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We gain tight rigorous bounds on the renormalisation fixed point for perioddoubling in families of unimodal maps with degree $4$ critical point. We use acontraction mapping argument to bound essential eigenfunctions and eigenvaluesfor the linearisation of the operator and for the operator controlling thescaling of added noise. Multi-precision arithmetic with rigorous directedrounding is used to bound operations in a space of analytic functions yieldingtight bounds on power series coefficients and universal constants to over $320$significant figures.

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4