Stochastic thermodynamics for classical non-Markov jump processes

Stochastic thermodynamics investigates energetic/entropic bounds in small systems, such as biomolecular motors, chemical-reaction networks, and quantum nano-devices. Foundational results, including the second law and thermodynamic uncertainty relations, predominantly rely on the Markov assumption -- neglecting history dependence of physical systems. However, while physicists recognise that the Markov assumption is dubious in real experimental setups, extending stochastic thermodynamics to general non-Markov systems has proven challenging due to their mathematical complexity. Here we establish the general theory of stochastic thermodynamics for arbitrary classical non-Markov jump processes. We introduce a key technique, called the {\it Fourier embedding}, which converts any non-Markov jump process into the Markov field dynamics of auxiliary Fourier modes. This approach yields the necessary and sufficient condition for time-reversal symmetry and enables the derivation of the second law for our non-Markov systems. Our framework accommodates diverse non-Markovian dynamics in realistic experimental setups and offers a guiding principle for physics-informed modelling of history-dependent fluctuations.