A mechanism for the superconducting transition in layered bulk substances, based on the existence of (non-plane wave) projective unitary representations (PURs) of their symmetry group and the generalisation of the theory of Cooper pairs to such representations, is proposed and investigated. In the continuum limit, the symmetry group appropriate to layering is a subgroup of the three-dimensional Euclidean group. It has an infinite family of PURs which can be characterised by an arbitrary real central charge in the commutators of momentum components within the layer. Quantum kinematics/dynamics in a sector with nonzero central charge (nontrivial PUR) is studied and the invariant one-particle Hamiltonian and equation of motion are shown to be formally those of a spinless charged particle in a magnetic field normal to the layers (pseudocharge in a pseudomagnetic field) irrespective of the actual charge or spin of the particle. The dielectric response of a gas of electrons in a sector with nonzero central charge is then worked out by standard Green function methods and the nature of the dynamically screened two-particle interaction determined. The influence of the pseudocyclotron frequency makes the effective potential attractive in a regime of wave number and frequency whose extent increases with increasing central charge, thus signalling instability towards the formation of (generalised) Cooper pairs. The general gap equation is then set up and, after a standard simplification, reduced to a form whose solution is known to lead to a critical temperature increasing monotonically with central charge. This model of high-temperature superconductivity is parity and time-reversal invariant and is in qualitative agreement with observations including those on the isotope effect. The fundamental differences between this model and the anion model are commented upon and speculative remarks offered on the physical origin and detection of the central charge.