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When President Bush appeared this month before a Catholic men's organization, he basked in the approval of an enthusiastic subset of Catholic voters that he and Sen. John Kerry covet - but which neither can definitively claim.

In an election year in which voters appear closely divided and a remarkable number have already chosen their candidates, polls show there are fewer ''undecideds'' than usual.

But ''look at any good study of undecideds and there's a big hunk of Catholics in that group. They're up for grabs,'' said John Green, a political scientist at the University of Akron in Ohio and an expert on the relationship between religious affiliation and voting.

Moreover, Catholics are thickly settled in key battleground states such as Ohio, Pennsylvania and Wisconsin, where a swing of relatively few popular votes can reap a rich reward of electoral votes.

Both candidates - Bush, a Republican and a Methodist, and Kerry, a Democrat and Roman Catholic - are trying to woo them.

Pollsters diagnosed a powerful dynamic at work in recent national elections: The degree to which a voter is religiously devout is a strong predictor of political preference.

Specifically, people who worship somewhere weekly - where is less important than how often - are much more likely to vote Republican than those who worship less frequently, or whose religious expression is more private, Green and other pollsters say.

There are several sizable exceptions to that rule: African-Americans, Latinos and Jewish voters, religiously observant or not, are solidly Democratic.

But elsewhere the pattern holds.

A recent Time magazine poll found that likely voters who considered themselves ''very religious,'' regardless of faith, favored Bush over Kerry by 59 percent to 35 percent. Those who considered themselves ''not religious'' favored Kerry 69 percent to 22 percent.

Those figures also held in the last presidential election, when so-called ''frequent attenders'' at worship favored Bush over Al Gore by 20 points, according to the Voter News Service.

As Green and Mark Silk put it late last year in Religion in the News, a journal published by Trinity College in Hartford, Conn., ''The Republican alliance of white Protestant and committed other Christians will once again face off against the Democratic coalition of minority faiths and the less committed.''

White, non-Latino Catholics, who make up the majority of the church's 62 million members, follow that pattern.

Regular Mass-goers like those in the Knights of Columbus, which comprised Bush's appreciative audience two weeks ago, are already in his camp.

They are cemented there by Bush's stand on critical cultural issues such as opposition to abortion and same-sex marriage, Green said.

Catholics who attend Mass irregularly or rarely have a preference for Kerry.

''But you've got a lot of Catholics in the middle,'' Green said. ''Independent Catholics are sometimes described as quintessential swing voters. Many feel conflicted. On social issues they're pulled toward Bush. On economic and social welfare issues, they're pulled to Kerry and the Democrats.''

Some measures indicate that Catholics overall are now closely divided between Bush and Kerry. At least one national pollster, John Zogby of Zogby International, gives Kerry a small but significant lead.

Among those still up in the air are many moderate Catholics, perhaps churchgoing, who feel conflicted about both candidates, Green said.

They don't like Kerry on abortion (he says he personally opposes it but supports a woman's right to have one), but they note that his economic plans and anti-death-penalty stance fit better in the Catholic Church's social justice teachings. At the same time, they don't like Bush's economic world view, but they back his position on big cultural issues, like his opposition to abortion and gay marriage.

''I have a lot of friends who are 'social justice' Catholics, and they're really conflicted,'' said Wayne Parent, a political scientist at Louisiana State University.

This was not always so.

Politically, Catholics used to be solidly Democratic. After their arrival from Ireland and other parts of Europe, they helped define the American immigrant experience. Coming of age as blue-collar families in big cities, they adopted traditional Democratic politics and became a pillar of Franklin Roosevelt's New Deal coalition formed in the 1930s and '40s.

Their high point as a cohesive voting group came in 1960, when 80 percent to 85 percent voted for Democrats in the election that made John F. Kennedy the nation's first, and so far only, Roman Catholic president, Green said.

But later years brought prosperity and upward mobility. As Catholics continued to make their way in significant numbers into the professions, corporate life and the suburbs, Republicanism became increasingly attractive, Green said.

Today Catholics overall - including Latinos and black Catholics - remain slightly Democratic. Most recently they voted for President Clinton twice, and then Al Gore in 2000.

But the Catholic monolith shattered long ago.

The ''Catholic vote'' is now so diverse ''I deny the existence of such a thing as the 'Catholic vote,' '' Zogby said.

That said, Zogby, Green and others believe that some Catholics - churchgoing or not - remain in play.