While you can play with cookies and incognito mode, the most definitive check is to check a third party site like ITA Matrix where there's no way of tracking you. Fares are not generated on the spot, there are fare filings and they are widely distributed through GDS systems along with fare bucket inventory. Fares have an 8-character fare basis code which uniquely identifies which fare you are actually booking. This gives you more info than simply looking at the booking class as multiple fares can share the same booking class. You can find the fare basis codes for the fares you are being quoted on united.com at the "View fare rules and restrictions" link when pricing an itin. ITA Matrix also shows fare basis codes when pricing itins which is why it is particularly useful to match against what you are seeing from united.com. Connecting itins can either be covered by a single fare, or there can be multiple fares involved for each segment (also known as broken fares) which will be indicated by multiple fare basis codes in a one-way direction.

If you want to definitively check whether or not there is fare bucket inventory for a particular flight, having an ExpertFlyer account is probably the way to go (or you can just post on FT and ask someone with an EF account to check inventory for you).