Pointwise convergence of multiple ergodic averages for mixing transformations

Claims Theorem 1.1: for every invertible mixing probability-preserving $T$ (no standardness, no mixing rate), every $n\ge2$ and bounded $f_1,\dots,f_n$, $\frac1N\sum_{k\le N}\prod_j f_j(T^{jk}x)\to\prod_j\int f_j$ almost everywhere along all $N$. Companions treat $n=3,4$ and three distinct nonzero slopes of either sign for mixing $T$. Lemma 2.5 of the principal manuscript also states almost-everywhere convergence of averages of at most three functions at distinct positive slopes on every standard system, derived from the companion's oscillation estimate. It does NOT settle the general question for $n\ge4$ on non-mixing systems, nor polynomial iterates.

Claims Theorem 1.1: for every invertible mixing probability-preserving $T$ (no standardness, no mixing rate), every $n\ge2$ and bounded $f_1,\dots,f_n$, $\frac1N\sum_{k\le N}\prod_j f_j(T^{jk}x)\to\prod_j\int f_j$ almost everywhere along all $N$. Companions treat $n=3,4$ and three distinct nonzero slopes of either sign for mixing $T$. Lemma 2.5 of the principal manuscript also states almost-everywhere convergence of averages of at most three functions at distinct positive slopes on every standard system, derived from the companion's oscillation estimate. It does NOT settle the general question for $n\ge4$ on non-mixing systems, nor polynomial iterates.