  Establishing a minimal microscopic model for cuprates is a key step towards
the elucidation of a high-$T_c$ mechanism. By a quantitative comparison with a
recent \emph{in situ} angle-resolved photoemission spectroscopy measurement in
doped 1D cuprate chains, our simulation identifies a crucial contribution from
long-range electron-phonon coupling beyond standard Hubbard models. Using
reasonable ranges of coupling strengths and phonon energies, we obtain a strong
attractive interaction between neighboring electrons, whose strength is
comparable to experimental observations. Nonlocal couplings play a significant
role in the mediation of neighboring interactions. Considering the structural
and chemical similarity between 1D and 2D cuprate materials, this minimal model
with long-range electron-phonon coupling will provide important new insights on
cuprate high-$T_C$ superconductivity and related quantum phases.
