Gibbs Measures on Subshifts of Finite Type: Five Equivalent Characterizations with Explicit Constants

We prove that five characterizations of Gibbs measures for H\"{o}lder potentials on topologically mixing subshifts of finite type are equivalent: the Jacobian condition, the classical cylinder-based Gibbs property, the eigenmeasure of the Ruelle transfer operator, the variational equilibrium state, and the minimizer of the large deviations rate function. The equivalence is established in a single theorem with explicit constants expressed in terms of the H\"{o}lder exponent, the potential norm, the alphabet size, and the mixing time. The proof yields explicit spectral gap estimates for the transfer operator via the Birkhoff cone contraction technique, Lipschitz stability of the Gibbs measure in Wasserstein distance under perturbation of the potential, and statistical limit theorems including a central limit theorem with Berry-Esseen bounds and a large deviations principle. This Part constitutes Part I of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.