
527 F.3d 577 (2008)
UNITED STATES of America, Plaintiff-Appellee,
v.
Frances Darlene DEDMAN, Defendant-Appellant.
No. 06-6124.
United States Court of Appeals, Sixth Circuit.
Argued: November 1, 2007.
Decided and Filed: May 29, 2008.
*581 ARGUED: Gregory A. Woosley, Lexington, Kentucky, for Appellant. Andrew Sparks, Assistant United States Attorney, Lexington, Kentucky, for Appellee. ON BRIEF: Gregory A. Woosley, Lexington, Kentucky, Roy Burl McCoy, Jr., Stoll Keenon Ogden, Lexington, Kentucky, for Appellant. Charles P. Wisdom, Jr., Assistant United States Attorney, Lexington, Kentucky, Elaine K. Leonhard, Assistant United States Attorney, Fort Mitchell, Kentucky, for Appellee.
Before: SILER, MOORE, and GILMAN, Circuit Judges.
MOORE, J., delivered the opinion of the court, in which SILER, J., joined. GILMAN, J. (pp. 603-06), delivered a separate dissenting opinion.

OPINION
KAREN NELSON MOORE, Circuit Judge.
This unusual case arises from the marriage between defendant Darlene Dedman's ("Dedman") adopted daughter (who was actually Dedman's cousin) and Dedman's adoptive father. Dedman appeals her conviction on the counts of conspiracy to defraud the United States Department of Defense, in violation of 18 U.S.C. § 286, and making material false statements to a federal agent, in violation of 18 U.S.C. § 1001. For these offenses she was sentenced to 27 months of imprisonment and *582 ordered to pay over $200,000 in restitution. The government alleged that Dedman orchestrated the marriage as part of a plan to collect her adoptive father's, John Watson's ("Watson"), military pension. Dedman argues that she could not be guilty of conspiracy because the marriage between her adopted daughter and her adoptive father was valid and, therefore, there was no false claim. Furthermore, Dedman asserts that if the marriage was statutorily invalid, then the marriage law is unconstitutional. Dedman also contends that the government failed to produce evidence sufficient to support her convictions. In addition, she maintains that the jury instructions regarding the marriage amounted to a directed verdict against her. Lastly, Dedman avers that the district court miscalculated the government's loss during her sentencing. Although both the government and the district court made some mistakes in their handling of this case, we conclude that none of the mistakes amount to reversible error. We therefore AFFIRM Dedman's convictions and sentence.

I. BACKGROUND

A. Factual Background
This case revolves around the disposition of Watson's annuity under the Survivor Benefit Program ("SBP"), a lifetime pension for "military members who have reached at least 20 years of full military service." Joint Appendix ("J.A.") at 107 (May 11, 2006, Trial Tr., Timothy Zelenak Test. at 25:20-21). As a veteran of three wars over twenty years, Watson was eligible for the SBP. When a pensioner dies, the funds are payable to a surviving spouse only if the spouse was married to the beneficiary for a year or longer. See 10 U.S.C. §§ 1447(7), (9) (defining "surviving spouse" as a widow who was married for a year or more immediately before the pensioner's death); 1450(a) (providing annuity benefits to eligible surviving spouses). In order to qualify for the annuity, the marriage between the military member and the spouse must have been valid in the state where the marriage took place. See 1 U.S.C. § 7 (defining marriage as "only a legal union" (emphasis added)); Bishop v. Oklahoma, 447 F.Supp.2d 1239, 1250 (N.D.Okla.2006) (applying state law to determine whether a marriage satisfies 1 U.S.C. § 7).
Dedman had a traumatic childhood, but in 1983, at age eighteen, she agreed to let Watson, a fifty-six-year-old family friend, adopt her as his daughter. In the early Nineties, Dedman first met her cousin Nelva Holland ("Holland") at Dedman's birth-father's funeral. The two women became friendly, and Holland moved in with Dedman's family with the understanding that Holland would help with the family's household chores while Dedman would help Holland with nursing-school expenses. It was around this time that Holland first got to know Watson, who was living with the Dedmans. According to Holland, it was also around this time that Dedman first mentioned to her Watson's pension.
At age nineteen, Holland tried to enroll in nursing school, but she discovered that the school required her to obtain health-insurance coverage. In order to help her enroll in school, the Dedmans adopted her so that she would be covered under Dedman's husband's health-care plan. In July 1996, Holland and Dedman had a dispute, resulting in the Dedmans kicking Holland out of their home. According to Holland's testimony, Dedman approached her after a short period of time with an offer: Holland would be allowed to return to the Dedman home but only if she agreed to marry Watson, who was then approximately sixty-nine years old. Holland agreed to the deal; Holland, Dedman, and Watson drove to Arkansas where Holland and Watson *583 married. According to Holland's testimony, the fact that she was now married to Watson had no impact on her relationship with Watson; the two remained friendly but there was no romantic or sexual involvement. One of Dedman's sons, however, testified that there was affection between Holland and Watson.
During Dedman's trial, the jury heard conflicting testimony regarding whether Holland and Watson were public with their relationship. Holland testified that after the wedding, Dedman told her to keep the marriage a secret, and Holland said that she and Watson never held themselves out as married. Two of Dedman's neighbors testified that Watson and Holland never appeared to be married. One of those neighbors testified that she learned about the marriage only when Dedman told her that she had "solved the situation with the annuity." J.A. at 172 (May 11, 2006, Trial Tr., Mary Beth Green Test. at 171:13-22). In contrast, although Dedman's husband claimed that he never knew why Holland and Watson were married, he thought that Holland and Watson were public about their marriage. However, Dedman's husband was not surprised to learn during his testimony that the neighbors were not aware of the marriage.
On December 28, 1997, Watson died. Shortly thereafter, Dedman and Holland applied for Watson's SBP benefits. Once the government started paying the annuity, the funds went straight to Dedman, who gave Holland some of the money. This system worked well until Dedman and Holland had another dispute in 2005. At the end of this dispute, Dedman accused Holland of having molested one of Dedman's children. Dedman proceeded to call government agencies to tell them that Holland was committing fraud. These calls prompted the investigation that concluded with the indictment incident to this appeal.
During the course of the government's investigation, Dedman told a federal investigator that she learned of the Holland-Watson marriage only in 2004. The investigator did not believe Dedman, in part because Dedman provided Watson's death certificate, which listed Holland as the spouse. The investigator was also suspicious of Dedman's defensive reaction to questions about the marriage. Upon cross-examination at trial, the investigator conceded that it was possible that Dedman meant that she first realized that the marriage was illegal in 2004, not that she first learned about the marriage at that time.

B. Procedural Background
On February 3, 2006, the government indicted Dedman for conspiracy to defraud the government in violation of 18 U.S.C. § 286. In March 2006, the government added an additional count for making material false statements to a government agent in violation of 18 U.S.C. § 1001. The government also charged Holland with the same conspiracy charge, but Holland pleaded guilty and testified in the hopes of getting probation.
On February 17, 2006, Dedman pleaded not guilty to both charges, and the trial began on May 11, 2006. During Dedman's trial, the government asked the district court to "take judicial notice of Arkansas law to the effect that marriage between a grandfather and a granddaughter, even if that relationship is created by adoption, is prohibited." J.A. at 234 (May 12, 2006 Trial Tr. at 131:20-24). Dedman's attorney objected to the request, but the district court took the requested judicial notice.
After both the government and Dedman presented their cases, the judge instructed *584 the jury. The district court gave the jury the elements of the conspiracy offense:
What the evidence in this case must show beyond a reasonable doubt is:

First: That two or more persons in some way or manner, came to a mutual understanding to try to accomplish a common and unlawful plan, as charged in the indictment;

Second: That the Defendant, knowing the unlawful purpose of the plan, willfully joined in it; and

Third: That the object of the unlawful plan was to defraud the government by obtaining the payment or allowance of a claim which is based on a false or fraudulent material fact.
J.A. at 52-53 (Instruction No. 11). The district court further explained the mental state that the jury was required to find in order to convict Dedman of conspiracy to defraud the government: "If you are convinced that there was a criminal agreement, then you must decide whether the government has proved that the defendant knowingly and voluntarily joined that agreement. To convict the defendant, the government must prove that she knew the conspiracy's main purpose, and that she voluntarily joined it intending to help advance or achieve its goals." J.A. at 55 (Instruction No. 13).
In accordance with the district court's earlier ruling that it would take judicial notice of the law of Arkansas, the district court instructed the jury that "I also have decided to accept as proved that a marriage between a grandfather and his adopted granddaughter is prohibited by Arkansas state law. You may accept these facts as true, but you are not required to do so." J.A. at 66 (Instruction No. 23).
On May 15, 2006, after over an hour of deliberation, the jury found Dedman guilty on both counts. On August 18, 2006, the district court sentenced Dedman. The district court concluded that Dedman owed to the government $230,990.93, but credited her with $21,760.00 as already returned. The court required Dedman to pay the difference, a total of $209,230.93. The district court also used that amount as the government's loss for enhancing Dedman's sentence under the U.S. Sentencing Guidelines Manual ("U.S.S.G.") § 2B1.1. Dedman's counsel objected to the district court's use of the $209,230.93 figure because it included withheld taxes, and he argued that the government never lost the money that it withheld. The district court disagreed with Dedman's counsel and concluded that the loss was greater than $200,000 for the purposes of U.S.S.G. § 2B1.1, and the district court therefore increased Dedman's base offense level by 12. The district court sentenced Dedman to 27 months of imprisonment for each of the counts, with the terms to run concurrently.
On August 25, 2006, Dedman filed a timely appeal. The district court had jurisdiction over the case pursuant to 18 U.S.C. § 3231. We have proper jurisdiction to hear this appeal under 28 U.S.C. § 1291.

II. THE MARRIAGE

A. Standard of Review
When a district court interprets state law, "a court of appeals should review de novo a district court's determination of state law." Salve Regina College v. Russell, 499 U.S. 225, 231, 111 S.Ct. 1217, 113 L.Ed.2d 190 (1991) (declaring the standard of review in a diversity case); see also Mike's Train House, Inc. v. Lionel, L.L.C., 472 F.3d 398, 413 (6th Cir.2006). "A matter requiring statutory interpretation is a question of law requiring de novo review, and the `starting point' for interpretation `is the language of the statute *585 itself.'" United States v. Caldwell, 49 F.3d 251, 251 (6th Cir.1995) (citation omitted) (quoting Consumer Prod. Safety Comm'n v. GTE Sylvania, Inc., 447 U.S. 102, 108, 100 S.Ct. 2051, 64 L.Ed.2d 766 (1980)). Similarly, "[w]here, in formulating instructions, the `district court engages in statutory construction as a matter of law, ... [the Sixth Circuit] review[s][the] conclusions de novo.'" United States v. Buckley, 934 F.2d 84, 87-88 (6th Cir.1991) (second, third, and fourth alterations in original) (quoting United States v. Brown, 915 F.2d 219, 223 (6th Cir.1990)).

B. The Significance of the Marriage
The government's primary argument at trial was that the marriage between Holland and Watson was prohibited by Arkansas law, and thus the marriage was void. See Ragan v. Cox, 208 Ark. 809, 187 S.W.2d 874, 876 (1945) (interpreting the predecessor provision to Arkansas's current marriage statute, ARK.CODE ANN. § 9-11-106, which declared marriage between grandparents and grandchildren "absolutely void" just as the current statute does, and concluding that "[s]uch marriages are not merely voidable, but void ab initio"). Dedman and Holland's SBP claims made on the basis of a void marriage were, therefore, the foundation of the false claim that the government alleged.
On appeal, however, the United States repeatedly asserts that the validity of the marriage does not actually matter because the government would still prevail even if the marriage was valid. The government claims that the marriage was a sham marriage; thus, the claims would be false even if the marriage was valid. This sham-marriage theory was not presented to the jury via instructions and the government cannot now use it to support the conviction.[1]
*586 "A crucial assumption underlying th[e] system [of trial by jury] is that juries will follow the instructions given them by the trial judge." United States v. Foster, 376 F.3d 577, 592 (6th Cir.) (alterations in original) (quoting Parker v. Randolph, 442 U.S. 62, 73, 99 S.Ct. 2132, 60 L.Ed.2d 713 (1979) (Rehnquist, J., plurality)), cert. denied, 543 U.S. 1012 (2004). Even if the jury believed the government's references to its sham-marriage theory during closing argument, that could not have served as a basis for the verdict if there were no corresponding instructions. "To the contrary, the jury might well have concluded that counsel's hollow argument ... was contrary to, and precluded by, the judge's statement of the law." United States v. Duncan, 850 F.2d 1104, 1118 (6th Cir. 1988), overruled on other grounds by Schad v. Arizona, 501 U.S. 624, 111 S.Ct. 2491, 115 L.Ed.2d 555 (1991). Therefore, we must assume that the jury did not apply the sham-marriage theory in reaching their verdict, and the government cannot invoke that theory to preserve the verdict. For that reason, we must now address the validity of the marriage.

C. Judicial Notice
Dedman first challenges her conviction by claiming that the district court erred when it took judicial notice of Arkansas law to conclude that state law forbids a marriage between a grandfather and his adopted granddaughter.[2] Although we hold that the district court did not err in its interpretation of Arkansas law, we determine that the district court erred in its use of judicial notice.
When the government requested judicial notice, and when the district court obliged, it appears that a subtle distinction in the law confused them. For over one hundred years, it was quite natural for judges to take judicial notice of statutes: "The law of any state of the Union, whether depending upon statutes or upon judicial opinions, is a matter of which the courts of the United States are bound to take judicial notice, without plea or proof." Lamar v. Micou, 114 U.S. 218, 223, 5 S.Ct. 857, 29 L.Ed. 94 (1885) (emphasis added). This was an evidentiary shortcut whereby courts could rely upon the law of other states without jumping through the evidentiary hoops of bringing the bound volumes into court and establishing their validity. We certainly followed that rule and took judicial notice of state statutes when necessary. See, e.g., Val Decker Packing Co. v. Corn Prods. Sales Co., 23 Ohio Misc. 162, 411 F.2d 850, 852 (6th Cir.1969) ("The Court was required to determine which statute was applicable. Having made that determination and dismissed the complaint, its judgment is subject to appellate review, and we have the right to consider all of the statutes and to apply the one which in our judgment is required under Ohio law. We may take judicial notice of the state statutes ...." (citing Lamar)).
What perhaps confused the district court and the parties in the instant case is that in recent years the terminology of *587 "judicial notice" has shifted. In several modern opinions, we have cabined the concept of "judicial notice" to facts alone. The effect of such a semantic move is largely minimal because judges are still entitledand indeed requiredto determine the applicable law, even if that law is the law of other states. See United States v. Wynn, 987 F.2d 354, 358 (6th Cir.1993) ("Our judicial system requires the prosecution to prove facts, not laws. Regarding the presentation of evidence, the phrase `judicial notice' applies to facts, not laws. On the other hand, judges, not juries, determine what is the law." (citations omitted)); see also Toth v. Grand Trunk R.R., 306 F.3d 335, 349 (6th Cir.2002) ("Thus, judicial notice is generally not the appropriate means to establish the legal principles governing the case."). Although we used to allow judicial notice of state law, now we consider that state law is simply a matter for the judge to determine.
The semantic adjustment is usually of little consequence, leaving judges with the same powers that they always had to determine and instruct the jury on the content of the applicable law. Nonetheless, in this case the shift in language appears to have had a detrimental impact. Had the district court properly applied our modern precedent, the court would have first determined the applicable law and then instructed the jury as to that law. See Wynn, 987 F.2d at 358 ("In contrast, legal elements, even when required for culpability, may properly be part of the district court's statement of the law when it instructs the jury."). Although the district court did determine and instruct the jury that "a marriage between a grandfather and his adopted granddaughter is prohibited by Arkansas state law," because the district court was confused with regards to judicial notice, the court also added to jury instruction number twenty-three the directive: "You may accept these facts as true, but you are not required to do so." J.A. at 66 (Instruction No. 23). Thus, the district court impermissibly gave the jury permission to ignore the court's determination of the applicable law.
Such a mistake, however, was not reversible error in the circumstances of this case. The legal determination that the district court empowered the jury to disregardthat Arkansas law forbids a marriage of the type at issue in her casewas a legal determination that went against Dedman. Had a juror disregarded the district court's conclusion of law, then the juror would not have voted to convict Dedman. Therefore, we conclude that the district court's error in taking judicial notice of Arkansas state law and then permitting the jury to reject "th[is] fact" was harmless.
The district court gave the jury the power to reject its determination of law in an apparent misapprehension of our Criminal Pattern Jury Instructions. According to the United States, the district court's instruction empowering the jury to reject its determination of state law was copied from Sixth Circuit Criminal Pattern Jury Instruction 7.19 ("I have decided to accept as proved the fact that ____, even though no evidence was presented on this point. You may accept this fact as true, but you are not required to do so."). There is little doubt in our minds that the district court selected that instruction because it was the only pattern instruction applicable to judicial notice. The Sixth Circuit Criminal Pattern Jury Instruction 7.19, however, is based upon Federal Rule of Evidence 201(g) ("In a criminal case, the court shall instruct the jury that it may, but is not required to, accept as conclusive any fact judicially noticed."), which is a rule that applies only to adjudicative facts and does *588 not cover a court's determination of law.[3] Indeed, the official commentary for the instruction states that the instruction "should be given whenever the court has taken judicial notice of a fact." Sixth Circuit Criminal Pattern Jury Instructions 7.19, 1991 Commentary (emphasis added). If Rule 201 is inapplicable, then its associated Criminal Pattern Jury Instruction is also inapplicable. The improper use of Criminal Pattern Jury Instruction 7.19 when there is a conclusion of law rather than an acknowledgment of fact might erroneously lead a juror to disregard an important legal determination by the district court. Accordingly, judges should take care to limit judicial notice and use of Criminal Pattern Jury Instruction 7.19 to matters of fact.

D. The District Court's Determination of Arkansas Law
Having concluded that it was entirely within the province of the district court to determine whether Arkansas law forbids a marriage between a grandfather and his adopted granddaughter, we must address whether the district court correctly concluded that such a marriage is indeed forbidden. Dedman argues that the district court misstated Arkansas law when it concluded that such a marriage is void, but we disagree. Although there is considerable ambiguity, the best reading of Arkansas law confirms the district court's interpretation. When courts interpret statutes, "[t]he statute is read as a whole and construed to give each word operative effect." Caldwell, 49 F.3d at 251 (reading several related statutes in conjunction to find a consistent meaning).
The Arkansas marriage statute states:
All marriages between parents and children, including grandparents and grandchildren of every degree, between brothers and sisters of the half as well as the whole blood, and between uncles and nieces, and between aunts and nephews, and between first cousins are declared to be incestuous and absolutely void. This section shall extend to illegitimate children and relations.
ARK.CODE ANN. § 9-11-106(a). Dedman claims that this statute, despite including grandparents and grandchildren, is not applicable to the instant case, because it uses the term "of every degree," which she asserts is a term of consanguinity.[4] The United States responds by pointing to the *589 Arkansas incest statute, which explicitly includes adopted grandchildren:
A person commits incest if the person, being sixteen (16) years of age or older, purports to marry, has sexual intercourse with, or engages in deviate sexual activity with another person sixteen (16) years of age or older whom the actor knows to be:
...
(5) A stepgrandchild or adopted grandchild.
ARK.CODE ANN. § 5-26-202(a). According to the United States, if Arkansas considers a relationship between a grandfather and his adopted granddaughter to be incestuous, then a marriage between the two must be void. The United States's interpretation, however, appears to be strained. For the United States to be correct, we would need to read the Arkansas marriage statute as saying that such a relationship is "incestuous and therefore absolutely void." Such an interpretation is certainly plausible. Equally plausible, however, is that the "and" in § 9-11-106(a) does not mean "therefore" but instead simply serves as a cross-reference to signify that the marriage statute does not preclude enforcement of the incest statute, should it apply.
We need not resolve this dispute regarding the intersection of the marriage and incest statutes because we hold that the Arkansas marriage statute, on its face, forbids a marriage between a grandfather and his adopted granddaughter. We base our conclusion on the use of the phrase "of every degree." We conclude, contrary to Dedman's assertion, that the use of the phrase "of every degree" does not limit the marriage ban to only those grandparents and grandchildren who are in a consanguineous relationship. The Arkansas legislature frequently uses "degree" with respect to relationships that are not consanguineous. For instance, several times the Arkansas legislature has referred to adopted relatives of different degrees. See ARK.CODE ANN. § § 4-59-201(11) (defining "relative" in a fraudulent-transfer statute as "an individual related by consanguinity within the third degree as determined by the common law, a spouse, or an individual related to a spouse within the third degree as so determined, and includes an individual in an adoptive relationship within the third degree" (emphasis added)); 9-27-355(a) (defining "relative" for the purpose of placing juveniles as "a person within the fifth degree of kinship by virtue of blood or adoption"); 9-28-402(18) (defining "relative" for the purpose of child-welfare-agency licensing as "a person within the fifth degree of kinship by virtue of blood or adoption"). Furthermore, there are numerous instances where the Arkansas legislature specifically references relationships of various degrees that are not consanguineous, often doing so alongside references to relationships of consanguinity. See, e.g., ARK. CONST. amend. XXIX, § 2 ("Husbands and wives of such officers, and relatives of such officers, or of their husbands and wives within the fourth degree of consanguinity or affinity,[5] shall likewise be ineligible." (emphasis added)); ARK. CONST. amend. LXXX, § 12 ("No Justice or Judge shall preside or participate in any case ... in which any party is related to him or her by consanguinity or affinity within such degree as prescribed *590 by law...."); ARK.CODE ANN. §§ 5-71-229(d)(3) (defining "immediate family" for purposes of a stalking statute as one of several things including "any person related by consanguinity or affinity within the second degree"); 6-11-102(c)(2) (barring any "person who is related within the fourth degree of consanguinity or affinity to any member of the board" from serving as Commissioner of Education); 6-15-102(d)(3) (barring any "person who is related within the fourth degree of consanguinity or affinity to any member of the State Board of Education or to the commissioner" from serving as assistant commissioner); 6-21-112(c)(3) (barring any "person who is related within the fourth degree of consanguinity or affinity to any member of the commission" from serving as the Director of the Division of Public School Academic Facilities and Transportation); 10-4-404(b) (barring the confirmation of a Legislative Auditor nominee if the nominee is "related in the second degree of consanguinity or affinity to any member of the General Assembly or a constitutional officer"); 10-4-409(c)(1) (barring any "person related to any member of the General Assembly or to the Legislative Auditor in the first degree of consanguinity or affinity" from employment in the Division of Legislative Audit); XX-XX-XXX (barring from city employment any person who "is related by blood or marriage in the third degree either to a member of the board of directors or to the city manager"); XX-XX-XXX (disqualifying any justice of the Supreme Court who "is related to any party within the third degree of consanguinity or affinity"); XX-XX-XXX(b) (barring anyone who "[i]s related to any party or attorney in the cause within the fourth degree of consanguinity or affinity" from serving as a petit juror); XX-XX-XXX(h) (barring any "person who is related within the fourth degree of consanguinity or affinity to any member of the board" from serving as the director of the Department of Workforce Education). The most that can be said in response is that occasionally the Arkansas legislature and courts have used "degree" in a manner that is ambiguous but does not contradict the conclusion that it is not a term of consanguinity. See ARK.CODE ANN. §§ 9-4-102(4); 9-9-211(b); 28-9-212(b); Ark. Code Judicial Conduct, Terminology (defining "third degree of relationship"). It is, therefore, eminently clear that the use of the phrase "of every degree" is not a term of consanguinity; instead, the Arkansas legislature has frequently used "degree" when discussing a variety of relationships, including adoptive ones.
Because we conclude that "of every degree" is not a term of consanguinity, Dedman has failed to establish that "of every degree" does not include adopted grandchildren. We conclude that the best interpretation of ARK.CODE ANN. § 9-11-106(a) is that "of every degree" bars marriage between grandparents and grandchildren of any kind, including those related by adoption. As the above discussion has shown, the Arkansas legislature is plainly aware of the variety of relationships that may exist in degrees, and in this case the legislature chose to include within the statute "grandparents and grandchildren of every degree." ARK.CODE ANN. § 9-11-106(a) (emphasis added). Thus, the legislature has used the most expansive language possible. Furthermore, by default Arkansas treats adoption as creating a relationship that is "as if the adopted individual were a legitimate blood descendant ... for all purposes including inheritance and applicability of statutes...." ARK.CODE ANN. § 9-9-215(a)(2). The Arkansas marriage statute includes no language that overturns that default. In the absence of any contrary evidence, the plain reading of § 9-11-106(a) bars a marriage between a *591 grandfather and his adopted granddaughter.
Accordingly, we conclude that the district court did not err when it determined that Arkansas law forbids such a marriage.

E. Dedman's Constitutional Challenge
Dedman contends that if Arkansas law forbids a marriage between a grandfather and his adopted granddaughter, then the Arkansas marriage statute is unconstitutional. Dedman bases her claim on the Supreme Court's precedent establishing the "fundamental importance" of the right to marry. Zablocki v. Redhail, 434 U.S. 374, 383, 98 S.Ct. 673, 54 L.Ed.2d 618 (1978) (striking down as unconstitutional a Wisconsin statute that barred those paying child support from getting married without court approval). The government responds by arguing that Dedman has no standing to challenge Arkansas's marriage statute.
We need not resolve such difficult questions; Dedman raises this constitutional challenge for the first time on appeal. At trial, Dedman's counsel objected only to the interpretation of the Arkansas marriage statute, not to its constitutionality. J.A. at 237 (Tr. at 134:8-10) ("And just so the record is clear, I would object to the Court taking judicial notice because that statute doesn't address a second generation adoption."). Because Dedman raises her objections on constitutional grounds only now on appeal, we review simply for plain error. Under that standard, we conclude that the district court did not commit plain error.
While constitutional challenges are typically reviewed de novo, when the argument was not raised at the district court "Sixth Circuit precedent requires application of the plain error standard." United States v. Barton, 455 F.3d 649, 652 (6th Cir.), cert. denied, ___ U.S. ___, 127 S.Ct. 748, 166 L.Ed.2d 579 (2006). Under Rule 52(b) of the Federal Rules of Criminal Procedure, we may consider "a plain error that affects substantial rights ... even though it was not brought to the court's attention." FED.R.CRIM.P. 52(b). In applying this rule:
First, we are to consider whether an error occurred in the district court. Absent any error, our inquiry is at an end. However, if an error occurred, we then consider if the error was plain. If it is, then we proceed to inquire whether the plain error affects substantial rights. Finally, even if all three factors exist,... we must decide whether the plain error affecting substantial rights seriously affected the fairness, integrity or public reputation of judicial proceedings.
United States v. Martin, 438 F.3d 621, 628 (6th Cir.2006) (quoting United States v. Thomas, 11 F.3d 620, 630 (6th Cir.1993)). We will, therefore, apply plain-error review to Dedman's constitutional argument.
We assume arguendo that Dedman has standing to advance her constitutional claims. Similarly, we also assume for purposes of this case that Arkansas's marriage statute is unconstitutional. If the statute were unconstitutional, then the district court would have committed error when it applied the statute; but even so, we could reverse only if the error were plain. In establishing whether an error is plain, the Supreme Court has held that "[a]t a minimum, [a] court of appeals cannot correct an error pursuant to Rule 52(b) unless the error is clear under current law." United States v. Olano, 507 U.S. 725, 734, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993); see also Johnson v. United States, 520 U.S. 461, 467, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997); United States v. Thomas, 11 F.3d 620, 630 (6th Cir.1993). The question before us then becomes *592 whether any such error is plain under current law.
We conclude that the district court did not commit plain error in applying the Arkansas marriage statute even assuming that the statute is unconstitutional. Although the Colorado Supreme Court has overturned a marriage statute that forbade a marriage between two adopted siblings not related by blood on the grounds that "prohibiting marriage between adopted children fails even to satisfy minimum rationality requirements," Israel v. Allen, 195 Colo. 263, 577 P.2d 762, 764 (1978), there have been hardly any other suggestions that such statutes are unconstitutional. In fact, the Arkansas Supreme Court has upheld the constitutionality of its incest statute on the basis that sexual relations between adopted family members are just as damaging to the family as sexual relations between those related by blood. Heikkila v. State, 352 Ark. 87, 98 S.W.3d 805, 807-08 (2003) (stating that the protection of the incest law "extends to step-relationships as well as blood relationships because sexual activity in step-relationships is equally disruptive of the family as would be sexual activity between blood relations"); Camp v. State, 288 Ark. 269, 704 S.W.2d 617, 619 (1986) ("[S]tepchildren and adopted children have been added to the crime of incest because society is as concerned with the integrity of the family, including step and adoptive relationships as well as those of blood relationships, and sexual activity is equally disruptive, whatever the makeup of the family."). Such arguments upholding the constitutionality of Arkansas's incest statute would be equally applicable to marriage statutes; if a sexual relationship between adopted family members would threaten the family, so too must marriage.
Any possible unconstitutionality of Arkansas's marriage statute is not "clear under current law." Olano, 507 U.S. at 734, 113 S.Ct. 1770. To be explicit, nothing we say today should foreclose any later court from determining the constitutionality of Arkansas's marriage statute,[6] but today we hold only that a ruling of unconstitutionality would not be clear from the precedent that exists today. Therefore, we hold that even if Dedman has standing to bring this constitutional challenge, and even if the district court committed error, any such error was not plain error. Dedman's constitutional challenge thus fails.

III. SUFFICIENCY OF THE EVIDENCE

A. Standard of Review
"We review a challenge to the sufficiency of the evidence by considering the evidence in the light most favorable to the prosecution to determine whether a rational trier of fact could have found that the essential elements of the crime were proven beyond a reasonable doubt." United States v. Spearman, 186 F.3d 743, 746 (6th Cir.), cert. denied, 528 U.S. 1033, 120 S.Ct. 560, 145 L.Ed.2d 435 (1999); see also Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979) ("Instead, the relevant question is whether, after viewing the evidence in the light most favorable to the prosecution, any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt."). "A defendant making such a challenge bears a very heavy burden." *593 Spearman, 186 F.3d at 746. "[I]t is well-settled that uncorroborated testimony of an accomplice may support a conviction in federal court." Id.

B. Sufficiency of the § 286 Evidence
Dedman claims that the government did not present sufficient evidence so that a rational trier of fact could have found beyond a reasonable doubt the essential elements for a conviction under 18 U.S.C. § 286. We disagree.
Section 286 provides: "Whoever enters into any agreement, combination, or conspiracy to defraud the United States, or any department or agency thereof, by obtaining or aiding to obtain the payment or allowance of any false, fictitious or fraudulent claim, shall be fined under this title or imprisoned not more than ten years, or both." 18 U.S.C. § 286. We have never before fully elaborated upon the elements of a violation of § 286. Our only attempt at identifying the elements that the government must prove was in an unpublished order one decade ago. In that case, we said that "[t]o sustain the conspiracy conviction under 18 U.S.C. § 286, the government must prove that: 1) there was a conspiracy to defraud the United States; 2) the defendant knew of the conspiracy and intended to join it; and 3) the defendant voluntarily participated in the conspiracy." United States v. Franklin, 168 F.3d 490 (6th Cir. Nov.10, 1998) (unpublished order) (citing United States v. Okoronkwo, 46 F.3d 426, 430-31 (5th Cir. 1995)). We believe that our attempt in Franklin was not sufficient because the first element tautologically restates the crime rather than separating out any individual elements. It was perhaps for that same reason that the Fifth Circuit, the circuit upon which we relied in Franklin, subsequently expanded upon the set of elements necessary to support a conviction under § 286. In United States v. Leahy, 82 F.3d 624, 633 (5th Cir.1996), the Fifth Circuit held that "[t]o obtain a conviction for conspiracy to defraud the United States under § 286, the government must prove beyond a reasonable doubt that the defendant entered into a conspiracy to obtain payment or allowance of a claim against a department or agency of the United States; that the claim was false, fictitious, or fraudulent; and that the defendant knew at the time that the claim was false, fictitious, or fraudulent."
The Fifth Circuit's Leahy approach is not the only one available. The Eleventh Circuit has also offered its interpretation: "This court will sustain a conviction for conspiracy to submit false claims to the United States, if the government proved the existence of an agreement to achieve an unlawful objective, the defendant's knowing and voluntary participation in the conspiracy, and the commission of an overt act in furtherance of it." United States v. Gupta, 463 F.3d 1182, 1194 (11th Cir.2006) (footnote omitted) (internal quotation marks omitted), cert. denied, ___ U.S. ___, 127 S.Ct. 2446, 167 L.Ed.2d 1132 (2007). With the exception of the addition of an overt-act requirement, the Eleventh Circuit's approach strikes us as suffering from the same malady of vagueness as our approach in Franklin.
After considering our sister circuits' efforts, we conclude that best approach is a hybrid of the Franklin and Leahy approaches. Accordingly, we clarify that the elements necessary for a conviction under § 286 are: (1) the defendant entered into a conspiracy to obtain payment or allowance of a claim against a department or agency of the United States; (2) the claim was false, fictitious, or fraudulent; (3) the defendant knew or was deliberately ignorant of the claim's falsity, fictitiousness, or fraudulence; (4) *594 the defendant knew of the conspiracy and intended to join it; and (5) the defendant voluntarily participated in the conspiracy.[7] Because the district court's instructions to the jury were not in contradiction with our clarification of the elements necessary to convict under § 286, we proceed to address Dedman's sufficiency-of-the-evidence claim.
There is no doubt that the government introduced sufficient evidence to sustain a conviction as to elements one, four, and five. The invalidity of the marriage satisfies the second element. As to element three, however, Dedman contends that the government failed to introduce sufficient evidence because it did not establish that she knew that the marriage was illegal and, therefore, she could have not known that the claim was false. The logical puzzle that Dedman provides us is this: if she knew that the marriage was illegal in Arkansas, why did she travel to Arkansas in the first place, if the marriage would have been legal in her home state of Kentucky? See KY.REV.STAT. ANN. § 402.010(1) (prohibiting marriage "between persons who are nearer of kin to each other by consanguinity, whether of the whole or half-blood, than second cousins").
Dedman's argument, while certainly creative, does not establish that the government failed to produce sufficient evidence to sustain her conviction. What Dedman's challenge asks us to determine is whether knowledge that a claim is false, fictitious, or fraudulent requires evidence establishing that Dedman knew that the claim was illegal. We conclude that actual knowledge of the illegality of a claim is not required; deliberate ignorance of the falsity of the claim is sufficient to establish knowledge of the false, fictitious, or fraudulent nature of the claim.
While the government was not required to produce evidence establishing that Dedman knew that the law forbade the type of claim she was making, the government was required to introduce evidence that she suspected that her claims were false yet did nothing to investigate the legality of her actions. Although it may seem contradictory that a defendant could not know that a claim was illegal but still believe it was likely she was violating the law, the scienter requirement for a conviction under the similar 18 U.S.C. § 287[8] demonstrates that such a state of knowledge is indeed possible. For instance, we have upheld convictions under § 287 where knowledge of falsity was not established at all, but where the government instead established deliberate ignorance. United States v. Holloway, 731 F.2d 378, 381 (6th Cir.1984) ("[T]his circuit has repeatedly *595 upheld the district court's knowledge instruction on the basis that it prevents a criminal defendant from escaping conviction merely by deliberately closing his eyes to the obvious risk that he is engaging in unlawful conduct. Accordingly, we hold that the instruction on deliberate ignorance was not erroneous." (citations omitted)). Other circuits have adopted a similar standard. United States v. Patient Transfer Serv., Inc., 413 F.3d 734, 742 (8th Cir.2005) ("There was evidence at trial that Wise actually knew that the billing practice being used at PTS was not proper or deliberately avoided investigating whether it was." (emphasis added)); United States v. Nazon, 940 F.2d 255, 258-59 (7th Cir.1991) ("You may infer knowledge from a combination of suspicion and indifference to the truth. If you find that a person had a strong suspicion that things were not what they seemed or that someone had withheld some important facts, yet shut his eyes for fear of what he would learn, you may conclude that he acted `knowingly' ...." (quoting approvingly from the district court's instructions)); United States v. Precision Med. Labs., Inc., 593 F.2d 434, 443-44 (2d Cir.1978) ("`(Y)ou may find Mr. Gall acted willfully and knowingly if you find that the claim was false and that Mr. Gall knew that the claims were false, or that he acted with a deliberate disregard of whether the claims were true or false; in other words, that he closed his eyes to the truth here. ...'" (quoting approvingly from the district court's instructions yet noting that the "willfully" language only increased the burden for the government even though it was perhaps unnecessary)).[9]
Although the cases regarding § 287 are only instructive to us when considering the requisite scienter for § 286, these § 287 cases demonstrate that deliberate ignorance can establish knowledge of falsity. In such cases, the defendant certainly does not know that what she is doing is illegal, but the defendant still has some sense that what she is doing is wrong and makes no effort to determine the propriety of her actions. Taking the evidence in the light most favorable to the prosecution, we believe that the totality of Dedman's actionsa pattern of skulking and schemingestablish that she had a sense that what she was doing was wrong, yet closed her eyes to that fact and proceeded with her plan. Accordingly, the government adduced sufficient evidence to show that Dedman knew or was deliberately ignorant of the falsity of her claims.
The government presented evidence sufficient that a rational factfinder could conclude that Dedman was at least deliberately ignorant of the falsity of her claim, if not possessing actual knowledge of the falsity. Although the reasons for holding the wedding in Arkansas, as opposed to Kentucky, might not be immediately clear, we disagree with the dissent's view that "[t]he likelihood that Dedman picked Arkansas knowing or even suspecting that the marriage would be void simply defies common sense." Dissenting Op. at 604. In fact, we believe that the trip to Arkansas is the government's strongest evidence in support of the conclusion that Dedman was deliberately ignorant of the falsity of her claims. Holding the marriage in Arkansas was plainly done to hide the existence of *596 the marriage from those in Kentucky who might know them. It is likely that, recognizing the absurdity and likely illegality of her claims, Dedman assumed that the marriage would be equally void and equally false in Arkansas as in Kentucky. Thus, Dedman likely viewed the choice between Arkansas and Kentucky as a choice between two locations in which the marriage would be illegal, and she picked the location that was farther from her home and her neighbors. In that case, a rational juror could believe that Dedman suspected the falsity of her claims and could view the trip as an attempt to assure secrecy for the relationship by procuring the illicit nuptials far from home, where the participants would not be known.
In addition to the location of the marriage, the government presented additional evidence that would support a finding of deliberate ignorance on the part of Dedman. For instance, the government presented testimony from both Holland and a family friend that suggested that Dedman carefully orchestrated the marriage in order to obtain money from the SBP, including a statement from a family friend about Dedman's comment that she had "solved the situation with the annuity." J.A. at 172 (Green at 171:13-22). Nothing presented during the trial controverted this damaging testimony. A rational juror could conclude that, because the marriage was Dedman's idea, she at least possessed some knowledge of the rules governing the SBP and thus understood the fraudulent nature of what she was trying to accomplish. Furthermore, the government provided uncontradicted evidence to establish that once Dedman turned on Holland, Dedman called various government agencies to inform them that Holland was defrauding the government. A rational juror could easily conclude that this indicates that if Dedman did not know that her SBP claim was false, she was at least pretty sure that it was false. In addition, when the government began to investigate the claims, Dedman invoked the phrase "a person of insurable interest," which is a term taken right out of the statute governing the SBP. J.A. at 189 (Quellhorst at 27:8-14); 10 U.S.C. § 1448(b). From that fact, a rational juror could certainly conclude that Dedman had more than a passing familiarity with the SBP's legal regime. It is, therefore, apparent to us, viewing the evidence in the light most favorable to the government, that a rational juror could certainly conclude beyond a reasonable doubt from the evidence presented that Dedman did possess knowledge of the falsity of the SBP claims.
The heart of the dissent's disagreement with our holding today is not over a question of law but of fact. The dissent claims that "there is no evidence that the defendant knew or was deliberately ignorant of what made her claim legally false." Dissenting Op. at 606. We disagree. As we have just indicated, there is no shortage of proof supporting a finding that Dedman was deliberately ignorant of the falsity of her claims, particularly when we take the evidence in the light most favorable to the prosecution, as we are bound to do. Furthermore, the dissent errs when it claims that Dedman's bad intent "provides no proof" supporting her conviction, id., because her bad intent is probative as to whether she acted in deliberate ignorance of the falsity of her claims. That she intended to collect the SBP funds and did so suspecting that her claims were false what could collectively be called her bad intentis what triggers the finding of deliberate ignorance and satisfies the third element of a § 286 offense.
Before moving on, it is worth clarifying the distinction between our holding here and the government's sham-marriage argument that we previously discussed. The *597 government, in its sham-marriage theory, claimed that a jury could convict Dedman for a violation of § 286 solely on the basis of her nefarious intent. In other words, the government claimed that she was guilty simply because Holland and Watson did not enter their marriage out of love; the government's argument did not depend upon a court finding the marriage invalid. Our conclusion in this section that there was sufficient evidence to establish that Dedman knew or was deliberately ignorant of the falsity of the claimand therefore sufficient evidence to convict Dedman does not rely on the government's theory. As we stated at the beginning of this section, a conviction under § 286 requires that the government prove five elements: (1) the defendant entered into a conspiracy to obtain payment or allowance of a claim against a department or agency of the United States; (2) the claim was false, fictitious, or fraudulent; (3) the defendant knew or was deliberately ignorant of the claim's falsity, fictitiousness, or fraudulence; (4) the defendant knew of the conspiracy and intended to join it; and (5) the defendant voluntarily participated in the conspiracy. Our focus in this section has been the third element; in contrast, the purpose of the government's sham-marriage theory was to make element two unnecessary. Our interpretation of the elements of § 286 requires both actual fraud (satisfied by the invalidity of the marriage) and knowledge or deliberate ignorance of the falsity or fraudulent nature of the claim (which we have discussed in this section). Dedman's intent to create a sham marriage helps to establish that she knew or was deliberately ignorant of the falsity of the claim. The government's sham-marriage theory, however, addresses only Dedman's intent, making it necessary for us to consider whether the marriage was valid. Accordingly, the fact that the marriage was a sham (i.e., the couple did not live or act as a married couple) might be probative of the knowledge or deliberate ignorance component of § 286 but would not be sufficient for us to uphold this conviction.
It should, therefore, be apparent that Dedman's conviction is in part the result of bad luck. Had Dedman, with the same intent, taken Holland and Watson to marry in Kentucky instead of Arkansas, the marriage would have been valid and there would have been no false claim. In fact, under the law we apply in this case, any person with nothing but the worst motives could enter into a marriage that lasts for over a year immediately preceding death and qualify for the SBP without triggering a violation of § 286; there could be no false claim as long as the marriage was valid. Such a hypothetical case, however, is not before us. We therefore conclude that there was sufficient evidence to support Dedman's conviction under § 286.

C. The Sufficiency of the § 1001 Evidence
Dedman also alleges that the government failed to produce sufficient evidence to sustain a conviction under 18 U.S.C. § 1001. That statute provides:
Except as otherwise provided in this section, whoever, in any matter within the jurisdiction of the executive, legislative, or judicial branch of the Government of the United States, knowingly and willfully
(1) falsifies, conceals, or covers up by any trick, scheme, or device a material fact;
(2) makes any materially false, fictitious, or fraudulent statement or representation; or
(3) makes or uses any false writing or document knowing the same to contain any materially false, fictitious, *598 or fraudulent statement or entry;
shall be fined under this title, imprisoned not more than 5 years. ...
18 U.S.C. § 1001(a). "A violation of § 1001 is comprised of five elements: (1) the defendant made a statement; (2) the statement is false or fraudulent; (3) the statement is material; (4) the defendant made the statement knowingly and willfully; and (5) the statement pertained to an activity within the jurisdiction of a federal agency." United States v. Lutz, 154 F.3d 581, 587 (6th Cir.1998).
In order to convict Dedman, the government provided evidence of three different statements that Dedman made: (1) a statement about Holland's location, (2) a statement about Holland's disability, and (3) a statement about when Dedman first learned about the marriage between Holland and Watson. When instructing the jury on the § 1001 claim, the district court told the jury that "[t]he government does not have to prove all three of these for you to return a guilty verdict on this charge. Proof beyond a reasonable doubt of any one of these ways is enough. In order to return a guilty verdict, all twelve of you must agree that at least one, and the same one, of these has been proved." J.A. at 70 (Instruction No. 27). Dedman now challenges the sufficiency of the government's evidence, claiming that the statement about Holland's location was not material, the statement about Holland's disability was not proved to be false, and the statement about when Dedman first learned about the marriage may have been a misunderstanding on Dedman's part. Even if Dedman is correct as to the comments regarding Holland's location and Holland's disability, we conclude that there was sufficient evidence for a jury to convict Dedman for false statements regarding when she first learned of the marriage.
If the government alleged three false statements but two lacked sufficient evidence, how do we know which false statement the unanimous jury used? We are certainly concerned that the jury may have relied upon an unsupported statement, but our precedent requires that we presume that the jury used the false statement that was supported by sufficient evidence, and that we uphold a conviction where there was sufficient evidence for at least one of the alleged false statements. In 1953, we addressed a similar issue when we concluded that when "[t]he indictment charged that the statement was false in five different respects," then "[i]f the Government's evidence proved that it was false in only one of the respects charged, it was nevertheless a false statement, and such proof would have sustained a verdict of guilty." Stevens v. United States, 206 F.2d 64, 66 (6th Cir.1953) (upholding a conviction under 18 U.S.C. § 1001). Stevens, however, did not resolve the issue at hand because it dealt with one statement that was in false in several different ways, while the instant case deals with several different allegedly false statements that were grouped into one count of the indictment. In 1970, the Supreme Court suggested that courts should uphold verdicts when multiple acts are alleged in one count and there is sufficient evidence as to one of the acts. Turner v. United States, 396 U.S. 398, 419-21, 90 S.Ct. 642, 24 L.Ed.2d 610 (1970) ("The general rule is that when a jury returns a guilty verdict on an indictment charging several acts in the conjunctive... the verdict stands if the evidence is sufficient with respect to any one of the acts charged." (footnote omitted)). Turner, however, was not dispositive of the issue before us, because the statute at issue, 26 U.S.C. § 4704(a), covered multiple forms of behavior that would trigger a violation (purchasing, selling, dispensing, *599 or distributing narcotics), and the indictment alleged acts under each; in contrast, in the case at hand, only one form of a violation was alleged under § 1001, but the indictment identified many possible false statements. Thus, Turner left open the exact question before us. In 1989, in United States v. Naserkhaki, 722 F.Supp. 242 (E.D.Va.1989), the Eastern District of Virginia concluded that a general verdict under § 1001 must be reversed if there is a chance that the jury may have relied upon insufficient factual grounds:
Since only one of defendant's misstatements satisfies Section 1001's materiality requirement, and since the jury rendered a general verdict, it is not possible to determine whether the verdict rests in whole, or in part, on the immaterial misstatement. Accordingly defendant's conviction is infected with error and cannot stand. Abundant authority requires that "a general verdict must be set aside if the jury was instructed that it could rely on any of two or more independent grounds, and one of those grounds is insufficient, because the verdict may have rested exclusively on the insufficient ground."
Id. at 249 (quoting Zant v. Stephens, 462 U.S. 862, 881, 103 S.Ct. 2733, 77 L.Ed.2d 235 (1983)). The Supreme Court, however, disagreed. In 1991, citing the same cases that Naserkhaki called its "abundant authority," compare id. with Griffin v. United States, 502 U.S. 46, 54-57, 112 S.Ct. 466, 116 L.Ed.2d 371 (1991), all eight participating Justices agreed that there was no basis for "set[ting] aside a general verdict because one of the possible bases of conviction was neither unconstitutional as in Stromberg [v. California, 283 U.S. 359, 51 S.Ct. 532, 75 L.Ed. 1117 (1931),] nor even illegal as in Yates [v. United States, 354 U.S. 298, 77 S.Ct. 1064, 1 L.Ed.2d 1356 (1957),] but merely unsupported by sufficient evidence." Griffin, 502 U.S. at 56, 112 S.Ct. 466; id. at 60-61, 112 S.Ct. 466 (Blackmun, J. concurring in the judgment). The reason that the Supreme Court found this untroubling was that, although jurors would not be well equipped to identify a legally deficient argument, "when they have been left the option of relying upon a factually inadequate theory," there is little danger, "since jurors are well equipped to analyze the evidence." Id. at 59, 112 S.Ct. 466. "[J]urors can rely on their own intelligence and experience to save them from relying upon a factually inadequate theory." United States v. Henning, 286 F.3d 914, 921 (6th Cir.2002). Thus, as long as there is no evidence to the contrary, we are to "presum[e] that jurors convicted on the factually sufficient theory." Id. at 922.
According to our precedent, if there was sufficient evidence to support one of the false statements that the government alleged, then we must presume that the jury relied on that false statement, and we must uphold the conviction. We conclude that there was ample evidence for a rational trier of fact to conclude beyond a reasonable doubt that Dedman was lying when she said that she found out about the marriage only in 2004. The testimony of Holland, Green, and the federal investigator all suggested that Dedman knew about the marriage from the moment of the nuptials, and that evidence was never controverted. Although the government investigator conceded that it was possible Dedman misunderstood the question and meant only that she first learned about the illegality of the marriage in 2004, the jury could have decided easily that Dedman understood the question. We, therefore, reject Dedman's claim that the evidence was insufficient to support *600 her conviction under § 1001.[10]

IV. THE JURY INSTRUCTIONS

A. Standard of Review
"`The standard on appeal for a court's charge to the jury is whether the charge, taken as a whole, fairly and adequately submits the issues and applicable law to the jury.'" Buckley, 934 F.2d at 87 (quoting United States v. Martin, 740 F.2d 1352, 1361 (6th Cir.1984)). A party must object to an instruction "before the jury retires to deliberate." FED.R.CRIM.P. 30(d). If the party fails to object, the instruction is reviewed only for plain error. United States v. DeJohn, 368 F.3d 533, 540 (6th Cir.2004). "Where, in formulating instructions, the `district court engages in statutory construction as a matter of law,... [the Sixth Circuit] review[s][the] conclusions de novo.'" Buckley, 934 F.2d at 87-88 (alterations in original) (quoting United States v. Brown, 915 F.2d 219, 223 (6th Cir.1990)).

B. Analysis
Dedman asserts that the district court improperly instructed the jury on its determination of Arkansas law and claims that the instruction amounted to a directed verdict against her. We find this argument to be without merit.
The very case that Dedman cites as support for her directed-verdict argument stated in no uncertain terms that "[t]he trial judge instructs the jury on the law applicable to the issues raised and, in appropriate circumstances, may comment on the evidence." United States v. Mentz, 840 F.2d 315, 319 (6th Cir.1988) (footnote *601 omitted) (holding that a trial court judge can instruct the jury on the law but must leave findings of fact to the jury). In the instant case, the district court did not decide facts for the jury but merely stated the law of Arkansas. That Arkansas law forbids such a marriage may have made it likely that a jury would convict Dedman, but it was not, as Dedman argues, equivalent to a judicial ruling that Dedman violated § 286. As we have already discussed, in order to obtain a conviction the government still needed to establish that: (1) the defendant entered into a conspiracy to obtain payment or allowance of a claim against a department or agency of the United States; (2) the claim was false, fictitious, or fraudulent; (3) the defendant knew or was deliberately ignorant of the claim's falsity, fictitiousness, or fraudulence; (4) the defendant knew of the conspiracy and intended to join it; and (5) the defendant voluntarily participated in the conspiracy. This jury instruction, which stated that "I also have decided to accept as proved that a marriage between a grandfather and his adopted granddaughter is prohibited by Arkansas state law," J.A. at 66 (Instruction No. 23), established only that the marriage in question was void. The natural consequence of the district court's determination of Arkansas law was that Dedman's claim was false, but the instruction did not establish any of the other necessary elements of the offense. Because Dedman did not object to the instruction at trial, we review for plain error and we conclude that the district court did not plainly err when it properly instructed the jury as to the applicable law.

V. THE LOSS CALCULATION

A. Standard of Review
Whether to apply a particular guideline provision is a legal question reviewed de novo. United States v. Triana, 468 F.3d 308, 321 (6th Cir.2006); United States v. Comer, 93 F.3d 1271, 1278 (6th Cir.), cert. denied, 519 U.S. 1033, 117 S.Ct. 595, 136 L.Ed.2d 523 (1996).

B. Analysis
Dedman contends that the district court erred when it calculated the government's loss by including the income taxes that were withheld from the SBP payments in addition to the net-of-tax payments actually received. The resolution of this question is important because if withheld taxes are not included, then the government's total loss would be below $200,000, and Dedman would receive an offense level addition of only ten levels. U.S.S.G. § 2B1.1(b)(1)(F). If, however, withheld taxes are included, then the total loss would exceed $200,000, and Dedman would receive an offense level addition of twelve levels. U.S.S.G. § 2B1.1(b)(1)(G). Whether withheld taxes can be considered part of the government's loss in calculating a sentence enhancement under U.S.S.G. § 2B1.1 is a question that neither this court, nor any other, appears to have addressed. As a matter of first impression, we conclude that the withheld taxes were properly included in the loss determination.
In resolving this novel issue, we construe the Guidelines. According to an application note to the 2005 Guidelines, "loss is the greater of actual loss or intended loss." U.S.S.G. § 2B1.1 cmt. 3(A). An application note defines "actual loss" as the "reasonably foreseeable pecuniary harm that resulted from the offense." U.S.S.G. § 2B1.1 cmt. 3(A)(i). In contrast, "intended loss" is the "pecuniary harm that was intended to result from the offense" and it includes "intended pecuniary harm that would have been impossible or *602 unlikely to occur." U.S.S.G. § 2B1.1 cmt. 3(A)(ii).
The Guidelines' comments on government benefits provide some further guidance. When addressing government benefits, an application note states that "loss shall be considered to be not less than the value of the benefits obtained by unintended recipients or diverted to unintended uses, as the case may be." U.S.S.G. § 2B1.1 cmt. 3(F)(ii) (emphasis added). This provision is helpful in two ways. First, by saying "not less than the value of benefits obtained," the Guidelines suggest that the dollar amount that the defendant receives serves as a minimum value for the government's loss, but that the government's loss can exceed that value. In this case, that permits the court to consider the withheld taxes. The government-benefits commentary also suggests that the "value of the benefits obtained" is inclusive of more than simply the sum added to the defendant's bank account. Because the SBP payments may be subject to income taxation, the only way a defendant could receive in excess of $200,000 is to file a claim for significantly more money. Thus, it can be said that Dedman received $230,000 worth of value even if she received only slightly more than $200,000 in checks; using the amount of the claim, rather than simply the net amount paid to her, reflects a better measure of the value of her true benefit. Cf. Triana, 468 F.3d at 323 ("[C]ourts have consistently held that in calculating loss, `substitution of defendants' gain is not the preferred method because it ordinarily underestimates the loss.'" (quoting United States v. Snyder, 291 F.3d 1291, 1295 (11th Cir.2002))).
That the amount pocketed by Dedman underestimates the government's true loss is further confirmed by considering the SBP's gross payments. The SBP may have only paid to Dedman an amount less than $200,000, but it also paid a considerable sum to state and federal tax agencies on Dedman's behalf. Dedman claims that withheld taxes should not be included in the government's loss calculation as though the withheld funds simply stayed with the SBP, ignoring the fact that the SBP paid the withheld taxes to the appropriate taxing agencies. Thus, the withheld taxes should be considered part of the government's loss. This analysis applies equally to Dedman if the withheld taxes were paid on Holland's behalf for any payments attributed as income to Holland.
To exclude the withheld taxes from the calculations of loss would create unfairness under the Guidelines. The Guidelines tie culpability to the length of punishment, and exclusion of withheld taxes would create the possibility that less culpable defendants would be punished more. See United States v. McBride, 362 F.3d 360, 375 (6th Cir.2004) (citing U.S.S.G. § 2B1.1). If withheld taxes were not included in the loss calculation, then someone who filed a false claim for $200,000.01 for some non-taxable benefit would be punished more than someone who filed a false claim for $230,000 that was subject to $30,000 in withheld taxes. Thus, not to include withheld taxes would be to break the Guidelines' intended correlation between intended loss and the amount of punishment.
Recently, in United States v. Coker, 514 F.3d 562 (6th Cir.2008), we dealt with a case where government prosecutors allowed the government loss attributed to one defendant to be reduced by the amount that the defendant paid in taxes on the fraudulent payments. That case does not alter our analysis in this case. In Coker, the government volunteered to reduce its loss by the amount of taxes that the defendant paid, so the loss calculation was never at issue in that case. Furthermore, Coker dealt with another defendant *603 in the same scheme, not the defendant who received a favorable loss calculation, so the validity of the loss calculation was not before the court. Most importantly, we acknowledged in that case that "[t]he prosecutor's decision to give Walsh a credit for the taxes he paid as `money returned to the victim' under U.S.S.G. § 2B1.1 comment (n.2(E)(i)) was not required by the Guidelines." Coker, 514 F.3d at 569 (emphasis added).
For all of these reasons, we therefore hold that withheld taxes are properly considered part of the government's loss under § 2B1.1. We affirm the district court's calculations of the amount of loss.

VI. CONCLUSION
In summary, we conclude that the government cannot rely on its sham-marriage theory because the district court did not instruct the jury as to that argument, and that necessitates that we determine the validity of the marriage between Watson and Holland. We determine that the district court correctly decided that the marriage was void under Arkansas state law. We further conclude that even if the Arkansas marriage statute were unconstitutional, it was not plain error for the district court to apply it. We also determine that there is sufficient evidence to support Dedman's convictions under 18 U.S.C. §§ 286 and 1001. We conclude that the district court's jury instruction regarding Arkansas law is not plainly erroneous. And finally, we hold that withheld taxes are properly considered part of the government's loss under U.S.S.G. § 2B1.1. For all of these reasons, we AFFIRM Dedman's convictions and sentence.
RONALD LEE GILMAN, Circuit Judge, dissenting.
The arranged marriage of Dedman's adoptive father to her adopted daughter was a sham if ever there was one. And because the marriage was void under Arkansas law, Holland (the adopted daughter) is obligated to repay the SBP benefits to which she was not entitled. In sum, the scheme cooked up by Dedman and Holland to continue receiving Watson's military pension benefits after his death carries the pungent aroma of "fraud" in the general sense of the word.
But the "ickiness" of the scheme (i.e., that a grandfather would marry his granddaughter) does not answer the question of whether Dedman violated 18 U.S.C. § 286. The fact that she "gamed the system," in other words, does not necessarily make her guilty of violating the statute.
We first need to be clear about what § 286 makes illegal. The majority has accurately set out the elements of the offense. (Maj. Op. at 593-94) I disagree, however, with the majority's conclusion that the government proved the third element of the offensethat Dedman either knew or was deliberately ignorant of the fact that the claim for benefits was false, fictitious, or fraudulent. In particular, I focus on the relationship between the second element of the offensethat the claim for benefits was in fact false, fictitious, or fraudulentand the knowledge component of the third element. To sustain a conviction under § 286, the government must produce sufficient evidence to prove that Dedman had knowledge (or was deliberately ignorant) of the very thing that made the claim false, fraudulent, or fictitious.
The majority explicitly states that the second element of the offense is "satisfied by the invalidity of the marriage" under Arkansas law, and I agree. (Maj. Op. at 597) But the majority's contention that Dedman either knew or was deliberately ignorant of the fact that the claim was false is based on nothing more than evidence indicating that the marriage between *604 Holland and Watson was in fact a "sham," entered into solely for the purpose of obtaining Watson's SBP benefits. As the majority explains, "the totality of Dedman's actionsa pattern of skulking and schemingestablish that she had a sense that what she was doing was wrong, yet closed her eyes to that fact and proceeded with her plan." (Maj. Op. at 595) This evidence indeed suggests that Dedman was intent on perpetrating a "fraud," but it has no probative force regarding whether Dedman knew that the marriage was illegal under Arkansas law. With no evidence indicating that Dedman knew or was deliberately ignorant of the fact that her claim for benefits was fraudulent because the marriage was illegalas opposed to being fraudulent because the marriage was a shamher conviction cannot be sustained.
One has to wonder why Dedman would take Holland and Watson to Arkansas to enter a marriage that she allegedly either knew or suspected was illegal in that state. As the majority acknowledges, Dedman would not have violated § 286 if Holland had been married in Kentucky, where such a marriage of those related only by adoption is legal. (Maj. Op. at 597) The likelihood that Dedman picked Arkansas knowing or even suspecting that the marriage would be void simply defies common sense. Instead, her most likely motive was to have the ceremony far away from home so that her neighbors would not learn of the marriage.
Dedman's ex-husband in fact testified at the trial that the wedding took place in Arkansas when Dedman went to pick up her ailing mother, who lived there. Holland also testified that Dedman had told her to keep the marriage secret "because [Dedman] didn't want [Watson] to look like a cradle robber and she didn't want me to look like a golddigger." As the only person present at the wedding who testified at trial, Holland said that she did not know that there was anything legally wrong with the marriage. Any suggestion that Dedman was deliberately ignorant of Arkansas law thus lacks evidentiary support and is a far cry from proof beyond a reasonable doubt.
This leaves only the question of whether proof of procuring a "sham" marriage can violate § 286 even if the marriage is otherwise legal. The Supreme Court answered that question in the negative in the analogous context of Social Security survivor benefits in Weinberger v. Salfi, 422 U.S. 749, 95 S.Ct. 2457, 45 L.Ed.2d 522 (1975). Weinberger involved a class-action challenge to the constitutionality of the Social Security Act's nine-month duration-of-relationship requirement for a widow to receive survivor benefits. Salfi had been married for only six months when her husband died, but she claimed that her marriage was otherwise valid and should entitle her to her deceased husband's Social Security benefits. A three-judge panel of the district court declared the nine-month requirement unconstitutional on the basis that the requirement "constitutes a presumption that marriages ... which did not precede the wage earner's death by at least nine months were entered into for the purpose of securing Social Security benefits." Id. at 767-68, 95 S.Ct. 2457 (quoting Salfi v. Weinberger, 373 F.Supp. 961, 965 (N.D.Cal.1974)).
In reversing the district court, the Supreme Court focused on Congress's intent to impose a prophylactic duration-of-relationship requirement as opposed to individual determinations regarding the good-faith intentions of purported "widows." The Court stated that the duration-of-relationship requirement "operates to lessen the likelihood of abuse through sham relationships entered in contemplation of imminent death." Id. at 780, 95 S.Ct. 2457. *605 It also noted that individual determinations are often flawed because of uncertainty that they "could effectively filter out sham arrangements, since neither marital intent, life expectancy, nor knowledge of terminal illness has been shown ... to be reliabl[y] determinable." Id. at 782-83, 95 S.Ct. 2457. Ultimately, the Court rejected the call for individual determinations of the validity of marriages and the good-faith intentions of the spouses, explaining as follows:
[T]he duration-of-relationship requirement represents not merely a substantive policy determination that benefits should be awarded only on the basis of genuine marital relationships, but also a substantive policy determination that limited resources would not be well spent in making individual determinations. It is an expression of Congress' [s] policy choice that the Social Security system, and its millions of beneficiaries, would be best served by a prophylactic rule which bars claims arising from the bulk of sham marriages which are actually entered, which discourages such marriages from ever taking place, and which is also objective and easily administered.
Id. at 784-85, 95 S.Ct. 2457. There is no reason to believe that the Supreme Court would reach a different conclusion in the analogous context of SBP benefits. I thus agree with the majority that Dedman's improper motives are irrelevant in proving that the claim was false, fictitious, or fraudulent.
But this inexorably leads me to the heart of the problem with the majority's analysis. Because there is no good-faith marriage requirement under the SBP, there are only two circumstances under which a surviving spouse such as Holland would not be entitled to SBP benefits: (1) where the marriage is invalid under state law, or (2) where a valid marriage does not meet the one-year durational requirement. Dedman's conviction is unsustainable because her intended fraudthe arrangement of a sham marriage between her adoptive father and her adopted daughterdid not disqualify Holland for SBP benefits. The only thing that would have made Dedman's conduct false, fraudulent, or fictitious under the SBP is proof beyond a reasonable doubt that she either knew or was deliberately ignorant of the fact that the marriage was invalid under Arkansas law.
Despite the majority's conclusion "that Dedman was deliberately ignorant of the falsity of her claims" (Maj. Op. at 596), it is unable to point to any evidence that Dedman suspected that the marriage in Arkansas would be invalid. The majority's statement that "[i]t is likely that ... Dedman assumed that the marriage would be equally void and equally false in Arkansas as in Kentucky," id. at 596-97, finds no support in the record and is in fact contrary to the evidence that exists on this key issue. First of all, the majority itself acknowledges that the marriage would have been perfectly legal in Kentucky. (Maj. Op. at 597) Second, Holland testified that she had no reason to believe that her marriage to Watson in Arkansas was invalid. There is no evidence to the contrary.
I am thus unpersuaded by the majority's ultimate conclusion that Dedman's "bad intent ... is what triggers the finding of deliberate ignorance and satisfies the third element of a § 286 offense." (Maj. Op. at 596) Dedman's "bad intent" was simply to orchestrate a sham marriage, but this provides no proof that she had any reason to believe that the marriage in Arkansas was invalid. In fact, her bad intent would have been the same regardless of whether the marriage took place in Kentucky (where it *606 would have been valid) or Arkansas. Moreover, Dedman had every reason to want the marriage to be valid in order for her to continue reaping the benefits of her adoptive father's military pension after he died.
The majority's conjecture, therefore, about Dedman's state of mind regarding the validity of the marriage is insufficient to support her conviction. In the end, my disagreement with the majority is fundamentally a dispute about what evidence is legally required to establish the third element of a § 286 offense where, as in the present case, there is no evidence that the defendant knew or was deliberately ignorant of what made her claim legally false. Although we are required to view the evidence in the light most favorable to the prosecution, I find the record bereft of such evidence here. Accordingly, I believe that Dedman has been wrongly convicted of this offense.
I now turn to another issue in this casethe amount of loss attributed to Dedman. On this question I agree with the result reached by the majority, but not with its reasoning. Including withheld taxes in the government's loss calculation will generally lead to punishing a defendant for a nonexistent loss because taxes that are withheld from a government payment are simply redirected to the United States Treasury. In Dedman's case, however, the money she obtained was from a specific program administered by an agency that undoubtedly has an annual budget appropriated by Congress. Paying SBP benefits to an improper recipient thus has a detrimental effect on the funds allotted to the SBP. Although the withheld taxes were redirected from the SBP to the Treasury, we have no way of knowing what happened to the money subsequently. The withheld taxes from the benefits received by Dedman and Holland were therefore "diverted to unintended uses" despite remaining under the control of the government. See U.S.S.G. § 2B1.1 cmt. 3(F)(ii) (explaining that in the context of a loss calculation for government benefits, "loss shall be considered to be not less than the value of the benefits obtained by unintended recipients or diverted to unintended uses, as the case may be").
In my opinion, of course, we should not even reach the amount-of-loss issue. We should instead reverse Dedman's conviction due to the government's failure to prove beyond a reasonable doubt that she either knew or was deliberately ignorant of the fact that Holland's marriage was illegal under Arkansas law. I therefore respectfully dissent.
NOTES
[1]  We are skeptical as to the validity of the government's sham-marriage theory, and as we will discuss later on, we believe that 18 U.S.C. § 286 requires a false, fictitious, or fraudulent claimthe invalid marriage, in this casein addition to knowledge of the falsity of the claim. The evidence that would support the government's sham-marriage theory might support a finding that Dedman believed her claims were false, but it says nothing about whether the claims were actually false. Regardless, at no point has the government identified a single case or statutory provision that supports its sham-marriage theory that a marriage entered into for improper motives would somehow, by itself, create a false claim under the SBP. The statute regulating the disbursement of the SBP requires only that a couple be validly married according to the state administering the marriage and that the marriage have lasted at least one year immediately before the pensioner's death; the statute says nothing about the couple's motives. See 10 U.S.C. § 1447(7); § 1450(a). Similarly, the Social Security Act defines "wife" and "husband," at least in part, as those married for "not less than one year immediately preceding [the application date]." 42 U.S.C. §§ 416(b) and (f). In the context of Social Security benefits, the Supreme Court has concluded that the time limit itself serves to deter sham marriages, see Weinberger v. Salfi, 422 U.S. 749, 767-85, 95 S.Ct. 2457, 45 L.Ed.2d 522 (1975), and at least one court has refused to add to the Social Security Act an unstated good-faith requirement. See Eisenhauer v. Mathews, 535 F.2d 681, 684 (2d Cir. 1976). Although Congress has chosen to deter sham marriages through a good-faith test in the immigration context, see 8 U.S.C. § 1154, for Social Security and military-pension benefits Congress has instead opted to use a length-of-marriage requirement. Given that the SBP statute requires a year of marriage immediately before death to deter sham marriages, it seems wholly inappropriate to supplement Congress's sham-marriage deterrent with our own judicially created deterrent of a good-faith standard.

The SBP statute does not provide a basis for the sham-marriage theory, and no courts have ever indicated that such a sham-marriage theory would support a conviction under § 286. We could find only dicta (and in only one case) that even suggests that the motive for a marriage might be a factor in determining eligibility for the SBP. See Munson v. United States, 30 Fed.Cl. 830, 834 (1994). Without grounding in either statute or caselaw, we are reluctant to conclude that there is any basis for the government's sham-marriage argument; under these circumstances, we believe that the government was required to establish that Holland and Watson's marriage was invalid.
[2]  For simplicity's sake, we refer to the Holland-Watson marriage as one between a grandfather and his adopted granddaughter. More specifically, however, what is in question is Watson's marriage to his adopted daughter's adopted daughter. Both parties and the district court have used this same shorthand, and we adopt it here as well.
[3]  Outside of our circuit, a leading treatise suggests that judicial notice of law is the appropriate semantic term for describing a court's determination of law, but even the treatise acknowledges that Federal Rule of Evidence 201 does not explicitly address judicial notice of law. See 21B CHARLES ALAN WRIGHT & KENNETH W. GRAHAM, JR., FEDERAL PRACTICE AND PROCEDURE § 5102.1 at 73 (2d ed.2005). Wright and Graham's treatise suggests that Rule 201 should apply "where the `fact' noticed is an element of the offense or cause of action. Where an element of a federal offense is a violation of state law, the court could properly notice that law for the purpose of instructing the jury what they must find to satisfy the element." WRIGHT & GRAHAM § 5103.1 at 109 (footnote omitted). Although judges are obligated to instruct the jury as to the applicable law, we disagree with the assertion that Rule 201 governs the instruction in the case at bar. The only case that Wright and Graham cite for their proposition says nothing about Rule 201 and instead is entirely consistent with our approach. See United States v. Clements, 588 F.2d 1030, 1037 (5th Cir. 1979) ("The determination of the applicable state law is a question of law to be determined by the court. A court may properly take notice of a state law. The court properly instructed the jury on the applicable law. No error has been shown." (citation and footnote omitted)).
[4]  According to Black's Law Dictionary, "consanguinity" is a blood relationship. BLACK'S LAW DICTIONARY 303 (6th ed.1990).
[5]  According to Black's Law Dictionary, "affinity" is a "connection existing, in consequence of marriage between each of the married persons and the kindred of the other." BLACK'S LAW DICTIONARY 59. A relationship of affinity is no more of a blood relation than that of adoption. In addition, "[d]egrees of relationship by affinity are computed as are degrees of relationship by consanguinity." Id.
[6]  We recognize that some of the rationales used to defend such laws have come under fire recently. For a general discussion of potential problems with laws that restrict relationships between consenting adults, even those that are related, see Note, Inbred Obscurity: Improving Incest Laws in the Shadow of the "Sexual Family," 119 HARV. L.REV. 2464 (2006).
[7]  The government does not need to prove an overt act under 18 U.S.C. § 286. Although an overt act is required under the general conspiracy statute, 18 U.S.C. § 371, the language of that statute specifically requires that "one or more of such persons do any act to effect the object of the conspiracy." 18 U.S.C. § 371. In contrast, when a statute does not require an overt act, we do not read that requirement into the statute. See Salinas v. United States, 522 U.S. 52, 63, 118 S.Ct. 469, 139 L.Ed.2d 352 (1997) (holding there is no overt-act requirement for the RICO conspiracy statute because "[t]here is no requirement of some overt act or specific act in the statute before us, unlike the general conspiracy provision"). Section 286, like the RICO conspiracy provision, does not require an overt act.
[8]  That statutory provision directs that: "Whoever makes or presents to any person or officer in the civil, military, or naval service of the United States, or to any department or agency thereof, any claim upon or against the United States, or any department or agency thereof, knowing such claim to be false, fictitious, or fraudulent, shall be imprisoned not more than five years and shall be subject to a fine in the amount provided in this title." 18 U.S.C. § 287.
[9]  The circuits, however, do not have unanimous agreement as to the appropriate standard. For instance, the Ninth Circuit sets quite a high standard and states that "[t]o be false, a claim must not only be inaccurate but consciously so." United States v. Barker, 967 F.2d 1275, 1278 (9th Cir.1991). In contrast, the Eighth Circuit has used the seemingly lesser standard of whether the defendant had the "intent to deceive." United States v. Martin, 772 F.2d 1442, 1444 (8th Cir.1985).
[10]  Dedman, however, may have had grounds for claiming that the second count of the indictment was duplicitous.

A duplicitous indictment is one that charges separate offenses in a single count. The overall vice of duplicity is that the jury cannot in a general verdict render its finding on each offense, making it difficult to determine whether a conviction rests on only one of the offenses or on both. Adverse effects on a defendant may include improper notice of the charges against him, prejudice in the shaping of evidentiary rulings, in sentencing, in limiting review on appeal, in exposure to double jeopardy, and of course the danger that a conviction will result from a less than unanimous verdict as to each separate offense.
Duncan, 850 F.2d at 1108 n. 4. We held in 1993 that "`Congress, in enumerating several different types of fraudulent conduct in Section 1001, did not create separate and distinct offenses,'" United States v. Hixon, 987 F.2d 1261, 1265 (6th Cir. 1993) (quoting United States v. UCO Oil Co., 546 F.2d 833, 838 (9th Cir. 1976)), and, thus, "it becomes proper to charge the different means, denounced disjunctively in the statute, conjunctively in each count of the indictment." Id. This meant not that multiple false statements could be alleged in a single count, but that the government could say within the same count that a particular statement was both a false statement and concealment by scheme, because either is a violation of § 1001. See id. Thus, we were too broad and misinterpreted Hixon when we later said that "multiple factual predicates for violation of a statute do[] not render the indictment duplicitous." United States v. Washington, 127 F.3d 510, 513 (6th Cir. 1997), cert. denied, 524 U.S. 940, 118 S.Ct. 2348, 141 L.Ed.2d 718 (1998). Separate allegedly false statements, as we have in the instant case, are not mere "multiple factual predicates" but rather are entirely separate offenses; even in Hixon, with statements made on four separate occasions, the government had the good sense to charge each statement as a separate violation of § 1001 in its own count. Hixon, 987 F.2d at 1264 n. 3. The government's indictment may have been duplicitous, but unfortunately for Dedman, "[f]ailure ... to raise the question prior to trial and verdict waived the vice of duplicity," United States v. Costner, 359 F.2d 969, 974 (6th Cir. 1966), and we cannot now reverse on that ground. See also United States v. Adesida, 129 F.3d 846, 849 (6th Cir. 1997) (holding that a duplicitous indictment can be dismissed only if the error is raised prior to trial; otherwise an instruction that requires unanimity among the jurors cures the error).
