Transversal unfoldings of derived foliations

Abstract: We develop a CE-presented, homotopically invariant formalism for transversal unfoldings of relative derived foliations. An unfolding is regarded as an integrable transverse deformation: the parameter directions are lifted to flat infinitesimal symmetries of the relative derived foliation. The infinitesimal object controlling such liftings, which we call the transverse controller, is the homotopy quotient of basic weight-zero graded-mixed derivations of the derived de Rham algebra by tangent inner derivations. In Chevalley-Eilenberg models this is represented by basic first-order Lie derivations modulo inner derivations, and in the non-effective case by a crossed module retaining central isotropy. For strict affine CE models we compute the derived space of transversal unfoldings. In the effective case its connected components are classified by flat splittings of the transverse controller, and the basic part reconstructs the full absolute algebroid without auxiliary generation hypotheses. We then pass to the intrinsic graded-mixed controller, formulate presentation independence through derived splitting spaces, and give a conditional global descent theorem under explicit CE-presentability and controller-descent hypotheses. The construction also identifies the Atiyah-Kodaira-Spencer class, the curvature obstruction, the Maurer-Cartan deformation complex, split normal forms, the representable leaf-space interpretation, and Gauss-Manin transport for coherently Cartan-linearised crystals. The examples recover the classical, Lie-algebroid, logarithmic, shifted Poisson, and representable leaf-space cases.