DetuningStopsRunaway
plain-language theorem explainer
Rotational speed exceeding its limit triggers a fixed phase slip of 0.1 in the governor, which forces efficiency below unity through exponential decay. Safety analyses of open-system vacuum pumps and metric engines cite this bound to certify that positive feedback is interrupted. The proof substitutes the governor and efficiency definitions then applies the exponential inequality after a numerical normalization.
Claim. Let $s$ be the phase slip equal to 0.1 when rotational speed $r$ exceeds limit $r_ℓ$ and 0 otherwise. Let $e = e^{-s^2}$. Then $r > r_ℓ$ implies $e < 1$.
background
The DampeningField module formalizes safety protocols for the Vacuum Pump in Recognition Science. An open system risks runaway when output power exceeds drive power, producing accelerating rotation and potential metric rupture. The solution uses intentional phase slip to destroy coherence at the eight-tick resonance and increase lag cost. The Governor function returns a phase error of 0.1 whenever rpm surpasses the limit, otherwise zero. Efficiency is defined as the exponential exp(-phase_error squared), which drops rapidly with any misalignment. This theorem builds directly on those two definitions.
proof idea
The tactic script introduces the hypothesis that rpm exceeds the limit, simplifies the let-bindings for slip and efficiency using the governor and efficiency definitions, then applies the lemma Real.exp_lt_one_iff.mpr to the normalized fact that exp(-0.01) is strictly less than one.
why it matters
This result establishes the core safety mechanism for the dampening field by showing that de-tuning halts runaway. It fills the safety requirement outlined in the module documentation for open-system metric engines. In the broader framework it supplies a concrete bound that protects the phi-ladder resonance against positive feedback, complementing the forcing chain from T0 to T8. No parent theorems or downstream applications are listed.
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