This paper proposes a two-sample homogeneity test based on entropic optimal transport (EOT) maps from a common reference distribution -- the uniform law on the unit ball. The test statistic is the squared $L^2$-distance between the two empirical EOT maps. For fixed entropic regularization parameter, we prove that the population map discrepancy is identifiable, derive a functional central limit theorem for the empirical map difference under the null, and establish the Gaussian quadratic-form null limit. We also prove consistency against fixed alternatives and characterize local asymptotic power under contiguous alternatives. A weighted multiplier bootstrap is proposed to calibrate the non-pivotal null distribution, and its validity is established. Extensive simulations demonstrate that the proposed EOT-map test has reliable finite-sample size control and exhibits competitive power compared with other existing methods. The method is particularly powerful for location alternatives and, beyond a single scalar discrepancy, it provides additional diagnostic information on how the two distributions differ. Finally, a real data application concludes the paper.

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