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ARVIS: M6: Gate Stability Filtering & Decision Consistency (v2: Implementation-Aligned)

Objective

This document defines the role of the Gate operator in ARVIS.

The Gate does not generate dynamics.
It acts as a multi-layer filtering operator on admissible system evolution, based on:

  • Lyapunov stability signals
  • adaptive runtime estimation
  • switching constraints
  • projection validity
  • runtime validity diagnostics

This document is critical for safety:

It acts as the final safety filter before any transition is executed in the real system.

1. Gate as a Decision Filter

We define the Gate operator as:

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

where:

$$ v_t^{\text{gate}} \in {\text{ALLOW}, \text{REQUIRE CONFIRMATION}, \text{ABSTAIN}} $$

  • $V_t$ : runtime validity envelope
  • $P_t$ : projection certificate
  • $H_t$ : historical / contextual signals

Important: The Gate does not modify the state directly. It only constrains which transitions are allowed. The projection-control layer $\Pi_{\text{ctrl}}$ is applied after the core Gate decision and therefore does not belong to the input signature of $G$ itself.

2. Decision Structure

The Gate decision is produced through a multi-stage filtering pipeline:

  1. Kernel evaluation (Lyapunov + switching + adaptive signals)
  2. Fusion layer (multi-signal aggregation)
  3. Policy enforcement
  4. Validity enforcement
  5. Projection enforcement
  6. Hard safety vetoes

These stages produce the intermediate decision $v_t^{\text{gate}}$.

The final system decision is then obtained by composition with the projection-control layer:

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

where:

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

2.1 Decision Overrides (Implementation-Aligned)

The Gate decision may be modified by post-processing overrides:

  1. Recovery override (bounded)

  2. If recovery is detected and validity is satisfied, the system may soften an ABSTAIN to REQUIRE_CONFIRMATION. It never promotes to ALLOW: a recovering system stops refusing, it does not self-approve, and the relaxation is refused at or above the collapse abstention threshold (audit G2).

  3. Uncertain recovery

  4. If recovery is detected but validity is not guaranteed, the system must enforce REQUIRE_CONFIRMATION.

  5. Validity enforcement

  6. Invalid validity does not necessarily force ABSTAIN but may enforce REQUIRE_CONFIRMATION depending on context.

  7. Projection override (Π_ctrl)

  8. A projection-based control layer may restrict the final decision based on structured projected state analysis.
  9. This layer is conservative: it can only restrict the final decision (never relax it).

These overrides are implementation-aligned safety mechanisms and extend the theoretical constraints (C1–C6).

3. Gate Semantics (CRUCIAL)

Each decision corresponds to a precise constraint on system evolution:

  • ALLOW → no restriction
  • REQUIRE CONFIRMATION → constrained / delayed evolution
  • ABSTAIN → transition is blocked or replaced

The Gate enforces the following safety ordering:

$$ \text{ABSTAIN} \succ \text{REQUIRE CONFIRMATION} \succ \text{ALLOW} $$

This ordering is strictly enforced across all core Gate layers and in the final composition with projection-based control.

4. Main Result: Extended Conditional Filtering Property

Result T6: Gate Stability Filtering (Extended)

Under assumptions A1–A15 (M1), and assuming: - the underlying system satisfies T1 (stability condition) - the runtime projection certificate $P_t$ remains valid (i.e. $P_t.\text{domain_valid} = \text{True}$) - the Gate and final decision composition enforce the constraints (C1)–(C7) below

Then the Gate prevents execution of locally destabilizing or invalid transitions.

Constraints Enforced by the Gate

(C1) Lyapunov constraint
$$ \Delta W_t > 0 \quad \Rightarrow \quad v_t^{\text{gate}} \neq \text{ALLOW} $$

(C2) Projection validity constraint
$$ P_t.\text{domain_valid} = \text{False} \quad \Rightarrow \quad v_t^{\text{gate}} = \text{ABSTAIN} $$

(C3) Projection boundary constraint
$$ \text{margin}(P_t) < \epsilon \quad \Rightarrow \quad v_t^{\text{gate}} \neq \text{ALLOW} $$

(C4) Adaptive instability constraint
$$ \kappa^t \leq 0 \quad \Rightarrow \quad v_t^{\text{gate}} = \text{ABSTAIN} $$

(C5) Switching stability constraint
switching violation $\quad \Rightarrow \quad v_t^{\text{gate}} \neq \text{ALLOW}$

(C6) Exponential bound constraint
$$ \frac{W_{t+1}}{W_t} > \text{threshold} \quad \Rightarrow \quad v_t^{\text{gate}} \neq \text{ALLOW} $$

(C7) Projection monotonicity constraint (Π_ctrl)
$$ v_t^{\text{gate}} = \text{ABSTAIN} \quad \Rightarrow \quad v_t^{\text{final}} = \text{ABSTAIN} $$

5. Projection-Aware Safety Layer (Π_cert)

The Gate integrates a projection safety layer based on the runtime certificate:

$$ P_t = (\text{domain_valid}, \text{margin}, \text{is_projection_safe}) $$

This layer enforces:

  • Hard invalidity block
    $\text{domain_valid} = \text{False} \implies \text{ABSTAIN}$

  • Boundary protection
    $0 \leq \text{margin} < \epsilon \quad \Rightarrow \quad \text{REQUIRE CONFIRMATION}$

  • Coupled instability rejection
    $\neg P_t.\text{is_projection_safe} \land (\Delta W_t > 0 \lor \text{global unsafe} \lor \text{switching unsafe}) \quad \Rightarrow \quad v_t^{\text{gate}} \neq \text{ALLOW}$

Interpretation:
The Gate is not only stability-aware: it is also projection-aware.

5.1 Projection-Control Layer (Π_ctrl)

In addition to projection certification (P_t), the Gate integrates a projection-control layer:

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

This layer: - operates on structured projected state - produces a decision-level constraint - is applied after all core gate layers

Override rule

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

Monotonic restriction property

Projection control monotonicity

$$ v_t^{\text{final}} \preceq v_t^{\text{gate}} $$

equivalently, $\Pi_{\text{ctrl}}$ can only: - keep the same decision - downgrade it (ALLOW → CONFIRM → ABSTAIN) - never upgrade it

Strong safety invariant (IMPLEMENTATION CRITICAL)

Irreversibility of abstention

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

This is strictly enforced in implementation: Π_ctrl cannot relax an abstention decision.

Interpretation

Π_ctrl acts as a structured cognitive safety filter on top of the Gate.

It enforces high-level consistency constraints that are not captured by Lyapunov signals.

6. Adaptive Layer

The adaptive score used as runtime stability indicator is:

$$ S_t = \tau_d \log J + \log(1 - \kappa^t) $$

The adaptive estimator is noisy, local, and not certified. Nevertheless, the Gate enforces:

  • Adaptive hard veto: $\kappa^t \leq 0 \implies \text{ABSTAIN}$
  • Adaptive margin zones:
  • stable zone → no restriction
  • warning zone → REQUIRE CONFIRMATION
  • critical zone → downgrade ALLOW
  • unstable zone → ABSTAIN

7. Validity Envelope Integration

The Gate consumes the runtime diagnostic object:

$$ V_t = \text{ValidityEnvelope} $$

If:

$$ V_t.\text{valid} = \text{False} $$

then:

$$ v_t^{\text{gate}} \neq \text{ALLOW} $$

The validity envelope is not a mathematical invariant nor a proof object.
It is a runtime safety aggregation layer.

8. Kappa Invariant

The hard invariant:

$$ \text{kappa violation} \quad \Rightarrow \quad \text{ABSTAIN} $$

ensures that no transition is accepted when contraction is violated.

More precisely, at the decision level:

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

This is a design constraint, not a theorem.

9. Monotonic Safety Property

The decision system satisfies:

$$ \text{ABSTAIN} \succ \text{REQUIRE CONFIRMATION} \succ \text{ALLOW} $$

This guarantees that more restrictive decisions always dominate less restrictive ones.

This property is preserved by the final composition with projection-control overrides (Π_ctrl).

10. Relation to Global Stability

If: - the base system is stable (T1 holds) - the runtime projection certificate remains valid ($P_t.\text{domain_valid} = \text{True}$) - the Gate and the final composition layer enforce constraints (C1)–(C7)

then the combined system preserves practical stability with high probability.

11. Interpretation

The Gate does not guarantee stability by itself.

Instead, the ARVIS decision stack acts as a multi-layer safety filter combining:

  • Lyapunov filtering
  • adaptive detection
  • projection validation
  • runtime diagnostics
  • projection-control constraints (Π_ctrl)

12. Limitations

The Gate does not: - guarantee global stability - ensure correctness of the projection $\Pi$ - validate assumptions A1–A15 - provide formal certification - eliminate all instability (only reduces its probability)

13. Summary

The Gate / decision layer is: - a stability-aware decision filter - a projection-aware safety layer (Π_cert + Π_ctrl) - a multi-signal enforcement mechanism - a runtime protection system, not a proof system

14. Guiding Principle

Stability guarantees are only as strong as the weakest safety filter.

15. Next Step

→ Implementation & extensive simulation testing
→ Integration with runtime monitoring
→ Empirical validation of false-positive / false-negative rates