This paper presents a comprehensive communication theoretic model for the physical layer of a cell-free user-centric network, formed by user equipments (UEs), radio units (RUs), and decentralized units (DUs), uniformly spatially distributed over a given coverage area. We consider RUs equipped with multiple antennas, and focus on the regime where the UE, RU, and DU densities are constant and therefore the number of such nodes grows with the coverage area. A system is said scalable if the computing load and information rate at any node in the network converges to a constant as the network size (coverage area) grows to infinity. This imposes that each UE must be processed by a (user-centric) finite-size cluster of RUs, and that such cluster processors are dynamically allocated to the DUs (e.g., as software defined virtual network functions) in order to achieve a balanced computation load. We also assume that the RUs are connected to the DUs through a packet switching network, in order to achieve adaptive routing and load balance. For this model, we define in details the dynamic cluster formation and uplink pilot allocation. As a consequence of the pilot allocation and the scalability constraint, each cluster processor has a partial view of the network channel state information. We define the condition of ``ideal partial CSI'' when the channel vectors that can be estimated are perfectly known (while the ones that cannot be estimated are not know at all). We develop two attractive cluster-based linear receiver schemes for the uplink, and an uplink-downlink duality that allows to reuse such vectors as precoders for the downlink.