would cite this when proving zero-penalty stability certificates. It is realized as a direct sum of two rectified-linear terms applied to the load ratio computed from area and demand.
claim$P(rhoMin, A, D) := max(0, rhoMin - rho) + max(0, rho - 1)$ where $rho := D / bandwidth(A)$.
background
The Critical Recognition Loading module introduces the load ratio rho = demand / bandwidth(area) as the fraction of available recognition bandwidth consumed by demand. Healthy operation is confined to the narrow band rhoMin < rho < 1, with timing governed by the native 8-tick cadence and the 360-tick supervisory horizon arising from lcm(8,45). The upstream loadRatio definition supplies the runtime ratio: 'The runtime load ratio for a bounded recognition system.'
proof idea
This definition is realized directly as the sum of two max expressions that measure the positive deviations of the load ratio from the lower and upper bounds of the critical band.
why it matters in Recognition Science
This definition supplies the penalty term used in the criticalRecognitionLoading_certificate and forcedCriticalRecognitionLoading_certificate theorems, which establish that critical loading implies subcriticality, bounded attention, and sufficient z. It operationalizes the narrow sub-saturation band described in the module documentation and feeds the zero-penalty lemma loadPenalty_zero_of_critical.
scope and limits
- Does not establish that the penalty vanishes inside the critical band.
- Does not constrain the numerical value of rhoMin.
- Does not incorporate the supervisory tick horizon or lcm(8,45).
- Does not prove existence of a controller achieving zero penalty.
formal statement (Lean)
196def loadPenalty (rhoMin area demand : ℝ) : ℝ :=
proof body
Definition body.
197 max 0 (rhoMin - loadRatio area demand) + max 0 (loadRatio area demand - 1)
198