forcedCriticalRecognitionLoading_certificate
plain-language theorem explainer
The theorem shows that any controller state meeting the forced critical recognition loading condition stays subcritical with attention at most phi cubed, z at least phi to the 45, pulse ticks dividing supervisory ticks, and zero load penalty at the critical minimum. Recognition bandwidth control theorists would cite it to confirm stability inside the sub-saturation band on the native 8-tick cadence. The proof is a direct specialization of the general criticalRecognitionLoading_certificate to the fixed rhoCriticalMin case.
Claim. Let $s$ be a controller state satisfying the forced critical recognition loading condition. Then the demanded recognition rate is strictly below the bandwidth of the occupied area, the attention coordinate satisfies $s.attention ≤ ϕ^3$, the signature coordinate satisfies $ϕ^{45} ≤ s.z$, the native pulse period divides the supervisory horizon, and the load penalty evaluated at the critical minimum ratio is zero.
background
ControllerState packages the minimal runtime variables for the governor: area, demand, entropy, attention, signature, and z. The load ratio is defined as demand divided by the bandwidth of the area. IsForcedCriticalRecognitionLoading s holds precisely when IsCriticalRecognitionLoading rhoCriticalMin s, where rhoCriticalMin is the lower edge of the healthy operating band. IsSubcritical area demand asserts demand < bandwidth area. The module works inside the Recognition Science control setting that uses the 8-tick actuation cadence and the 360-tick supervisory horizon obtained from lcm(8,45). The upstream criticalRecognitionLoading_certificate supplies the parameterized version of the same conclusion for arbitrary rhoMin.
proof idea
The proof is a one-line wrapper that applies criticalRecognitionLoading_certificate directly to the hypothesis h. Because IsForcedCriticalRecognitionLoading s unfolds to IsCriticalRecognitionLoading rhoCriticalMin s, the general certificate specializes immediately to the required conjunction of subcriticality, attention bound, z lower bound, divisibility, and zero penalty.
why it matters
This declaration closes the specific forced-critical case inside the Critical Recognition Loading module, supplying the concrete stability certificate needed for the sub-saturation regime sketched in the module documentation. It inherits the 8-tick octave and lcm-derived supervisory horizon from the Recognition Science forcing chain (T7). No downstream uses are recorded yet, so the result functions as a terminal lemma for any later runtime or physics deployment theorem that assumes the forced critical condition.
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