Partner deals, teams-of-8 cross-imps. I'm afraid I'm going to stick you with our auction; this actually isn't a particularly good hand for our methods. Don't bother telling me how your methods would have done better, I know they would have done, but then you wouldn't have had this problem. The opponents don't bid.
Partner You
1S 2C
2H 3D
3S 4D
4H 4S
5H ?
1S = 5 card majors
2C = forcing to 2S ONLY
2H = natural, 54 or better in the majors, would tend to rebid 2H with 6-4 on anything except a real sub-minimum opener
3D = FSF (2S and 3S would both have been NF)
3S = would like to have a 6th spade, but may be stuck for a bid. 3H would definitely show a 5th heart, so a 5422 with no diamond stop may have to bid 3S (but AKxxx Kxxx xx Kx would probably bid 4C and most 5431s would bid 3NT even without a diamond stop if they didn't have good spades)
4D = slam try in spades, likely to have 3-card support but may be very strong, fairly balanced with a doubleton spade and nothing else to bid e.g. 2335 with poor diamonds. 4NT here would have been natural, invitational so this definitely shows spade support.
4H = cue bid.
4S = nothing else to say
5H = cue bid. Extra values, denies the CK or a singleton diamond (or DA). 5S instead would be a general try, probably focussing on your trump support (it would not be asking for a diamond control).
Well, you are worth a slam. But 6S or 6NT? Partner is most likely to be 6421 or possibly 5422 (but then he will have more high cards and definitely have good spades).
You generally don't play 5NT as choice of slams, though it's a little difficult to think of any other meaning for it. But even if you try 5NT, partner won't necessarily realise you are offering spades and NT, he'll think it's spades or clubs.
I'll even go a step further: I'll tell you partner's hand:
KQ109xx
AK7x
x
xx
Now, which slam would you prefer to play in?

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