ARVIS: M0: System Boundary & Mathematical Scope Definition
Objective
This document defines the exact mathematical boundary of the ARVIS system.
It establishes:
- what parts of ARVIS are formally modeled
- what parts are observable but not yet formalized
- what parts are heuristic / engineering layers
- the mapping between code objects and mathematical objects
This is the root document ensuring that:
No claim exceeds its formal scope.
1. System Overview
ARVIS is modeled as a discrete-time hybrid dynamical system with: - fast state dynamics - slow adaptation dynamics - switching regimes - bounded disturbances - observable outputs
Execution Semantics Clarification
The discrete-time index $t$ represents a logical system transition.
In the implemented system, a transition from $t$ to $t+1$ may require multiple internal runtime execution steps (scheduler ticks).
However: - the mathematical model treats each transition as atomic - stability guarantees apply only to completed transitions
Formally:
$$ t \rightarrow t+1 \quad \Longleftrightarrow \quad \text{pipeline fully executed} $$
Intermediate execution states (partial pipeline execution, scheduling, preemption) are not part of the mathematical model.
2. Mathematical Core (IN SCOPE)
The current formal system is defined by:
State Variables
| Symbol | Description | Type |
|---|---|---|
$x_t$ |
Fast state (risk, stability signals) | $\mathbb{R}^n$ |
$z_t$ |
Slow state (adaptation, memory) | $\mathbb{R}^m$ |
$q_t$ |
Switching mode | finite set $\mathcal{Q}$ |
$w_t$ |
Disturbance / perturbation | bounded |
System Dynamics
$$ x_{t+1} = f_{q_t}(x_t, z_t, w_t) $$
$$ z_{t+1} = z_t + \eta \, g_{q_t}(x_t, z_t, w_t) $$
$$ q_{t+1} = \sigma(q_t, \xi_t) $$
Lyapunov Structure
$$ W_q(x, z) = V_q(x) + \lambda | z - T_q(x) |^2 $$
Stability Condition (T1)
$$ \frac{\log J}{\tau_d} + \log(1 - \kappa_{\mathrm{eff}}) < 0 $$
with:
$$ \kappa_{\mathrm{eff}} = \alpha - \gamma_z \eta L_T $$
Important Scope Clarification
The system defined above represents an abstract mathematical model.
It is not the full cognitive system.
In particular, the model assumes that each transition $(t \to t+1)$ is computed as a fully completed transformation, even if the runtime system executes this transformation incrementally.
In particular: - the projection ( \Pi ) from real inputs to ( (x, z, q, w) ) is external to this model - stability guarantees apply only to the projected dynamics
No guarantee is made on the original input space.
3. Code → Math Mapping
3.1 Core Signals
| Code Object | Math Object | Role | Status |
|---|---|---|---|
RiskSignal |
component of $x_t$ |
fast state | IN MODEL |
| Stability metrics | component of $x_t$ |
fast state | IN MODEL |
SlowState |
$z_t$ |
slow dynamics | IN MODEL |
SwitchingRuntime |
$q_t$ proxy |
switching logic | APPROXIMATION |
| Disturbance inputs | $w_t$ |
perturbations | PARTIAL |
3.2 Derived Components
| Code Object | Math Interpretation | Status |
|---|---|---|
target_map |
$T_q(x)$ |
IN MODEL |
composite_lyapunov |
$W_q(x,z)$ |
IN MODEL |
global_stability_observer |
bound estimator | RUNTIME MONITOR (no formal guarantee) |
switching_params |
$J, \tau_d, \kappa$ |
IN MODEL |
3.3 Cognitive Layer
| Code Object | Math Status |
|---|---|
symbolic_state |
OUT OF MODEL |
| context reasoning | OUT OF MODEL |
| memory semantics | OUT OF MODEL |
| LLM outputs | OUT OF MODEL |
3.4 Decision Layer
| Code Object | Math Status |
|---|---|
| decision gates | HEURISTIC |
| action selection | HEURISTIC |
| policy enforcement | HEURISTIC |
| fallback logic | HEURISTIC |
The decision layer (Gate, Confirmation, Execution) is not part of the dynamical system, but acts as a supervisory filtering layer over admissible transitions.
The runtime scheduler, which orchestrates process execution and resource allocation, is also outside the mathematical model. It ensures that transitions are executed consistently with the atomic step assumption but does not alter the modeled dynamics.
4. Scope Classification
4.1 Mathematically Defined (Abstract Model)
- Fast state dynamics $
x_t$ - Slow state $
z_t$ - Composite Lyapunov $
W$ - Stability condition T1
- Switching bounds (idealized)
4.2 Partially Formalized
- Switching mechanism $
q_t$ - Disturbances $
w_t$ - Runtime stability monitoring
- Mapping from signals → state
4.3 Not Formalized (Yet)
- Cognitive reasoning
- Symbolic representations
- Multi-agent interactions
- Decision policies
- LLM uncertainty
5. Critical Gap (Primary Risk)
The main theoretical gap is the projection:
$$ \Pi : \mathcal{C} \rightarrow (x, z, q, w) $$
This projection is: - not formally defined - not guaranteed to be injective - potentially lossy - not proven Lipschitz globally
This implies: - distinct cognitive states may map to identical projected states - Lyapunov stability may hold in the projected space while failing in the original space
All stability guarantees are conditional on the projection being well-behaved.
This is currently the main limitation of the system.
6. Formal Guarantee Scope
Current guarantees apply ONLY to:
The abstract hybrid system defined in Section 2.
They DO NOT apply to: - full cognitive pipeline - decision making - symbolic reasoning - external model outputs
7. Required Next Steps
To extend guarantees to the full system: 1. Formalize projection operator ( \Pi ) 2. Define bounded cognitive disturbance model 3. Integrate symbolic state into slow dynamics 4. Prove robustness (ISS) w.r.t. perturbations 5. Align runtime signals with theoretical variables
8. Principle of Valid Claims
A claim is valid ONLY if: 1. It corresponds to a defined mathematical object 2. It is covered by a result 3. The code implements that object faithfully 4. The assumptions are satisfied in practice
9. Final Statement
ARVIS currently provides: - a valid stability core for a hybrid system - a partial mapping to a cognitive pipeline - a runtime monitoring layer consistent with theory
It does NOT yet provide: - full cognitive system guarantees - formal verification of decision layers - certified robustness to arbitrary inputs
10. Guiding Rule
Expand the mathematical domain until it matches the real system: not the opposite.