The document simply states that if you use adaptive mode and if CR3 or paging mode is different between the cores, the cache is not shared, and the cores "compete" for the cache. It does not show how the partition works.
The easiest way to implement this is to statically reserve half the data caching methods for each processor. This essentially assigns half the data cache statically to each processor.
Alternatively, they can add an extra bit to the virtual tag of each cache line to distinguish which processor the line belongs to. This will allow the use of a dynamic cache partition. This is better suited to describe "competing" for the cache than a static partition.
If you really need to know, you can create some micro tests to make sure that these schemes are actually used.
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