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Mr. Hammond explained that saying “no further itemization is required” meant that the campaign was not afraid of the FEC.

“You’re basing this off the FEC would prefer that we comply with their rules. But their rules do not have the backing of law,” Mr. Hammond told The Times.

Mr. Gingrich’s campaign has paid or promised $500,000 in mystery money to 45 employees and the candidate for unspecified reimbursements. It has reported $210,000 more in checks to fundraising consultants and others that mix unspecified reimbursements with compensation, making it impossible to know how much a person was paid.

The campaign committee spent $800,000 directly with travel companies, including for the use of a private jet, so major expenses such as airfare aren’t included in the money paid to people supposedly to reimburse for travel. Hotel expenses are notably largely absent, meaning they likely are included.

Unaccounted-for cash

The biggest recipients of unaccounted-for cash are Mr. Gingrich at $220,000; campaign director Michael Krull at $55,000 to $70,000; and Mr. Hammond at $48,000. Cushman Enterprises, the firm of Mr. Gingrich’s daughter, Jackie Gingrich Cushman, was paid more than $40,000 for “fundraising consulting/travel” and “travel,” on top of $20,000 in straight income.

All of those purported costs are in addition to other major expenses that also were paid by staffers and reimbursed, but for which the campaign did list the underlying receipts.

By comparison, the Mitt Romney campaign — which is four times as large — reimbursed 120 employees $130,000 in unspecified expenses. Mr. Romney’s brother, Scott, by far is the largest recipient at $12,000.

In a sense, Mr. Gingrich’s accounting is a curious inversion of what has transpired with finances connected to Mr. Romney, his chief rival. Business associates who want to give to a super PAC supporting Mr. Romney have formed shell corporations so donor disclosures show only a cryptically named company, without saying where that company got its money.

On the Gingrich side, the black box is on the side of money going out. But the Romney super PAC is not officially connected to the candidate — and Gingrich staffers function as middlemen only if, of course, the money is actually being paid to others. Without the actual payments being included in disclosures, campaign lawyers noted, it is impossible to know if any number of campaign finance laws are being broken.

Reimbursements aside, donated dollars also have gone to Mr. Gingrich and associates for goods provided — but The Times found chronic accounting problems and questionable purposes there, too. The campaign paid $70,000 to Gingrich Productions, Mr. Gingrich’s wife’s company, for Web hosting — but it also paid $60,000 to Rackspace, a major Web-hosting company, over the same time period.

Websites typically have only one host. By comparison, Mr. Romney’s campaign paid between $26,000 and $57,000 to Amazon, another major Web host, records indicate.

Both campaigns had additional expenses for design and other online-related services, but hosting charges are usually separate. While the Gingrich campaign also purchased the Newt.org domain from his wife’s company, the hosting charges did not represent that, a separate $8,400 deal.

Additionally, Mr. Gingrich personally sold to his campaign a mailing list of names of supporters for nearly $50,000 — but a mandated disclosure of significant personal assets months before didn’t include any such list among Mr. Gingrich’s valuable possessions. The watchdog group Citizens for Ethics in Washington filed a complaint alleging the mailing list must have belonged to one of Mr. Gingrich’s business groups and he was selling something that didn’t belong to him to convert campaign donations into personal cash.

The campaign never reported paying the money to Mr. Gingrich in its expenditure reports to the FEC, letting word slip in a newspaper interview instead. A spokesman later acknowledged that the campaign failed to list the payment in its third-quarter reports.

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