Thermodynamics Reconstructed Through the Unified Structural Equation of Entropy

Entropy has historically been treated as a thermodynamic quantity, later reinterpreted in statistical mechanics, information theory, quantum information, and dynamical systems. Despite their diverse origins, these entropy concepts share a common structural essence: the disappearance of bias, the spreading of distributions, and the irreversible loss of distinguishability. Building on this shared structure, the unified structural equation of entropy expresses entropy evolution using four minimal parameters: the initial value, dispersion rate, dispersion resistance, and an externally induced promotion term. This paper reconstructs thermodynamics from this general entropy framework. The second law is reformulated as the special case with no promotion, heat is reinterpreted as the energetic cost of dispersion, free energy is expressed as projected energy under entropic constraints, and non-equilibrium thermodynamics is rewritten in terms of the structural parameters $D$, $\gamma$, and $\alpha$. This reconstruction positions thermodynamics as a special projection of a more general mathematical theory of irreversibility.