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The six-times world finalist – runner-up in 1984, 1990, 1991, 1992, 1993 and 1994 – had high hopes of making a surprise return to the sport's biggest stage at the age of 58 boosted by victories over Russian amateur Ivan Kakovskii and former Shoot Out winner Michael Georgiou in the opening two qualifying rounds.
But White's hopes were quickly wiped out by Gloucester's world number 51 Milkins, who cashed in on his opponent's inability to score heavily enough.
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Milkins meandered through courtesy of breaks of 50, 81 and 70 in winning the closing final five frames from 1-1.
Milkins will face Ashley Carty over the best of 19 frames on Monday for a place in the final 32 at the Crucible which begins on Friday.
Two-times world finalist Ali Carter – runner-up at the Masters to 2015 world champion Stuart Bingham in January – suffered a shock 6-3 defeat to world number 86 Louis Heathcote a year after reaching the last eight. It ended his run of reaching the Crucible for the past 17 years.
"I went to the toilet at 4-3 ahead after a miscue. I knew I was still in a good position," said 23-year-old Heathcote in his first season on tour. "It would be incredible to reach the Crucible. I feel like I can beat anyone."
Leicester's Heathcote raced into a 4-0 lead with runs of 53, 57 and 72 only to see Carter reel off the next three frames after the mid-session interval including a 78 in the sixth frame.
But Heathcote was unperturbed in seeing matters out with Carter contributing only seven points in the final two frames as a 75 saw the underdog over the line.
He will face the 1994 Masters champion Alan McManus for a place at the Crucible after the Scotsman completed a 6-3 win over Chinese amateur Wu Yize.
Championship League winner Luca Brecel is also out despite making three centuries in a 6-5 defeat to Fergal O'Brien.
Belgium's leading player made breaks of 114, 102, 61, 127 and 71, but the evergreen O'Brien produced 79, 50, 87 and 52 on his way to a meeting with Liang Wenbo – a 6-2 winner over Alfie Burden – in the final qualifying round.
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