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on Matrix multiplications and AM-GM inequalities (AlphaEvolve problem 47)

The statement

For positive-semidefinite $d \times d$ matrices $A_1, \ldots, A_n$ and any unitarily invariant norm $|||\cdot|||$ (including the operator norm and Schatten $p$ -norms) and $m \leq n$ , define

C(n,m,d) \coloneqq \inf \frac{ \frac{1}{n^m} \sum_{j_1, j_2, \ldots, j_m = 1}^{n} |||A_{j_1}A_{j_2}\ldots A_{j_m}|||}{ \frac{(n-m)!}{n!} \sum_{\substack{j_1, j_2, \ldots, j_m = 1 \\ \text{all distinct}}}^{n} |||A_{j_1}A_{j_2}\ldots A_{j_m}|||}

where the infimum is taken over all matrices $A_1,\dots,A_n$ and invariant norms $|||\cdot|||$ . What is $C(n,m,d)$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.