Sussex took control against Derbyshire at Hove, but a defiant unbeaten fifty from Wayne Madsen halted their progress.
The second division leaders reduced the visitors to 73 for 5 after Sussex had piled up 607 for 8, their highest total against Derbyshire.
Wayne Madsen held them up and was unbeaten on 79 at stumps on day two as he put conditions in perspective, but Derbyshire are still 429 runs behind on 178 for 6.
They were soon in trouble as Indian left-armer Jaydev Unadkat, who is back at Sussex for the run-in, took 3 for 19 in a skilful five-over opening spell.
Opener Harry Came edged Unadkat’s third ball to first slip where Tom Alsop took a good, low catch while Brooke Guest drove lavishly at another out-swinger but straight to backward point.
It was 19 for 3 in the seventh over when Fynn Hudson-Prentice struck with his fifth ball, which straightened enough to pin left-hander Luis Reece, and Derbyshire lost their fourth wicket on 24 when Tom Haines took a fine diving catch low to his left to remove skipper David Lloyd, after Unadkat pushed one across his defences.
Madsen and Aneurin Donald added 49 for the fifth wicket but they were parted when off-spinner Carson, who earlier made a career-best 97, nipped one through Donald’s defensive push with his fifth ball.
Madsen found another ally in Anuj Dal and they put on 84 for the sixth wicket in 24 overs but Carson made another important breakthrough just before stumps when Dal (45) was caught by Alsop at short leg off the back foot.
Sussex skipper Simpson had earlier led the successful charge to secure the maximum five batting bonus points – for getting to 450 inside 110 first innings overs – with an early assault on Derbyshire pacemen Daryn Dupavillon and Zak Chappell.
Simpson had already taken three lovely offside fours off Dupavillon’s opening over of the day when Alsop chipped Chappell’s loosener, at the start of the second over, straight to short mid-wicket after he had added just a single to his overnight 69.
Alsop swished his bat in annoyance at his mistake but Simpson, who had resumed on 25, was in no mood to waste an opportunity to score quick runs.
Two extra cover fours off Chappell were followed by a straight driven four off Dupavillon that took him to a 68-ball fifty and another booming straight four, this time off Chappell, to reach 60.
Hudson-Prentice (8) edged Anuj Dal’s medium pace to keeper Guest, but Carson brought up Sussex’s 450 in the 108th over by slamming left-arm spinner Jack Morley high past mid off for four.
Carson swung Morley over the short legside boundary for the first of his three sixes, the final one over mid-wicket as he went past his previous highest score of 87.
Simpson eventually holed out to long off on 121, after facing 167 balls and hitting a six and 13 fours. He also reached 10,000 first-class career runs in the process.
And Carson’s bid for a maiden first-class hundred ended when he was bowled swinging at a full ball from Dupavillon. His excellent effort came from 125 balls, with six fours besides his trio of sixes, and Sussex’s declaration immediately followed.
Derbyshire batter Aneurin Donald said: “I think it’s pretty obvious that we are far behind in the game and for us it is a matter now of pride and trying to scrap for everything we can get over the last two days.
“We are gutted to be in the position we find ourselves in but we only have ourselves to blame. When we batted today a few of our top order nicked off against an international-class attack early on but Sussex are top of the table for a reason.
“Wayne [Madsen] has shown that you can get runs out there. So that’s what we have to focus on as a team, just try to get as many runs on the board as we can over the weekend and see where that takes us.”
