In order to win the Republican presidential nomination on the first ballot, Donald Trump has to win approximately 62% 56% [see update] of the remaining delegates. With a few Cruz-friendly states still on the horizon, that means Trump has to win big in New York and in California. So far he’s on track to get a big delegate haul out of his home state, but a new Field poll in California suggests that his prospects for striking gold on the West Coast are slim:

Trump leads Cruz in California 39 percent to 32 percent among likely Republican voters, according to the poll. Kasich is running a distant third.

But in Los Angeles County and areas of inland California, Cruz, the senator from Texas, holds an advantage. The regional differences are significant because California Republicans award nearly all of their 172 delegates by congressional district, three delegates each to the winner of each district. Losing even a handful of districts could hurt Trump’s effort to amass the 1,237 delegates necessary to secure the nomination.

“The result that we got in this poll does not benefit Trump’s chances,” said Mark DiCamillo, director of the poll. “If these regional differences persist, the delegate allocations will be more divided (among candidates).”

Interestingly, Field finds a strong correlation between Trump voters and those who voted for Arnold Schwarzenegger:

Likely Republican voters who say they voted for Schwarzenegger now support Trump over Cruz by a nearly three to one margin, according to the poll. Republicans who voted for Tom McClintock or some other candidate are far more likely to support Cruz. Schwarzenegger himself has endorsed Kasich this year.

The correlation offers both a parallel and a demonstration of a lack of long-term memory among California Republicans. Schwarzenegger launched his gubernatorial bid based largely on his celebrity and was hailed as a savior for the GOP as the state slipped into deep-blue status. He won in the recall election that booted Gray Davis from office, and at first put forward a center-Right agenda. As soon as Schwarzenegger got his political nose bloodied, however, he threw in with Democrats and governed as a center-Left politician.

The poll shows that California may become a real battleground, even if New York does not. Cruz has an 11-point lead in Los Angeles County (40/29), which has 18 of California’s 53 Congressional districts, in whole or in part (via Moe Lane). Cruz also leads in the Central Valley by nine, 42/33, which would cover another eight districts or so. That puts Cruz in reach of 78 delegates of California’s 172 in the winner-take-all-by-CD primary, which would be enough to keep Trump from a first-ballot nomination.

Perhaps a bit more curiously, Cruz — vying to be the first Latino atop a major-party ticket — doesn’t have much of an advantage among his own ethnic demographic. He only leads Trump 35/32, within the margin of error, while John Kasich gets 19%. That could cramp Cruz’ chances in California, the LA Times reports, unless Cruz begins to heavily campaign on that basis in the districts where it will matter:

There are 14 California congressional districts — out of 53 — where at least 20% of registered Republicans are Latinos, according to Paul Mitchell, vice president of Political Data Inc. The average Latino share of the GOP electorate in those districts is 34%.

“Cruz has an advantage in these districts for sure,” Mitchell says.

But Mike Madrid, a Republican consultant and the grandson of Mexican immigrants, isn’t so sure.

Cruz’s position on illegal immigration, after all, isn’t much different than Trump’s. True, unlike Trump, he hasn’t called Mexicans who came here illegally rapists, drug dealers and violent criminals. But, like Trump, he advocates deporting them and building a border wall.

“Cruz has never been a favorite of the Latino community,” Madrid says. “He’s getting support only because they hate Trump so much. Your choice is Trump or Cruz? Good luck….

“Cruz needs to get out of Huntington Beach and start campaigning in Huntington Park. That’s the only way he wins. For the first time in his life, he has to campaign among Latino Republicans and embrace his Latino heritage.”

It wouldn’t be the first time in his life, but it might be the only time in the lives of Latino Republicans in California that their votes matter for the GOP nomination, given the late scheduling of the state’s primary. It’s encouraging to Cruz that Kasich hasn’t gotten much support in this equation, but he’ll need to find a way to inspire Latinos in the LA and Central Valley areas to come out and vote. Cruz needs to demonstrate strength in this demographic or else risk the embarrassment that Hillary Clinton experienced with her performance among women in Wisconsin. If Cruz can do that, he may not win the battle in California, but he’ll gain enough delegates to give himself an opportunity to win the war in Cleveland.

One other key to watch will be the gender gap. Field shows Trump up 17 among men and Cruz up four among women. Cruz managed to neutralize that advantage in Wisconsin and crushed Trump. If he can do the same in California, it’s possible that Cruz can win the battle — and that might be enough to convince the rest of the delegates to throw in with him on an earlier ballot in Cleveland.

Update: Some delegate math from ConservativeLA:

I’ve seen both 56% and 62% bandied about, but CLA makes a good case for the former. I’ve edited the opening paragraph accordingly.