  In cuprates, the strong correlations in proximity to the antiferromagnetic
Mott insulating state give rise to an array of unconventional phenomena beyond
high temperature superconductivity. Developing a complete description of the
ground state evolution is crucial to decoding the complex phase diagram. Here
we use the structure of broken translational symmetry, namely $d$-form factor
charge modulations in (Bi,Pb)$_2$(Sr,La)$_2$CuO$_{6+\delta}$, as a probe of the
ground state reorganization that occurs at the transition from truncated Fermi
arcs to a large Fermi surface. We use real space imaging of nanoscale
electronic inhomogeneity as a tool to access a range of dopings within each
sample, and we definitively validate the spectral gap $\Delta$ as a proxy for
local hole doping. From the $\Delta$-dependence of the charge modulation
wavevector, we discover a commensurate to incommensurate transition that is
coincident with the Fermi surface transition from arcs to large hole pocket,
demonstrating the qualitatively distinct nature of the electronic correlations
governing the two sides of this quantum phase transition. Furthermore, the
doping dependence of the incommensurate wavevector on the overdoped side is at
odds with a simple Fermi surface driven instability.
