Continuous Reformulation of Boltzmann Statistics and Geometric Partition Functions

We introduce in this paper a spectral parametrization of Boltzmann-Gibbs statistical mechanics, in which the canonical weighting is preserved while the energy spectrum is described by a functional law $E_{n}=E_{0} f(n)$. This framework provides a scale-organized representation of canonical ensembles and enables the analysis of thermodynamic properties under nonuniform spectral structures. Rather than modifying the entropy or probability law, the approach presented promotes the spectral organization itself as a geometric degree of freedom. This leads to a unified description of additive, multiplicative, and power-law spectra within the standard canonical formalism.