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The "dialogue of multipoles" matched asymptotic expansion for small blackholes in the presence of compact dimensions is extended to the Post-Newtonianorder for arbitrary dimensions. Divergences are identified and are regularizedthrough the matching constants, a method valid to all orders and known asHadamard's partie finie. It is closely related to "subtraction ofself-interaction" and shows similarities with the regularization of quantumfield theories. The black hole's mass and tension (and the "black holeArchimedes effect") are obtained explicitly at this order, and a Newtonianderivation for the leading term in the tension is demonstrated. Implicationsfor the phase diagram are analyzed, finding agreement with numerical resultsand extrapolation shows hints for Sorkin's critical dimension - a dimensionwhere the transition turns second order.Comment: 28 pages, 5 figures. v2:published versio

Matched asymptotic expansion for caged black holes: regularization of the post-Newtonian order