用李雅普诺夫函数控制神经网络状态，解决低精度硬件中的数值溢出问题。

Low-precision neural networks are attractive for resource-constrained hardware, but fixed-point arithmetic introduces failure modes that are often hidden by idealised quantisation models. In particular, two's-complement overflow wrapping can corrupt hidden activations by changing both their magnitude and sign, leading to unstable numerical error propagation and severe accuracy degradation. This paper proposes a Lyapunov-stabilised quantisation framework for low-precision neural networks operating under hardware-style wrapping arithmetic. The hidden-state energy is monitored through a layerwise Lyapunov function, and a monotone projection is applied to enforce bounded and non-increasing state evolution across depth. The method is evaluated on MNIST using a compact patch-based transformer under post-training quantisation and quantisation-aware training with fixed-point bit-widths from 4 to 16 bits. Monte Carlo results show that unconstrained wrapped quantisation-aware training collapses to near-chance accuracy across 6-16 bits, with activation overflow rates exceeding 11%. In contrast, the proposed monotone Lyapunov projection suppresses activation overflow to below 0.012% and restores stable low-precision learning, achieving 86.55% accuracy at 12 bits. These results demonstrate that Lyapunov-based state control can act as a hardware-aware stabilisation mechanism for reliable fixed-point neural inference and training.