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Mr. Gingrich’s presidential campaign operations are nearly identical to those of the nonprofit he left to pursue the bid, American Solutions, which Ms. Burkett, who lives in small-town Georgia, received $320,000 over 18 months for helping run.

The accounting practices that are especially fast and loose for a presidential campaign began with the older group, which raised $50 million between 2007 and 2010, but spent most of it on fundraising expenses. It spent $7 million on payments to travel-specific companies, including $3.2 million for a private jet, in addition to reimbursing staffers $1 million for unspecified travel expenses.

It paid $6 million in salary to about 30 staffers, the core crew of which is now on the campaign or super PAC.

The nonprofit’s finances also showed that the favored Gingrich explanation for payments can mean just about anything: Disbursements described simply as “travel” include $100,000 for a speaker’s fee and $650 at a Washington, D.C., bar.

Not afraid of the FEC

While the advent of super PACs has lifted contribution limits and outside groups have repeatedly pushed the boundaries of the new system, the campaign’s effort to hide money stands out because the argument for relaxed campaign finance laws was always that full disclosure, not dollar caps, was the antidote to corruption.

“Now that those players have gotten what they wanted, they’re unwilling to do the disclosure, and Gingrich is Exhibit A,” said Paul S. Ryan, FEC program director of the Campaign Legal Center.

Despite overseeing unprecedented amounts of money, the FEC’s ability to enforce any of its rules is weak, and the political will to do so is at a low — both of which the Gingrich campaign’s boldness seems predicated on.

The commission is made up of three Republicans and three Democrats, and many of its votes have resulted in ties. Additionally, the three Republicans do not seem to be interested in taking action against candidates from either party, Mr. Ryan said.

“The Gingrich campaign may be banking on the three Republican commissioners being just as hostile to campaign finance disclosure laws as the Gingrich campaign apparently is,” he said.

The FEC staff whose letters have been ignored would need four of the commissioners to vote to approve an investigation, then to accept its findings and, if necessary, a settlement that would resolve the issue in exchange for a fine and the required information. Though very rare in the past, a campaign could refuse and the dispute would go to court, where it could argue that FEC rules, as built up over decades of precedent and guidance, have all along been an incorrect interpretation of federal statute.

Why Mr. Gingrich would desire the distraction of taking a stand to reshape campaign finance law is a mystery — though the willingness to take the risk to hide the source of the money also gives the appearance that the truth could be unsavory.

“In past elections, the last thing a candidate would want to be in the news for is apparent violation of campaign finance laws,” Mr. Ryan said.

Mr. Gingrich, a student of history, may be predicting that noncompliance will simply be ignored.

“The FEC has rendered itself a nonentity,” Mr. Ryan said. “They’re the cops in the doughnut shop.”