OK, now this is weird since I found some tips on how to solve this issue, since some people claim it happens when the phone tries to sync with Gmail. Here is my suggestion.
First I followed a suggestion that I should delete all cache and other files from the contact storage app which was a
BAD IDEA since it deleted all my phone contacts (which were backed up, thankfully, and the Google contacts were intact). So don't do this
The weird part is this:
Then I imported all my backed up phone and Gmail contacts into Outlook, edited them all
by removing stuff from the Notes parts (except the information the phone had previously put in, like links for Facebook contacts), wiped all my Gmail contacts, uploaded the new CSV made in Outlook and synced my phone.
I have done the sync at least 15 times since.
Not even one force close of the android.process.acore.
EDIT: it's been a week since this procedure - still no FC
EDIT 2: some 2 weeks now - still no FC
EDIT 3: over three months - no FC of the acore process