  The strongly interacting Hubbard model on the square lattice is reduced to
the low energy Plaquette Boson Fermion Model (PBFM). The four bosons (an
antiferromagnon triplet and a d-wave hole pair), and the fermions are defined
by the lowest plaquette eigenstates. We apply the Contractor Renormalization
method of Morningstar and Weinstein to compute the boson effective
interactions. The range-3 truncation error is found to be very small, signaling
short hole-pair and magnon coherence lengths. The pair-hopping and magnon
interactions are comparable, which explains the rapid destruction of
antiferromagnetic order with emergence of superconductivity, and validates a
key assumption of the projected SO(5) theory. A vacuum crossing at larger
doping marks a transition into the overdoped regime. With hole fermions
occupying small Fermi pockets and Andreev coupled to hole pair bosons, the PBFM
yields several testable predictions for photoemmission, tunneling asymmetry and
entropy measurements.
