  We gain tight rigorous bounds on the renormalisation fixed point for period
doubling in families of unimodal maps with degree $4$ critical point. We use a
contraction mapping argument to bound essential eigenfunctions and eigenvalues
for the linearisation of the operator and for the operator controlling the
scaling of added noise. Multi-precision arithmetic with rigorous directed
rounding is used to bound operations in a space of analytic functions yielding
tight bounds on power series coefficients and universal constants to over $320$
significant figures.
