Emergent physical time from a cycle count: formulation of mechanics and thermodynamic

This work presents a formulation of mechanics and thermodynamics in which the fundamental evolution parameter is not physical time 𝑡, but a cycle count 𝑁. Time is introduced as a derived quantity through a local frequency 𝑓, via the relation 𝑑𝑡=𝑑𝑁/𝑓, enabling a systematic rewriting of temporal derivatives and physical equations in terms of the cycle parameter. From this postulate, kinematic quantities (velocity and acceleration), dynamical variables (momentum, force, work, and energy), and thermodynamic relations (first and second laws, entropy, and temperature) are reconstructed step by step. A distinction is established between a homogeneous regime with constant frequency, where the formalism is exactly equivalent to standard physics, and a non-homogeneous regime with variable frequency, where additional terms arise and modify both dynamics and thermodynamic evolution. A variational formulation is developed directly in cycle space, defining a frequency-dependent Lagrangian and allowing the application of the Noether's theorem to analyze the relation between symmetries and conserved quantities. It is shown that conservation laws are recovered in the homogeneous case, while in the variable-frequency regime they reflect the non-uniformity of the local evolution rate. The formalism thus provides a coherent mathematical framework in which time can be interpreted as an emergent quantity associated with a local frequency field, maintaining full consistency with classical physics in the appropriate limit and offering a basis for extended interpretations in non-homogeneous regimes.