  We study polynomial summation over unit circles over finite fields of odd
characteristic, obtaining a purely algebraic integration theory without
recourse to infinite procedures. There are nonetheless strong parallels to
classical integration theory over a circle, and we show that the super Catalan
numbers and closely related rational numbers lie at the heart of both theories.
This gives a uniform analytic meaning to these up to now somewhat mysterious
numbers.
  Our derivation utilises the three-fold symmetry of chromogeometry between
Euclidean and relativistic geometries, and we find that the Fourier summation
formulas we derive in these two different settings are closely connected.
