ARVIS: M1: Assumptions
Objective
This document defines all assumptions required for the mathematical analysis of the ARVIS abstract system.
These assumptions apply only to the projected dynamical system defined in M0.
They do not necessarily hold for the full cognitive pipeline.
1. General Principle
All guarantees in ARVIS are conditional:
They hold only if the assumptions below are satisfied.
No runtime certification of these assumptions is claimed.
2. Structural Assumptions
A1: Measurability
For all modes $q \in \mathcal{Q}$:
- $
f_q(x, z, w)$ is measurable - $
g_q(x, z, w)$ is measurable
A2: Local Regularity
Functions $f_q$, $g_q$ are locally Lipschitz:
$$ |f_q(x_1, z_1, w_1) - f_q(x_2, z_2, w_2)| \leq L_x |x_1 - x_2| + L_z |z_1 - z_2| + L_w |w_1 - w_2| $$
3. Boundedness Assumptions
A3: Bounded Disturbances
$$ |w_t| \leq W_{\max} $$
A4: Bounded Initial Conditions
$$ |x_0| \leq X_{\max}, \quad |z_0| \leq Z_{\max} $$
4. Lyapunov Assumptions
A5: Positive Definiteness (class-K form)
There exist class-K functions $\alpha_1, \alpha_2$ such that, on the
projected domain:
$$ \alpha_1(|x|) \leq V_q(x) \leq \alpha_2(|x|) $$
Why the class-K form (audit M2, campaign MATH-A M4). The quadratic
form $c_1\|x\|^2 \leq V_q(x) \leq c_2\|x\|^2$ previously stated
here is UNSATISFIABLE by the implemented $V$: the constructive
candidate (arvis/math/lyapunov/lyapunov.py) is a clamped convex
combination of normalized axes, hence of order $\|x\|$ near the
origin, and no finite $c_2$ bounds a linear function by a quadratic
there. The class-K form is the standard weakening that the implemented
$V$ satisfies on the unit hypercube (with $\alpha_i$ linear), and
the Lyapunov results of M2/M6/M7 that only need positive definiteness
and radial monotonicity carry through with it. Results that genuinely
require quadratic bounds (exponential-rate statements) apply to the
quadratic family (arvis/math/lyapunov/quadratic_lyapunov.py,
$V_q(x) = x^\top P_q x$), which satisfies the quadratic form by
construction and remains available as the paper-aligned variant.
A6: Decrease Condition
There exist ( \alpha > 0 ), ( \gamma_w > 0 ) such that:
$$ V_q(x_{t+1}) - V_q(x_t) \leq -\alpha |x_t|^2 + \gamma_w |w_t|^2 $$
A7: Target Map Regularity
$$ |T_q(x_1) - T_q(x_2)| \leq L_T |x_1 - x_2| $$
A8: Slow-Fast Coupling
$$ |g_q(x, z, w)| \leq \gamma_z \big( |x| + |z| + |w| \big) $$
5. Time-Scale Separation
A9: Small Adaptation Rate
$$ 0 < \eta \leq \eta_{\max} $$
with ( \eta_{\max} ) sufficiently small for stability analysis.
6. Switching Assumptions
A10: Average Dwell-Time
$$ N_\sigma(t_0, t) \leq N_0 + \frac{t - t_0}{\tau_d} $$
A11: Mode Compatibility
$$ W_{q'}(x, z) \leq J \cdot W_q(x, z) $$
7. Effective Contraction Condition
A12: Contraction Condition
$$ \kappa_{\mathrm{eff}} = \alpha - \gamma_z \eta L_T $$
The system is contractive only if:
$$ \kappa_{\mathrm{eff}} > 0 $$
This condition is not guaranteed a priori and must be estimated or enforced.
8. Disturbance Compatibility
A13: Bounded Influence
$$ |f_q(x, z, w) - f_q(x, z, 0)| \leq \gamma |w| $$
9. Interface Assumptions (Critical)
A14: State Observability (Approximate)
The system produces observable quantities ( \hat{x}_t ) such that:
$$ |\hat{x}_t - x_t| \leq \epsilon $$
This is an approximation, not a guaranteed observer.
A15: Projection Assumption (Weak Form)
There exists a projection:
$$ \Pi : \mathcal{C} \rightarrow (x, z, q, w) $$
such that: - outputs are bounded - projection is locally regular in nominal conditions
However: - Π is not assumed injective - Π is not guaranteed globally Lipschitz - Π may lose information
All guarantees apply only to the projected system.
10. Hidden Engineering Assumptions
The implementation implicitly assumes: - normalized signals - finite-dimensional representations - bounded numerical errors - stable measurement pipeline
These are not formally proven.
11. Assumption Violations
If any assumption is violated: - theoretical guarantees do not hold - stability claims become invalid - runtime behavior may diverge from model
12. Summary
All ARVIS guarantees are: - conditional - model-level (projected system) - dependent on A1–A15
13. Key Limitation
The main limitation is:
the lack of formal characterization of the projection ( \Pi )