concurring in part and dissenting in part:

Although I concur with the majority’s holding affirming Adams’ conviction on the second count relating to the 2000 Form 1040, I respectfully dissent from the holding vacating his conviction on the first count relating to the Form 1040X. Because Adams has not raised a sufficiency of the indictment claim, and because the evidence is sufficient to support the jury’s verdict on this count, I would affirm the jury’s conviction as to the Form 1040X.

The majority admits that Adams frames his appeal in terms of “sufficiency of the evidence,” yet makes the significant leap to construe his argument as challenging the sufficiency of the indictment on Count I. One searches in vain to find a sufficiency of the indictment claim in Adams’ briefing. Rather, Adams is clearly challenging the district court’s denial of his Rule 29 motion for judgment of acquittal. His argument on Count I is framed in terms of his entitlement to an acquittal based on the fact that a “reasonable legal argument” existed that the jurat on the 1040X did not include the 1999 Schedule C. In other words, Adams is arguing that there was insufficient evidence to convict him of making a false statement on his amended tax return because he did not swear to the truth of the Schedule C. He does not argue that the indictment was insufficient as to that claim. Importantly, the Government only responded to the actual claim that Adams made regarding the insufficiency of the evidence to support his conviction; we simply do not know how the Government might have responded to a claim that the indictment was insufficient. Since Adams has not properly challenged the sufficiency of the indictment in his opening brief, this argument is waived. See, e.g., Edwards v. Johnson, 209 F.3d 772, 776 n. 1 (5th Cir. 2000); Yohey v. Collins, 985 F.2d 222, 224-25 (5th Cir. 1993); Lindsey v. Prive Corp., 161 F.3d 886, 894 n. 5 (5th Cir. 1998).1

Therefore, rather than reviewing the sufficiency of the indictment de novo, we *654should review the denial of the motion for judgment of acquittal as Adams has requested. We review the district court’s denial de novo, affirming the jury’s verdict “if a reasonable trier of fact could conclude from the evidence that the elements of the offense were established beyond a reasonable doubt, viewing the evidence in the light most favorable to the verdict and drawing all reasonable inferences from the evidence to support the verdict.” United States v. Ragsdale, 426 F.3d 765, 770-71 (5th Cir. 2005).

To support a conviction for filing a false tax return under 26 U.S.C. § 7206(1), evidence must be sufficient to show that: (1) the defendant wilfully made and subscribed to a materially false tax return; (2) the return contained a written declaration that it was made under penalties of perjury; and (3) the defendant did not believe that the return was true as to every material matter. See United States v. Loe, 262 F.3d 427, 435 (5th Cir. 2001). When Adams filed his Form 1040X he swore it was true, correct, and complete. By swearing to the completeness of the amended return Adams verified that the Form 1040X, combined with the attached 1999 Form 1040, represented the changes necessary to render a full accounting of his income for 1999. Whether Adams was required by the IRS to attach the 1999 Form 1040 is immaterial. The crucial fact is that Adams did attach this return. By doing so he necessarily swore that it, along with the Form 1040X, accounted for his total 1999 income. However, evidence at trial showed Adams continued to omit $277,551.46 in gross receipts. This money should have been reported on Line 1 of Schedule C of the attached 1999 Form 1040. The Form 1040X plus the 1999 Form 1040 could not have reflected his entire 1999 income, and thus when Adams swore that the information was “complete” he wilfully made a false statement. Therefore, there was sufficient evidence for the jury to find that Adams was guilty of violating 26 U.S.C. § 7206(1), and the jury’s verdict should be affirmed as to Count I.

Accordingly, I concur with the majority’s holdings in Part II.A, B, D, and E, but respectfully dissent from Part II.C.

Defendant-Appellant Jon Dale Adams was convicted on two counts of filing false income tax returns in violation of 26 U.S.C. § 7206(1). He appeals his conviction, contending that he is entitled to acquittal because the government improperly charged him with one count and, in any event, that there was insufficient evidence to support the jury’s verdict. In the alternative, Adams claims that the district court abused its discretion in denying him a new trial. We vacate Adams’s conviction as to one count, affirm his conviction as to the other count, and remand for re-sentencing.

I. FACTS AND PROCEEDINGS

Adams is the former owner of Secrets Cabaret strip club and Stardust Oasis bar in Jackson, Mississippi. He filed a federal income tax return for 1999 (the “1999 Form 1040”) that failed accurately to reflect at least $450,000, being proceeds of the sale of the strip club in October 1999. Adams signed the inaccurate return and mailed it to the IRS in June 2000. Then, in early 2001, Adams hired accountant and former IRS agent Perry Smith in what Adams’s counsel describes as an effort to *636“get his taxes straight.” On February 27, 2001, with Smith’s guidance, Adams completed, signed, and mailed a Form 1040X “Amended U.S. Individual Income Tax Return” (the “Form 1040X”), which amended the 1999 Form 1040 to report an additional $450,000 in gross income from the sale of the strip club. The Form 1040X included a jurat signed by Adams, which stated:

Under penalties of perjury, I declare that I have filed an original return and that I have examined this amended return, including accompanying schedules and statements, and to the best of my knowledge and belief, this amended return is true, correct, and complete.

IRS Form 1040X instructs taxpayers to “[ajttaeh only the supporting forms and schedules for the items changed.” In this case, the supporting form was Adams’s Form 4797, which he properly appended to the Form 1040X. In addition, however, Adams also included an unsigned copy of his 1999 Form 1040 together with a copy of his 1999 “Schedule C” — “Profit or Loss From Business” — that had been originally submitted with, and was considered part of, the 1999 Form 1040. The parties agree, and IRS instructions and Treasury regulations confirm, that Adams was under no obligation to submit the 1999 Form 1040 and Schedule C with the Form 1040X. In fact, he was affirmatively instructed not to do so.

In August 2001, again with Smith’s assistance, Adams completed, signed, and mailed his 2000 Form 1040 tax return (the “2000 Form 1040”). On that return, he reported gross receipts of $400,423.

In 2002, the IRS assigned Agent Jerry Porter to investigate whether Adams had violated tax statutes in reporting his 1999 and 2000 income. On February 22, 2007, a grand jury indicted Adams, charging him with two counts of filing false income tax returns in violation of 26 U.S.C. § 7206(1). This offense is distinct from tax evasion, which “requires proof of an intention to evade or defeat a tax, whereas § 7206(1) penalizes the filing of a false return even though the falsity would not produce tax consequences.”1

Count I of the indictment relates to the Form 1040X. It specifically alleges that Adams did not believe the amended return was true “in that the ... Form 1040X reported gross receipts on Line 1 of Schedule C of the 1999 U.S. Individual Income Tax Return Form 1040 attached thereto as $227,415.21, whereas, as he then and there well knew and believed, he had gross receipts substantially in excess of the amounts reported.... ” The government contends that Adams had continued to understate his 1999 income by omitting $277,551.46 in gross receipts from the Form 1040X.

Count II of the indictment relates to the 2000 Form 1040. It specifically alleges that on this return Adams had “reported gross receipts on Line 1 of Schedule C as $400,423.00, whereas, as he then and there well knew and believed, he had gross receipts in excess of the amounts reported....”2 The government contends that the 2000 Form 1040 understated Adams’s income by $66,013.

A jury convicted Adams on both counts. He subsequently filed a Rule 29 motion for judgment of acquittal, or in the alternative, for a new trial pursuant to Rule 33. The district court denied the motion and this timely appeal followed.

*637II. ANALYSIS

A. Standards of Review

We review de novo the district court’s denial of a post-trial motion for judgment of acquittal.3 In reviewing for sufficiency of the evidence, we must affirm the jury’s verdict “if a reasonable trier of fact could conclude from the evidence that the elements of the offense were established beyond a reasonable doubt, viewing the evidence in the light most favorable to the verdict and drawing ah reasonable inferences from the evidence to support the verdict.”4 Here, however, Adams’s challenge to his Count I conviction is an attack on the sufficiency of the indictment.5 When, as here, the defendant has challenged the sufficiency of the indictment in district court, we review the indictment’s sufficiency de novo,6 taking its allegations *638as true.7 We review the district court’s denial of a motion for new trial for abuse of discretion.8

B. Applicable Law

We cannot emphasize too strongly the framework for considering this case: It is not a criminal tax evasion case;9 it is not a criminal failure-to-file case.10 It is a perjury case — specifically, the filing of a tax return, under penalty of perjury, that contains false statements, which is criminalized in 26 U.S.C. § 7206C1).11 Under that section, the government has the burden of proving that: “1) the accused willfully made and subscribed to a tax return, 2) the return contained a written declaration that it was made under penalties of perjury! [ ] 3) the accused did not believe that the return was true as to every material matter[,]”12 and 4) that the return was false as to a material matter.13 Here, Count I of the indictment pinpoints precisely the sole material matter that the government, via the grand jury, insists made Adams’s Form 1040X false: the amount of gross receipts reported on the copy of the Schedule C from his 1999 income tax return that he included with the Form 1040X. Count II of the indictment also pinpoints precisely the alleged material false statement: the reported gross receipts on the Schedule C from his 2000 income tax return included with that same 2000 return.

C. Motion for Acquittal: Count I— False Form 1040X

1. Relationship of the Form 1040X to the 1999 Schedule C

An amended tax return, Form 1040X, can give rise to liability for filing a false tax return in the same manner as can any other tax return.14 Additionally, a Schedule C can give rise to liability because “section 7206(1) requires the same duty of honest reporting on schedules as it requires for entries on the Form proper.” 15 This liability attaches because “schedules, when appropriate, become integral parts of’ the forms to which they relate and “are incorporated therein by reference.”16

*639A copy of Adams’s 1999 Schedule C had been appended to his original 1999 Form 1040. The government, however, did not seek to indict Adams for making false statements on his 1999 Form 1040, nor could it have, because by the time the government indicted Adams, the applicable statute of limitations barred charges of making a false statement on his 1999 Form 1040.17 But, if Adams had been so indicted, we would first have asked the relatively straight-forward question whether the 1999 Schedule C was an integral part of the 1999 Form 1040. As we have previously answered that question in the affirmative,18 we know that a falsity on the Schedule C could have served as the basis for a charge of filing a false 1999 return.

The crime charged in Count I, however, was submitting a false Form 1040X, specifically the Form 1040X to correct Adams’s return for the 1999 tax year. Moreover, the indictment based this charge only on the falsity of Adams’s 1999 Schedule C, a document that IRS instructions and Treasury regulations indicate Adams need not have and should not have submitted with the Form 1040X.19 The government nevertheless contends that Adams may be held criminally liable for statements on the 1999 Schedule C because, by signing the jurat on his Form 1040X, he swore to the truth of “accompanying schedules.” The IRS argues that this includes the 1999 Schedule C that, despite instructions not to attach it to the Form 1040X, Adams had voluntarily attached. Adams counters that the Form 1040X’s jurat cannot be read as extending to the 1999 Schedule C, so that, as a matter of law, he cannot be held criminally liable for any false statements that may appear in that attached schedule. Here is why we agree with Adams.

The jurat on the Form 1040X states that Adams swears that he “examined this amended return, including accompanying schedules and statements, and to the best of my knowledge and belief, this amended return [no mention of schedules or statements] is true, correct, and complete.”20 By specifying that the signer’s examination extends to the amended return and all attachments while limiting the signer’s assurance of truth, correctness, and completeness to just the amended return, the jurat’s language makes a clear distinction between that which Adams examined and that which he swore was true — a classic example of the maxim Inclusio unius est exclusio alterius. In the common case, this distinction is insignificant because accompanying schedules and statements are generally considered integral parts of the return to which the jurat applies.21 In this case, though, Adams’s 1999 Schedule C— which did constitute an integral part of his 1999 Form 1040 — does not constitute an integral part of the Form 1040X: It was *640not a supporting schedule for the item being amended, and it was attached only as a “courtesy.”22 In signing the jurat on the Form 1040X, Adams swore that he had examined, inter alia, the copy of the 1999 Schedule C, but swore to the veracity of only the amended return (and of any accompanying documents integral to the amended return, a set of documents that does not include the Schedule C from his 1999 Form 1040). As the 1999 Schedule C cannot be incorporated by reference into the Form 1040X,23 it is a leap too far for the government to bootstrap the falsity in the time-barred 1999 Schedule C as the basis for a criminal charge of submitting a false Form 1040X.24 We again emphasize that this is a perjury case, not a tax-evasion case. In a tax-evasion case, Adams’s alleged failure to report taxable receipts would be relevant, irrespective of which form(s) tended to demonstrate the tax deficiency. But, when, as here, the government alleges perjury, the specific false statement charged is the very essence of the offense. In such cases, we must hold the government to proving the particular falsity it alleges even if the evidence should demonstrate that the defendant made other false statements for which liability could have properly attached.

We reject the overly expansive reading that the government would have us give to the jurat. Taking the government’s theory to its logical conclusion would result in amending taxpayers incurring liability virtually every time they include a copy of an original tax return with their amended returns. After all, by the time a taxpayer files a Form 1040X, he already knows that his original return is erroneous. Indeed, that is the very reason for filing the amended return. Still, for completeness, taxpayers frequently attach copies of the originals to the amended returns.25 Under the government’s theory that the Form 1040X’s jurat extends to the truthfulness of not only the Form 1040X and integral schedules but to any attached copies of previous tax returns as well, the amending taxpayer would be vouching for the accuracy of the very tax return that he is recognizing as erroneous in his voluntary amendment. This result defies all logic: It cannot be the case that the amending taxpayer is swearing to the veracity of each statement contained in the admittedly *641deficient original return that he is attempting to correct as well as in every attachment to that original return. Thus, the ambit of the amended return’s jurat cannot be read so broadly as to cover statements in the original return and its schedules and attachments.26

Having determined that by signing the jurat of the Form 1040X, Adams did not expose himself to potential liability for false statements contained in the 1999 Schedule C, we next assess the sufficiency of the indictment.

2. Sufficiency of the Indictment

The Federal Rules of Criminal Procedure require that the indictment be “a plain, concise and definite written statement of the essential facts constituting the offense charged.” Fed.R.Crim.P. 7(c)(1). The indictment is sufficient if it alleges every element of the crime charged and in such a way as to enable the accused to prepare his defense and to allow the accused to invoke the double jeopardy clause in any subsequent proceeding. When reviewing the indictment, we must keep in mind that the law does not compel a ritual of words and that an indictment’s validity depends on practical, not technical, considerations. And the starting place for any determination of whether the charged conduct is proscribed by a criminal statute is a reading of the language of the charging instrument and the statute itself27

An indictment that describes the offense by tracking the relevant statute’s unambiguous language is generally sufficient; yet, “that general description ‘must be accompanied with such a statement of the facts and circumstances as will inform the accused of the specific offense, coming under the general description, with which he is charged.’ ”28

In this case, we interpret Adams’s challenge as asserting that the indictment fails to allege an offense under 26 U.S.C. § 7206(1) because it alleges a falsity in a document that cannot serve as the basis for filing a false Form 1040X itself. We agree with Adams: Here, the devil is in the details. The government specifically alleged a falsity only on the 1999 Schedule C, and when the government sets forth “only one particular kind of falsity,” it has the burden of convicting on solely that *642falsity.29 Thus, the Adams indictment’s statement of the “facts and circumstances” of the offense is deficient; it does not adequately or accurately state the element of falsity.

Yet, this determination does not end our inquiry. If an indictment fails to allege an element, we review the failure for harmless error, provided that, as here, the defendant properly raised the issue at the district court.30 Under the harmless error standard, we ask “whether the error affects substantial rights,” that is “whether it appears ‘beyond a reasonable doubt that the error complained of did not contribute to the verdict obtained.’ ”31 Relevant to this inquiry are the two primary functions of an indictment: “that it (1) provides notice of the crime for which the defendant has been charged, allowing him the opportunity to prepare a defense; and (2) interposes the public into the charging decision, such that a defendant is not subject to jeopardy for a crime alleged only by the prosecution[.]”32

After careful review, we cannot say that it appears beyond a reasonable doubt that the indictment’s error did not contribute to Adams’s conviction on Count I. The indictment, on its face, sends the clear message that Adams need only mount a defense against the accusation that he falsely “reported gross receipts on Line 1 of Schedule C of the 1999 U.S. Individual Income Tax Return Form 1040 attached thereto as $227,415.21.” Instead, the government apparently meant to accuse Adams of submitting an amended tax return that, in combination with his previous 1999 Form 1040 and accompanying Schedule C, under-reported gross income by continuing to refrain from reporting substantial gross receipts derived from his businesses in 1999. In an indictment sufficient to pursue that theory, the government could have made the straight-forward allegation that Line 1, Column C of the Form 1040X, “adjusted gross income,” was false because it failed to account for— omitted — significant taxable 1999 receipts.33 Although the indictment surely put Adams on notice that the government was charging him with willfully under-reporting gross receipts for 1999 in the filing of the 1040X amended return, reasonable *643doubt exists regarding his opportunity to prepare a defense. It is not difficult to imagine that Adams would defend against an alleged falsity on the 1999 Schedule C in a different manner than he would defend against a falsity on the Form 1040X. For one thing, Adams had completed the 1999 Schedule C without the assistance of a paid tax preparer; accountant Smith prepared Adams’s Form 1040X. If Adams had been relieved of the burden of concentrating on the 1999 Schedule C, he might have had an enhanced opportunity to negate the offense’s willfulness element by shifting blame to Smith, e.g., “My accountant told me the Form 1040X was fíne.” This assertion is far more improbable if Adams is under the impression that he must defend the veracity of the 1999 Schedule C, which he completed at a time when he had no one at whom he could point the proverbial finger. This distinction alone is sufficient to hold that it does not appear beyond a reasonable doubt that the error in the indictment did not contribute to the jury’s verdict on Count I. We decline to further hypothesize how the error affected Adams’s opportunity to prepare for trial, and we do not reach the issue of the error’s affect on the indictment’s second purpose, that of ensuring that grand jurors, not just prosecutors, make charging decisions based on probable cause.34

We conclude that the jury convicted Adams on an erroneous indictment, and we cannot say that the error was harmless. Thus, the conviction on Count I must be vacated.35 Nothing we have stated, however, precludes the government, in its prosecutorial discretion, from re-indicting Adams in the proper form.36

*644Furthermore, we would have reached this result regardless whether we (1) focused on the substance of Adams’s argument, i.e., assessed the indictment’s sufficiency, or (2) took the dissent’s more technical approach — glossing over the fact that the government charged Adams with committing a crime for which he could not be convicted — and focused on a sufficiency-of-the-evidence inquiry.

Under the dissent’s approach, we would have asked if, when viewing the evidence and drawing all inferences in the light most favorable to the verdict, the record contained evidence on which a reasonable jury could have determined that Adams swore the 1999 Schedule C was “true correct, and complete.” Our analysis demonstrates that Adams could not have sworn to the truth of the 1999 Schedule C as a matter of law. For this reason, the district court should never have asked a jury to consider the issue; no reasonable jury could have convicted Adams on Count I.

The dissent, in contrast, would hold that there is sufficient evidence to support Adams’s conviction on Count I because he swore that the Form 1040X was complete when in fact it was not because the Form 1040X, combined with the attached copy of the 1999 Form 1040 and accompanying Schedule C, did not represent a full accounting of Adams’s 1999 income. Yet, the dissent either ignores or loses sight of the fact that the government has never charged Adams with failing to submit a complete return. The government instead charged that Adams made a specific false statement on line 1 of the Schedule C. The government’s election to charge Adams in this manner forecloses our making a post-hoc determination that there is sufficient evidence that Adams’s return simply was not complete. If we were to do that, we would affirm a conviction based on how the government wishes it had charged Adams, not on how it did charge him. We cannot be so quick to give the government this break, and we reiterate that when the government alleges “only one particular kind of falsity,” it must convict solely on that falsity.37 We repeat for emphasis that the government had multiple opportunities to charge Adams properly for a section 7206(1) violation. Even a federal court of appeals is in no better position than is the government to conclude that there exists sufficient evidence for a conviction on an uncharged crime.

D. Motion for Acquittal: Count II— False 2000 Form 1040

1. Government’s Burden

' The government did not present direct proof that Adams’s 2000 gross receipts were greater than the $400,423 that he reported. Instead, the government employed an indirect method of proof — one known as the “bank deposits and cash expenditures method.”38

Under this method, all deposits to the taxpayer’s bank and similar accounts in a single year are added together to determine the gross deposits. An effort is made to identify amounts deposited that are non-taxable, such as gifts, transfers of money between accounts, repayment of loans and cash that the taxpayer had in his possession prior to that year that was deposited in a bank during that year. This process is called “purification.” It results in a figure called net taxable bank deposits.

The government agent then adds the amount of expenditures made in cash, for example, ... cash [a] doctor received from fees, did not deposit, but gave to his wife to buy groceries. The total of *645this amount and net taxable bank deposits is deemed to equal gross income.39

The government then compares this gross income figure to the income that the taxpayer reported. “In asking the jury to rely on this analysis, as a basis for deciding that the taxpayer willfully understated] his true income, the government necessarily relies on circumstantial evidence.”40

In these cases, courts require the government to meet well-established criteria — in addition to the indirect calculation of gross receipts — without which it fails, as a matter of law, to establish guilt beyond a reasonable doubt. Specifically, a court should not allow a case to reach a jury unless the government establishes “with reasonable certainty” two components designed to ensure that the expenditures and deposits represent taxable income for the year in question.41 First, the government must demonstrate with reasonable certainty the taxpayer’s cash on hand at the beginning of the tax year in question.42 Second, the government has the burden of proving that the receipts in question were taxable. Supreme Court precedent permits the government to satisfy this second burden in either of two ways: “It can show that there is a likely taxable source of the unreported income, Holland v. United States, 348 U.S. 121, 138, 75 S.Ct. 127, 99 L.Ed. 150 (1954), or it can negate all possible nontaxable sources of income, United States v. Massei, 355 U.S. 595, 78 S.Ct. 495, 2 L.Ed.2d 517 (1958).”43 Because of the difficulties associated with proving a negative, the government proves “reasonable certainty” of the lack of non-taxable income sources by demonstrating that it performed an “adequate investigation that did not disclose non-taxable sources.”44

The “government must prove a full and adequate investigation” that “establishes] a guarantee of essential accuracy in the circumstantial proof at trial as an element of the government’s burden of proving *646guilt beyond a reasonable doubt.”45 The government need not, however, “conjecture about and either prove or negate every conceivable defense once it has met its own burden, explored every reasonable avenue and investigated every lead furnished by the taxpayer.”46 Each investigation’s adequacy “necessarily turns on its own circumstances. The critical question is whether the investigation was sufficient to support the inference that the unexplained excess in deposits was in fact attributable to currently taxable income.”47

2. Gross Receipts

We are satisfied that the government provided the jury with sufficient evidence that in 2000 Adams received gross receipts greater than the $400,423 that he reported. Agent Porter testified how his investigation yielded gross receipts of $466,436, and the government introduced the relevant supporting data into evidence. Although on appeal Adams denies having had year-2000 receipts of $466,436, he does not directly attack the government’s calculations, concentrating instead on the sufficiency of cash-on-hand and source-of-receipts evidence.

a. Cash on Hand

The government’s calculations were based on the assumption that Adams had no cash on hand at the beginning of 2000. Given the government’s $466,436 calculation, if Adams had at least $66,013 cash on hand on January 1, 2000, the government would be unable to bear its burden because the unreported deposits could have come from cash on hand rather than from year-2000 receipts. The government advances that it investigated as much as it reasonably could and thus met its duty, particularly given that Adams was in control of key facts. For example, the government learned from Adams that he maintained a safe deposit box which could have contained currency. Accordingly, Agent Porter verified the box’s existence and looked at records that reflected the frequency of Adams’s visits to the box. Yet, these actions did not provide direct evidence of cash on hand; Agent Porter had no way of knowing what was actually inside the box. When a defendant provides some facts that corroborate a “hoarding” claim, the government cannot sit idle and rely “solely upon a natural disinclination to believe that large sums of money are ever cached away.”48 In this case, the government did not sit idle; it investigated the safe-deposit-box lead. Adams was free to present his own witnesses to testify about his cash hoard, and the jury was free to discredit Agent Porter’s testimony. But, requiring anything more of the government’s investigation into hoarding would allow defendants to defeat tax prosecutions in virtually every *647case in which a hoarding defense is mounted.49

The government set forth at least a dozen factors supporting the conclusion that Adams had no cash on hand. Tracing Adams’s financial history from 1996, each factor tended to show that he had no cash on hand in January 2000. For example, in a 1996 bankruptcy petition, he claimed to have only $128.32 in cash; and when Adams updated that figure in 1999, he did not claim any additional cash on hand. Further, Adams filed a motion in the bankruptcy court in 1999 stating that he had no money to obtain counsel in a pending lawsuit.

Adams responds that although these and other factors may be sufficient to establish no cash on hand for the average taxpayer, he is not the average taxpayer because he had just sold Secrets Cabaret for $450,000, and thus was wealthy. Adams asserts that “[h]is actions in bankruptcy court and his dealings with his lawyer cannot make him poor when the numbers show him to be rich.”50 Adams specifically references the $450,000 proceeds from the Secrets Cabaret sale, but this is money that could not have constituted cash on hand because it was used to purchase a certificate of deposit and was no longer held in cash on January 1, 2000. Adams also represents that because he withdrew $87,134 from one of his bank accounts in 1999, that money still could have been on hand in 2000. The government countered -with testimony that Adams was spending more money than he made in 1999, and offered evidence sufficient for the jury to conclude that Adams had exhausted the withdrawal in covering his significant ongoing debts.

Agent Porter’s investigation was adequate. He took detailed steps, beginning with Adams’s 1996 tax returns, to reconstruct Adams’s cash-flow for the several years leading up to the tax year covered by his 2000 tax return; and the agent followed relevant leads to the extent possible. On the discrete facts of this case, these actions meet the reasonable-certainty standard for submitting evidence to a jury and provide sufficient support for a finding, beyond a reasonable doubt, of no cash on hand. For Adams’s theory to prevail, the jury would have to ignore his bankruptcy filings, previous tax filings, and other evidence that pointed to his having no cash on hand. The jury was under no obligation to do so.

b. Likely Taxable Source/Negating Non-Taxable Sources

Adams insists the government stipulated that lawful activity at the Stardust Oasis was the sole source of his 2000 gross receipts, so that the government had the burden of establishing the Stardust Oasis as the likely source of unreported receipts. Unable to point to an actual stipulation, Adams directs us to (1) the district court’s decision to prohibit the government from introducing testimony regarding receipts *648from a video store that Adams might have owned, and (2) the district court’s statement that its “understanding” was that the government had stipulated that Stardust Oasis and Secrets Cabaret were the only sources of Adams’s income. Assuming ar-guendo that it did in fact stipulate that it would not rely on other income sources, the government asserts that (1) it was under no obligation to prove the likely income source because it adequately established the absence of non-taxable sources, and (2) even if it were required to prove the likely source, it did so.

Agent Porter testified that he investigated and found no non-taxable income sources. This investigation included searching for non-taxable sources such as loans, gifts, and inheritances. Agent Porter reviewed Adams’s check registers, deposit slips, and other bank documents for any indication of non-taxable receipts, and he found none. Moreover, Adams provided Agent Porter with no non-taxable leads that he neglected to pursue. Adams fails to explain with particularity how Agent Porter’s non-taxable source investigation was deficient; and once it conducts a thorough investigation, the government need not “negate all possible non-income sources of the deposits, particularly where the source of the income is uniquely within the knowledge of the taxpayer.”51 We conclude that Agent Porter’s investigation into non-taxable sources was sufficiently thorough.

Given this conclusion, the government was under no obligation to demonstrate an adequate investigation into likely income sources. Proving both (1) lack of nontaxable income and (2) lack of cash on hand is sufficient to establish that gross receipts for the year in question were taxable.52 In so ruling, we are not relieving the government of any portion of its burden. Even after the trial court confirms the adequacy of the government’s investigation, the jury still may not find a defendant guilty unless the government establishes each element of the section 7206 offense beyond a reasonable doubt.53 That is precisely what the jury found in this case, and Adams presents no justification for our disturbing that finding. Accord*649ingly, Adams is not entitled to acquittal on Count II.

E. Motion for New Trial

Adams asserts that the trial court committed reversible error in making rulings of two categories: (1) denial of a mistrial after Agent Porter testified as to Adams’s mental state, and (2) evidentiary rulings regarding Agent Porter’s investigation report.

A trial court may grant a Rule 33 motion for new trial “if the interest of justice so requires.”54 “[Mjotions for new trial are not favored, and are granted only with great caution.”55

1. Rule 704(b): Testifying as to Mental State

Federal Rule of Evidence 704(b) states:

No expert witness testifying with respect to the mental state or condition of a defendant in a criminal case may state an opinion or inference as to whether the defendant did or did not have the mental state or condition constituting an element of the crime charged or of a defense thereto. Such ultimate issues are matters for the trier of fact alone.

Soon after Agent Porter took the stand as an expert witness, the following exchange took place:

Q: And after you completed your investigation of the defendant, did you come to a conclusion?

A: Yes, sir.

Q: What was that conclusion?

A: After completing the investigation, I concluded that the defendant, under penalties of perjury, willfully filed a 1999 Form 1040X and a 2000 Form 1040, knowing that it was false in that the — it was false as to a material matter.

Defense counsel immediately objected pursuant to Rule 704(b) and requested a mistrial, insisting that “willfully” is an ultimate issue for the jury. The trial court sustained the objection but denied the motion for mistrial, offering the following curative instruction:

[Y]ou heard testimony from the witness generally stating that he had concluded that the defendant acted willfully in filing a false claim. The rules do not permit an expert witness to give an opinion about what ... a criminal’s mental state is. So I’m going to strike that portion of his testimony from the record. I’m going to instruct you not to consider that opinion. Okay? I’m instructing you that you cannot consider the expert’s opinion as to what he thinks Mr. Adams’ mental state is.

Adams contends that the trial court abused its discretion in refusing to grant a mistrial on this ground. The government responds that Agent Porter’s statement was at most “borderline,” and that even if the testimony were inappropriate, the court’s instruction sufficiently cured the issue.

Agent Porter’s statement violated Rule 704(b), but the trial court’s curative instruction adequately remedied the error.56 “A prejudicial remark may be ren*650dered harmless by curative instructions to the jury. Moreover this Court gives great weight to the district court’s assessment of the prejudicial effect of an objectionable remark.”57 Here, the district court was satisfied that there was “no significant possibility of a substantial impact on the verdict.” The court further noted that given the weight of the evidence, any error was harmless.58 The district court acted well within its discretion. This was a four-day trial with extensive testimony that would have allowed the jury to reach a decision as to willfulness. Even though it is true that willfulness is inherently not susceptible to proof by direct evidence and that Agent Porter’s statement was likely the only direct testimony that the jury heard about Adams’s mental state, the court was within the bounds of its discretion in striking the testimony and instructing the jury accordingly.

2. Evidentiary Rulings on Agent Porter’s Report

Adams urges that the district court’s rulings on Agent Porter’s investigation report warrant a new trial. Adams claims that the court erred by (1) refusing to permit defense counsel, on cross-examination, to elicit testimony from Agent Porter that Adams had told him several things beneficial to his case, and (2) failing to require the government to produce the entire report at trial as Jencks Act material. The district court did not abuse its discretion in making these rulings.

a. Cross-Examination of Agent Porter

Agent Porter’s report recounts various statements that Adams made during two interviews in 2002. At trial, defense counsel wanted to ask Agent Porter about Adams’s statements that (1) he did not operate Secrets Cabaret or the Stardust Oasis in 2000; (2) $400,000 deposited in 2000 represented proceeds generated by the club in previous tax years; and (3) he kept a large amount of cash- — -totaling at least $200,000 — in a safe deposit box and in a safe at his home. The district court granted the government’s motion in limine on this issue, and restricted defense counsel to establishing that Adams told Agent Porter that he had cash on hand at the beginning of 2000.

Adams submits three theories why he should have been able to elicit these hearsay statements. First, Adams contends that without this information, the jury had no way of assessing whether the government satisfied its burden of conducting a full investigation of all leads.59 Under the circumstances of this case, this argument is unconvincing. Through the testimony of others and the argument of counsel, the jury was aware of *651Adams s theories that, (1) in 2000, the establishments were non-operational, (2) he maintained a safe deposit box, and (3) the $400,000 was from a different tax year. We reject Adams’s contention that, without the benefit of his own hearsay statements, the jury had insufficient information to evaluate the government’s investigation.

Second, Adams asserts that under Federal Rule of Evidence 705, he has the right to cross-examine an expert on any information, even hearsay information, that the expert used to reach his opinion. Under this rule, an expert witness may rely on reports that would be inadmissible as evidence, but, on cross-examination, the expert may be required to disclose the facts or data underlying his opinion.60 “While revealing the basis for an expert’s opinion is allowed, such disclosure does not facilitate the admission of otherwise inadmissible evidence.”61 Limitations apply to this disclosure such that “the evidence should be admitted only for the limited purpose of discrediting or impeaching the testifying expert and the jury should be carefully instructed about its restricted use.”62 Further, disclosure on cross-examination is subject to Federal Rule of Evidence 403’s prejudice/probative value balancing test.63 The district court viewed Adams’s statements as “unreliable and un-redeemable” hearsay that Adams wanted admitted for their truth rather than to discredit Agent Porter or to show that he was on notice of potential leads. This determination — a clear indication that the trial court found that the intended cross-examination did not meet the prejudice/probative value test — did not constitute abuse of discretion.

Third, Adams contends that the “rule of completeness” requires admission' of the expert’s entire report. The Federal Rule of Evidence addressing completeness, Rule 106, does not apply to a witness’s testimony at trial.64 As the government did not introduce any portion of Agent Porter’s report into evidence, Rule 106 is inapplicable.65 Neither does the common law rule of completeness provide support for Adams. This rule allows “parties to present evidence explaining, varying, or contradicting a witness’s portrayal of the conversation”; the rule does not “provid[e] carte blanche to do so by any means desired.” 66 Adams had ample opportunity to explain his position, and the trial court was within its discretion in refusing to give Adams such “carte blanche.”

We take this opportunity to re-emphasize that “ ‘trial judges retain wide latitude’ to limit reasonably a criminal defendant’s right to cross-examine a witness ‘based on concerns about, among other things, harassment, prejudice, confusion of the issues, the witnesses] safety, or inter*652rogation that is repetitive or only marginally relevant.’ ”67 We reject each of Adams’s claims of error relating to his cross-examination of Agent Porter.

b. Jencks Act

“The Jencks Act requires that the government provide the defendant with witness statements that relate to the subject matter-on which the witness has testified.” 68 Failure to comply with this requirement, coupled with a statement that was “highly useful,” even “invaluable,” to the defense’s cross-examination of a key witness, warrants reversal.69 At the close of Agent Porter’s testimony, the defense asked that the government produce his report as Jencks material. The government asserted that it was under no obligation to produce the report because it consisted of only two classes of items: (1) those unrelated to Agent Porter’s testimony and (2) those related but merely dupli-cative of his testimony and the other material produced.

After an in camera review of the report, the district court required the government to produce a redacted version.70 Proceeding section by section through Agent Porter’s report, the trial court required production of those statements that were (1) within the scope of direct examination and (2) not “merely a summary of what [had] been produced, with no reflection of his analysis.” The court also required production of statements with impeachment value and, erring on the side of caution, required production of any statements “if it appeared that there was an intermingling of Agent Porter’s analysis” between that which would be discoverable and that which would not.

We have previously approved of a district court’s decision to redact an agent’s report. In United States v. Medel, also a tax case, we determined that an agent’s report could be Jencks material but that the district court did not abuse its discretion in ordering production of only a portion of the report.71 The Medel court redacted “summaries of summaries,” other duplicative information, and portions dealing with the agents’ evaluation of the strength of their case and suggestions for rebutting the taxpayer’s defenses.72 According to Adams, the redacted portions of the report here contain a rebuttal of Agent Porter’s finding that Adams had no cash on hand. Adams views the report as demonstrating that Agent Porter’s actual conclusions were contrary to those he related at trial. Our review satisfies us that the district court’s redactions served their intended purpose — the removal of unrelated or duplicative information — and that the district court acted within its discretion.73 Adams’s challenge fails.

*653We affirm the district court’s decision to deny Adams a new trial.

III. CONCLUSION

We vacate Adams’s conviction on Count I. We affirm Adams’s conviction on Count II, but remand for re-sentencing on that count as the sole count of conviction.

VACATED in part; AFFIRMED in part; and REMANDED.