  In order to clarify the high-$T_c$ mechanism in inhomogeneous cuprate layer
superconductors, we deduce and find the correlation strength not revealed
before, contributing to the formation of the Cooper pair and the 2-D density of
state, and demonstrate the pairing symmetry in the superconductors still
controversial. To the open questions, the fitting and analysis of the diverging
effective mass with decreasing doping, extracted from the acquired
quantum-oscillation data in underdoped YBCOO$_{6+x}$ superconductors, can
provide solutions. Here, the results of the fitting using the extended
Brinkman-Rice(BR) picture reveal the nodal constant Fermi energy with the
maximum carrier density, a constant Coulomb correlation strength
$k_{BR}$=$U/U_c$>0.90, and a growing Fermi arc from the nodal Fermi point to
the isotropic Fermi surface with an increasing $x$. The growing of the Fermi
arc indicates that a superconducting gap develops with $x$ from the node to the
anti-node. The large $k_{BR}$ results from the $d$-wave MIT for the pseudogap
phase in lightly doped superconductors, which can be direct evidence for
high-$T_c$ superconductivity. The quantum critical point is regarded as the
nodal Fermi point satisfied with the BR picture. The experimentally-measured
mass diverging behavior is an average effect and the true effective mass is
constant. As an application of the nodal constant carrier density, to find a
superconducting node gap, the ARPES data and tunneling data are analyzed. The
superconducting node gap is a precursor of $s$-wave symmetry in underdoped
cuprates. The half-flux quantum, induced by the circulation of $d$-wave
supercurrent and observed by the phase sensitive Josephson-pi junction
experiments, is not shown due to anisotropic or asymmetric effect appearing in
superconductors with trapped flux. The absence of $d$-wave superconducting
pairing symmetry is also revealed.
