Server main memory is increasingly heterogeneous, incorporating technologies such as zswap, CXL memory expanders, and emerging phase-change memories (PCM). To decide where to place pages, operating systems have to choose between efficient but coarse-grained access bits, out-of-band profiling, or costly runtime instrumentation. This paper proposes memory fit, a hardware abstraction that provides precise and accurate page placement information to an operating system. Memory fit measures how well a page "fits" each type of memory in the system. Memory fit separates the mechanism of measuring affinity between pages and memories from the OS policy of choosing which pages to move when. The paper describes soft trace matching (STM), a computationally and space efficient algorithm to compute memory fit. The key technical insight behind STM is that it can estimate memory fit by comparing accesses to a page with a learned model of "good" traces for a memory. We have implemented memory fit and STM in a BOOMv3 RISC-V processor running Linux 6.6. On extended GUPS, memory fit placements recover 2.2-3.8 times as much of the performance gap to optimal placement as Linux TPP. For Redis with equal DRAM and PCM capacities, placements selected by memory fit recover 80% of the performance gap, versus 19% for TPP, in a placement replay. Memory fit requires only 64 bits of state per 4 KiB page and can be updated on every main-memory access with only a 1.2% overhead of OS kernel notification processing.

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