EASTERBROOK, Circuit Judge.

Indiana's Bureau of Motor Vehicles will not register or transfer title to a car or other motor vehicle unless the buyer furnishes information that includes a Social Security number. 140 Ind. Admin. Code § 6-1-2. For corporations and similar entities, by contrast, Indiana wants a federal employer identification number (EIN). It is possible to obtain an EIN without having a Social Security number. The Internal Revenue Service will issue an employer identification number to anyone who has an individual taxpayer identification number (ITIN), and it will issue an ITIN to anyone who wants one. Persons who cannot obtain Social Security numbers — including not only aliens whose visas do not allow them to work in the United States, but also aliens who lack authority to be in the United States at all — can have an ITIN for the asking and use it for many financial transactions.

Omar Duran Lagunes and four colleagues established a business to help people without Social Security numbers navigate the process of titling vehicles in Indiana and obtaining license plates. For each client, Duran's service used a client's individual taxpayer identification number to obtain an employer identification number, registered a limited liability company named after the client (so John Doe received "John Doe LLC"), and submitted in the LLC's name the required paperwork and fees. The service used clients' real names and addresses. Each client paid about $350, which included the fees remitted to the Bureau of Motor Vehicles. Indiana issued the titles and licenses as requested; the state has never suggested that holding title to a personal vehicle through an LLC violates any rule of state law. Nor has the Internal Revenue Service stated that it is improper to obtain an employer identification number for use by an entity that will own property but not generate income.

But the United States Attorney for the Northern District of Indiana procured an indictment charging Duran and colleagues with two federal crimes. Count One alleges that defendants conspired, in violation of 8 U.S.C. § 1324(a)(1)(A)(v)(I), to violate 8 U.S.C. § 1324(a)(1)(A)(iii) and (iv) by shielding unauthorized aliens from detection and encouraging them to reside in the United States. It also alleges that defendants violated 18 U.S.C. § 2 by aiding and abetting the violation of § 1324(a)(1)(A); this does not add anything, so we do not mention § 2 again. Count Two alleges that defendants conspired to commit mail or wire fraud, in violation of 18 U.S.C. § 1349. All defendants were convicted of both counts and have been sentenced to imprisonment as short as 24 months (Evelyn Rivera Borrero and Yalitza Exclusa-Borrero) and as long as 84 months (Duran).

The charge of fraud could have been a simple one. Indiana taxes the sale of motor vehicles. An application for transfer of title must report the price at which the sale occurred and include the appropriate tax. Duran and the other defendants inserted false prices, such as $100 or $200, into the forms and remitted sales tax less than state law required — or so the indictment charged. Defendants used the means of interstate commerce (including the Internet) to acquire tax identifiers and create LLCs, and Indiana used the mails to send registration papers and license plates, so a financial fraud comes within the scope of the mail-fraud statute. See Schmuck v. United States, 489 U.S. 705, 109 S.Ct. 1443, 103 L.Ed.2d 734 (1989).

But at trial the prosecutor did not emphasize Indiana's financial loss. To convict of mail or wire fraud, the jury must find that false statements injured a victim by depriving it of "money or property". See 18 U.S.C. § 1341. The jury instructions allowed a conviction to rest on a conclusion that defendants caused the state to retitle the vehicles, even if it did not suffer a financial loss. That is possible only if vehicle titles and license plates are "property" from the perspective of Indiana, the scheme's victim. The prosecutor favored this approach so that he could emphasize a second fraud: defendants falsely reported that all applicants had insurance, putting bogus policy numbers in the forms submitted on behalf of clients who lacked insurance while knowing that the state did not check up. False reports of insurance coverage played a role in the state's willingness to transfer titles and issue license plates but did not cost it anything out of pocket.

The United States' appellate brief concedes that treating titles and licenses as "property" is a legal error. Cleveland v. United States, 531 U.S. 12, 121 S.Ct. 365, 148 L.Ed.2d 221 (2000), holds that state and municipal licenses, and similar documents, are not "property" in the hands of the public agency. This circuit had reached a similar conclusion in Toulabi v. United States, 875 F.2d 122, 125 (7th Cir. 1989). Until the case reached this court no one — not the prosecutor, not the judge, and not defense counsel — recognized that the principal theory on which the mail fraud count was indicted, and the jury was instructed, was legally defective. But the defendants' appellate brief relied on Cleveland, and the prosecutor acknowledged the error.

The United States has asked us to affirm the fraud convictions anyway, observing that the record contains evidence from which a jury could have found that the defendants defrauded Indiana out of "money" — the tax it could have collected had the selling prices been reported honestly. Moreover, the prosecutor observes that counsel for three of the five defendants approved the jury instructions. Yet although waiver blocks these three defendants from using the error in the instructions as a reason to reverse (for the other two, plain-error review is available), all five defendants requested a judgment of acquittal under Fed.R.Crim.P. 29. If, as they contend, they have been convicted of acts that the law does not make criminal, they are entitled to that relief notwithstanding the legal errors that contaminated the jury instructions. See Bousley v. United States, 523 U.S. 614, 118 S.Ct. 1604, 140 L.Ed.2d 828 (1998).

Whether they have been so convicted is uncertain. The instructions permitted the jury to convict on the theory that title papers and licenses are "property" from Indiana's perspective, with which the state parted because of false statements about insurance. If that was the jury's sole ground of conviction, then defendants are entitled to acquittal. But the instructions also permitted the jury to convict on the theory that by misstating the selling prices defendants defrauded Indiana out of money that should have been paid as sales tax. If that was the jury's ground of conviction, then the judgments should be affirmed.

Unfortunately, the instructions did not require the jury to choose among theories of culpability. The jury may have relied exclusively on the registration-papers-as-property theory, exclusively on the sales tax theory, or on some mixture of frauds and understandings of "property". We just don't know. That makes it impossible for us to evaluate the sufficiency of the evidence, because the jury rather than the judge decides which evidence to believe. The jury might have believed the evidence about false statements concerning insurance, leading to a verdict that only the registration-papers-as-property theory could support, and disbelieved the evidence about false statements concerning purchase prices. Or the reverse. Or it could have accepted all of the prosecution's evidence and all legal theories of culpability.

Griffin v. United States, 502 U.S. 46, 112 S.Ct. 466, 116 L.Ed.2d 371 (1991), holds that if the evidence is sufficient to convict on one theory, and insufficient to convict on another, then the conviction should be affirmed. The court must assume that the jury followed its instructions and discounted the theory unsupported by the evidence. But when the instructions allow a jury to convict on two theories, one of which is legally insufficient, then the court must remand for a new trial, because a jury that followed its instructions might have convicted on the invalid ground while disdaining the proper one. See Yates v. United States, 354 U.S. 298, 77 S.Ct. 1064, 1 L.Ed.2d 1356 (1957), which like several similar decisions was reaffirmed in Griffin, 502 U.S. at 51-56, 112 S.Ct. 466. Remand is the right outcome here as well. The legal error in the jury instructions prevents the court from knowing what evidence the jury credited, and without that knowledge it is impossible to evaluate the motion for a judgment of acquittal. A properly instructed jury could have convicted the defendants on this record, however, so the Double Jeopardy Clause does not foreclose a new trial, if the prosecutor chooses to press on. Contrast Burks v. United States, 437 U.S. 1, 98 S.Ct. 2141, 57 L.Ed.2d 1 (1978).

As for Count One, which charged defendants with shielding unauthorized aliens and encouraging them to remain in the United States: there are multiple problems, not least of which is that the prosecutor did not prove that defendants knew their clients to be unauthorized aliens or recklessly disregarded that fact. At least four categories of persons could have found defendants' service attractive: (1) aliens who are in the United States "in violation of law" (a qualification in each clause of § 1324(a)(1)(A)), and thus unable to obtain Social Security numbers; (2) aliens who are authorized to be in the United States but not authorized to work (many kinds of visa do not authorize employment) and thus unable to obtain Social Security numbers; (3) persons who have Social Security numbers but do not want to reveal them to state officials, either because they are concerned about privacy or because they object on principle to all use of Social Security numbers; (4) persons who want to carry on a trade or business through an LLC and own a car as part of that business (many lawyers and other professionals do business as LLCs or P.C.s). Category 1 can be further divided into (a) aliens the United States does not know about; (b) aliens the United States knows about and is trying to remove; and (c) aliens the United States knows about and has decided, as an exercise of prosecutorial discretion or the parole power, not to remove.

Assisting persons in categories 2, 3, and 4 does not violate § 1324(a)(1)(A) on anyone's understanding. Only category 1(a) could be covered by § 1324(a)(1)(A)(iii), which makes it a crime to "shield from detection" an alien who is in the United States "in violation of law". Categories 1(b) and 1(c) cannot be shielded from detection because, by hypothesis, the United States knows about them. And only categories 1(a) and 1(b) are unambiguously covered by § 1324(a)(1)(A)(iv), which makes it a crime to "encourage ... an alien to ... reside in the United States" when residence is "in violation of law". The application of clause (iv) to aliens in category 1(c) is problematic; the United States' brief does not address that subject.

The United States contends that defendants either knew or recklessly disregarded (the two mental states specified by § 1324(a)(1)(A)) the fact that their clients were aliens not authorized to live in the United States. But having neglected to prove that the defendants' clients were aliens, the prosecutor did not supply an adequate basis for the jury to infer that they recklessly disregarded the risk that their clients were in the United States "in violation of law". Defendants' advertising in Spanish-language publications, and the use of Spanish in their own operations, does not allow an inference that they knew, or were reckless about, the unlawful presence of their clients in the United States. Many persons who read or speak Spanish are citizens of this nation; some Spanish-speaking aliens are in categories 1(c) or 2. The record does not offer a basis for estimating what portion of defendants' clientele was in categories 1(a) and 1(b).

Let us assume that the defendants were recklessly indifferent to the likelihood that some of their clients were aliens in the United States "in violation of law". Conviction under clause (iii) still would be impossible, because defendants did not attempt to "shield from detection" any of their clients. Quite the contrary, they provided state officials with their clients' real names and addresses. If as the prosecutor insists only an unauthorized alien would want to title a car through an LLC, then defendants were providing state officials with a list of aliens who should be removed, and the state had only to turn it over to federal officials (or federal officials could have asked for it). Putting the names and addresses of aliens into official files does not shield anyone from detection — nor does it conceal or harbor unauthorized aliens, the other two ways to violate clause (iii).

This leaves clause (iv), which we now quote in full. It says that any person who "encourages or induces an alien to come to, enter, or reside in the United States, knowing or in reckless disregard of the fact that such coming to, entry, or residence is or will be in violation of law" commits a felony. The prosecutor did not charge defendants with encouraging or inducing anyone to come to or enter the United States; the prosecution relies entirely on a contention that the defendants' service encouraged or induced aliens to "reside in" the United States by making it easier for them to own cars. Of course it wasn't defendants who offered the means for persons without Social Security numbers to own cars; Indiana did that. All defendants did was roll into one service a series of steps that anyone could have performed independently.

If making it easier to own a car by packaging multiple public services into a bundle is enough to induce or encourage an alien to "reside" in the United States, and thus violate § 1324(a)(1)(A)(iv), what else qualifies? An alien goes to the corner store and tells the proprietor that he lacks a visa but needs bread or meat; the proprietor sells him food. An alien goes to a medical clinic and tells the physician that he lacks a visa but needs help for high blood pressure; the doctor treats him. Has the grocer or doctor violated § 1324(a)(1)(A)(iv)? Without food or medical care, the alien must leave the United States; selling food or medical care is much more potent in facilitating residence than bundling a set of services that help someone put a car's title in the name of an LLC. Yet at oral argument counsel for the United States denied that the grocer or doctor would violate § 1324(a)(1)(A)(iv).

If the defendants' bundles violate federal law, what of the titling service that Indiana offers? What, indeed, of drivers' licenses and vehicle titles that some states, including Illinois, provide to self-declared unauthorized aliens? These states believe that, given their inability to remove aliens (only the national government can do that), it is best to provide documents so that drivers can be identified when they violate traffic laws or are involved in accidents, and so that mandatory-insurance requirements can be enforced. Has every legislator who voted for such a law, the governor who signed it, and every public employee who processes an unauthorized alien's request for a driver's license committed a federal felony? How about a desk clerk at the Chicago Transit Authority who sells a transit card, allowing an alien to use the bus and subway system (which like a car can facilitate holding a job that the alien cannot lawfully perform)? How about the dealer who sells an alien a car?

The list of goods and services that will help an unauthorized alien continue to reside in the United States is long. If defendants have violated § 1324(a)(1)(A)(iv), presumably all of these other persons have done so as well. Yet the Supreme Court held in Plyler v. Doe, 457 U.S. 202, 102 S.Ct. 2382, 72 L.Ed.2d 786 (1982), that states are required to provide some public services, including education, to unauthorized aliens. If public bodies do not violate § 1324(a)(1)(A)(iv) by providing services to unauthorized aliens, however, why would it be a felony for a private business to combine several governmental offerings into one package?

At oral argument counsel for the United States denied that any of these illustrations (sales of food, provision of drivers' licenses to declared unauthorized aliens, and so on) violates federal law, but he could not draw a defensible line between defendants' conduct and these illustrations. He observed, for example, that none of the illustrations entails use of an LLC. So what? Section 1324(a)(1)(A)(iv) does not make the use of LLCs essential to a violation. Nor does the statute declare it to be a federal felony to use LLCs that lack a business, as opposed to personal, objective (another proposed distinction). None of the prosecutor's stopping points has anything to do with the text of § 1324(a)(1)(A)(iv).

These convictions can be sustained only if the provision of any sufficiently valuable service — food, medicine, transportation — to an unauthorized alien is a felony because it helps the alien "reside" in the United States. That would take the statute beyond a sensible understanding; the Rule of Lenity, if nothing else, would forbid it.

Consider the one thing most likely to induce an unauthorized alien to reside in the United States: employment. It is unlawful to employ an alien whose visa does not authorize work. 8 U.S.C. § 1324a. But the penalties for violating this statute are fines, § 1324a(e), and imprisonment capped at six months, § 1324a(f). What sense could it make to set six months as the maximum punishment for employing an unauthorized alien, while allowing imprisonment for five years under § 1324(a)(1)(B)(ii) for someone who makes it just a little easier for the same alien to get clear title to the car used to drive to work? It is far better to read § 1324(a)(1)(A)(iv) as inapplicable to the provision of goods and services that are attractive to unauthorized aliens, legally residing aliens, and citizens alike.

The convictions on Count One are reversed, and the convictions on Count Two are vacated. We remand for entry of a judgment of acquittal on Count One. The United States Attorney must decide whether to reprosecute Count Two on a charge shorn of any allegation that title papers and licenses are "property" from Indiana's perspective. Pending the new trial (if one occurs), all defendants are entitled to be released on bail. Some of them may already have served more time than would be appropriate following a conviction on Count Two alone. (That is why we entered a judgment of reversal on the day this case was argued, while telling the parties than an opinion would follow.)