Deterministic Spiral-Time Governance for Safe Execution in the Presence of Hallucinations in LLM-Controlled Climbing and Walking Robots

Large Language Models (LLMs) are increasingly integrated into robotic autonomy stacks for semantic planning and high-level decision support. In climbing and walking robots, long-horizon deployment is constrained by hallucination drift, retrieval instability, and non-deterministic reasoning variance, which can propagate into physically unsafe locomotion behavior. Weintroduce a deterministic external governance layer based on a Spiral-Time operator formalism. Interaction history is embedded into a triadic state ψ(t) = t+iϕ(t)+jχ(t), where ϕ(t) encodes contextual coherence and χ(t) = ∂tϕ(t) captures temporal torsion associated with abrupt divergence. A scalar instability functional ∆Φ(t) deterministically regulates memory writes, retrieval widening, verification mode, and safe fallback switching. Crucially, we define hallucination operationally in robotics as falsifiable inconsistency between LLM-issued claims/commands and a ground-truth oracle derived from simulator state. We provide (i) a formal supervisor model, (ii) a deterministic gating algorithm, and (iii) a discrete Lyapunov-based boundedness/ISS argument for the governor state under bounded measurement noise. A reproducible simulation protocol is specified for climbing and walking tasks with controlled perturbations and statistical evaluation (seeds, confidence intervals, ablation tests). Importantly, the proposed framework does not modify the underlying LLM or its claim-generation mechanism. Instead, it operates at the execution interface, selectively f iltering and constraining the propagation of generated claims into physical actions. In addition to the synthetic stochastic evaluation, we provide a minimal physicsgrounded transfer validation using a MuJoCo-based quadruped environment. All governor parameters are kept identical to the synthetic setting without retuning. The embodied validation reproduces the qualitative reduction in unsafe actions and deterministic mode switching behavior under contact-rich dynamics, while leaving the underlying hallucination statistics unchanged. The framework is model-agnostic and does not modify LLM weights, offering auditability and predictable behavior suitable for safety-sensitive legged robot deployments.