We develop an explicit plane-wave (PW) × photon-Fock approach to the quantized, velocity-gauge Pauli–Fierz Hamiltonian for periodic, first-principles calculations: because the quantized vector potential is spatially uniform, the light–matter coupling reduces to an operator-valued shift of the crystal momentum, K→ K+Â/c, and the existing plane-wave machinery of ordinary solid-state DFT is reused essentially unchanged. This PW×Fock construction puts first-principles cavity QED of periodic solids on the same footing as the Fock-space coupled-cluster and configuration-interaction methods already developed for molecules, opening hitherto inaccessible systems—materials in a linear or chiral cavity, in particular—to first-principles plane-wave calculations. We illustrate the method with monolayer graphene in linear and chiral cavities, obtaining a polarization-selective Haldane gap together with the corresponding density-of-states, real-space, and circular-dichroism optical signatures. A full Brillouin-zone Berry-curvature calculation confirms the topological character of the chiral-cavity gap directly: the occupied manifold carries a quantized Chern number that steps through a non-monotonic sequence, C=1→3→-1→1→2→-1→1, with two narrow, sign-reversed windows—a property of extended, vacuum-dressed matter with no counterpart in a finite molecular system.