Constraint Embedding Dependence and Branch-Dependent Gauge Structure in Presymplectic Hamiltonian Systems with Application to Vector–Tensor Gravity

We analyze the role of constraint embedding in presymplectic Hamiltonian systems and its impact on the classification of first- and second-class constraints. In vector–tensor gravity with a non-minimal curvature coupling of the form ₄ A_ A^ R, we show that kinetic mixing between metric and vector degrees of freedom induces a deformation of the Dirac constraint algebra. This deformation leads to a branch-dependent gauge structure governed by the degeneracy properties of the kinetic Hessian matrix. Using a 3+1 ADM decomposition and Dirac’s constrained Hamiltonian formalism, we derive the full constraint structure of the theory and identify the conditions under which gauge symmetry is preserved or dynamically broken. Our results demonstrate that gauge structure in such systems is not absolute, but depends on the embedding properties of the constraint surface in phase space, leading to distinct degenerate and non-degenerate dynamical branches.