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)2CuO6+δ 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 Δ as a proxy for local hole doping. From the Δ dependence of the charge modulation wave vector, 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 wave vector on the overdoped side is at odds with a simple Fermi-surface-driven instability. © 2019 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the »https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.