arXiv:2604.04235v2 Announce Type: replace Abstract: Safety filters based on control barrier functions (CBFs) and high-order control barrier functions (HOCBFs) are commonly implemented through quadratic programs (QPs), which may become infeasible under conflicting safety and actuator constraints. For linear systems with affine safety functions, we exploit repeated input-space directions to obtain an explicit polyhedral feasibility domain, linear-program certificates for regional feasibility and final-gain selection, and conditions identifying gain-independent obstructions and the minimum common final gain. A compatible QP metric yields an exact saturation-based filter and, together with the regional certificate, persistent feasibility and safety. For a single exterior ellipsoidal CBF with unrestricted inputs, we also derive an exact, gain-independent linear matrix inequality (LMI) test. Double-integrator and aircraft examples illustrate the results.