The Yankees pose for a photo after clinching a postseason...

The Yankees pose for a photo after clinching a postseason berth at Yankee Stadium on Tuesday. Credit: Kathleen Malone-Van Dyke

Nobody seemed to understand exactly why the Yankees stood only one victory away from a playoff spot before Tuesday night’s game against the White Sox. The math was complicated enough that MLB had to sort through all the wild-card permutations before letting the team officially know a day earlier, something to do with so many contenders playing head to head during this final week.

This we do know, however. After yet another tumultuous season debating the merits of Brian Cashman’s team-building, Aaron Boone’s managerial ability and the fundamental flaws of this $320 million roster, the defending AL champs now have to be considered as dangerous a deep October threat as anyone still alive in the derby — if not more so.

It was Jose Caballero, the deadline-acquired former Rays antagonist, who officially punched the Yankees’ postseason ticket Tuesday with a two-out, two-strike, nine-pitch walk-off RBI single in the ninth inning that finally beat the White Sox, 3-2, before a crowd of 38,318 in the Bronx.

Before that, the embattled Anthony Volpe — who had ceded playing time this month to Caballero — sparked the rally with a leadoff single and eventually scored the tying run when reliever Brandon Eisert airmailed ball four to Cody Bellinger.

These Yankees, as always, are built on prodigious power, as their MLB-leading 263 homers will attest — 31 more than the next-closest Dodgers before Tuesday. But they dispatched the Sox without going deep Tuesday, and instead showed a resourcefulness that has surfaced more in recent weeks to nail down the franchise’s eighth playoff trip in the past nine seasons.

“It’s been a challenging year, no question about it,” said Boone, who joins the Dodgers’ Dave Roberts as the only two managers to reach the postseason in seven of his first eight years at the helm. “But at my core, as we got here to the final month, I feel like our best baseball was absolutely ahead of us — and hopefully even still is ahead of us.

“I think we’re a really good club. It doesn’t guarantee anything, but I’ll certainly take our chances up against anyone.”

The Blue Jays should be worried, too, because the Yankees could be standing atop the AL East by the time the dust clears Sunday night. Toronto’s lead is now down to one game (but technically two with the tiebreaker edge) after losing Tuesday’s series opener to the Red Sox, who are desperate to stay alive in the wild-card hunt. The feisty Rays are next up at Rogers Centre, while the Yankees have the White Sox and Orioles to finish up. “I want to win every game,” Aaron Judge said of the division chase. “I want it really bad.”

Amid the postgame party, Judge was serenaded with “M-V-P” chants by his teammates as they sprayed him with a mixture of champagne and beer. But the Captain — who was walked three times Tuesday, twice intentionally — leaned on his capable supporting cast for the clinching victory.

Jazz Chisholm Jr., one of only four players to reach the 30-homer, 30-steal threshold this season, went 3-for-3 and scored the Yankees’ first run, thanks to a leadoff walk and stealing second base.

“We’re not here just to get into the wild card,” Chisholm said. “We’re here to win the division and I feel like we’re very close to that.”

It’s been a long climb back from the Yankees’ 20-31 plunge that stretched from June 13 to Aug. 10, a span where they were among MLB’s worst-performing teams. But they’re 27-12 since — the best record in the majors over that period — and now have a real shot at not only the AL East title, but the top seed. Fortunately, Judge’s flexor tendon injury hasn’t stopped him from powering an offense that had the AL’s top OPS (.827) since the Aug. 11 resurgence, along with the most runs (220).

Despite subtracting Juan Soto, last year’s third-place finisher for MVP, the Yankees replaced that plate swagger — and saved $765 million — with the $5 million bargain Trent Grisham (33 homers after nine a year ago), former MVP Cody Bellinger (his 97 RBIs are second only to Judge’s 105) and the revelation Ben Rice (.823 OPS is third). Throw in Chisholm’s 30-30, with Giancarlo Stanton staying healthy enough for 21 homers and a .919 OPS, and the Yankees’ intimidation factor is at a season-high.

The Yankees aren’t just out-slugging opponents, they were suffocating them, too, as the rotation’s 2.81 ERA since Aug. 11 was the best in the majors before Tuesday night, well below the Guardians’ 3.22. It’s been quite a turnaround. On Aug. 10, the third-place Yankees trailed Toronto by as many as 6 1⁄2 games, and only a half-game separated them from the Guardians for the final wild-card spot.

Now, the Yankees have the Blue Jays within reach, after the first taste of what could be many sips of champagne to follow. “The expectation here is to win,” Bellinger said. “And this is the first step. We’re playing really well, and ultimately we know we got a lot more to do. We’re in it, that’s the biggest goal, and we just got to keep going.”

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