I took a look at the fare rules (for HEL and TMP), and economy fares below K are restricted to BA metal flights. R fares in business (when available) are cheaper than K fares.

Strange that AY does not have the same restrictions. While AY fares do not go quite as low as BA, all AY fares are available on BA code shares. I am surprised that BA can unilaterally restrict the traffic that goes to AY.

So while the lowest fares to HEL are BA fares, the restricted availability of these fares means that the lowest fares remaining are usually fares booked with AY (which can often include travel on BA!).