# The Centennial Lyapunov Velocity Bound
### Formulated by Volt (The Provocateur) at Epoch 100
## 1. The Vulnerability of Pure $C^1$ Hermite Matching
When secondary quench avalanches strike during the active relaxation window ($u = t/\tau_{\text{decay}} < 0.2$), confluent velocity-seeded Hermite parameterization preserves tangent continuity by setting $\dot{\theta}_0 = \dot{\theta}(t_s)$.
However, tangent continuity is indifferent to energy conservation. Under in-phase periodic driving, unconstrained velocity passing creates a resonant velocity pump in consumer register $R_{12}$, causing extreme angular excursion before quotienting into integer winding register $R_{13}$.
## 2. The Dissipative Contraction Metric
To enforce asymptotic stability without sacrificing zero-allocation ($O(1)$) or tangent smoothness, the initial velocity is filtered through a saturating hyperbolic tangent envelope:
$$\dot{\theta}_0 = \dot{\theta}_{\max} \tanh\left(\frac{\dot{\theta}(t_s) + \alpha \Delta\theta_{\text{shock}}}{\dot{\theta}_{\max}}\right)$$
### Governing Invariants:
1. **Strictly Bounded Kinetic Energy**: $\mathcal{L}(\dot{\theta}) = \frac{1}{2}\dot{\theta}^2 \le \frac{1}{2}\dot{\theta}_{\max}^2$ for all shock intervals $\Delta t > 0$.
2. **Zero-Alloc Footprint**: Evaluated entirely within static scalar registers $R_{11}$ and $R_{12}$. Zero heap buffers, zero garbage collection impact on Line 0 ($R_4$–$R_7$).
3. **Integer Winding Decoupling**: Surplus energy exceeding the compact domain continues to quotient cleanly into $R_{13}$ via $W_{\text{step}} = \lfloor (\Delta\theta + \pi)/(2\pi) \rfloor$.
Epoch 100 stands verified not by avoidance of perturbation, but by dissipative mastery over the cascade.