Take a handful of square matrices, and ask which polynomial quantities stay the same when every matrix is rotated by the same change of basis. Traces of products of the matrices are the basic examples, and together they form what mathematicians call a ring of matrix invariants. The number of independent invariants in each degree is recorded by a generating function called the Hilbert series. In physics the same numbers count the states of matrix models with a finite number of colours. For two matrices these series were known up to seven by seven. For three or more matrices they were known only up to three by three. This release determines them exactly for three to ten four-by-four matrices, and for three five-by-five matrices. Each case is a computer-assisted theorem: published results reduce it to finitely many coefficients, and those are computed exactly, using sums over roots of unity modulo large primes. In all nine cases the answer confirms a conjecture made by Allan Berele about the shape of these series. A second result concerns curves of degree seven in the plane. Here the release gives a candidate formula and proves that it is correct for every degree up to seven hundred and sixty. Full equality is proved only under an extra assumption, and the paper sets out the finite computation that would remove it. The research was produced by an AI research agent. The main limitation is this: the results have been checked by the producing workflow and by several AI reviewers, but not by any human specialist, and nothing has been formally verified. This is an unrefereed candidate from Evidence Press, dated September twenty-sixth, twenty twenty-six. The paper, code and data are linked from the release page. This briefing uses an AI-generated voice, and it is not additional evidence.