Adaptive networks couple the evolution of node states to the evolution of the interactionsbetween them. In fast-adapting phase oscillator networks, a slow-manifold reduction of a pairwise microscopic model can generate effective higher-order terms in the phase dynamics. Weask whether this higher-order structure survives the dense-graph continuum limit, and whetherit matters if one first reduces and then passes to the continuum, or first passes to the continuumand then reduces. We prove well-posedness and discrete-to-continuum convergence for the unreduced and first-order reduced models, and we construct the continuum slow manifold directlyin a Banach-space setting. Along admissible equal-cell step approximations, the two routes givethe same first-order continuum vector field, including the same pairwise correction and tripletoperator, up to controlled O(ε2) remainders. A continuum mixed-derivative criterion then showsthat, for suitable coupling functions, the resulting triplet operator is genuinely nonpairwise inthe smooth bounded-kernel class. Thus the higher-order term is not a finite-network artefact,but persists in the macroscopic continuum description considered here