ESPPU INPUT: C${}^3$ within the "Linear Collider Vision"

The Linear Collider Vision calls for a Linear Collider Facility with a physics reach from a Higgs Factory to the TeV-scale with $e^{+}{e}^{-}$ collisions. One of the technologies under consideration for the accelerator is a cold-copper distributed-coupling linac capable of achieving high gradient. This technology is being pursued by the C${}^3$ collaboration to understand its applicability to future colliders and broader scientific applications. In this input we share the baseline parameters for a C${}^3$ Higgs-factory and the energy reach of up to 3 TeV in the 33 km tunnel foreseen under the Linear Collider Vision. Recent results, near-term plans and future R&D needs are highlighted.