POSNER, Circuit Judge.

This case, in the form in which it comes to us (an appeal from a contempt judgment against a company called SonCo Holdings), has a small head but a long tail. It began in 2000 as a suit by the SEC, filed in a federal district court in Indiana, charging fraud in violation of federal securities law. The court appointed a receiver, Joseph Bradley, to take charge of the' assets and distribute them among the victims of the $31 million fraud. Bradley went hunting for the assets and found that some of them had been used to acquire oil and gas leases in Texas in the name of a sham corporation called Branson Energy Texas.

Some of these leases, referred to as the "Hull-Silk" leases, were in their "secondary term." An oil and gas lease typically specifies two periods. In the first, the "primary term," the lessee is required to produce oil (we omit "and gas," to simplify) from the leased wells. If he fails to produce, the lease terminates and the wells revert to the lessor, except that the lessee can stave off reversion by paying an annual "delay rental," provided he starts producing by the end of the primary term. In the secondary term, failure to produce (other than temporarily) triggers immediate reversion. Midwest Oil Corp. v. Winsauer, 159 Tex. 560, 323 S.W.2d 944, 946 (1959); Cobb v. Natural Gas Pipeline Co., 897 F.2d 1307, 1309 (5th Cir.1990); Owen L. Anderson et al., Hemingway Oil and Gas Law and Taxation §§ 6.2-6.3, pp. 217-23 (4th ed.2004). Thus, during the primary term, the lease is equivalent to an option.

Although the lessee is the producer in the sense that it is "his" oil that is flowing from the wells, Texas law distinguishes between the lessee of the wells and the operator, who extracts the minerals from them. The lessee receives the cash flow from the sale of the oil and pays the operator to extract it. Operators are regulated by the Texas Railroad Commission. The same person or firm can be both lessee and operator. ALCO Oil & Gas Co. was both the lessee and operator of the Hull-Silk leases when in 2002, after the SEC suit had been filed and the receiver had been appointed, it assigned the leases to "BET," as the Branson corporation was known. ALCO remained the operator of the leases.

BET, remember, was a tool of the fraudsters, and its rights (and therefore the receiver's rights after he took over BET) in its properties, including the Hull-Silk leases, were contested. Among the entities that claimed to have valid legal interests in the leases was SonCo Holdings, which filed a claim (the precise nature of which is unclear, along with much else in this case) with the receiver in 2006. After protracted negotiations SonCo came to terms with the receiver in January 2010 and the district court entered an "agreed order" specifying the terms of settlement. The agreed order is the focus of this appeal because it is SonCo's violation of the order that precipitated the imposition of the sanction from which it appeals to us.

The order required SonCo to pay the receiver $580,000 for an assignment of the Hull-Silk leases, and this part of the order was carried out. (SonCo actually paid the receiver $600,000 for the leases, the additional $20,000 being a penalty for delay, and we'll use the larger figure rather than $580,000.) But there was more to the order, because the settlement was tripartite, the third party to it being ALCO, the operator of the Hull-Silk wells. The wells had been unproductive, in part because of freeze orders entered by the district court to prevent the dissipation of BET's assets. Reversion of the leases to the lessor (or lessors—we don't know whether there was more than one), because they were in their secondary term yet not producing, had been staved off only by an order by the district court. But as the operator of the leases ALCO had been compelled to post a $250,000 cash bond with the Texas Railroad Commission to assure payment of any costs that the Commission might impose on ALCO for failing as the operator of the wells to take proper measures to conserve oil and gas and prevent or remedy environmental damage from its operations. See Texas Natural Resources Code §§ 91.103, .104, .1041, .1042, .105, .142. ALCO could get its $250,000 back only if it was replaced as operator and the new operator posted an equivalent bond that would replace ALCO's.

The reason the receiver was interested in ALCO's operation of the leased wells—and the reason therefore for ALCO's inclusion in the agreed order—was that the $250,000 for ALCO's bond had come in part from the defrauded investors; for remember that ALCO had been hired as the operator by BET, though there is no suggestion of wrongdoing by ALCO. Regarding ALCO's bond the agreed order therefore provided that "SonCo shall obtain a bond . . . that shall replace Alco's bond so that Alco and the Receiver may obtain the release of its bond paid for with defrauded investor funds" (emphasis added). Thus the $250,000 would not stop with ALCO if it ceased to be the operator of the wells—not all $250,000 at any rate; some (we haven't been told how much) would be added to the receiver's assets because it had come from victims of the fraud.

ALCO was desperate to relinquish its position as operator because it anticipated mounting liabilities to the Texas Railroad Commission and perhaps other entities, with little prospect of offsetting revenues. The agreed order as we just saw required SonCo to replace ALCO's $250,000 bond with its own bond. The order also, according to the receiver's, ALCO's, and the district judge's interpretation, required SonCo to replace ALCO as operator, making SonCo both the lessee, and thus owner of the oil, and the operator of the wells. The replacement would require the permission of the Texas Railroad Commission but the grant of that permission was expected to be pro forma, provided that SonCo demonstrated, presumably by posting the replacement bond, its financial ability to shoulder the costs, referred to earlier, of conservation and of curing environmental violations. Texas Natural Resources Code § 91.107; cf. id. § 52.026.

But more than a year after the agreed order was issued, SonCo still had failed to post the bond that would replace ALCO's bond; it had sent the Commission a cashier's check for $250,000 but had timed it to arrive on the last business day prior to the extended deadline set by the district judge. And it had failed to obtain the Commission's authorization to operate the wells. It had applied for that authorization too at the last minute, and the application was incomplete; it was a fair inference that SonCo never actually intended to become the operator. (Since the authorization to operate the wells was never given, presumably the $250,000 check for an operator's bond was never cashed.)

On motion by the receiver and ALCO, the judge held SonCo in contempt of the agreed order and as a sanction ordered it to return the Hull-Silk leases to the receiver. But the judge allowed the receiver to keep the $600,000 that SonCo had paid him for the leases. The judge also awarded the receiver and ALCO more than $22,000 in attorneys' fees, an award not challenged in this appeal.

SonCo returned the leases as ordered to the receiver, who assigned them to another company, Wilson Operating Company, which is unconnected with any of the parties. Wilson in turn assigned them to still another unrelated party.

Because the receiver no longer has the leases, he and ALCO argue that SonCo's appeal is moot. Reversing the contempt judgment, they argue, would mean returning the Hull-Silk leases to SonCo, and the receiver cannot do that because he no longer has them; Wilson's assignee has them. True, SonCo has filed a lis pendens in a Texas court against the leases. It hopes the lis pendens will enable it to wrest them back from the current lessee (Wilson's assignee). If valid, the lis pendens is constructive notice that SonCo is litigating the receiver's, and hence Wilson's, claim to be empowered to assign the leases. Texas Water Rights Commission v. Crow Iron Works, 582 S.W.2d 768, 771 (Tex.1979). "A filed lis pendens is constructive notice of the underlying lawsuit, and a prospective buyer is on notice that he acquires any interest subject to the outcome of the pending litigation." World Savings Bank, F.S.B. v. Gantt, 246 S.W.3d 299, 303 (Tex.App.2008). But the lis pendens was mentioned for the first time in SonCo's reply brief, and is discussed there for all of three sentences. The receiver has said nothing about it. Its bearing on the appeal is opaque. We ignore it.

The argument that the appeal is moot because there is no way (ignoring the effect of the lis pendens) to revest SonCo with the leases interprets SonCo's challenge to the contempt judgment too narrowly. SonCo argues that it didn't violate the agreed order, and therefore should not have been sanctioned. It would (or says it would) prefer to have the oil leases back, but failing that it wants its $600,000 back. That would change what it considers an unjustified sanction into a rescission of the receiver's assignment of the leases to it, thus restoring the parties to the positions they occupied before the agreed order was entered.

In response, the receiver argues that, desiring as he does to wind up the receivership, he has distributed the receivership assets to the various creditors and no longer has $600,000 to give back to SonCo. But what he actually has said is that "the Receiver's funds have been distributed or spoken for as part of court-ordered distributions." The words we've italicized imply that the receiver still has some assets in hand. Maybe substantial assets. On August 4, 2011, he had $882,000; a week later the district court permitted him to distribute $625,000; since then he has paid out another $15,000; that leaves $242,000 plus however much of ALCO's $250,000 bond gets added to the receivership assets rather than returned to ALCO. There are other claimants to the receivership assets, but there is no evidence that if the contempt judgment were reversed SonCo would be unable to receive any of the money it paid the receiver for the leases that it has lost, probably forever despite the lis pendens.

So the appeal is not moot. But in the alternative the receiver and ALCO ask us to dismiss it as frivolous and impose sanctions on SonCo under Fed. R.App. P. 38. We're about to see that the appeal is not frivolous, and so the motion to dismiss on the alternative ground is also denied and we turn to the merits, where there are two issues: whether SonCo violated the agreed order and if so whether the sanction was justified.

The agreed order was poorly drafted. Nowhere does it say in so many words that SonCo is to become the operator of the Hull-Silk leases. But it requires SonCo to replace ALCO's operator's bond, and this implies that SonCo was also required to take the necessary steps to become the operator because the bond it was required to replace was an operator's bond. If SonCo replaced ALCO's bond with its own bond, the implication would be that SonCo had become the operator in place of ALCO.

The agreed order further states that "Alco will provide to SonCo . . . execution of necessary documents relating to operations. . . [and] execution of any other documents necessary for SonCo's operation of the Hull-Silk" leases (emphasis added), and that SonCo shall have a "breathing space" (originally 90 days, later extended to nearly 18 months) in which no one can sue it or file claims against it (except claims by state or federal authorities to protect public health or safety) or impose sanctions on it—and it is to use this period to "prepare certain leases for operation as SonCo sees fit, and work out a plan of action with the Texas Railroad Commission" (emphasis added).

From the receiver's standpoint, it is true, the important thing was not that SonCo actually be the operator of the Hull-Silk leases but that a new operator's bond be posted so that the receiver could get his hands on some of the $250,000 that ALCO would regain when its bond was replaced. From ALCO's standpoint the important thing was that it be released as operator because it was continuing to accrue liability for the environmental and other costs that the wells had created. We assume therefore that having become the lessee of the wells pursuant to the agreed order SonCo could have complied with it by engaging another oil company to be the operator rather than by becoming the operator itself. But it didn't do that either. We conclude that it did violate the order, although there has been no determination of why it did so—what game it was and is playing.

Our conclusion that SonCo violated the order may seem inconsistent with the principle that only the violation of an unambiguous court order (or as the cases frequently say, with the redundancy beloved of lawyers and judges, a "clear and unambiguous" court order) can be punished as a contempt of court. SEC v. Hyatt, 621 F.3d 687, 692 (7th Cir.2010); Prima Tek II, L.L.C. v. Klerk's Plastic Industries, B.V., 525 F.3d 533, 542 (7th Cir.2008); Goluba v. School District of Ripon, 45 F.3d 1035, 1037 (7th Cir.1995); Salazar ex rel. Salazar v. District of Columbia, 602 F.3d 431, 442 (D.C.Cir.2010); In re Grand Jury Investigation, 545 F.3d 21, 25 (1st Cir.2008). We said the order was poorly drafted, and meant it wasn't clear.

But we've seen that it contains strong hints that SonCo was to be the operator, as when it said, in a passage we quoted earlier, that SonCo was required to execute "documents necessary for SonCo's operation of the Hull-Silk" leases. And context, which can disambiguate, Goluba v. School District of Ripon, supra, 45 F.3d at 1038; see FDA v. Brown & Williamson Tobacco Corp., 529 U.S. 120, 132, 120 S.Ct. 1291, 146 L.Ed.2d 121 (2000); Brown v. Gardner, 513 U.S. 115, 118, 115 S.Ct. 552, 130 L.Ed.2d 462 (1994); Lawrence B. Solum, "District of Columbia v. Heller and Originalism," 103 Nw. U.L.Rev. 923, 974 (2009), does so in this case. SonCo had to become the operator (or hire a substitute) in order to comply with the explicit requirement that it replace ALCO's bond and enable ALCO to resign as operator, a result the order was also clearly intended to bring about—if it didn't, ALCO would have gained nothing from the settlement that the order approved, and it was a party to the settlement.

So the order was violated; but was the sanction proper? Judges have inherent authority to impose sanctions for misconduct by litigants, their lawyers, witnesses, and others who participate in a lawsuit over which the judge is presiding. United Mine Workers of America v. Bagwell, 512 U.S. 821, 831, 114 S.Ct. 2552, 129 L.Ed.2d 642 (1994); Chambers v. NASCO, Inc., 501 U.S. 32, 43-50, 111 S.Ct. 2123, 115 L.Ed.2d 27 (1991). Usually the sanction is a fine, an award of attorneys' fees, or some other monetary exaction, and is simply called a "sanction," and no particular procedures, including specification of the burden of proof, are prescribed for determining whether misconduct warranting a sanction has occurred. But if the judge terms the misconduct giving rise to a punitive sanction (as distinct from a compensatory one—the domain of civil contempt, discussed below) "contempt of court," he brings into play (if he is a federal judge) rules and a statute that cabin his discretion. A federal court is empowered, so far as bears on this case, to "punish by fine or imprisonment, or both, at its discretion, such contempt of its authority. . . as Disobedience . . . to its lawful. . . order." 18 U.S.C. § 401. The court may act summarily if the contempt was committed in the judge's presence and he saw or heard it, Fed.R.Crim.P. 42(b), but that is not this case. In cases of contempt involving violation of an order rather than acting up in the judge's presence, the judge must among other things appoint a lawyer (normally the U.S. Attorney, see Young v. United States ex rel. Vuitton et Fils S.A., 481 U.S. 787, 107 S.Ct. 2124, 95 L.Ed.2d 740 (1987)) to prosecute a charge of criminal contempt. Fed. R.Crim.P. 42(a)(2). And the contemnor cannot be punished with a jail sentence of six months or more unless he is convicted by a jury. Codispoti v. Pennsyvania, 418 U.S. 506, 512, 94 S.Ct. 2687, 41 L.Ed.2d 912 (1974); Bloom v. Illinois, 391 U.S. 194, 88 S.Ct. 1477, 20 L.Ed.2d 522 (1968).

Jail is irrelevant to this case, as is the judicial power to fine or imprison a civil contemnor in order to coerce his obedience to a court order (for example, an order to turn over a deed to a plaintiff), rather than to "punish" him, the latter being the domain of section 401 and Rule 42. But monetary sanctions for a civil contempt can also be compensatory rather than coercive, as in this case. United Mine Workers v. Bagwell, supra, 512 U.S. at 829, 114 S.Ct. 2552. The rules and statute that we cited do not apply to civil contempts, though notice and an opportunity to be heard are required. We'll see that the judge treated SonCo's misbehavior as a civil contempt.

A large body of case law holds that civil contempt must be proved by clear and convincing evidence, FTC v. Trudeau, 579 F.3d 754, 763 (7th Cir.2009); Goluba v. School District of Ripon, supra, 45 F.3d at 1037; FTC v. Lane Labs-USA, Inc., 624 F.3d 575, 582 (3d Cir.2010); Southern New England Tel. Co. v. Global NAPs Inc., 624 F.3d 123, 145 (2d Cir.2010); Eagle Comtronics, Inc. v. Arrow Communication Laboratories, Inc., 305 F.3d 1303, 1314 (Fed.Cir.2002); 11A Charles Alan Wright et al., Federal Practice and Procedure § 2960, p. 380 and n. 59 (2d ed.1995) (and cases cited there), though it is in tension with the Supreme Court's insistence on a presumption in favor of the less onerous standard of preponderance of the evidence in federal civil cases. Grogan v. Garner, 498 U.S. 279, 286, 111 S.Ct. 654, 112 L.Ed.2d 755 (1991); Herman & MacLean v. Huddleston, 459 U.S. 375, 387-91, 103 S.Ct. 683, 74 L.Ed.2d 548 (1983); see Harrods Ltd. v. Sixty Internet Domain Names, 302 F.3d 214, 225-26 (4th Cir. 2002). The opinion in Herman & MacLean, holding that preponderance is the appropriate standard for securities fraud, notes, as instances in which the presumption is rebutted and the higher standard of clear and convincing evidence is imposed, proceedings to terminate parental rights, involuntary commitment proceedings, and deportation proceedings. 459 U.S. at 389, 103 S.Ct. 683. We have criticized the tendency of federal courts to impose the higher standard unless a statute makes it the standard, Doll v. Brown, 75 F.3d 1200, 1203-04 (7th Cir.1996), and it is difficult to see why civil contempt should require a higher standard of proof than securities fraud; it has little in common with the examples given in Herman & MacLean of the exceptional cases governed by the higher standard. But we need not try to solve this puzzle in the present case.

What we can't glide over is the confusing overlap between sanctions for contempt on the one hand and sanctions, often indistinguishable from those for contempt, criminal or civil, that a judge can impose in the exercise either of the inherent judicial power that we mentioned earlier or under the authority granted by a rule or statute, such as Rules 11 and 37 of the civil rules or 28 U.S.C. § 1927, to punish misconduct by a lawyer or litigant. The judge can impose such sanctions summarily (though notice usually is required before imposition, to give the lawyer or litigant a chance to defend himself), at least if the sanction takes the form of a payment to the opposing party to compensate him for the consequences of the sanctionable misconduct; if the sanction is purely punitive, and severe, additional process may be required. Mackler Productions, Inc. v. Cohen, 146 F.3d 126, 129-30 (2d Cir.1998); see Eisenberg v. University of New Mexico, 936 F.2d 1131, 1136-37 (10th Cir.1991); Donaldson v. Clark, 819 F.2d 1551, 1559 n. 10 (11th Cir.1987) (en banc). But there is no general requirement of proof by clear and convincing evidence.

The judge in our case used the magic word "contempt" to characterize SonCo's violation of the agreed order; he obviously thought the violation deliberate, rather than the product of a misunderstanding of the order. But he did not think that he was fining SonCo, which would have made this a case of criminal contempt; he thought he was compensating the receiver and ALCO. That is why we said he was treating SonCo's misbehavior as a civil contempt.

The justification he offered for taking away SonCo's leases but not requiring the receiver to return the $600,000 that SonCo had paid the receiver for them, thus in effect imposing a $600,000 sanction on SonCo, was that "that money must be used to compensate the attorneys for Alco and the Receiver . . . [and] also . . . to compensate Alco for the harms caused by SonCo's noncompliance with [the agreed order]. . . . Those uses of the $600,000 will make the Receiver and Alco whole and will replenish funds that should have been returned to defrauded investors but instead have been dipped into as a result of SonCo's contempt of court."

Since the judge intended the remedy he was ordering to be compensatory, he did not have to call the misconduct giving rise to the order "contempt." Had he not called it contempt it would not have had to be proved by clear and convincing evidence. But he called it contempt, so it had to be if the cases that impose that standard survive Herman & MacLean and Grogan, yet he didn't mention the burden of proof. But SonCo is not objecting to that; it's arguing that the remedy isn't really compensatory, but rather punitive, so that the judge actually found SonCo guilty of criminal contempt without complying with the procedural rules governing such contempts.

A judge has to justify the sanctions he imposes. FTC v. Trudeau, supra, 579 F.3d at 770-71; Autotech Technologies LP v. Integral Research & Development Corp., 499 F.3d 737, 752 (7th Cir.2007); Mid-American Waste Systems, Inc. v. City of Gary, 49 F.3d 286, 293 (7th Cir.1995); FTC v. Kuykendall, 371 F.3d 745, 763 (10th Cir.2004) (en banc). Since the judge in this case intended the sanction (perhaps better termed "remedy") to be compensatory, he had to explain what it was compensating for; and he did not do that. For what costs had the receiver and ALCO incurred as a consequence of SonCo's violation of the agreed order, besides the $22,000 in attorneys' fees that the judge directed SonCo to pay? He didn't say. It might seem obvious that ALCO lost the $250,000 in bond money that it would have recovered had SonCo become the operator of the Hull-Silk leases or hired a substitute to operate them. But no; the judge ordered the new lessee, Wilson Operating Company (this was before Wilson assigned the leases), to pay $250,000 to the receiver, to be divided with ALCO. The unstated but inescapable premise was that Wilson's assignee was going either to operate the leases or hire an operator, and in either event would replace ALCO's bond. Without replacement the Texas Railroad Commission would not return ALCO's bond money, which was the guaranty of ALCO's ability to cover at least some of the conservation and environmental costs to which its operation of the wells might give rise.

Maybe ALCO incurred additional costs by virtue of having remained the operator for more than a year after the agreed order was entered. But we cannot find any estimate of those costs anywhere, or for that matter any estimate of the costs incurred by the receiver as a consequence of SonCo's contempt, beyond the attorneys' fees separately compensated for by SonCo. We haven't even been told what the receiver got for assigning the leases to Wilson. It is possible that most—maybe all—of the $600,000 loss that the judge's order imposes on SonCo is a form of punitive damages that would require recharacterizing the finding of civil contempt as a procedurally irregular finding of criminal contempt.

So while we affirm the judge's order insofar as it determines that SonCo willfully violated the agreed order, we vacate the sanction, and remand. The judge on remand will have three options: reimpose the sanction he imposed, upon demonstrating that it is a compensatory remedy for a civil contempt after all; impose a different, or perhaps no, sanction whether for civil contempt or for misconduct not characterized as contempt; or proceed under the rules governing criminal contempts.

AFFIRMED IN PART, REVERSED IN PART, AND REMANDED.