I. INTRODUCTION The microscopic mechanism that generates high temperature superconductivity in the cuprates continues to be controversial. One set of proposals is based on the analogy with heavy fermion metals where a superconducting dome is observed surrounding the quantum critical point (QCP) that arises as antiferromagnetism is suppressed by an external parame- ter such as pressure. 1 In this case the pairing glue arises from the exchange of the soft longitu- dinal antiferromagnetic fluctuations in the vicinity of the QCP. In the cuprates doping plays the role of the external parameter and there are several proposals for the nature of the QCP that appears near optimal doping involving fluctuations in various order parameters e.g. nematic, 2 d-density wave 3 and orbital currents 4 in addition to antiferromagnetism. 5 A sec- ond set goes back to Anderson’s very early proposal that the strong singlet nearest neighbor correlations in the 2-dimensional Heisenberg antiferromagnet generates pairing when doped holes are introduced. The advocates of this resonant valence bond (RVB) mechanism point to the strong asymmetry in the cuprate phase diagram between the physical behavior on the under- and overdoped sides of optimal doping and the QCP. This contrasts strongly with the symmetric dome observed in heavy fermions. Further the highly anomalous physical proper- ties that characterize the pseudogap phase at underdoping are associated with a short range spin liquid in the cleanest cuprate materials, e.g. YBa$_{2}$Cu$_{4}$O$_{8}$ and HgBa$_{2}$CuO$_{4+}$$_{x}$. Nonethe- less strong correlations and the absence of a broken translational symmetry in the pseudogap phase have proved to be formidable obstacles to constructing a comprehensive microscopic RVB theory for underdoped cuprates. For more details see several recent reviews. 6–9 Several years ago we proposed a 2-dimensional array of weak coupled 2-leg Hubbard ladders as an example of a model where occurs a truncation of the full Fermi surface to pockets associated with hole or electron doping in a system without broken symmetry. 10 Subsequently this model led to a phenomenological ansatz for the propagator in underdoped cuprates starting from a renormalized mean field description of an undoped RVB spin liquid insulator. 11 This phenomenological propagator has been recently used successfully to fit a range of experiments covering many anomalous properties of the pseudogap phase. 12–18 In this paper we extend our earlier analysis to the case of an array of lightly doped 4-leg Hubbard ladders with an onsite weak interaction. Our goal is to construct a tractable 2- dimensional model with a partially truncated Fermi surface in which d-wave pairing arises 2