I encountered a rare situation that really got me thinking about ranges on flops of three cards of one value. I have included a specific hand history below but am interested in the broader point rather than that specific hand

In generic terms, we raise preflop and see a flop heads up IP. The flop is all the same value - e.g. TTT. When checked to what do we do with various parts of our range? The answer will vary slightly based on many dynamics but I couldn't really think of situations where I can have any checkback range against competent opponents, which means I have to bet quads when I have them?

On TTT I would be betting all kinds of missed hands, boats etc so any checkback looks crazy strong right?

I'm not worried that villain folds in the hand below but obviously am never going to build up a big enough sample to test any theory against.

Please tell me if I've taken the concept too far!

***** Hand History for Game 1111111111 ***** (Party)
Tourney Hand NL Texas Hold'em
Table Turbo (Real Money)
Seat 7 is the button
Seat 4: Hero ( 3550 )
Seat 5: Player5 ( 3183 ) - VPIP: 22, PFR: 17, 3B: 11, AF: 1.8, Hands: 127
Seat 9: Player9 ( 4235 ) - VPIP: 54, PFR: 2, 3B: 6, AF: 1.2, Hands: 46
Seat 7: Player7 ( 2532 ) - VPIP: 15, PFR: 12, 3B: 5, AF: 3.4, Hands: 699
Player9 posts small blind [80].
Hero posts big blind [160].
** Dealing down cards **
Dealt to Hero [ As Th ]
Hero posts ante of [20].
Player5 posts ante of [20].
Player9 posts ante of [20].
Player7 posts ante of [20].
Player5 folds
Player7 folds
Player9 calls [80]
Hero raises [340]
Player9 calls [340]
** Dealing Flop ** [ Tc, Ts, Td ]
Player9 checks
Hero bets [335]
Player9 folds
Hero wins 1415 from main pot

I could definitely have a check back range on these flops. I want to check my monsters so I'm also happy checking back hands with outs to a monster (hands like AJ+/KJ+/QJ). It prevents you from getting x/r and can entice a turn stab from villain.

Regardless if you're balancing or not, this is obv an exploitable 54/2 opponent looking to hit a monster and gii. Check back and if he hits a 6 outer you stack him. He has no clue what you're doing anyway, not paying attention to the table at all.

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I'm surprised he's ever folding here, must have like 7 high. I don't really mind betting here expecting him to chase a lot of overs or call down all his ace high, build that pot!

I like to bet here something ridiculous like 5-10% of the pot just to build the pot and occasionnally be check raised vs pocket pairs.