The Temporal Non-Constructibility Theorem in Classical Dynamical Systems

This work establishes a rigorous classical theorem concerning the observability of time in closed autonomous dynamical systems. Using only the standard framework of classical mechanics, smooth finite-dimensional state spaces, autonomous vector fields, and algebras of observables, it proves that the external time parameter t that appears in the equations of motion has no invariant empirical meaning for an internal observer.

The analysis formalizes internal observation through restricted algebras of accessible observables and represents empirical evolution as equivalence classes of parametrized curves modulo smooth monotonic reparametrizations. These histories preserve ordering, recurrence, orientation, and all inter-observable correlations while eliminating dependence on the absolute rate at which the trajectory is traversed.

The Temporal Non-Constructibility Theorem states that for any autonomous classical system (M, F), any monotonic reparametrization τ = ψ(t) yields observable histories γ and γ ∘ ψ⁻¹ in the same equivalence class. No internal measurement, nor any ratio of observable rates, can distinguish one parametrization from another. The parameter t therefore functions as a gauge label rather than an observable physical magnitude.

The result applies to conservative Hamiltonian systems, dissipative autonomous systems, and gravitational two-body dynamics. It clarifies the precise sense in which classical evolution is geometric rather than temporal, provides a rigorous operational foundation for time-independent formulations such as Jacobi’s principle, and aligns with relational approaches without invoking spacetime, relativity, statistical mechanics, or quantum theory.

This theorem identifies the boundary between the representational structure of classical mechanics and what an internal observer can empirically access. It shows that, within autonomous classical systems, time as an external parameter is not reconstructible from within the dynamics.

Note. The corresponding relativistic and quantum results are developed in the companion papers Temporal Non-Constructibility in General Relativity (Zenodo, DOI: 10.5281/zenodo.17765323) and Quantum Temporal Non-Constructibility (Zenodo, DOI: 10.5281/zenodo.17772357) respectively.