Now that the president has declared Hillary Clinton to be the Democratic presidential nominee, there is little else to do perhaps except wait for the other Democratic hopefuls to acknowledge the fact and deliver their concession speeches. Then the Junior Senator for New York can set about picking her vice presidential running mate before the election begins in earnest.

As disappointing as it may be for the other Democratic candidates, not to mention the rank and file, who may feel cheated out of their rare moment of party democracy, and the MoveOn.org crowd, who are looking for a good ill-tempered scuffle with the party establishment, the president is probably talking good sense.

According to his reading of the situation, “She’s got a national presence and this is becoming a national primary … And therefore the person with the national presence, who has got the ability to raise enough money to sustain an effort in a multiplicity of sites, has got a good chance to be nominated.”

Karl Rove, who knows a thing or two about getting people elected, has come to the same conclusion. On his farewell tour of conservative talk shows, he said, “I think she’s likely to be the nominee.”

What both the president and his Svengali may have added is that the same is true of Mayor Giuliani. He, too, enjoys national prominence and name recognition, and this familiarity has allowed him to edge ahead of his rivals.

It is not true, as Tom Vilsack, the former governor of Iowa, suggested on Sunday, that the rest of the country have yet to discover what New Yorkers already know about the mayor. We live in the age of television, e-mail, and the internet, and you would have to be an Amish hermit to have escaped the fact that Mr. Giuliani has a colorful past.

Like Mrs. Clinton, Mr. Giuliani looks like a winner. And, like Mrs. Clinton, he has wasted little time scrapping with rivals within his own party and every moment fighting the general election as if he were already the candidate.

The MoveOn.org’s Swiftboating of General Petraeus was a case in point. While Mr. Thompson was getting out of his pajamas and Mr. Romney was making an ad attacking his fellow Republicans, Mr. Giuliani was fighting the good fight and embarrassing Mrs. Clinton into putting distance between herself and the left of the Democratic Party.

The rest of the field may take some comfort in Mark Twain’s dictum that “A lie is half way round the world before the truth has its boots on,” but they have their work cut out to catch up with the two New Yorkers.

Which leads to the second point made by the president and Mr. Rove about Mrs. Clinton, that in their judgment she will not win the White House. They base their argument upon the large numbers of negative opinions being expressed against her.

As Mr. Rove explained, “She’s going into the general election with, depending on what poll you look at, in the high forties on the negative side, and just below that on the positive side, and there’s nobody who has ever won the presidency who started out in that kind of position.”

To which this piece of anecdotal evidence may be summoned in her favor: two solid Democratic women, certain voters and energetic fundraisers both, are sat on a sofa before dinner on New York’s Upper East Side last week. First woman: “I can’t stand her. I like Edwards, but I suppose I will have to vote for her.” Second woman: “I agree, but what other choice is there? I like Obama, but I will end up voting for her.”

In polls, both women would count as “negatives” — and both will be voting for her.

Which brings us to Mrs. Clinton’s choice of vice president. Only two of her contenders would be sure to enhance the ticket, but of those Mr. Edwards is busy burning his boats. He hopes that by winning early in Iowa and New Hampshire, even by a whisker, he can leverage his way into the top slot. It is a long shot, but if he fails Mrs. Clinton will owe him nothing.

Unlike the other Democrats in the race, he barely conceals his contempt for what he considers to be a Clinton stitch-up. As Mr. Rove will tell you, Mr. Edwards would have been a formidable Democratic candidate in 2004, but he had to make do with second place.

Mr. Obama would be an asset to the ticket. He suggests freshness and idealism where Mrs. Clinton offers experience and pragmatism. But his election would demand a leap of faith when most of the electorate have their minds firmly on their pocketbooks. And while voters this time might seriously contemplate electing the first woman president, far too many cannot bring themselves even to tell pollsters that they are not yet prepared to vote for an African-American.

Which brings us to Evan Bayh. The Senator for Indiana is a longstanding friend of the Clintons, both husband and wife, and a prominent member of the Clintons’ favorite party vehicle, the Democratic Leadership Council. He voted for the war but against Donald Rumsfeld. He does not favor a precipitate withdrawal from Iraq. He is a winning Democrat in a deep red state. This week he endorsed Mrs. Clinton, suggesting that, if the Democrats are to win, their candidate “must be experienced and seasoned, must be smart and must be tough.” In response, Mrs. Clinton said, looking into his eyes, “The really hard work starts after you win. And that’s when I will need the guidance of someone I admire so much.”

It is still early days, but it would be well worth putting a wager on Mr. Bayh while the odds are still long.