Gradient-Stabilized Recursive Filtering (GSRF)

Recursive systems—whether in control loops, optimisation engines, or autonomous agents—exhibit a class of failure modes that conventional safeguards cannot reliably prevent. The core issue is structural: when a system's output feeds back into its own inputs, small errors can compound exponentially.

Runaway gradients occur when optimisation processes update parameters without bounds, leading to values that grow unboundedly large or oscillate violently. In recursive systems, this is not an edge case but a predictable failure mode under specific conditions.

Recursive amplification compounds errors across iterations. A 1% deviation in iteration n can become a 50% deviation by iteration n+10 if the system lacks inherent stability constraints. This is why systems that "usually work" fail catastrophically under pressure.

Delayed failure detection is endemic to recursive architectures. By the time monitoring systems register anomalous behaviour, the underlying state may already be irrecoverable. Probabilistic safeguards, which trigger on statistical anomalies, often activate too late or not at all when the system drifts gradually rather than failing abruptly.

Optimisation loops in safety-critical systems present a specific risk: they are designed to seek extrema, but without deterministic constraints, they cannot distinguish between beneficial optimisation and runaway optimisation toward destructive states. Alignment—ensuring a system pursues intended goals—is necessary but insufficient. A well-aligned system can still exhibit recursive instability if its underlying dynamics are unbounded.

In safety-critical systems, probabilistic safeguards provide statistical guarantees that hold on average. But averages do not prevent the single catastrophic failure that destroys a turbine, crashes an aircraft, or corrupts a financial system. Deterministic constraints that guarantee zero overshoot behaviour, by contrast, provide guarantees that hold in every execution, without exception.

GSRF is a zero-overshoot safety filter for recursive and safety-critical systems. It enforces deterministic constraints before execution, not after failure detection. The framework is designed to maintain stability with zero overshoot under recursion.

Gradient bounding ensures that update magnitudes remain within predefined envelopes. Rather than allowing gradients to grow without limit and then attempting to detect runaway states, GSRF constrains the rate of change structurally. The system cannot produce unbounded updates because the update mechanism itself is bounded.

Recursive depth constraints limit how far feedback can propagate through the system before encountering a stabilising checkpoint. This prevents the exponential error amplification that characterises unconstrained recursion.

Stability enforcement prior to execution means that GSRF validates system state and proposed actions against stability criteria before they are committed. GSRF acts as a deterministic, zero-overshoot safety filter: actions that would violate stability constraints are rejected or modified before they affect system state.

Prevention of uncontrolled system divergence follows from the above. By bounding gradients, limiting recursion depth, and filtering decisions prior to execution, GSRF eliminates the pathways through which recursive systems typically fail.

The framework operates on mathematical guarantees rather than empirical heuristics. The stability bounds are derived analytically and provide deterministic bounds within the defined envelope, bounded-by-construction under stated assumptions. This is what distinguishes a deterministic safety layer from probabilistic monitoring.

An interactive demonstration compares bounded (GSRF) versus unbounded (EMA) filtering on synthetic signals.

Clarity about scope prevents misapplication:

• Not a plug-and-play library. GSRF requires system-specific analysis to configure stability bounds and integration points. There is no generic "install and run" implementation.
• Not a SaaS product. GSRF is a framework deployed through tailored engineering work, not a hosted service.
• Not a general AI alignment solution. GSRF addresses recursive stability, not value alignment, goal specification, or intent interpretation. These are separate problems.
• Not probabilistic or heuristic-based. The guarantees GSRF provides are deterministic. If probabilistic monitoring is sufficient for your application, GSRF may be more rigorous than necessary.

GSRF is designed for systems where failure is not tolerable and where recursive dynamics create instability risk. Zero overshoot behaviour is essential in safety-critical control systems where even momentary excursions beyond acceptable bounds can cause irreversible damage:

• Autonomous control systems — including vehicle control, flight systems, and process control where feedback loops must remain stable under all conditions
• Robotics — particularly systems with learned controllers or adaptive behaviours that could exhibit runaway optimisation
• Optimisation engines — any system that iteratively improves toward an objective and risks divergence if unconstrained
• Reinforcement learning pipelines — where reward-seeking behaviour must be bounded to prevent unsafe policy development
• Industrial automation — control systems governing physical processes where instability has immediate physical consequences
• Financial or operational decision systems — automated trading, resource allocation, or scheduling systems where recursive optimisation must be bounded
• Safety-critical software environments — any context where deterministic guarantees are required by regulation, liability, or operational necessity