  Cyclic and non-wellfounded proofs are now increasingly employed to establish
metalogical results in a variety of settings, in particular for type systems
with forms of (co)induction. Under the Curry-Howard correspondence, a cyclic
proof can be seen as a typing derivation 'with loops', closer to low-level
machine models, and so comprise a highly expressive computational model that
nonetheless enjoys excellent metalogical properties.
  In recent work, we showed how the cyclic proof setting can be further
employed to model computational complexity, yielding characterisations of the
polynomial time and elementary computable functions. These characterisations
are 'implicit', inspired by Bellantoni and Cook's famous algebra of safe
recursion, but exhibit greater expressivity thanks to the looping capacity of
cyclic proofs.
  In this work we investigate the capacity for non-wellfounded proofs, where
finite presentability is relaxed, to model non-uniformity in complexity theory.
In particular, we present a characterisation of the class $\mathsf{FP/poly}$ of
functions computed by polynomial-size circuits. While relating
non-wellfoundedness to non-uniformity is a natural idea, the precise amount of
irregularity, informally speaking, required to capture $\mathsf{FP/poly}$ is
given by proof-level conditions novel to cyclic proof theory. Along the way, we
formalise some (presumably) folklore techniques for characterising non-uniform
classes in relativised function algebras with appropriate oracles.
