Skip to content

ARVIS: Formal Theory ↔ System Correspondence

Research-grade Mapping Document: Implementation-Aligned (M0–M12)

0. Purpose

This document establishes a formal correspondence between:

  • the theoretical framework introduced in ARVIS: Two-Timescale Hybrid Cognitive Control with Lyapunov Stability Guarantees,
  • and the current implementation of the ARVIS Cognitive Operating System.

It serves three critical roles:

  • Traceability: mapping each mathematical construct to runtime artifacts
  • Auditability: clarifying which theoretical assumptions are enforced, approximated, or unverified
  • Structural clarity: separating the proven theoretical core, the implementation-aligned extensions, and the decision-theoretic algebra layer

1. Formal Model vs Runtime Representation

1.1 Fast State (Contracting Subsystem)

Theory
$x_t \in \mathbb{R}^n$
Fast contracting dynamics governed by $f_q$.

Implementation
ctx.scientific.lyapunov.cur_lyap, ctx.scientific.lyapunov.prev_lyap

Interpretation
The implementation does not represent $x_t$ explicitly. Instead, it operates on a compressed Lyapunov-relevant projection of the state.
Thus, the system is energy-driven rather than state-driven. Stability is monitored directly in Lyapunov space.

→ This constitutes a model reduction: $x_t \mapsto W_t$

1.2 Slow State (Adaptive Subsystem)

Theory
$z_t \in \mathbb{R}^m$ with slow adaptation:
$z_{t+1} = (1-\eta)z_t + \eta T(x_t)$

Implementation
ctx.scientific.lyapunov.slow_state, ctx.scientific.lyapunov.symbolic_state

Extension
The implementation introduces a symbolic / cognitive state layer that is: - non-metric, - not part of the original Lyapunov formulation, - interacting with decision and control.

Thus:
$z_t^{\text{impl}} = (z_t^{\text{metric}}, z_t^{\text{symbolic}})$

1.3 Switching System

Theory
$q_t \in Q$ (discrete switching mode)

Implementation
ctx.scientific.switching (switching_runtime, switching_params, switching_metrics)

Interpretation
Switching is implemented as a constraint-evaluated runtime process including dwell-time estimation, switching frequency monitoring, and stability condition evaluation.

Switching directly influences decision filtering (Gate). It is therefore both a dynamical component and a decision constraint source.

1.4 Disturbance Model

Theory
$w_t$ bounded (scalar disturbance)

Implementation
collapse_risk, uncertainty, conflict_pressure

Interpretation
Disturbance is lifted to a multi-dimensional cognitive disturbance space:
$w_t \in \mathbb{R}^k$ with $k > 1$.
This strictly generalizes the scalar disturbance model in the theory.

2. Lyapunov Layer (Core Equivalence)

2.1 Composite Lyapunov Function

Theory
$W_q(x,z) = V_q(x) + \lambda |z - T(x)|^2$

Implementation
CompositeLyapunov.W, CompositeLyapunov.delta_W

Status
Implemented in CompositeLyapunov, with one deliberate divergence on the decisive path: the gate consumes the FAST energy V alone on complete quadruples (the measured full-W variant relaxed verdicts and broke the registered ISS bound; kept as the v0 design by decision DM-C1, M10 section 10.5, full wiring re-posed at DM4). W and delta_W are computed and published as observability.

2.2 Lyapunov Increment

Theory
$\Delta W_t = W_{t+1} - W_t$

Implementation
ctx.scientific.composite.delta_w

Extension
Computed under partial observability and fallback-compatible. Introduces robust observability without requiring full state access.

2.3 Temporal Extension (Implementation only)

ctx.scientific.composite.delta_w_history
Used for: - empirical stability monitoring (M10, six campaigns executed; sections 10-16) - global stability guards

3. Switching Stability (Result ↔ Runtime)

3.1 Theoretical Condition

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

3.2 Runtime Evaluation

switching_lhs, kappa_eff, ctx.switching_metrics

3.3 Interpretation

The system performs online verification of the theoretical inequality.
Unlike the pure theoretical model, this condition is not only observed but actively enforced via decision filtering (Gate operator).

4. Projection Layer (Runtime Certification: M3.3)

4.1 Theoretical Projection

$\Pi : \mathcal{C} \to (x_t, z_t, q_t, w_t)$

4.2 Implemented Projection

$\Pi_{\text{cert}} : \mathcal{O}_{\text{runtime}} \to P_t$
with $P_t = (\text{domain_valid}, m_t, \text{is_projection_safe}, \text{level})$

4.3 Interpretation

The projection layer currently acts as a runtime certification operator, not a full mathematical projection. It ensures bounded domain, deterministic validation, and compatibility with Gate enforcement.

5. Decision System (Algebraic Layer: M11–M12)

5.1 Decision Space

$$ V = {\text{ALLOW}, \text{REQUIRE_CONFIRMATION}, \text{ABSTAIN}} $$

with the order:

$$ \text{ALLOW} \preceq \text{REQUIRE_CONFIRMATION} \preceq \text{ABSTAIN} $$

5.2 Algebraic Structure

$(V, \wedge)$ where $v_1 \wedge v_2 := \min_{\succ}(v_1, v_2)$ forms a totally ordered meet-semilattice.

Since $V$ is finite and totally ordered, it also forms a complete lattice.

5.3 Decision Operators

  • Gate: $G : \mathcal{S} \times \mathcal{P} \to V$
  • Projection-control: $\Pi_{\text{ctrl}} : \mathcal{S} \to V$

5.4 Final Decision Rule

$$ v_t = G(s_t, P_t) \wedge \Pi_{\text{ctrl}}(s_t) $$

5.5 Interpretation

The decision layer is a lattice-based composition whose monotonicity and non-relaxation are pinned by tests (the trace grammar and the sanctioned-relaxation allowlist, campaign GATE-SEM). The inputs feeding that lattice remain partly heuristic (regime estimation, signal coercions); the lattice constrains how verdicts compose, not where the signals come from.

6. Dual Stability Structure

ARVIS implements two orthogonal stability layers:

6.1 Energy Stability (Lyapunov)

Driven by $W_t$, $\Delta W_t$, $\widehat{\kappa}_t$ and enforced by the Gate operator $G$.

6.2 Structural Stability (Projection)

Driven by projection signals and enforced by $\Pi_{\text{ctrl}}$.

6.3 Composition

$$ v_t = v_t^{\text{gate}} \wedge v_t^\pi $$

ensures that no structural violation can be bypassed by energy signals.

7. Control Layer (Closed-Loop: M7)

Implementation
- adaptive epsilon
- exploration modulation

Key invariant
$W_t \uparrow \implies u_t \downarrow$

8. Runtime Enforcement Layer

Unlike the theoretical model, ARVIS actively enforces constraints at runtime through: - Gate filtering (M6) - Projection certification (M3.3) - Projection-control filtering (M11) - Algebraic fusion (M12)

9. Robustness & Fault Tolerance

The implementation includes: - fallback execution - safe defaults - degraded modes

Result: graceful degradation instead of instability.

10. Validity Domain

All guarantees apply only within the valid operating region $\mathcal{O}_{\text{valid}}$, where: - projection is valid, - assumptions A1–A15 hold, - perturbations are bounded.

11. Theoretical Coverage vs System Scope

Covered with stated limits (see M13 and VERSIONING for the axes not evaluated) - Lyapunov monitoring on the projected fast energy (the composite full-W wiring is deferred, DM-C1/DM4) - Hybrid switching: measured and disclosed always (switching_safe_measured); enforced in the enforce posture; the default posture is monitor-only, and ALLOW requires the live MEASURED adaptive margin instead (DM-G1/DM-G2bis) - Practical stability (M8): theory-level; the noise and mode axes are not evaluated at runtime

Partially Formalized - Decision algebra (M12) - Projection-control (M11)

Not Fully Proven - Confidence modeling - Full cognitive disturbance model - Adaptive control optimality

12. Research Implications

ARVIS is a strict extension of the theoretical model.

  • The theory defines → a stable dynamical core
  • The system adds → a decision-theoretic safety architecture

13. Roadmap Toward Full Formalization

  • Formalize $\Pi_{\text{ctrl}}$ as an operator with guarantees
  • Extend ISS theory to the cognitive disturbance space
  • Formalize the decision algebra in control-theoretic terms
  • Prove stability under adaptive control laws

14. Conclusion

The ARVIS implementation: - preserves the theoretical stability core - ✔ introduces a decision lattice enforcing safety constraints - ✔ integrates energy and structural stability - carries runtime empirical validation (M10, six executed campaigns on two published corpora; the projection layer additionally holds offline Phase A evidence per the M3 appendix)

Final Insight

ARVIS is not only a Lyapunov-stable system, but a lattice-controlled hybrid cognitive system where: - stability is measured (Lyapunov), - safety is enforced (Gate + $\Pi_{\text{ctrl}}$), - decisions are structured algebraically (M12).