ARVIS: M4: Adaptive Effective Stability Estimation
Objective
This document defines the first adaptive extension of the ARVIS stability core.
It introduces: 1. an online estimator of the effective contraction coefficient, 2. an adaptive stability condition, 3. a runtime interpretation of local stability margins.
This extension remains consistent with the existing ARVIS core:
$$ \kappa_{\mathrm{eff}} = \alpha - \gamma_z \eta L_T $$
and the switching stability condition:
$$ \frac{\log J}{\tau_d} + \log(1 - \kappa_{\mathrm{eff}}) < 0 $$
while making the theoretical model responsive to observed runtime trajectories.
1. Motivation
The current ARVIS core uses a static effective contraction parameter $\kappa_{\mathrm{eff}}$.
This is mathematically clean but operationally limited because:
- the true local contraction may vary over time,
- switching behavior may evolve across regimes,
- perturbation intensity can change,
- projection quality may fluctuate within the validated domain.
A static $\kappa_{\mathrm{eff}}$ serves as a solid baseline, but is insufficient as the final runtime quantity.
2. Local Observed Decay Model
We model the observed Lyapunov evolution locally as:
$$ W_{t+1} \leq (1 - \kappa_t) W_t + \xi_t $$
where: - $W_t \geq 0$ is the composite Lyapunov energy, - $\kappa_t \in [0,1)$ is the local effective decay coefficient, - $\xi_t \geq 0$ aggregates mismatch, noise, switching jumps, and projection residuals.
Ignoring perturbation for the first estimator yields:
$$ \widehat{\kappa}t^{\text{raw}} = 1 - \frac{W{t+1}}{W_t} \quad \text{for } W_t > \varepsilon $$
3. Clipped Estimator
To avoid pathological values, we define:
$$ \widehat{\kappa}t^{\text{clip}} = \mathrm{clip}\left(1 - \frac{W{t+1}}{W_t}, \kappa_{\min}, \kappa_{\max}\right) $$
with $0 \leq \kappa_{\min} < \kappa_{\max} < 1$.
This clipping prevents division instability, spurious explosive values, and estimates greater than 1.
4. Smoothed Adaptive Estimator
The online estimate is computed recursively:
$$ \widehat{\kappa}_{t+1} = (1 - \rho) \widehat{\kappa}_t + \rho \widehat{\kappa}_t^{\text{clip}} $$
where $\rho \in (0,1]$ is the smoothing factor.
- Small $\rho$ → more conservative and smoother estimate
- Large $\rho$ → faster adaptation to observed changes
5. Adaptive Small-Gain Condition
We define the adaptive switching margin:
$$ S_t = \frac{\log J_t}{\tau_d(t)} + \log(1 - \widehat{\kappa}_{\mathrm{eff}}(t)) $$
Interpretation: - $S_t < 0$ → locally stable regime - $S_t \approx 0$ → marginal regime - $S_t > 0$ → locally unstable regime
This is not yet a full theorem of adaptive global stability. It is a practical runtime criterion directly aligned with the static ARVIS core.
6. Conservative Runtime Principle
The adaptive estimator must satisfy the following safety properties: 1. Always bounded output, 2. Monotone degradation when energy grows, 3. Conservative behavior near $W_t = 0$, 4. Explicit “estimation unavailable” state when data are insufficient.
7. Scope of Claim
At this stage, ARVIS does not claim: - a full theorem of adaptive global exponential stability, - convergence of the estimator itself, - correctness under arbitrary switching.
ARVIS only claims:
the existence of a bounded, observable, and implementation-aligned adaptive estimator of the local effective contraction rate.
8. Next Mathematical Step
The next theorem-level goal is to prove that if: 1. $\widehat{\kappa}_{\mathrm{eff}}(t)$ tracks the true $\kappa_t$ with bounded error, 2. switching remains sufficiently slow (dwell-time compatible), 3. perturbations remain bounded,
then the adaptive version of the ARVIS small-gain condition implies practical exponential stability on the validated domain.