Overnight rain created soft conditions which made for a feast of birdies.
But two-time major victor Jordan Spieth proved a good score was still possible, the 2015 champion carding five birdies and two bogeys in a 69 to finish one over par overall.
Winning the U.S. Open is something he has thought about many times, the 26-year-old from Southport said.
"We're breaking down all kinds of barriers", Harman quipped. "It was downwind to where I knew if I hit it solid, I could definitely get it there. I feel like my game has been good enough to compete in the majors this year", Thomas said.
Below Harman on 12 under, three players are exhausted on 11 under - Thomas, Brooks Koepka and England's Tommy Fleetwood.
Nobody in the top 16 on the leaderboard has won a major.
Also at 11 under was Tommy Fleetwood, who was poised to take the lead until his pitch came up short, his next shot went 60 feet down a slope on the back of the 18th green and he turned potential birdie into bogey and settled for a 68.
The nearest challengers to Harman include a man who made history on Saturday with the lowest score to par in U.S. Open history. Morgan shot 77 and, remarkably, still had the lead at 4-under 212.
Thomas was already on course for a share of the lead as he stepped up to the 18th having carded nine birdies against two bogeys in a spectacular round.
Thomas had 300 yards to the hole for his second shot at the 18th.
"I had no idea (I was close to a record) in terms of the relation to par", Thomas was quoted as saying by BBC Sport. Still, without Miller on the broadcast, we were deprived of his opinions on Thomas's achievement, and forced to wonder just what line of reasoning he might take in an effort to maintain his personal fiefdom, as Miller was famous for dropping in his accomplishment at any possible moment. It's nearly like 'Well, whatever I do for the rest of the year, it's going to be a good year'.
"It was obviously an awesome day", he said on Saturday. "I'm looking forward to my shot at it". It's not going to be an easy day, for sure.
"It's the best I've struck the ball in a long time. I was trying to get it in the best spot to make four". Si Woo Kim is alone at nine under.
The low-scoring sets up a wide open final Sunday, with the world's top three ranked players - No.1 Dustin Johnson, No.2 Rory McIlroy and No.3 Jason Day - all missing the cut on Friday.
Action concludes on Sunday evening, with another first-time major victor likely to be crowned.
Matsuyama, the world number four, had to settle for a one-under-par 71 which left him on six under, six off the pace. In fact you have to look down to 17th place on the leaderboard before you find a major victor in South African Louis Oosthuizen, the 2010 British Open champion.
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