ARVIS: M1: Result Inventory
Objective
This document defines the complete structure of theoretical results for ARVIS.
It organizes:
- all results
- their dependencies
- proof hierarchy
- scope of each result
This is the blueprint of the formal proof.
1. Structure Overview
The proof is decomposed into: 1. Fast subsystem stability 2. Slow-fast coupling control 3. Switching behavior 4. Global stability under switching 5. Robustness to disturbances (ISS) 6. (Future) adaptive extension
2. Dependency Graph
Results depend on each other as follows:
T1 → T2 → T3 → T4 → T5
where: - T1: Fast stability - T2: Slow tracking - T3: Composite decrease - T4: Switching stability - T5: Global exponential stability
3. Results Definitions
T1: Fast Subsystem Stability
Statement
Under assumptions A1–A6, for each mode $q$:
$$ V_q(x_{t+1}) - V_q(x_t) \leq -\alpha |x_t|^2 + \gamma_w |w_t|^2 $$
Interpretation - Fast dynamics are dissipative - Stability holds in absence of large disturbances
T2: Slow Tracking Bound
Statement
Under A7–A9:
$$ |z_t - T_q(x_t)| \leq C \eta $$
for some constant $C > 0$
Interpretation
- Slow state tracks the target manifold
- Error is proportional to the time-scale parameter $\eta$
T3: Composite Lyapunov Decrease
Statement
Under A1–A9:
$$ W_q(x_{t+1}, z_{t+1}) - W_q(x_t, z_t) \leq -\kappa_{\mathrm{eff}} W_q(x_t, z_t) + \gamma |w_t|^2 $$
with:
$$ \kappa_{\mathrm{eff}} = \alpha - \gamma_z \eta L_T $$
T4: Stability Under Switching
Statement
Under A10–A11:
$$ W_{q_{t+1}}(x_t, z_t) \leq J \cdot W_{q_t}(x_t, z_t) $$
and switching frequency bounded by the dwell-time constraint.
T5: Global Exponential Stability
Statement
Under all assumptions A1–A12, if:
$$ \frac{\log J}{\tau_d} + \log(1 - \kappa_{\mathrm{eff}}) < 0 $$
then:
$$ W(t) \leq C \cdot e^{-\beta t} W(0) $$
for some $C > 0$, $\beta > 0$.
T6: Input-to-State Stability (ISS)
Statement
Under A1–A13:
$$ W(t) \leq C e^{-\beta t} W(0) + \gamma \sup_{k \leq t} |w_k|^2 $$
T7: Observed Stability Consistency
Statement
Under A14:
$$ |\hat{W}(t) - W(t)| \leq \epsilon $$
T8: Projection Consistency (CRITICAL)
Statement
Under A15:
$$ |\Pi(c_1) - \Pi(c_2)| \leq L_\Pi |c_1 - c_2| $$
4. Proof Strategy
Each result will be proven using: - Lyapunov methods - singular perturbation arguments - switching system theory - ISS framework
5. Key Risks
The weakest links are:
1. Projection operator $\Pi$
2. Disturbance modeling
3. Switching approximation
4. Observability error
6. Minimal Valid Core
The minimal provable core is:
T1 + T3 + T4 + T5
Everything else extends it.
7. Extension Plan
Next additions: - Adaptive dwell-time result - Contraction-based stability - Robust switching adaptation
8. Next Step
→ M3_1_cognitive_state_model.md
Where we formalize: - cognitive variables - projection operator - mapping guarantees