  The determination of the coil-globule transition of a polymer is generally
based on the reconstruction of scaling laws, implying the need for samples from
a rather wide range of different polymer lengths $N$. The spectral point of
view developed in this work allows for a very parsimonious description of all
the aspects of the finite-size coil-globule transition on the basis of the
first two Rouse (cosine) modes only, shedding new light on polymer theory.
Capturing the relevant configuration path features, the proposed approach
enables to determine the state of a polymer without the need of any information
about the polymer length or interaction strength. Importantly, we propose an
experimental implementation of our analysis that can be easily performed with
modern fluorescent imaging techniques, and would allow differentiation of coil
or globule conformations by simply recording the positions of three discernible
loci on the polymer.
