Those who use or are familiar with my preference for cuebidding techniques will know that 2NT is often a cuebid that denies good trumps and that 3NT is often a Serious 3NT "cuebid" showing contextually serious values.
Well, what about 4NT as a cuebid?
A recent problem posed to me suggests that 4NT may, in fact, also be a general cuebid. The problem posed: You hold x-AJ10xx-xxx-Qxxx. Partner opens 1H, vulnerable against not vulnerable. Your RHO overcalls 2D. You (not my choice) buid 4H. This is passed to partner, who cuebids 4S. Now what?
Using 4NT by Responder as RKCB makes little sense. Responder knows too little about what Opener has in mind. So, it seems that 4NT should be a cuebid of some variety. But, what does this cuebid show?
It seems that 4NT should be a "convenience bid," showing something of note that is not otherwise capable of description. In some auctions, perhaps 4NT should show a cuebid that is too high to make. Thus, if the trump fit was diamonds, and 4NT was not a viable option, then 4NT might show the missing heart control but inability to make a club cuebid. That seems like a rare auction.
This auction is not quite as rare. We agree hearts, making 4NT not viable as a contract. 4NT is only available as a cue after the 4S cue, which bypassed nothing. 5C was an option, but that would show a club control. 5D would also be an option, but that would seem to deny a club control while showing a diamond control. What, then, should 4NT show?
On the one hand, perhaps 4NT should be "serious," showing two controls. This would make 5C show a club control but deny a diamond control; 5D would show a diamond control and deny a club control. This gains little, as 5D after 5C would be Last Train and could resolve that issue.
Perhaps the better solution is for 4NT to be, in this auction, what 4NT was originally described as showing -- a convenience bid. Sure, 5C and 5D are still available, but Responder cannot show a spade control. In my opinion, then, 4NT should probably show a spade control in this auction.
So, if one partner cuebids 4S after heats is agreed, and if 4NT as the next call cannot logically be ace-asking, then it seems that 4NT should be used to show the unavailable cuebid, most often a spade cue. The inference, then, of a 5C cue would be to show a club control but deny a spade control. Similarly, 5D would seem to show a diamond control and deny any other side-suit control.
However, in one specific auction type, this gets interesting. Suppose Responder is known to be weak, but he may have three controls. E.g., 1H-2D-4H-P-4S-P-?
How to show all three controls that he has?
First, we assume that Responder will not cooperate (usually) without a top heart honor.
4NT shows a spade control. If Opener bids 5C, showing a club control, he infers the need for a diamond control. Responder will be able to cue 5D.
4NT shows a spade control, part II. If Opener bids 5D, showing a diamond control and denying a club control, Responder can accept with a spade control, a heart control, and a club control.
So far, we have covered both hands with both major controls and a minor control.
If Responder has a club control and a heart control, he can bid 5C and accept a 5D Last Train to Clarksville probe. Hence, he can "show" the hand with a heart control and both minor controls.
The sole remaining three-control hand is both minor controls and a spade control. With that hand, Responder can make a somewhat systemic 5D cuebid, showing these three controls (all side controls).