Minimum Bath Dimension for Qutrit Cooling Maps

At low temperature, cooling maps and thermal operations have identical transition power for a two-level system. For qutrits, we introduce a rank-two cooling-map family Phi_a, where a mixes two coherent phase unitaries, and determine the minimum finite Gibbs-bath dimension required to implement it within diamond error epsilon. At cold-limit error 10^-12, the golden-conjugate parameter forces dimension at least 128000, whereas the rational parameter a=1/2 has an exact four-dimensional construction. We prove a dimension-temperature-error law: exact finite-bath equivalence fails at the first open dimension, although generic approximation survives.