Embedding calculus for parallelized manifolds

We study a variant of the embedding functor $(M,N)$ that incorporates homotopical data from the frame bundle of the target manifold $N$. Given a parallelized $m$-manifold $M$ and an $n$-manifold $N$ equipped with a section of its $m$-frame bundle, we define a modified embedding functor $\stackrel{~}{}(M,N)$ that interpolates between the standard embedding and a reference framing. Using the manifold calculus of functors, we identify the Taylor tower of $\stackrel{~}{}(M,N)$ with a mapping space of right modules over the Fulton-MacPherson operad. We prove a convergence theorem under a codimension condition, establishing a weak equivalence between $\stackrel{~}{}(M,N)$ and its Taylor approximation. Finally, under rationalization, we describe the derived mapping space in terms of a combinatorial hairy graph complex, enabling computational access to the rational homotopy type of the space of embeddings.