CLEVELAND — The Twins’ realistic hopes of repeating as American League Central division champions vanished weeks ago. On Thursday, it became official.

The Twins were mathematically eliminated from winning the division with 3-2, 10-inning loss to the Cleveland Guardians on Thursday afternoon at Progressive Field. Their only hopes of making it to the postseason rest on them qualifying as Wild Card team, and the loss tied them with the Detroit Tigers for the final playoff spot.

The Twins dropped three of four games in Cleveland, and all of the losses were late heartbreakers. After being walked off on Wednesday night, Twins relievers bent but did not break during the middle of Thursday’s game, stranding two runners every inning between the fifth and seventh. Louie Varland and Griffin Jax then got through the eighth and ninth innings quickly.

But the Guardians broke through in the 10th, walking the Twins off on an Andrés Giménez single that brought home automatic runner José Ramírez.

That came after the Twins were unable to do anything with their opportunity in the top of the inning despite having the automatic runner on board, plus Matt Wallner, who was hit by a pitch, and Manuel Margot, who drew a walk. With just one out and the bases loaded, Twins stars Byron Buxton and Carlos Correa were unable to convert those base runners to runs.

The Twins were outhit 11-3, with Guardians pitchers retiring 14 hitters in order from the fifth through ninth innings. The Twins scored their only runs in the fifth after Kyle Farmer led off the frame with a double. Willi Castro later walked, and both came home to score on the same play, a Margot single was bobbled by center fielder Lane Thomas.

Margot’s single gave the Twins the lead, but it was their last hit of the day. They had been trailing since the first inning, when starter Simeon Woods Richardson surrendered a solo home run to designated hitter Kyle Manzardo.

Woods Richardson maneuvered through 4⅔ innings and did not allow another run, making way for newly acquired Cole Irvin in the fifth.

The Guardians tied it off Irvin in the sixth on a Brayan Rocchio sacrifice fly after the Guardians had loaded the bases on a single just past the reach of second baseman Kyle Farmer, an infield hit to shortstop Castro and a bloop single.

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