Finite observability, conservation laws, and state-dependent effective laws: a Ludwig-style perspective on climate physics

Abstract Many physical theories are formulated in terms of continuous fields, infinite resolution and exact trajectories, while empirical access to complex systems is inherently finite. This tension is particularly acute in climate physics, where multiscale dynamics, effective closures and historically evolving observations constrain what can be operationally defined. Building on Günther Ludwig’s axiomatic programme, we develop a framework in which physical meaning is grounded in finite observation and statistically stable regularities. Climate states are represented as probability distributions over observables constrained by conservation laws, rather than as points in phase space. Within this setting, we argue that the governing laws of climate physics are necessarily state-dependent effective laws. This necessity is understood in an operational sense: under finite resolution and regime-limited empirical access, no state-independent effective closure can be validated across distinct climate regimes, even if the underlying dynamical equations remain universal. Abrupt transitions, long intrinsic timescales and regime dependence are therefore properties of transition statistics rather than deterministic trajectories. By placing climate physics in direct analogy with statistical mechanics, we identify it as part of a broader class of physical theories in which probability, conservation and effective description are foundational.