[deception5]

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We really want to see a showdown HU here.

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Why? What range are you putting him on?

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These pretty much represent the two thoughts going on in my mind at the time...

My 3-bet is pretty much telling him I have AK/KQ/AsJs/KJ so his cap is either KJ/J8, 2 spades, QT, T9, a set, or an oddly played AK/KQ (both of which I'd discount). Of these hands I'm thinking:

Behind:

KJ (6): Drawing to 3 outs.

J8 (0-9 depending on cc range): Drawing to 5 outs.

2 spades, probably something like AJ/AT/A9/QJ/QT/Q9/JT/J9 with AJ/AT less likely with no 3-bet (probably 8 on average, far more if we consider worse hands): Drawing dead.

T9 (0-16 depending on whether he'd cc T9o - or even T9s for that matter): Dead

Set: Dead, but KK/JJ are unlikely, so probably only 3 combinations to worry about.

Ahead:
QT - no spades (9): 8 outs
QT - one spade (6): 15 outs
KQ (8): 1/4 of the time he has a redraw to the flush.

Tied
AK (6): 1/3 of the time he has a redraw to the nut flush, the rest we split.

So really we're hoping for an overplayed KQ or QT. If we check and he bets, we can rule out QT since there was no purpose in capping the flop if he's not taking the free card (although there is a possibility that QT with one spade would bet the turn when checked to).