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WASHINGTON — Republicans in and out of Washington savaged President Obama’s proposed budget for the 2013 fiscal year as a broken promise on deficit reduction made worse by a series of gimmicks that overstate what budget savings there are.

The theatrics of budget day are time-honored. The party that does not hold the White House decries “magic asterisks” and “Enron accounting” in the president’s budget, then declares it dead on arrival. But the indignation that greeted Mr. Obama’s budget on Monday seemed to be amplified by the presidential election, debt crises already rocking Europe and the sheer size of the additional debt envisioned by the president’s plan: $6.7 trillion through 2022.

“The president has abandoned his role as leader of the nation,” declared Senator John Barrasso, Republican of Wyoming.

Senator Bob Corker, Republican of Tennessee, accused the president of mocking the American people. Senator Ron Johnson, Republican of Wisconsin, asked why the president “is even seeking a second term” after his “true failure of leadership.”

At the heart of Republican objections is accounting. Mr. Obama boasted of $4 trillion in deficit reduction over the next 10 years in his proposal. Republican budget writers on Capitol Hill saw a fraction of that, as little as $300 billion.

At issue is how the government projects spending and deficits. Of the $4 trillion in deficit reduction noted by the White House, $3 trillion would come from tax increases and spending cuts. Another $900 billion would come from domestic spending caps agreed to with Republicans last year to resolve the impasse over raising the nation’s statutory borrowing limit.

But if Congress and the president did nothing but continue current policies, Senator Jeff Sessions of Alabama, the ranking Republican on the Budget Committee, said the accumulated debt over the next 10 years would be only $300 billion higher than the debt Mr. Obama is projecting if all his policies are adopted.

In large part, that is because automatic cuts to military and nonmilitary programs totaling $1.2 trillion are already set to go in force in 2013. The Obama budget assumes those cuts will not happen. The president also assumes that sharp cuts to reimbursement rates for doctors treating Medicare patients will not ever go into force. Republicans say that effectively negates $522 billion over 10 years.

Republicans also protest that President Obama is “saving” nearly $1 trillion by not spending over the coming decade what the United States has spent each year on wars in Iraq and Afghanistan. True, a Republican administration may have unwound those wars more slowly than the Obama administration has, but Republican budget writers say that no one believes combat costs would have stayed steady over the next 10 years. Therefore, counting savings is misleading, Republicans say, since the savings come from money that would never have been spent.

As Senate Budget Committee Republicans count it, the $1.2 trillion in automatic spending cuts that Mr. Obama wants to negate, the $522 billion in Medicare physician reimbursement costs and the war savings from spending that was never going to happen should be subtracted from the $3 trillion in new deficit reduction that the White House is presenting. That leaves about $300 billion in real deficit reduction over the coming decade.

A senior administration official conceded the Republican point that the $3 trillion in new deficit reduction the budget claims is offset by eliminating the across-the-board cuts scheduled to take place next year, but he dismissed its importance. Whether the White House says it is offering $3 trillion in new savings and $1 trillion in previously agreed-upon savings or $2 trillion in new savings on top of last year’s $2 trillion, the total is the same: $4 trillion in deficit reduction.

The official also said that the budget does assume the cuts to Medicare reimbursement never take place, but that the deficit each year is adjusted upward to reflect that cost.

Democrats also dismissed Republican charges that war savings is budget gimmickry. Even some leading Republicans, like Representative Paul D. Ryan of Wisconsin, chairman of the House Budget Committee, and Senator Jon Kyl of Arizona, have used the same approach, and Republicans on the House and Senate Armed Services Committees are considering canceling automatic military spending cuts by declaring that war savings will more than make up for the lost deficit reduction.

Republican indignation on Monday came in marked contrast to their response to George W. Bush’s budget proposals, which did not count costs for the wars in Iraq or Afghanistan, for the same Medicare physician payment adjustments, or for the annual cost of staving off an expansion of the alternative minimum tax.

If the Obama White House had budgeted for 2013 and beyond the way Mr. Bush had, its deficit forecast for 2022 would have been $167 billion, or 0.7 percent of the economy, one Obama administration official said. Instead, because White House budgeters are adjusting for such costs, the deficit is forecast to be 2.8 percent of the economy that year, $704 billion.

Senator Rob Portman, Republican of Ohio, who was White House budget director for some of those Bush budgets, called the Obama budget “beyond the line.” Asked about the gimmicks the Bush budget office used, Mr. Portman responded, “His are at a new level.”

The one charge the White House has no defense against is that with the new budget, Mr. Obama has broken his 2009 promise to cut the deficit in half in his first term. The deficit that year was a record $1.4 trillion. The deficit in fiscal 2012 will total $1.3 trillion.

The Republican National Committee released an online advertisement to greet Budget Day with images of a toddler at play as her mother frets about the debt she will be saddled with by Mr. Obama’s recklessness.

“President Obama, you broke your promise,” the narrator scolds. “I’ll never forget that.”