A Foundational Framework for Intrinsic Temporal Functionals: Definition, Invariance, and Non-Reducibility in Deterministic Dynamical Systems

This work introduces a foundational framework for the formulation and analysis of intrinsic temporal functionals associated with deterministic dynamical systems. A notion of Internal Time is defined as an additive, monotone, and reparameterization- invariant functional accumulated along system trajectories, independent of external temporal parameters or coordinate choices. Structural properties of such functionals are established, including invariance under time reparameterization and state-space transformations, as well as non-reducibility to external parametric time, relativistic proper time, entropy, or information-theoretic measures. The framework is formu- lated in a minimal and abstract setting, without reliance on physical clocks, stochastic assumptions, or domain-specific models. The results are intended to clarify the con- ceptual status of intrinsic temporal observables and to provide a stable mathematical reference for subsequent investigations in dynamical systems and related areas.