Reading further, the 'sysctl' call seems to only be for BSD (https://www.freebsd.org/cgi/man.cgi?sysctl(3)). I could find the man page for sysctl for BSD but not Linux. There is a _sysctl in Linux (http://man7.org/linux/man-pages/man2/sysctl.2.html), but it's use is discouraged and it doesn't provide the necessary information.

Now I suspect that the aforementioned GNU coreutils 'uname' implementation is only for non-Linux systems, as none of the underlying system calls are relevant on Linux. I expect if one compiled that uname on Linux, 'uname -p' would emit 'unknown'.

Meaning I still don't know how to get a 'uname -p' result on Linux (without invoking uname -p).