ARVIS: M5: Adaptive Stability Integration

Objective

This document formalizes how the adaptive stability layer is integrated into the ARVIS runtime system.

It describes: 1. how the adaptive effective contraction estimate is produced, 2. how it is exposed to observability, 3. how it affects gating and control decisions, 4. what claims are justified at the system level, 5. what remains outside the current proof scope.

This document serves as the operational bridge between: - the adaptive mathematical extension of the stability core (M4), - the actual runtime behavior of the ARVIS cognitive pipeline.

1. Position in the ARVIS Formal Stack

The ARVIS documentation stack now includes: - the formal boundary of the modeled system, - the abstract hybrid state structure, - the theorem inventory, - the projection operator and its validation protocol, - the validated projection domain, - the adaptive effective contraction estimator (M4).

This document extends that stack by specifying how adaptive stability becomes an active runtime signal inside the implemented system.

2. From Static Stability to Adaptive Runtime Stability

The original ARVIS core relies on a static quantity:

$$ \kappa_{\mathrm{eff}} = \alpha - \gamma_z \eta L_T $$

and the switching stability condition:

$$ \frac{\log J}{\tau_d} + \log(1 - \kappa_{\mathrm{eff}}) < 0 $$

This forms the theorem-level baseline.

The adaptive runtime layer introduces an observed quantity:

$$ \widehat{\kappa}_{\mathrm{eff}}(t) $$

derived directly from the evolution of the composite Lyapunov energy along actual runtime trajectories.

This creates two distinct but complementary layers:

2.1 Static Stability Layer

Used for: - theorem statements, - parameter admissibility, - baseline small-gain reasoning.

2.2 Adaptive Runtime Stability Layer

Used for: - online observability, - instability detection, - runtime control modulation, - gate-level self-protection.

The adaptive layer does not replace the static layer: it extends it operationally.

3. Adaptive Runtime Estimation Pipeline

The adaptive runtime pipeline is:

$$ o_t \ \rightarrow\ \Pi(o_t)\ \rightarrow\ (x_t, z_t, q_t, w_t)\ \rightarrow\ W_t\ \rightarrow\ \widehat{\kappa}_{\mathrm{eff}}(t) $$

where: 1. $o_t$ is an observable cognitive input, 2. $\Pi$ is the implemented projection operator, 3. $W_t$ is the composite Lyapunov energy, 4. $\widehat{\kappa}_{\mathrm{eff}}(t)$ is the smoothed adaptive contraction estimate.

This pipeline is implemented. The projection layer it feeds carries offline Phase A evidence (M3 appendix, tests/math/); its runtime empirical validation is the M10 protocol, executed on two registered corpora (campaigns 1 and 2, see the M10 report and its section 16 for the honest criterion-by-criterion reading).

4. Adaptive Stability Quantities

The runtime adaptive layer exposes the following key signals:

4.1 Adaptive Effective Contraction

$$ \widehat{\kappa}_{\mathrm{eff}}(t) $$

Interpretation: - Larger positive values → stronger local contraction - Near-zero values → marginal behavior - Non-positive values → local instability risk

4.2 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

4.3 Adaptive Runtime Regime

A categorical label: - stable - marginal - unstable - unavailable

This regime is an operational abstraction, not a formal theorem statement.

4.4 Role in Gate Decision

The adaptive estimate does not directly determine the decision, but contributes to the Gate filtering process through:

fusion signals,
instability detection,
decision overrides.

It acts as a secondary stability signal supporting the Lyapunov core.

5. Runtime Integration Points

Adaptive stability is integrated into the ARVIS runtime at three levels:

5.1 Observability Level

Adaptive quantities are exposed through the runtime observability interface (contraction estimate, switching margin, regime label).
Purpose: inspectability, traceability, and timeline persistence.

5.2 Gate Level

Adaptive stability participates in gate-level decision shaping: - A locally unstable adaptive margin may trigger an abstention-style veto, - A marginal adaptive margin may escalate a permissive decision toward confirmation.

The gate is no longer driven solely by static or fast-state criteria: it now reacts to observed trajectory-level instability.
This is the first level of adaptive self-protection.

5.3 Control Level

Adaptive stability modulates runtime control behavior: - Unstable regime → reduced exploration and increased conservatism, - Unstable regime → lowered effective risk tolerance.

ARVIS now dynamically adjusts its control posture based on observed stability.
This is the first level of adaptive self-regulation.

6. System-Level Interpretation

At the current stage, ARVIS can be described as:

a cognitive runtime system equipped with an explicit Lyapunov-based adaptive stability observer whose outputs actively influence both gating and control decisions.

This is stronger than a purely theoretical stable system, a purely monitored system, or a purely heuristic architecture.
It marks the beginning of a closed-loop cognitive stability architecture.

7. What Is Justified Now

Valid claims at this stage: - ARVIS exposes a runtime adaptive estimate of effective contraction, - ARVIS computes an adaptive switching margin aligned with its static small-gain core, - ARVIS uses adaptive stability information in gate behavior, - ARVIS uses adaptive stability information in control modulation, - ARVIS therefore implements a first closed-loop adaptive stability mechanism.

Claims not yet justified: - Full formal theorem of adaptive global exponential stability, - Proof that the adaptive estimator converges to the true local contraction, - Proof of optimality of the adaptive control modulation, - Proof that adaptive gate/control coupling preserves all desirable task-level behaviors, - Universal claims over arbitrary runtime trajectories outside the validated domain.

8. Relation to the Existing Proof Skeleton

This document extends M2 as follows: - M2 provides the locally valid proof skeleton on the validated projection domain, - M5 adds a runtime adaptive layer operating on top of that domain.

The static proof skeleton remains the mathematical baseline, while the adaptive layer provides online, regime-sensitive enforcement and modulation.

9. Current Proof Boundary

The proof boundary remains limited to: - the validated projection domain, - bounded perturbation assumptions, - current empirical switching behavior, - currently implemented runtime adaptation rules.

The adaptive layer is currently justified as implementation-aligned, empirically exercised, and mathematically motivated, but not yet fully theorem-closed.

10. Runtime Invariants Introduced by Adaptive Integration

The adaptive layer is designed to preserve the following invariants:

I1: No adaptive crash
Adaptive estimation must fail-soft and never break the pipeline.

I2: Bounded adaptive estimate

$$ 0 \leq \widehat{\kappa}{\mathrm{eff}}(t) \leq \kappa{\max} $$

(whenever available)

I3: Conservative instability response
If the adaptive regime is unstable, the system must become more conservative, never less.

I4: Margin-aware gate behavior
Positive adaptive margin must not silently bypass restrictive gating.

I5: Observability consistency
When available, adaptive values must be externally inspectable.

11. Remaining Gaps Before Stronger Claims

Major remaining gaps: 1. Persistence of adaptive stability signals in the timeline, 2. Broader runtime benchmark coverage, 3. Stronger characterization of adaptive regime transitions, 4. Theorem-level adaptive dwell-time analysis, 5. Proof that gate/control coupling preserves practical stability under repeated feedback.

12. Next Recommended Steps

Expose adaptive stability signals in the timeline/trace,
Benchmark adaptive regime transitions on broader scenarios,
Formalize adaptive gate and control invariants,
Extend the proof skeleton toward adaptive practical stability,
Prepare a theorem-grade statement for adaptive switching.

13. Final Statement

ARVIS now includes an adaptive runtime stability layer that: - estimates effective contraction online, - computes a local adaptive switching margin, - exposes stability state to observability, - influences both gate and control behavior.

This represents a meaningful transition from

“a system that has a stability theory”

to

“a system that observes and actively uses its own stability at runtime.”

This is not yet the final theorem of adaptive global stability.
It is the first implementation-aligned closed-loop adaptive stability integration layer.