POSNER, Circuit Judge.

This case is before us for the second time, the Supreme Court having vacated the judgment, which we had affirmed, and remanded the case to us for reconsideration. Black v. United States, ___ U.S. ___, 130 S.Ct. 2963, 177 L.Ed.2d 695 (2010).

The defendants—senior executives of Hollinger International—had been convicted by a jury of three counts of mail and wire fraud, in violation of 18 U.S.C. §§ 1341 and 1342, and defendant Black had also been convicted of obstruction of justice, in violation of 18 U.S.C. § 1512(c). The judge had sentenced Black to a total of 78 months in prison, Atkinson and Boultbee to 24 and 27 months, and Kipnis to probation with six months of home detention.

The three fraud counts (which we'll treat as two, because two of the three relate to transactions with the same company, APC) were submitted to the jury under two theories: that of a scheme of fraudulent appropriation of money to which Hollinger was legally entitled (we'll call this "pecuniary fraud"), and that of a scheme to deprive Hollinger of the latter's "intangible right of honest services," 18 U.S.C. § 1346, amending sections 1341 and 1342. The first theory required that the defendants have obtained a pecuniary benefit at the expense of Hollinger; the second did not; and because the jury returned a general verdict on the fraud counts, we cannot be absolutely certain that it found the defendants guilty of pecuniary fraud as well as, or instead of, honest-services fraud.

After we affirmed, the Supreme Court held that the latter form of fraud requires proof that the defendant(s) received a bribe or kickback, as otherwise section 1346 would be unconstitutionally vague. Skilling v. United States, ___ U.S. ___, 130 S.Ct. 2896, 2931, 177 L.Ed.2d 619 (2010); see United States v. Cantrell, 617 F.3d 919, 921 (7th Cir.2010); United States v. Urciuoli, 613 F.3d 11, 17-18 (1st Cir. 2010). That was not proved here and so the defendants could not lawfully be convicted of honest-services fraud. But if it is not open to reasonable doubt that a reasonable jury would have convicted them of pecuniary fraud, the convictions on the fraud counts will stand. Hedgpeth v. Pulido, ___ U.S. ___, 129 S.Ct. 530, 531-32, 172 L.Ed.2d 388 (2008) (per curiam); see Neder v. United States, 527 U.S. 1, 15-16, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999); United States v. L.E. Myers Co., 562 F.3d 845, 855 (7th Cir.2009); United States v. Cappas, 29 F.3d 1187, 1192 (7th Cir.1994); United States v. Jackson, 196 F.3d 383, 386 (2d Cir.1999). "An instructional error arising in the context of multiple theories of guilt no more vitiates all the jury's findings than does omission or misstatement of an element of the offense when only one theory is submitted." Hedgpeth v. Pulido, supra, 129 S.Ct. at 532 (emphasis in original).

The case would still have to be remanded to the district court for resentencing unless it was reasonably certain that the judge would have imposed the same sentences even if the charge of honest-services fraud had not been submitted to the jury. Suppose no reasonable jury would have failed to find pecuniary fraud. Nevertheless that same jury, having been instructed on honest-services fraud, might have found the defendants guilty of honest-services fraud as well. The judge, if she thought that, would then have been incorrectly sentencing the defendant for two crimes rather than one. She might think honest-services fraud the more serious crime, or at least that it made the defendants' conduct more reprehensible and so merited heavier overall sentences.

We begin with defendant Black's argument that the submission of that charge to the jury contaminated his conviction of obstruction of justice, and that therefore he is entitled to a retrial on the obstruction count as well as on the fraud counts. He was charged with having concealed or attempted to conceal documents "with the intent to impair the [documents'] integrity or availability for use in an official proceeding," in violation of 18 U.S.C. § 1512(c)(1). There was compelling evidence that he knew that the acts that later formed the basis of the fraud charges against him and his codefendants were being investigated by a grand jury and by the SEC. In the midst of these investigations Black with the help of his secretary and his chauffeur removed 13 boxes of documents from his office, put them in his car, was driven home, and helped carry them from the car into his house.

He later returned the boxes; and copies of the documents were available to the government before the boxes were removed; but it was material to the investigation whether Black had had copies in his office. For that would mean that he had received them and might know they were material to the government's investigation. Furthermore, the boxes may have contained documents, of which there were no copies, that he'd removed before returning the boxes. That is speculation; but the possibility of such tampering helps to explain why the obstruction statute does not require proof of obstruction, as distinct from intent to obstruct, in order to convict. The usual consequence of an obstruction of justice is not that a guilty person is acquitted but that the government expends additional resources to prevent the effort at obstruction from succeeding, as in our case of United States v. Wells, 154 F.3d 412, 414-15 (7th Cir.1998), where the defendant's lie about the proceeds of his robbery sent the police on a wild goose chase. Similarly, concern that a suspect may be concealing material documents incites additional investigative efforts by the government. See United States v. Tankersley, 296 F.3d 620, 623-24 (7th Cir.2002).

Thus, as we explained in a portion of our first opinion not disturbed by the Supreme Court and therefore the law of the case, the obstruction of justice statute does not require proof of materiality unless the alleged obstruction takes the form of a lie that could not be expected to have any effect on the justice process. United States v. Buckley, 192 F.3d 708, 710 (7th Cir.1999). Being able to deny the materiality of a document is a common reason for concealment. So it is enough for conviction that a document was concealed in order to make it unavailable in an official proceeding. See, e.g., United States v. Senffner, 280 F.3d 755, 762 (7th Cir.2002); United States v. Phillips, 583 F.3d 1261, 1263-64 (10th Cir.2009); United States v. Carson, 560 F.3d 566, 584 (6th Cir.2009); United States v. Lessner, 498 F.3d 185, 197-98 (3d Cir.2007). The evidence that the boxes were removed in order to conceal documents from the government investigators was compelling, even though Black's secretary loyally testified that Black intended to remove the documents to a temporary office that she would set up for him in her home because he had to vacate his office at Hollinger within ten days. Her testimony was inconsistent with his having put the boxes in his car (not hers, which was at the scene) and taken them to his home rather than to hers. There was also evidence that in removing the boxes he tried to avoid the surveillance cameras in his office building—unsuccessfully.

In any event, the sufficiency of the evidence to convict Black of obstruction is no longer an open question; and since the jury was separately instructed on obstruction, the fact that it received an erroneous instruction on another count would ordinarily be irrelevant. United States v. Holzer, 840 F.2d 1343, 1349 (7th Cir.1988). But Black argues that had the jury not been told it could convict him of honest-services fraud, it might well have acquitted him of obstruction of justice. He appeals to cases in which convictions on counts on which the jury was properly instructed were reversed because a count that was later dismissed was so inflammatory that it created a "prejudicial spillover." E.g., United States v. Lazarenko, 564 F.3d 1026, 1042-44 (9th Cir.2009); United States v. Edwards, 303 F.3d 606, 639-40 (5th Cir. 2002); United States v. Vebeliunas, 76 F.3d 1283, 1293-94 (2d Cir.1996). These cases are in superficial tension with our decision in Holzer and also in United States v. Schwartz, 787 F.2d 257, 264 (7th Cir.1986), but those are decisions about misjoinder, and hold that the time to sever a trial because of a prejudicial spillover from one count to another is before the trial begins. If a count is submitted to the jury under an instruction apt to poison the jury's consideration of other counts as well, the defendant may be entitled to a new trial.

But this is not such a case. The theory of honest-services fraud submitted to the jury was esoteric rather than inflammatory; the evidence of such fraud was a subset of the evidence of pecuniary fraud; and the evidence of obstruction of justice was very strong. No reasonable jury could have acquitted Black of obstruction if only it had not been instructed on honest-services fraud. It would still have been the case that Black had known he was being investigated for fraud. He could not have known that years later the Supreme Court would invalidate one of the fraud charges. And if he were clairvoyant, he would have known that the other fraud charge—pecuniary fraud—would not be invalidated.

At argument Black's lawyer posed the following amusing hypothetical in an attempt to use the error in the fraud instruction to undermine his client's conviction for obstruction of justice: Suppose the Justice Department launches an investigation of a man suspected of having an affair with Minnie Mouse, and while the investigation is under way the man burns his Disney comics. Although an "official investigation" was pending (and capable of being obstructed) when he destroyed the comics, this could not be construed as an obstruction of justice, because the crime under investigation did not exist and therefore he could not have acted with the corrupt intent necessary for guilt of obstruction. Black's lawyer hoped by this hypothetical case to persuade us that the jury would have interpreted his client's intent in removing the documents differently had it known that the honest-services fraud under investigation at the time was not a crime; it would have been more willing to credit his innocent explanation for his action and conclude that he had not acted with corrupt intent. But Black was not under investigation for an obviously nonexistent crime, such as carnal knowledge of a fictional mouse; he was under investigation for conventional pecuniary fraud as well as honest-services fraud, and besides it was not obvious at the time of his removing the documents that honest-services fraud was a nonexistent crime. Hundreds of persons must have been convicted of it before the Supreme Court, years after Black's act, narrowed it to cases in which the defendant receives a bribe or a kickback. Black had also to fear—and just as acutely—being prosecuted for pecuniary fraud, as of course he was, and the elements of that crime are unchanged from when he acted.

So the conviction for obstruction will stand. The two fraud counts present a stronger case for ordering a new trial (and for all the defendants, not just Black). The first of these counts concerns a subsidiary of Hollinger called APC, which owned a number of small community newspapers that it was in the process of selling. When it had only one left—a weekly community newspaper serving Mammoth Lake, California (population 7,093 in 2000, the year before the fraud)— defendant Kipnis, Hollinger's general counsel, prepared and signed on behalf of APC an agreement to pay the other defendants, plus another Hollinger executive, a total of $5.5 million in exchange for their promising not to compete with APC for three years after they stopped working for Hollinger. The money was paid. Neither Hollinger's audit committee, which was required to approve transactions between Hollinger's executives and the company or its subsidiaries (such as APC) because of the conflict of interest, nor Hollinger's board of directors, was informed of this transaction.

That Black and the others might start a paper in Mammoth Lake to compete with APC's tiny newspaper there was a ridiculous idea; no one would pay them to promise not to do something they obviously would never want to do. But they argued that really the $5.5 million represented management fees owed them, and that they had characterized the fees as compensation for granting covenants not to compete in the hope that Canada, where a substantial percentage of the management fees had been generated, might not treat the fees as taxable income. Although Hollinger is a large, sophisticated, public corporation, no document was found to indicate that the $5.5 million expenditure was ever approved by the corporation or credited to the management-fees account on Hollinger's books. The checks were drawn on APC even though there was evidence that the defendants had no right to management fees from that entity, and the checks were backdated to the year in which APC had sold most of its newspapers, the purpose being—or so the jury could find—to make the richly compensated covenants not to compete seem less preposterous.

The evidence was certainly sufficient to prove a pecuniary fraud, see Cleveland v. United States, 531 U.S. 12, 19-20, 121 S.Ct. 365, 148 L.Ed.2d 221 (2000); United States v. Orsburn, 525 F.3d 543, 546 (7th Cir.2008); United States v. Henningsen, 387 F.3d 585, 589-90 (7th Cir.2004), and the jury was correctly instructed on the elements of such a fraud. But it was also instructed that it could convict the defendants upon proof that they had schemed to deprive Hollinger and its shareholders of their right to the defendants' honest services. This instruction did not require the jury to find that the defendants had taken any money or property from Hollinger; all it had to find was that in failing to disclose the recharacterization of the management fees to the audit committee and the board, they had failed to render honest services to Hollinger and had done so in an effort to obtain a private gain. That was a good instruction before the Supreme Court ruled that honest-services fraud requires proof of a bribe or kickback, but no longer; and the question is therefore whether a reasonable jury might have convicted the defendants of depriving the company of their honest services for private gain but not have convicted them of pecuniary fraud.

That is unlikely, but no stronger assertion is possible. Although the defendants did not deny having sought a private gain, they contended that it was intended to be a gain purely at the expense of the Canadian government, not at the expense of Hollinger because (they contend) Hollinger owed them the money; and they were not accused of defrauding the Canadian government, only of defrauding Hollinger. There was plenty of evidence that Hollinger did not owe them $5.5 million in management fees, but the evidence was not conclusive, while all that the jury had to find in order to convict them of honest-services fraud was their failure to level with the board and the audit committee, which was irrefutable.

Had they disclosed that the recharacterization of management fees would net them a higher after-tax income, the board might have decided that the addition to their income warranted a reduction in the size of the fees. "A man is none the less cheated out of his property, when he is induced to part with it by fraud/ because he gets a quid pro quo of equal value. It may be impossible to measure his loss by the gross scales available to a court, but he has suffered a wrong; he has lost his chance to bargain with the facts before him." United States v. Rowe, 56 F.2d 747, 749 (2d Cir.1932) (L. Hand, J.); see also Ranke v. United States, 873 F.2d 1033, 1039-40 (7th Cir.1989). The defendants had a duty of candor to the board in the conflict-of-interest situation in which they found themselves, and by violating that duty they caused Hollinger to make false filings with the SEC, and they did so for their private gain. That was a solid honest-services case before the Supreme Court weighed in, but not a solid pecuniary-fraud case. The government did not argue, as it might have done, by analogy to cases such as United States v. Richman, 944 F.2d 323, 330 (7th Cir.1991), and United States v. Gole, 158 F.3d 166, 168 (2d Cir.1998), that even if the defendants were owed the fees, they had obtained them fraudulently, as when an employee who is owed $100 by his employer forges a check to himself for the amount and thus fraudulently appropriates money owed him. Cf. Edwards v. State, 49 Wis.2d 105, 181 N.W.2d 383, 387-88 (1970); State v. Self, 42 Wn.App. 654, 713 P.2d 142, 144 (1986).

What we're calling the second fraud count involves payments to the defendants (via Hollinger) of $600,000 in connection with Hollinger's sale to two companies, Forum and Paxton, which to simplify we'll treat as one, of community newspapers; the $600,000 was allegedly compensation for the defendants' promising not to compete with these newspapers after the sale. The defendants don't contend that the money represented management fees owed by Hollinger. The contention rather is that the money was compensation for bona fide covenants not to compete. The contention is implausible because these are small newspapers and the defendants could have no interest in going into competition with them as individuals—for the covenants bind them, not Hollinger or any other company that might want to enter the community-newspaper business. The owners of Forum-Paxton testified that they didn't request such a covenant. Their testimony was not only disinterested but was supported by the clowning note that David Radler, an executive of Hollinger, wrote to the defendants, in which he said that Forum-Paxton had "asked for a 5-year non-compete from Conrad [Black] and me covering not only the states wherein they purchased assets but those states that border the said states. This would leave us only Alaska, Wyoming and Louisiana for us to continue our activities.... I have been assured there is [sic] suitable accommodations four [sic] our new headquarters in Casper, Wyoming."

What makes the contention that the $600,000 was compensation for covenants not to compete additionally and decisively unbelievable is that there are no covenants. The defendants concede that none was prepared, but attribute the omission to innocent mistake. The concession fatally undermines their challenge to the convictions on this count. Either the failure to prepare covenants was an innocent mistake—in which event the defendants could no more be convicted by a reasonable jury of honest-services fraud than of pecuniary fraud, because a merely careless withholding of services owed a principal by an agent was never criminal fraud under the honest-services provision (or any other provision) of the mail and wire fraud statutes, United States v. Cochran, 109 F.3d 660, 667-68 (10th Cir.1997); see also United States v. Sorich, 523 F.3d 702, 707-08 (7th Cir.2008); United States v. Bloom, 149 F.3d 649, 654-55 (7th Cir.1998)—or no covenants were intended, and the fees were part of the purchase price of the newspapers, owed to Hollinger and stolen by the defendants. No reasonable jury could have acquitted the defendants of pecuniary fraud on this count but convicted them of honest-services fraud.

The defendants argue that maybe the jury believed that the absence of the covenants was an innocent mistake but convicted them because they failed to disclose the payments to the board. The failure to disclose is mentioned in passing in the information, but the evidence at trial, and the closing arguments, focused on whether the absence of a written covenant was merely an oversight or instead proof of pecuniary fraud.

The jury acquitted the defendants on two other counts related to covenants not to compete with Forum-Paxton. But in those instances the fees went to Hollinger, and it is Hollinger that issued covenants not to compete, not the defendants, and Hollinger was a far more plausible entrant into Forum-Paxton's markets than the defendants, as individuals, were. The only rational explanation for the split verdict is that the jury believed that the $600,000 that the defendants received from Forum-Paxton without covenants not to compete, unlike the other transactions with that company, was proceeds of a plain-vanilla pecuniary fraud—and only a pecuniary fraud. For had the jury believed that a failure to disclose the fees for promising not to compete with the little newspapers was honest-services fraud, it would have convicted the defendants on all the fraud counts, because the defendants disclosed those fees neither to the board nor to the shareholders; and the jury didn't do that.

When to this logical point are added the absence of a written record of a $600,000 transaction, the disinterested testimony by the newspapers' buyers that they did not request covenants not to compete, Radler's implicit boast that the covenants were fabrications, and the absence of an economic reason for them (because the defendants had no conceivable interest in becoming individual publishers of small community newspapers), the evidence of pecuniary fraud is so compelling that no reasonable jury could have refused to convict the defendants of it.

To sum up, the convictions on the APC count are reversed, and the convictions on the Forum-Paxton count and the obstruction of justice count are affirmed. The sentences are vacated and the case remanded for resentencing, as well as for trial, limited however to the APC count.

But although the defendants are entitled to a new trial on that count, the entitlement is moot unless the government decides to retry them. The government may wish instead, in order to conserve its resources and wind up this protracted litigation, to dismiss the APC count and proceed directly to resentencing. The judge could consider at the resentencing hearing the evidence that had been presented at the original trial concerning APC in determining what sentences to impose on the two counts (the $600,000 fraud involving Forum-Paxton and, with respect to Black, obstruction of justice as well) of which the defendants were properly convicted. 18 U.S.C. § 3661. "A jury's verdict of acquittal does not prevent the sentencing court from considering conduct underlying the acquitted charge, so long as that conduct has been proved by a preponderance of the evidence." United States v. Watts, 519 U.S. 148, 157, 117 S.Ct. 633, 136 L.Ed.2d 554 (1997) (per curiam); see also United States v. Quintero, 618 F.3d 746, 755 (7th Cir.2010); United States v. Duncan, 400 F.3d 1297, 1304-05 (11th Cir.2005). (And there was no acquittal on the APC count, just an error warranting—barely—a retrial.) But of course it is for the government to determine, not us, how to proceed on remand.

AFFIRMED IN PART, REVERSED IN PART, AND REMANDED.