M7: Closed-Loop Adaptive Stability Result

1. Objective

This document establishes the closed-loop stability behavior of ARVIS along the full cycle:

Projection → Dynamics → Gate → Control → Dynamics

It formalizes:

how the system reacts to instability,
how control adapts based on stability signals,
under which conditions the full closed-loop system remains practically stable.

2. Closed-Loop System Definition

We extend the hybrid system from M1 with control and decision:

$$ x_{t+1} = f_{q_t}(x_t, z_t, u_t, w_t) $$

$$ z_{t+1} = z_t + \eta \, g_{q_t}(x_t, z_t, u_t, w_t) $$

with control input:

$$ u_t = C(v_t^{\text{final}}, \kappa^t, W_t) $$

and decision coming from the Gate:

$$ v_t^{\text{gate}} = G(W_t, \Delta W_t, \kappa^t, P_t, V_t, H_t) $$

$$ v_t^{\text{final}} = \min_{\succ} \big(v_t^{\text{gate}}, v_t^{\pi}\big) $$

where:

$v_t^{\pi}$ : projection-control decision (Π_ctrl)

The projection-control layer is defined as:

$$ v_t^{\pi} = \Pi_{\text{ctrl}}(x_t, z_t, H_t) $$

This layer produces a decision-level constraint applied after Gate evaluation.

3. Control Law (from implementation)

The current implementation defines the control action as:

$$ u_t = (\epsilon_t, \text{exploration}_t) $$

with two main modulations:

3.1 Confidence-driven control

$$ \epsilon_t = \epsilon_0 \cdot f(\text{confidence}) $$

3.2 Lyapunov modulation

$$ \epsilon_t \leftarrow \epsilon_t \cdot \phi(\Delta W_t) $$

$$ \text{exploration}_t \leftarrow \text{exploration}_t \cdot \psi(\Delta W_t) $$

where:

$\phi(\Delta W_t) \leq 1$ and decreases when instability increases ($\Delta W_t > 0$),
$\psi(\Delta W_t) \leq 1$ and decreases when instability increases.

Note: exact functional forms of $\phi$ and $\psi$ are implementation-specific.

strong_decrease → epsilon *= 0.8
soft_decrease   → epsilon *= 0.9

4. Closed-Loop Mechanism

The system implements three nested stability feedback loops:

Loop 1: Stability feedback
$W_t \to \Delta W_t \to v_t$
(Gate reacts to instantaneous energy variation)
Loop 2: Adaptive estimation feedback
$W_t \to \kappa^t \to v_t$
(Gate reacts to long-term contraction/adaptation estimate)
Loop 3: Control modulation feedback
$v_t^{\text{final}}, W_t \to u_t \to f_{q_t}(\cdot)$ (Control reduces aggressiveness when Gate signals caution or when energy increases)
Loop 4: Projection-control feedback (NEW)
$(x_t, z_t) \to v_t^{\pi} \to v_t^{\text{final}}$
(structured projection constrains final decision independently of Lyapunov signals)

5. Main Result: Closed-Loop Practical Stability

Result T7: Closed-Loop Practical Stability

Under assumptions:

A1–A15 (M1)
projection validity on $\mathcal{O}_{\text{valid}}$ (M3)
Gate stability preservation (M6: T6, including Π_ctrl constraints)
bounded control modulation ($\epsilon_t, \text{exploration}_t$ remain bounded)

then:

The ARVIS system is practically stable in closed loop.

Formal statement:

There exist constants $C, \beta, \gamma > 0$ and a bounded control-induced perturbation $\epsilon_{\text{ctrl}}$ such that:

$$ W(t) \leq C \, e^{-\beta t} \, W(0) + \gamma \sup_{k \leq t} |w_k|^2 + \epsilon_{\text{ctrl}} $$

where:

$\epsilon_{\text{ctrl}}$ accounts for the residual perturbation introduced by bounded control actions.

6. Stability Mechanism (Key Insight)

The system guarantees the following reactive behaviors:

6.1 Instability reaction

If $\Delta W_t > 0$:

Gate → restricts decisions (CONFIRM or ABSTAIN)
Control → reduces exploration and action scale ($\epsilon_t \downarrow$, exploration$_t \downarrow$)
Π_ctrl → may further restrict decisions based on structured state

6.2 Adaptive reaction

If $\kappa^t \leq 0$ (contraction fails or reverses):

Gate → ABSTAIN or strong CONFIRM
Control → strong reduction of aggressiveness
Π_ctrl → enforces consistency constraints even when Lyapunov signals are borderline

Additionally, if a hard kappa violation is detected:

$$ \kappa\text{-violation} \Rightarrow v_t^{\text{gate}} = \text{ABSTAIN} $$

This enforces a strict safety barrier independent of recovery signals.

6.3 Global protection

If instability accumulates ($\sum \Delta W_t > \text{threshold}$):

Global guard triggers
System evolution is blocked or heavily slowed

6.4 Projection-Control Reaction

The projection-control layer enforces additional constraints:

If projected state indicates high-risk configuration → downgrade decision
If structured inconsistencies are detected → REQUIRE_CONFIRMATION or ABSTAIN

$$ v_t^{\text{final}} = \min_{\succ}(v_t^{\text{gate}}, v_t^{\pi}) $$

Strong invariant

Irreversibility of abstention

$$ v_t^{\text{gate}} = \text{ABSTAIN} \Rightarrow v_t^{\text{final}} = \text{ABSTAIN} $$

7. Critical Property: Self-Stabilization

The system satisfies the negative feedback invariant:

$$ W_t \uparrow \quad \Longrightarrow \quad u_t \downarrow $$

This invariant is explicitly exposed at runtime via:

ctx.extra["closed_loop_feedback"]

which encodes:

energy increase detection
expected control reduction

Interpretation:
Any increase in Lyapunov energy automatically reduces system aggressiveness (self-stabilizing behavior).

Additionally:

$$ (x_t, z_t) \text{ unsafe} \quad \Longrightarrow \quad v_t^{\text{final}} \downarrow $$

The system is not only energy-stabilized, but also structure-stabilized via Π_ctrl.

8. Stability Invariants (Runtime Enforcement)

The current implementation enforces the following invariants:

I1: Bounded control
$0 < \epsilon_t \leq \epsilon_{\max}$
I2: Conservative reaction
If unstable → $u_t$ becomes more conservative
I3: No positive feedback loop
There is no mechanism such that $W_t \uparrow \Rightarrow u_t \uparrow$
→ This is critical for stability
I4: Gate dominance
If $v_t^{\text{final}} = \text{ABSTAIN}$ → no risky evolution is allowed
I5: Kappa Dominance (Hard Constraint)
If contraction fails → system cannot proceed
(enforced after fusion)
I6: Projection monotonicity (Π_ctrl)
Π_ctrl cannot relax a restrictive decision
I7: Abstention irreversibility
$v_t^{\text{gate}} = \text{ABSTAIN} \Rightarrow v_t^{\text{final}} = \text{ABSTAIN}$

9. Final Statement

The ARVIS closed-loop system: Projection → Dynamics → Gate → Control → Dynamics: is practically stable under the validated assumptions and implementation constraints.

The combination of Lyapunov-based Gate decisions, projection-control constraints (Π_ctrl), and adaptive control modulation creates a multi-layer self-stabilizing feedback architecture that:

detects instability early,
restricts decisions conservatively,
reduces control aggressiveness proportionally,
prevents positive feedback loops.

This constitutes the final runtime stability guarantee of the ARVIS cognitive architecture, closing the loop between formal hybrid stability theory and implemented adaptive behavior.

This architecture combines: - energy-based stabilization (Lyapunov) - structural stabilization (Π_ctrl) - adaptive modulation (control layer)