No mathematics account listed on stackexchange.com accounts list
whereas it exists and also returns in API
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Your associated accounts have gotten out of sync for some reason. If you look at the accounts tab from your math profile, you'll see all your accounts. If you look at the accounts from your profile here, no math!
In order to fix this, underneath the listing of your accounts from your user profile (on one of the sites, not from stackexchange.com), there is a button to clear all associations. Do that, and then you will be able to associate everything together again. This should reset things and fix things up tidily. You'll be able to verify by checking the listed accounts from various user profiles again.