OPINION

BLOCK, District Judge:

James Francis Murphy appeals the district court's judgment convicting him of interfering with the administration of the tax laws, presenting fictitious financial instruments, and presenting false claims to the United States; and sentencing him principally to four years' imprisonment. For the following reasons, we vacate the fictitious financial instrument convictions, affirm the remaining convictions, and remand.

I

A. Murphy's Interactions with the IRS

In 2001, Murphy, an osteopath, began diverting income from his medical practice to a trust. The result was a drastic reduction in income reported on his personal income tax returns. Murphy's adjusted gross income went from approximately $100,000 in 2000 to $12,476 in 2001, $5,064 in 2002 and less than $0 in 2003. He did not file a personal return at all in 2004 or 2005. The trust, meanwhile, reported income of between $700,000 and $1 million, but virtually all of it was offset by claimed deductions.

The IRS opened an audit of Murphy and his trust in 2006. Though initially cooperative, Murphy eventually stopped communicating with the IRS examiner, who referred the case for collection.

In February 2008, Murphy contacted the IRS in an attempt to settle his tax liability. As payment, he offered four "bonded promissory notes." Collectively, the notes purported to satisfy Murphy's tax liability for 2003-2005 and his trust's tax liability for 2003. Murphy was the "maker" of the notes, each of which was payable "to the order of" the Secretary of the Treasury and the IRS examiner "for credit to" the Department of the Treasury and the IRS. Each note was "secured" by a "private discharging and indemnity bond" and "private offset bond" in the possession of "[Secretary of the Treasury] Mr. Henry M. Paulson, Jr., holder in due course."

With the notes, Murphy sent to the IRS a copy of a "certified deposit order" directing Secretary Paulson to "settle all obligations" with the IRS by means of an "authorized setoff from prepaid exemption account." Murphy's "payment" was prefaced by the following explanation:

The IRS's collection efforts continued. In response to the agency's attempt to levy his personal assets, Murphy submitted personal returns for 2003-2007. The returns for 2003 and 2004 were blank. Those for 2005-2007 claimed overpayments of approximately $462,000, $460,000 and $314,000, respectively; each of those returns lists the full amount of the reported overpayment as the "[a]mount ... you want refunded to you" and provides routing and account numbers for that purpose. No money was refunded to Murphy.

Murphy continued to correspond with the IRS over the remainder of 2008. None of the correspondence included any legitimate form of payment.

B. Murphy's 2008 Conviction

Meanwhile, in February 2008, Murphy attempted to bypass security at San Diego International Airport by presenting fraudulent diplomatic papers. He was arrested and charged with making false statements to federal officers and related crimes. He proceeded to trial, waiving his right to a jury, and testified in his own defense. District Judge Thomas Whelan found Murphy guilty and sentenced him to time served and three years' supervised release.

C. The Present Proceedings

In June 2012, Murphy was charged with (1) one count of interfering with the administration of the tax laws, in violation of 26 U.S.C. § 7212(a); (2) four counts of presenting fictitious financial instruments, in violation of 18 U.S.C. § 514; and (3) three counts of presenting false claims to the United States, in violation of 18 U.S.C. § 287. The § 7212(a) count was premised on a series of nine discrete acts occurring between September 2000 and the date of the indictment. The § 514 counts charged Murphy with presenting fictitious documents "purporting to be an actual security or financial instrument issued under the authority of the United States"; each count concerned one of the "bonded promissory notes" submitted to the IRS in February 2008. The three § 287 counts corresponded to Murphy's personal tax returns for 2005-2007.

Murphy moved, inter alia, to dismiss the § 7212(a) count as time-barred and/or duplicitous. The district court denied both branches of the motion.

The case proceeded to trial. Murphy offered character witnesses to attest to his honesty and also testified in his own defense. Evidence of Murphy's 2008 conviction for making false statements to federal officers was introduced during cross-examination of one of the character witnesses and again during cross-examination of Murphy.

Murphy moved for a judgment of acquittal, which was denied. The district court then instructed the jury. As pertinent here, Murphy raised no objections to the instructions given.

The instructions on the § 7212(a) count informed the jury that it "must agree that the defendant committed at least one of the [nine acts listed in the indictment] and must agree on which listed act he or she committed." Although the instructions did not make reference to the dates of the various acts, the verdict form asked the jury to state whether it found that Murphy has "committed at least one of the act or acts charged after June 21, 2006, with all of us agreeing unanimously as to at least one act after that date."

With respect to the § 514 counts, the district court told the jury that the government had to prove, inter alia, that "[t]he defendant passed, uttered or made or possessed a specified security that was false or fictitious." It did not state that the specified security had to purport to be issued "under the authority of the United States," or explain what that phrase meant.

Finally, the instruction on the § 287 counts required the government to prove that Murphy knowingly presented a claim to the United States that was false, fictitious or fraudulent as to a material fact. Although the instruction emphasized that the claim had to be "against" the United States, it did not inform the jury that an attempt to reduce tax liability would not constitute a claim.

During summation, defense counsel argued that Murphy had a good-faith belief in the legality of his conduct. In rebuttal, the government asked the jury to consider Murphy's credibility. In that regard, the prosecution offered the following:

Defense counsel did not make a contemporaneous objection, but moved for a mistrial after summations had concluded. He characterized the prosecution's rebuttal as implying that Judge Whelan had explicitly found Murphy not credible, and "us[ing] Judge Whelan, or a federal district court judge, as vouching for their case."

The district court denied the motion, but gave the jury the following curative instruction:

A few hours later, the jury found Murphy guilty on all counts. With respect to the § 7212(a) count, it unanimously found that Murphy had committed at least one act of interference within six years of the indictment.

Murphy subsequently made various post-trial motions. He renewed his motions for dismissal of the § 7212(a) count, for a judgment of acquittal, and for a mistrial. In addition, he moved for a new trial based on claimed inadequacies in the district court's instructions on the § 514 and § 287 counts. The district court denied all motions and sentenced Murphy principally to four years' imprisonment.

On appeal, Murphy challenges each of his convictions. He argues that the evidence was insufficient to support the § 514 convictions and, alternatively, that the district court erred by failing to instruct the jury on an element of that offense. He likewise claims instructional error with respect to the § 287 charges because the district court did not instruct the jury that an attempt to reduce tax liability is not a "claim" within the meaning of the statute. He argues that the § 7212(a) charge was untimely and duplicitous. Finally, he argues that all of the convictions were tainted by improper references to his 2008 conviction during the government's rebuttal summation.

As set forth in Part II, we conclude that the evidence was sufficient to preclude a judgment of acquittal on the § 514 counts. Because, however, it was not so overwhelming that it negated the prejudice flowing from the lack of any instruction that the financial instruments in question had to be issued "under the authority of the United States," we remand for a new trial. We address the remainder of Murphy's arguments in Part III.

II

18 U.S.C. § 514 provides, in pertinent part, that

The indictment somewhat simplified this statutory string of alternatives, alleging that Murphy

The government concedes that the theory of guilt alleged in the indictment is controlling.

As noted, Murphy argues that there was insufficient evidence to support the convictions and, alternatively, that the district court's instructions omitted an element of the offense. Both arguments center on the element that the instruments in question must purport to have been "issued under the authority of the United States."

A. Sufficiency of the Evidence

Murphy's motion for judgment of acquittal preserves his sufficiency challenge. See United States v. Navarro Viayra, 365 F.3d 790, 793 (9th Cir. 2004). We review the district court's denial of that motion de novo. See United States v. Goyal, 629 F.3d 912, 914 (9th Cir. 2010). Substantively, our review requires us "to determine 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.'" United States v. Nevils, 598 F.3d 1158, 1163-64 (9th Cir. 2010) (en banc) (quoting Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979)).

The "bonded promissory notes" that Murphy sent to the IRS purport to be "negotiable instrument[s], tendered lawfully by [Murphy] ("Maker")" as evidence of "debt[s] to the Payee." As the government points out, the notes refer to then-Secretary of the Treasury Henry Paulson, but the most prominent of these references makes the notes payable to his order. Another identifies him as the "co-payee." In addition, they are "for credit to" the Treasury and the IRS. Thus far, Murphy's notes appear to be what notes typically are: promises by the maker (Murphy) to pay the payee (Secretary Paulson) various debts (the tax liabilities of Murphy and his trust). Nothing about that arrangement suggests that they were issued by the United States.

The documents accompanying the notes, however, shed at least some light on how Murphy expected the notes to serve as payment for his tax liabilities. His cover letter to the IRS explained that the notes evidenced that he was "accepting for value each of the most recent offers to expand funds under Public Policy you have sent." His "certified deposit order" then directed Secretary Paulson to deposit the accepted offers and credit them against an "authorized setoff from prepaid exemption account."

Murphy's dense legalese makes it difficult to decipher his meaning. However, the evidence — at least under the standards of Jackson v. Virginia — does suggest that Murphy, through a convoluted series of accounting devices, ultimately sought to pay his taxes from an account created for him, and held on his behalf, by the United States government.1 Thus construing the evidence, a rational jury could find beyond a reasonable doubt that the notes were issued under the authority of the United States.

B. Instructional Error

Since Murphy did not object to the district court's jury instructions, we review them only for plain error. See United States v. Alferahin, 433 F.3d 1148, 1154 (9th Cir. 2006). For plain-error review to result in reversal, "[t]here must be an `error' that is `plain' and that `affect[s] substantial rights.'" United States v. Olano, 507 U.S. 725, 732, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993) (quoting Fed. R. Crim. P. 52(b)). "Moreover, Rule 52(b) leaves the decision to correct the forfeited error within the sound discretion of the court of appeals, and the court should not exercise that discretion unless the error `seriously affect[s] the fairness, integrity or public reputation of judicial proceedings.'" Id. (quoting United States v. Young, 470 U.S. 1, 15, 105 S.Ct. 1038, 84 L.Ed.2d 1 (1985)).

The government concedes that the district court's omission of any mention of "under the authority of the United States" was both error and plain. The third prong "in most cases ... means that the error must have been prejudicial." Olano, 507 U.S. at 734, 113 S.Ct. 1770. Failure to submit an element of the offense to the jury is potentially prejudicial because "[t]he Constitution gives a criminal defendant the right to demand that a jury find him guilty of all the elements of the crime with which he is charged." United States v. Gaudin, 515 U.S. 506, 511, 115 S.Ct. 2310, 132 L.Ed.2d 444 (1995). It is not necessarily prejudicial, however, see Neder v. United States, 527 U.S. 1, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999), because "it remains possible the jury made the necessary finding," United States v. Perez, 116 F.3d 840, 847 (9th Cir. 1997) (en banc) (citation and internal quotation marks omitted). The likelihood that this happened increases with the strength of the government's evidence. See Alferahin, 433 F.3d at 1158 ("Other cases have also upheld convictions rendered on incomplete or erroneous jury instructions, but like Neder, these cases have relied on the existence of `strong and convincing evidence' that the missing element of the crime had been adequately proved by the prosecution." (quoting Perez, 116 F.3d at 848)).

Although the evidence that the "bonded promissory notes" were issued under the authority of the United States is sufficient to avoid a judgment of acquittal, it is not sufficient to convince us that "the jury made the necessary finding," Perez, 116 F.3d at 847, or that it would have done so had it been asked. As noted above, the notes themselves — even as elucidated by Murphy's cover letter — are ambivalent, and the ambiguity was not resolved by other evidence or counsels' summations. The government faults Murphy for failing to offer evidence or argument that the bonds did not purport to be issued under the authority of the United States, but it was not Murphy's burden to do so. Rather, it was the government's burden to convince the jury, beyond a reasonable doubt, that all elements of the crime were present. We are not satisfied that it did so, and therefore conclude that Murphy has demonstrated prejudice.

We further conclude that we should exercise our discretion to correct the error. The fourth prong of plain-error analysis requires us to decide whether "the greater threat to the integrity and fairness of judicial proceedings would arise from the reversal of a conviction on flawed jury instructions rather than from affirming an imperfect verdict." Alferahin, 433 F.3d at 1159. Here, too, the strength of the evidence is a factor. See Johnson v. United States, 520 U.S. 461, 470, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997) (affirming denial of new trial because evidence of materiality, an element of the offense, was "overwhelming").

Murphy was undeniably denied his constitutional right to have all elements of the crime submitted to the jury, while the government was concomitantly relieved of its "obligation to prove every element beyond a reasonable doubt." Perez, 116 F.3d at 847. These are serious concerns, going to the very heart of a criminal proceeding.2 On the other side of the ledger, the government's evidence that the notes purported to be issued under the authority of the United States was not overwhelming, leaving the nature of the notes open to debate. As a result, the verdict of a properly instructed jury is not a foregone conclusion. In these circumstances, we think allowing the convictions to stand poses a greater threat to the fairness and integrity of the proceeding and, therefore, vacate them and remand for a new trial on those counts.

III

With respect to the other convictions, some of Murphy's claims of error were preserved; others were not. Regardless of the standard of review, all are without merit.

A. § 287 Charges

It was not error for the district court not to instruct the jury that an attempt to reduce tax liability is not a "claim" within the meaning of 18 U.S.C. § 287. "[T]he filing of a false tax return pursuant to a scheme to obtain an unjustified tax refund is sufficient to establish a violation of presenting a false claim against the United States." United States v. Miller, 545 F.2d 1204, 1212 n. 10 (9th Cir. 1976), abrogated on other grounds by Boulware v. United States, 552 U.S. 421, 128 S.Ct. 1168, 170 L.Ed.2d 34 (2008). Murphy's argument that he asked the IRS to credit the amounts due to his tax liabilities for other years is belied by the record: the returns in question included routing and account numbers for payment to Murphy, and an IRS agent testified that, had the returns been accepted, they would have resulted in payments to Murphy. In any event, there is no question that Murphy falsely claimed that the United States owed him money; how he planned to spend it is irrelevant. Cf. United States v. Jackson, 845 F.2d 880, 882 (9th Cir. 1988) ("Our focus must be on the substance of the transaction, the disbursement of government funds, and not on the timing or form of the entry in the government's accounting ledgers.").

B. § 7212(a) Charge

The § 7212(a) charge was timely. A charge of tax evasion under 26 U.S.C. § 7201 "is timely so long as it is returned within six years of an affirmative act of evasion," even if the evasion first began outside the period. United States v. DeTar, 832 F.2d 1110, 1113 (9th Cir. 1987). We agree with the Fourth Circuit that the same rule applies to interference with administration of the tax laws under § 7212(a). See United States v. Wilson, 118 F.3d 228, 236 (4th Cir. 1997) ("The limitations period for a violation of § 7212(a) ... begins to run on the date of the last corrupt act."). The jury found that Murphy committed at least one act of interference less than six years before the indictment was returned, and we reject the argument that earlier acts of interference should immunize Murphy from liability for a crime occurring within the limitations period.

The § 7212(a) charge was not duplicitous because the nine discrete acts of interference alleged in the indictment "merely state[d] multiple ways of committing the same offense." United States v. Arreola, 467 F.3d 1153, 1161 (9th Cir. 2006) (citing United States v. UCO Oil Co., 546 F.2d 833, 835 (9th Cir. 1976)). In any event, the district court's special unanimity instruction ensured that the jury did not find Murphy guilty "without having reached a unanimous verdict on the commission of a particular offense." UCO Oil Co., 546 F.3d at 835. Because the district court listed the charged acts of interference immediately after instructing the jury that the government had to prove that Murphy "acted corruptly [and] with the intent to impede or obstruct the due administration of justice," it is not plausible that the jury unanimously found that Murphy committed a particular act without also agreeing that he did so with the requisite intent.

C. Rebuttal Summation

Finally, with regard to the government's rebuttal summation, Murphy's 2008 conviction for making false statements to federal officers was relevant to his credibility, a matter placed squarely in issue by Murphy's testimony that he held a good-faith belief that his actions were legal. The prosecutor's comments accurately recounted the facts leading up to the conviction, and she disclaimed what Murphy calls the "clear implication" that the jury should defer to Judge Whelan's assessment of Murphy's credibility. Finally, even if the government's rebuttal summation had been improper, we are satisfied that it was harmless in light of the district court's curative instruction and the numerous other bases for questioning Murphy's bona fides.

IV

For the foregoing reasons, we vacate the § 514 convictions and affirm the others. Accordingly, we vacate the district court's judgment and remand for a new trial on the § 514 counts, as well as for resentencing upon either the completion of the new trial or the government's election to dismiss those counts.

AFFIRMED in part, VACATED in part, and REMANDED.