  We analytically derive from first physical principles the functional
dependence of wireless channels on the RIS configuration for generic (i.e.,
potentially complex-scattering) RIS-parametrized radio environments. The
wireless channel is a linear input-output relation that depends non-linearly on
the RIS configuration because of two independent mechanisms: i)
proximity-induced mutual coupling between close-by RIS elements; ii)
reverberation-induced long-range coupling between all RIS elements.
Mathematically, this "structural" non-linearity originates from the inversion
of an "interaction" matrix that can be cast as the sum of an infinite Born
series [for i)] or Born-like series [for ii)] whose $K$th term physically
represents paths involving $K$ bounces between the RIS elements [for i)] or
wireless entities [for ii)]. We identify the key physical parameters that
determine whether these series can be truncated after the first and second
term, respectively, as tacitly done in common cascaded models of
RIS-parametrized wireless channels. Numerical results obtained with the
physics-compliant PhysFad model and experimental results obtained with a RIS
prototype in an anechoic (echo-free) chamber and rich-scattering reverberation
chambers corroborate our analysis. Our findings raise doubts about the
reliability of existing performance analysis and channel-estimation protocols
for cases in which cascaded models poorly describe the physical reality.
