Why does wc and stat produce different results for /proc/[pid]/cmdline?

I am trying to understand why wc and stat report different things for /proc/[pid]/cmdline. wc says my shell's cmdline file is 6 bytes in size:

$ wc --bytes /proc/$$/cmdline
6 /proc/10425/cmdline

stat says the file is 0 bytes in size:

$ stat --format='%s' /proc/$$/cmdline
0

file agrees with stat:

$ file /proc/$$/cmdline
/proc/10425/cmdline: empty

cat gives this output:

$ cat -vE /proc/$$/cmdline
-bash^@

All of this is on Linux rather than on any other *nix OS. Do the stat and wc programs have a different algorithm for computing the number of bytes in a file?