  We relate duality mappings to the "Babbage equation" F(F(z)) = z, with F a
map linking weak- to strong-coupling theories. Under fairly general conditions
F may only be a specific conformal transformation of the fractional linear
type. This deep general result has enormous practical consequences. For
example, one can establish that weak- and strong- coupling series expansions of
arbitrarily large finite size systems are trivially related, i.e., after
generating one of those series the other is automatically determined through a
set of linear constraints between the series coefficients. This latter relation
partially solve or, equivalently, localize the computational complexity of
evaluating the series expansion to a simple fraction of those coefficients. As
a bonus, those relations also encode non-trivial equalities between different
geometric constructions in general dimensions, and connect derived coefficients
to polytope volumes. We illustrate our findings by examining various models
including, but not limited to, ferromagnetic and spin-glass Ising, and Ising
gauge type theories on hypercubic lattices in 1< D <9 dimensions.
