InCriticalBand
The critical band predicate asserts that a recognition system operates in the critical loading regime when its load ratio lies strictly above a forced minimum threshold but below saturation. Control theorists working on recognition bandwidth or semantic condensation would cite this predicate when proving stability or zero-penalty lemmas. The definition is supplied directly as the conjunction of the two strict inequalities that bound the regime.
claimThe critical band condition holds precisely when the load ratio satisfies $r < R < 1$, where $r$ is the forced lower edge from phase-transition geometry, $R = R_{dem}/R_{max}$ is the ratio of demanded recognition rate to the maximum bandwidth of the active region.
background
The module treats the load ratio $R = R_{dem}/R_{max}$ as the central control variable for bounded recognition regions. Healthy operation is required to remain inside the narrow sub-saturation interval $r < R < 1$, with the actuator running on the native eight-tick cadence while stability is assessed on the 360-tick supervisory horizon fixed by lcm(8,45). Upstream structures supply the J-cost calibration, ledger factorization, and eight-tick phase definition that fix the underlying recognition costs and periodicity.
proof idea
The definition is introduced directly by stating the conjunction of the lower-bound inequality and the strict upper bound below saturation on the load ratio.
why it matters in Recognition Science
This predicate supplies the core structural condition for the control theorem on recognition bandwidth and semantic condensation. It is invoked by the downstream results that establish implication to subcritical operation, negation of overload, and vanishing load penalty. The predicate closes the gap between the forced lower edge (from phase-transition geometry) and the saturation limit, supporting the eight-tick octave and the overall Recognition Science forcing chain.
scope and limits
- Does not fix a numerical value for the minimum load ratio.
- Does not address dynamics of crossing the band boundaries.
- Does not incorporate entropy or attention constraints from the condensation gate.
- Does not prove non-emptiness of the interval for given area and demand.
formal statement (Lean)
80def InCriticalBand (rhoMin area demand : ℝ) : Prop :=
proof body
Definition body.
81 rhoMin < loadRatio area demand ∧ loadRatio area demand < 1
82
83/-- The forced lower edge of the control band from the phase-transition geometry. -/