I must admit that I missed this one on the rec.games.bridge post as a possible continuation. I must change the hand slightly for a 15-17 NT.
You hold AQx-Ax-Jxxx-KQJx. You open a meaty 1NT. Partner transfer to spades (2H), and RHO doubles this. You complete the transfer (2S), and partner surprises you by bidding 3NT.
It seems that partner has five spades but wasted values in hearts (from his perspective). RHO must hold cruddy hearts (long) and justification. Most likely, this cause was ordained because he holds a red two-suiter, longer hearts, and he was planning a two-bid auction if needed. (Diamonds his likely second suit because my clubs are too good.)
Could we have a slam in this auction? What if partner holds something like K10xxx-KQx-x-Axxx? Twelve is easy that way. Make his hearts KJ10, same result -- slam makes. 6C makes if he actually has Kxx in hearts.
So, how do you move, if you elect to not pass 3NT? It seems that "empathetic splinter" theory works here. However, the parameters are strange.
You assume, from partner's 3NT call, that you must have the heart Ace to invite slam. Thus, the "top control" suit is known. What is not known, then, is which suit will provide the weakness and which will provide the secondary fit. 4C would seem to indicate the secondary fit potential (5 of seven critical cards in spades, clubs, and the top heart).
So, this is a strange new Empathetic Splinter matrix, the third beast.
Type One: The necessary shortness is established, so we jump into the secondary fit suggestion, making the fourth suit the top-control suit. E.g., 1NT-P-2C-P-2H-P-2S-P-4C. Here, hearts must be xxxx because the heart stiff is the known downgradable value. 4C then isolates clubs as the source for slow control values and diamonds as the Ace-only suit.
Type Two: The necessary second suit is known, so we jump in the suit for the empathetic splinter. E.g., partner shows invitational with spades and clubs. 4D, then, might agree spades, clubs as the secondary fit, diamonds isolated as the xxxx suit, and hearts as the Ace-only suit.
Type Three: The necessary Ace-only suit is known, so we jump into the secondary fit potential. The example above.
Notice that Type One and Type Three always involve a jump into the secondary fit, whereas Type Two involves a jump into the hole (where the stiff is needed). You could easily have different defaults as to what you jump into, as long as these are agreed in advance. For example, you could have a scheme where you always jump into the "strongest" of positions between the two unknowns when one is known. This would change your agreement as to Type Two that you jump into the "Ace only" suit, because the "Ace Only" suit is in a sense a "stronger" holding than the "xxxx" suit.
I believe that these are the only three possibilities. My defaults are as above.