  We consider a standard one-dimensional Brownian motion on the time interval
$[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$.
We show that the resulting processes can be expressed explicitly in terms of
shifted Legendre polynomials and the original Brownian motion, and we use these
representations to prove that the processes converge weakly as $N\to\infty$ to
the zero process. This gives rise to a polynomial decomposition for Brownian
motion. We further study the fluctuation processes obtained through scaling by
$\sqrt{N}$ and show that they converge in finite dimensional distributions as
$N\to\infty$ to a collection of independent zero-mean Gaussian random variables
whose variances follow a scaled semicircle. The fluctuation result is a
consequence of a limit theorem for Legendre polynomials which quantifies their
completeness and orthogonality property. In the proof of the latter, we
encounter a Catalan triangle.
