
544 F.3d 149 (2008)
UNITED STATES of America, Appellee,
v.
TRIUMPH CAPITAL GROUP, INC., Frederick W. McCarthy, Lisa A. Thiesfield, Ben F. Andrews, Defendants,
Charles B. Spadoni, Defendant-Appellant.
Docket No. 06-4970-cr.
United States Court of Appeals, Second Circuit.
Argued: March 28, 2008.
Decided: September 25, 2008.
*152 John H. Durham, Deputy United States Attorney for the District of Connecticut (William J. Nardini, Nora R. Dannehy, Assistant United States Attorneys, Of Counsel, on the Brief), for Appellee.
Richard M. Asche, Litman, Asche & Gioiella, LLP (Russell M. Gioiella, Litman, Asche & Gioiella, LLP, on the Brief), New York, NY, for Defendant-Appellant.
Before: POOLER, HALL, Circuit Judges, and GLEESON, District Judge.[1]
JOHN GLEESON, United States District Judge:
Charles B. Spadoni appeals from a judgment of the United States District Court for the District of Connecticut convicting him of racketeering, racketeering conspiracy, bribery, wire fraud and obstruction of justice and sentencing him principally to a 36-month term of imprisonment in connection with a political bribery scheme. Spadoni challenges the sufficiency of the evidence of his intent to offer a bribe, and argues that the government unconstitutionally suppressed material exculpatory and impeaching evidence. He also claims that the evidence was insufficient to establish the intent element of his obstruction of justice charge, and that the jury was improperly instructed on that element. While we find the evidence sufficient to sustain the jury's verdicts, we reverse and remand for a new trial on the racketeering, racketeering conspiracy, bribery and wire fraud counts because we conclude that the government suppressed material exculpatory and impeaching evidence. However, because we conclude that the district court properly instructed the jury on the obstruction of justice charge, we affirm Spadoni's conviction on that count, but we vacate the sentence and remand for resentencing on that count.

BACKGROUND

A. The Trial Evidence

In July of 1997, Paul J. Silvester, who was at the time the Deputy Treasurer of the State of Connecticut, was appointed Treasurer to fill a midterm vacancy. The *153 Connecticut Treasurer's Office manages billions of dollars in state pension funds, and the Treasurer has the authority to make investment decisions for these funds. At the time Silvester joined the Treasurer's Office, a Boston-based private equity firm, Triumph Capital Group, Inc. ("Triumph"), was managing some of the state pension fund's investments.
Silvester was friends with Spadoni, a politically active lawyer in Connecticut. In 1997, Silvester helped Spadoni get hired as Triumph's general counsel.

1. The Campaign Contribution Bribe

In 1998, Silvester, who had been an interim appointment as Treasurer, decided to run for a full term and needed to raise funds for his campaign. As state law prohibited investment firms doing business with the Treasurer's Office from contributing to the race for Treasurer, Silvester instead attempted to raise money for the Connecticut Republican Party, which agreed to contribute 70% of the amount Silvester raised to Silvester's campaign.
Silvester told Spadoni that it would be "helpful" to his campaign if Triumph could raise money for the Connecticut Republican Party, Trial Tr. vol. 5, 115, June 19, 2003, and Triumph accordingly made a $100,000 commitment to the party. Additionally, Silvester told Spadoni that he would like to have one of his employees, Lisa Thiesfield, serve as his full-time campaign manager, but she was not permitted to perform campaign activities during her work hours at the Treasurer's Office. Spadoni subsequently informed Silvester that Triumph hired Thiesfield for an assignment unrelated to state business, paying her a $20,000 to $25,000 fee. This allowed Thiesfield to quit her Treasurer's Office job and work as a full-time campaign manager.
Silvester lost the election, and Spadoni promptly approached him to propose an investment deal with Triumph. Silvester decided to invest $150 million of state pension funds with Triumph before he left office, an amount comparable to the size of Triumph's previous contracts with the state. Silvester thought the deal made good business sense, but would "probably not" have agreed to this deal if Triumph had not donated $100,000 to the Connecticut Republican Party and hired Thiesfield. Trial Tr. vol. 5, 156.

2. The Consulting Contracts Bribe

Before the $150 million investment agreement was finalized, Silvester asked Spadoni to pay Thiesfield and Christopher Stack, a close associate of Silvester, a 1% finder's fee even though they had not in fact acted as finders in connection with the deal.
Thiesfield was unemployed due to her resignation from the Treasurer's Office to serve as Silvester's campaign manager, and Silvester was concerned about finding employment for her and for his other staffers. Stack was not an employee of the Treasurer's Office, but he had acted as a finder on previous investment opportunities with the state in the past.[2] Silvester told Spadoni that Stack's company, Collegiate Capital, was in trouble, and that he wanted to help Stack out of his financial difficulty.
Spadoni told Silvester that he would discuss the proposed finder's fee arrangement with Triumph's owner, Frederick McCarthy. At trial, Silvester testified that Spadoni subsequently informed him *154 that McCarthy "did not want to pay the fee as a finder's fee," but wanted to wait until Silvester was out of office and then "work it out at that point," which Silvester understood to mean that he and Triumph would later devise some arrangement that would be economically identical to the finder's fee he had proposed. Trial Tr. vol. 5, 161-62.[3] Silvester testified that this influenced him to increase the amount of funds being invested from $150 million to $200 million in order to increase the amount that Stack and Thiesfield would receive, but he did not discuss with Spadoni this decision to increase the value of the contract.
On November 12, 1998, Silvester executed a $200 million investment contract with Triumph-Connecticut II, L.P., a partnership that Triumph created to invest these funds. At some point Stack and Thiesfield, each using a wholly-owned LLC, executed a consulting contract with Triumph which would pay each $1,000,000 over three years. The government claimed at trial that these were sham contracts under which Stack and Thiesfield assumed virtually no duties and were paid regardless of any results, contrary to industry practice.
At trial, the date on which Stack's contract was signed was the subject of dispute. Stack testified that his contract was signed on November 11, 1998, a day before the investment contract was executed, and that his contract was postdated to appear that it was executed after Silvester left office in January of 1999. Stack testified that he remembered that this was the date his contract was signed because he found tickets to a matinee he had attended on that date, and he remembered that he signed the contract on the same day that he attended the matinee. Visitor logs from the law firm offices where Stack testified that he signed the contract confirmed that a meeting involving Stack, Spadoni and McCarthy took place on that date. Stack also testified that he received a draft contract by fax shortly before the signing, and Triumph's records reflect that Triumph sent a fax to Stack on November 9, 1998.
However, on several occasions during the course of his cooperation with the government, Stack told prosecutors that his consulting contract was signed at the end of December in 1998, between Christmas and New Year's Day. When he was first interviewed by the government, Stack made no mention of the November 11, 1998 meeting at the law firm and instead stated that he first learned of a possible consulting contract while vacationing in Tortola after Triumph's investment contract with Connecticut was signed. According to Stack, Silvester and Thiesfield were also vacationing with him in Tortola, and Silvester told him that he would later get a consulting contract.[4] Stack also told the government that by the time the consulting contract was eventually signed, the Triumph-Connecticut investment contract had already been fully funded, which did not happen until December 4, 1998.
Although Silvester was not present for the signing of Stack's consulting contract, his trial testimony indicated that the consulting contract was signed after the investment contract between Triumph and Connecticut. Specifically, Silvester testified that after he and Stack returned from *155 Tortola, which was after the investment contract was signed, Spadoni told him that McCarthy was "favorably inclined" to enter into a contract with Stack. Trial Tr. vol. 6, 59, June 20, 2003. He also testified that he and Stack discussed Stack's arrangements to sign the contract in mid-December 1998 or later, and that it was his understanding that Stack's consulting contract was signed in December 1998 or in January 1999.[5]

3. Obstruction of Justice

Shortly after Silvester left office, the investments he made late in his term came under scrutiny. On May 25, 1999, a grand jury subpoena was served on Triumph-Connecticut II, L.P. calling for the production of records related to the investment contract. Triumph-Connecticut II, L.P. complied with this subpoena.
Shortly after the subpoena was served, Spadoni told Silvester about it. He said that Triumph did not believe its consulting contracts with Stack and Thiesfield were covered by the subpoena, but that Triumph's lawyers anticipated more subpoenas in the future. Spadoni spoke to Silvester again and told him that an attorney had advised him to destroy documents not called for by subpoena in anticipation of further subpoenas, and had recommended specialized deletion software to remove them from his computer. He also asked Silvester to destroy any copies he might have of a contract between Silvester's new employer, Park Strategies, and a company owned by Ben Andrews, an eventual co-defendant of Spadoni on unrelated charges.
On July 13, 1999, the grand jury issued another subpoena, which led Triumph to produce the Stack and Thiesfield consulting contracts. On December 29, 1999, the grand jury issued an additional subpoena, which led to the production of backup tapes from Triumph's computer networks.
On April 11, 2000, the grand jury subpoenaed a Triumph laptop computer assigned to Spadoni. An FBI forensic computer examiner testified at trial that his inspection of the laptop revealed that a copy of the commercial document deletion software "Destroy-It!" was installed on the computer on June 21, 1999, and used to delete files in a directory named "Triumph" on June 23, 1999. On December 28, 1999, the software was used to delete two files in a directory named "LAT, LLC," which was the name of Thiesfield's wholly-owned company.
Among the documents deleted from the laptop were files named "Stack Contract" and "LAT Contract." These files were accessed on November 16, 1998, and at the time they had last been modified on November 10, 1998. The forensic examiner testified that "Stack Contract" was also accessed on November 23, 1998 and May 31, 1999, but he could not determine whether this file was accessed or modified between November 23, 1998 and December 31, 1998. "LAT Contract" was accessed on June 3, 1999 and December 31, 1999. The document deletion software was also used to remove files called "Park Strategies Agreement," "Engagement Letter," and others apparently unrelated to this case.
At one point after the investigation began, Triumph's comptroller, Robert Trevisani, discussed with Spadoni how to destroy computer files securely, and remarked, "if we were trying to hide something, we could use a program like CleanSweep." Trial Tr. vol. 7, 217, June 24, 2003. Spadoni informed Trevisani *156 that the program he needed "would be Destroy-It!." Id.
Stack approached the government early in its investigation and offered to reveal Silvester's dealings in exchange for immunity. Silvester subsequently pled guilty to criminal conduct involving five funds doing business with the State of Connecticut. He pled guilty to bribery with respect to three of those funds, and to mail fraud with respect to Triumph and another fund.

B. The Procedural History

Triumph, Spadoni, Thiesfield, McCarthy, and Andrews were charged in a superseding indictment on January 9, 2001, and Spadoni was tried along with Triumph in June and July of 2003.[6] At trial, the district court instructed the jury that it could return a conviction on the obstruction of justice charge if it found that "in the defendant's mind, his or its conduct had the natural and probable effect of obstructing or interfering with the grand jury proceeding." Trial Tr. vol. 16, 123, July 10, 2003. In giving this charge, the district court denied Spadoni's request to insert the words "he knew that" after "the defendant's mind."[7]
On July 16, 2003, the jury acquitted Spadoni of the charges related to the campaign contributions bribe, but convicted him of bribery and mail fraud in connection with the consulting contracts bribe and of obstruction of justice. It found the racketeering act based on the campaign contributions bribe not proven, but found the racketeering acts based on the consulting contracts bribe and the obstruction of justice counts proven, and convicted Spadoni of racketeering and racketeering conspiracy.
Spadoni moved for a judgment of acquittal or a new trial, challenging the sufficiency of the evidence supporting his convictions on all counts. On September 16, 2005, the district court denied this motion with respect to the bribery, wire fraud, and obstruction of justice counts but found Spadoni entitled to a judgment of acquittal on the racketeering and racketeering conspiracy charges. United States v. Spadoni (Spadoni I), No. 00-CR-217 (EBB), 2005 WL 2275938, at *11 (D.Conn. Sept. 16, 2005). The district court later reinstated the conviction on the racketeering and racketeering conspiracy charges. United States v. Spadoni (Spadoni III), No. 00-CR-217 (EBB), 2006 WL 2771642, at *2 (D.Conn. Sept.11, 2006).[8]
Spadoni also moved for a new trial based on claims that the prosecution suppressed material exculpatory and impeaching evidence in violation of Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), and Giglio v. United States, 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972). After the trial, Spadoni's counsel acquired from Silvester a *157 set of notes Silvester had handwritten for his attorneys, Hubert Santos and Hope Seely. These notes, which had been created to assist Santos in conducting plea negotiations on Silvester's behalf, had also been typewritten verbatim by Santos's office. Spadoni's counsel drafted an affidavit for Silvester to sign, and represented to the district court that Silvester edited the affidavit and offered to testify to its truth if subpoenaed but refused to sign it due to a fear that it could adversely affect his community confinement status. The unexecuted affidavit indicates that Silvester's attorney met with the government in an attorney proffer and that he conveyed to the government the substance of Silvester's notes. An attorney proffer apprises the government of what the client would be able to provide by way of testimony, and is often the first step in a defendant's effort to obtain a cooperation agreement with the government. The unexecuted Silvester affidavit also states that he personally conveyed the same information to the government when he began cooperating.
Silvester's notes detail corrupt activity by Silvester in connection with several entities. They also include a version of Silvester's interaction with Spadoni that is somewhat different from Silvester's trial testimony. The notes, written in Silvester's voice, state:
I told Charlie [Spadoni] to pay a finder to Stack and Theisfield [sic] and then I wanted Triumph to hire Elizabeth and Mike MacDonald also. He said he would take it up with Fred [McCarthy]. He came back and said they would not pay any finder or offer employment to anyone connected to me in exchange for the deal. Charlie said it would be quid pro quo and he could not advise his boss to agree to it. Charlie said that Fred was sympathetic to the situation of certain staffers and they would be as helpful as possible after I left office but that it would have to be arms length and make sense for Triumph. I said fine.
Gioiella Decl. Ex. A, June 13, 2005.
In response to Spadoni's Brady motion, the government produced, for the first time, notes taken by FBI Special Agent Charles E. Urso during the attorney proffer. These notes state, in part:
  PS told CS-told
  pay Stack + Lisa T.-fees-
  CS would not pay finder-
  Employee-MM-EW-
  CS told could [not] pay related to a deal.
  McCarthy sympathetick [sic] to staff-
  arms length needed to make sense-
Gov.'s Resp. to Gioiella Decl. Attach. A.[9]
The government argued that it never possessed Silvester's handwritten or typewritten notes. It further argued that the information in those notes and in Urso's notes was not material exculpatory or impeaching information because it was consistent with Silvester's pretrial interviews with the government, his grand jury testimony, and his trial testimony. The relevant FBI report summarizing Silvester's pretrial interview with the government reads, in part:
Silvester proposed to Spadoni that he would like Stack and Thiesfield to be paid as finders.... Silvester told Spadoni he wanted them to be paid a point, split between the two. Spadoni said he would take it up with McCarthy.
Spadoni related that it was a problem (legal reasons) when Silvester was in office but he would be glad to sit with *158 them after he was out of office. Silvester understood that Spadoni would start the process of hiring Thiesfield and Stack during negotiations.
Id. Attach. B. Before the grand jury, Silvester described the situation similarly, saying that after he proposed the finder's fee to Spadoni, Spadoni checked with McCarthy, "came back, said that they would rather sit with them separately after I was out of office and work it with them at that time." Grand Jury Tr. 58-59, June 13, 2000.
Silvester's statements in pretrial interviews and before the grand jury were consistent with his trial testimony. At trial, Silvester said that Spadoni indicated that Triumph "did not want to pay the fee as a finder's fee" but instead "wanted to wait until [Silvester] was out of office and then sit with those folks and work it out at that point." Trial Tr. vol. 5, 161. Silvester testified that he took this to mean "that they would sit and work it out on terms similar to the economics that [he and Spadoni] had discussed." Id. at 162.
On September 7, 2006, the district court denied Spadoni's Brady motion, finding that the government did not possess Silvester's notes and that the information contained in both Urso's and Silvester's notes was not materially different from the content of Silvester's pretrial interviews, grand jury testimony and trial testimony. United States v. Spadoni (Spadoni II), No. 00-CR-217 (EBB), 2006 WL 2595574, at *5-*6 (D.Conn. Sept.7, 2006).
On October 25, 2006, the district court sentenced Spadoni principally to concurrent 36-month terms of imprisonment on all counts and a $50,000 fine. This appeal followed.[10]

DISCUSSION

A. The Legal Standards


1. Motion for a Judgment of Acquittal

Federal Rule of Criminal Procedure 29 provides that after a jury verdict, "the court on the defendant's motion must enter a judgment of acquittal of any offense for which the evidence is insufficient to sustain a conviction." Fed. R.Crim.P. 29(a); see also Fed.R.Crim.P. 29(c)(1) (allowing for such motions after jury verdicts). "A defendant challenging the sufficiency of the evidence supporting a conviction faces a `heavy burden.'" United States v. Glenn, 312 F.3d 58, 63 (2d Cir.2002) (quoting United States v. Matthews, 20 F.3d 538, 548 (2d Cir.1994)). A court may overturn a conviction on this basis "only if, after viewing the evidence in the light most favorable to the Government and drawing all reasonable inferences in its favor," it finds that "`no rational trier of fact' could have concluded that the Government met its burden of proof." Glenn, 312 F.3d at 63 (quoting United States v. Morrison, 153 F.3d 34, 49 (2d Cir.1998)). "`[T]he relevant question is whether ... any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.'" Glenn, 312 F.3d at 63 (quoting Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979)). A court's analysis considers "`the evidence in its totality,' and the Government `need not negate every theory of innocence.'" Glenn, 312 F.3d at 63 (quoting United States v. Autuori, 212 F.3d 105, 114 (2d Cir.2000)).
In order to "`avoid usurping the role of the jury,'" Autuori, 212 F.3d at 114 (quoting United States v. Guadagna, 183 F.3d 122, 129 (2d Cir.1999)), courts must *159 "`defer to the jury's assessment of witness credibility and the jury's resolution of conflicting testimony'" when reviewing the sufficiency of the evidence. Glenn, 312 F.3d at 64 (quoting United States v. Bala, 236 F.3d 87, 93-94 (2d Cir.2000)); see also, e.g., Autuori, 212 F.3d at 114 ("We may not substitute our own determinations of credibility or relative weight of the evidence for [those] of the jury."). The relevant inquiry is "`whether upon the evidence, giving full play to the right of the jury to determine credibility, weigh the evidence, and draw justifiable inferences of fact, a reasonable mind might fairly conclude guilt beyond a reasonable doubt.'" Autuori, 212 F.3d at 114 (quoting United States v. Mariani, 725 F.2d 862, 865 (2d Cir.1984)).
Where a fact to be proved is also an element of the offense, however, "it is not enough that the inferences in the government's favor are permissible." United States v. Martinez, 54 F.3d 1040, 1043 (2d Cir.1995). A court "must also be satisfied that the inferences are sufficiently supported to permit a rational juror to find that the element, like all elements, is established beyond a reasonable doubt." Id. "[I]f the evidence viewed in the light most favorable to the prosecution gives `equal or nearly equal circumstantial support to a theory of guilt and a theory of innocence,' then `a reasonable jury must necessarily entertain a reasonable doubt.'" Glenn, 312 F.3d at 70 (quoting United States v. Lopez, 74 F.3d 575, 577 (5th Cir.1996)).
We review a district court's denial of a motion for a judgment of acquittal de novo, "applying the same standard of sufficiency as the district court." United States v. Florez, 447 F.3d 145, 154 (2d Cir.2006).

2. Motion for A New Trial

Federal Rule of Criminal Procedure 33(a) provides that "[u]pon the defendant's motion, the court may vacate any judgment and grant a new trial if the interest of justice so requires." Fed. R.Crim.P. 33(a). "Generally, a motion for a new trial `should not be granted unless the trial court is convinced that the jury has reached a seriously erroneous result or that the verdict is a miscarriage of justice.'" Smith v. Carpenter, 316 F.3d 178, 183 (2d Cir.2003) (quoting Atkins v. N.Y. City, 143 F.3d 100, 102 (2d Cir.1998)). A district court has "broad discretion ... to set aside a jury verdict and order a new trial to avert a perceived miscarriage of justice." United States v. Ferguson, 246 F.3d 129, 133 (2d Cir.2001) (quoting United States v. Sanchez, 969 F.2d 1409, 1413 (2d Cir.1992)). Though a district court is entitled to "weigh the evidence and in so doing evaluate for itself the credibility of the witnesses," Sanchez, 969 F.2d at 1413 (internal quotation marks omitted), it "must strike a balance between weighing the evidence and credibility of witnesses and not `wholly usurp[ing]' the role of the jury," Ferguson, 246 F.3d at 133 (quoting Autuori, 212 F.3d at 120). While courts have "broader discretion to grant a new trial under Rule 33 than to grant a motion for acquittal under Rule 29," they "nonetheless must exercise the Rule 33 authority `sparingly' and in `the most extraordinary circumstances.'" Ferguson, 246 F.3d at 134 (quoting Sanchez, 969 F.2d at 1414).
We review the district court's denial of a new trial motion for abuse of discretion, e.g., United States v. Rivas, 377 F.3d 195, 199 (2d Cir.2004), but we review the district court's factual findings only for clear error, United States v. Imran, 964 F.2d 1313, 1318 (2d Cir.1992).


*160 B. The Sufficiency of the Evidence of the Consulting Contracts Bribe

Spadoni argues that there is insufficient evidence to support the jury's verdict finding him guilty of bribery and wire fraud with respect to the consulting contracts bribe or the jury's finding that the corresponding racketeering act was proved. Specifically, he claims that the government failed to produce sufficient evidence to enable a reasonable jury to conclude, beyond a reasonable doubt, that Spadoni intended to influence Silvester to increase the size of the Triumph-Connecticut contract. We disagree.
Spadoni argues that the government never produced any evidence that he and Silvester ever discussed increasing the size of the investment contract or that Spadoni had any indication that Silvester was considering such an increase. It is true that Silvester did not testify that he explicitly indicated that he would increase the size of the investment contract. However, he did testify that he asked Spadoni for contracts by which Triumph would pay Silvester's associates "finder's fees," which would be a percentage of the total investment contract amount. According to Silvester, Spadoni told him that while Triumph could not pay finder's fees to Silvester's associates at that time, Triumph would be happy to "work it out" when Silvester left office. Trial Tr. vol. 5, 161. Silvester testified that he interpreted this response as an agreement to work out the details of an arrangement economically identical to a finder's fee.
The jury could have credited this testimony and inferred that Spadoni agreed to percentage-based contracts in order to induce Silvester to maximize the return to his associates by increasing the size of the investment contract. Additionally, the jury heard evidence indicating that Stack's contract was signed a day before the investment contract was executed; that the Stack and Thiesfield consulting contracts were fraudulently postdated until after Silvester left office; and that  just like Silvester's proposed finder's fee arrangement  the Stack and Thiesfield contracts assigned them no real duties but paid the two of them 1% of the amount of the investment contract. The jury could reasonably have credited this evidence and found it to support the inference not only that Spadoni was aware that the consulting contracts were corrupt, but also that he intentionally structured them so as to conform to Silvester's proposal and to give Silvester an incentive to increase the size of the investment contract.
Spadoni argues that even if the evidence was sufficient to warrant a reasonable jury's finding that he likely possessed the requisite intent, it was not sufficient to support this conclusion beyond a reasonable doubt. However, viewed in the light most favorable to the government, the foregoing evidence strongly indicates Spadoni's guilt. Spadoni's efforts to obstruct the investigation evidence a consciousness of guilt that further supports the jury's verdicts. See United States v. Robinson, 635 F.2d 981, 986 (2d Cir.1980) (finding evidence of obstruction of the government's investigation admissible to show consciousness of guilt); see also United States v. Malpiedi, 62 F.3d 465, 467 (2d Cir.1995) (similar). Accordingly, the jury's verdicts were supported by sufficient evidence.[11]


*161 C. The Brady/Giglio Motion

Spadoni argues that the district court abused its discretion in denying his motion for a new trial on the ground that the government unconstitutionally suppressed material exculpatory and impeaching evidence. We agree.
The government has a duty to disclose all material evidence favorable to a criminal defendant. E.g., United States v. Madori, 419 F.3d 159, 169 (2d Cir.2005) (citing Brady v. Maryland, 373 U.S. 83, 87, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963)); see also Giglio v. United States, 405 U.S. 150, 154-55, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972) (applying Brady to material that can be used to impeach prosecution witnesses). When the government violates this duty and obtains a conviction, it deprives the defendant of his or her liberty without due process of law. E.g., Rivas, 377 F.3d at 199.
"A Brady violation occurs when the government fails to disclose evidence materially favorable to the accused." Youngblood v. West Virginia, 547 U.S. 867, 869, 126 S.Ct. 2188, 165 L.Ed.2d 269 (2006). Evidence that is not disclosed is suppressed for Brady purposes even when it is "known only to police investigators and not to the prosecutor." Kyles v. Whitley, 514 U.S. 419, 438, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995). Evidence is favorable if it is either exculpatory or impeaching. See, e.g., Strickler v. Greene, 527 U.S. 263, 281-82, 119 S.Ct. 1936, 144 L.Ed.2d 286 (1999). Evidence is material if "there is a reasonable probability that, had the evidence been disclosed to the defense, the result of the proceeding would have been different." Youngblood, 547 U.S. at 870, 126 S.Ct. 2188 (internal quotation marks omitted). However, a "`showing of materiality does not require demonstration by a preponderance of the evidence that disclosure of the suppressed evidence would have resulted ultimately in the defendant's acquittal,'" id. (quoting Kyles, 514 U.S. at 434, 115 S.Ct. 1555), but only a "`showing that the favorable evidence could reasonably be taken to put the whole case in such a different light as to undermine confidence in the verdict,'" Youngblood, 547 U.S. at 870, 126 S.Ct. 2188 (quoting Kyles, 514 U.S. at 435, 115 S.Ct. 1555). The assessment of materiality is made in light of the entire record. United States v. Agurs, 427 U.S. 97, 112, 96 S.Ct. 2392, 49 L.Ed.2d 342 (1976).
In denying Spadoni's Brady motion, the district court concluded that the government did not possess Silvester's unsigned affidavit and therefore did not suppress its contents. Spadoni II, 2006 WL 2595574, at *5. This factual finding was not clearly erroneous. See Imran, 964 F.2d at 1318 ("We will not disturb the district court's findings of fact in conjunction with a Rule 33 motion unless the findings are clearly erroneous.").
However, it is undisputed that the government possessed Agent Urso's proffer notes, and that it did not disclose them *162 until it responded to Spadoni's post-trial motion. The district court found the contents of the notes, as well as of the unsigned affidavit, not to be materially different from the contents of Silvester's statements in the disclosed FBI interview report, in his grand jury testimony, and at trial. With due respect for the district court, we cannot agree with this determination.
The district court found that Agent Urso's proffer notes were not materially inconsistent with Silvester's other testimony based on its opinion that "[i]n every iteration of the events ..., Spadoni's initial response was to decline Silvester's request to pay finder's fees to Stack and Thiesfield." Spadoni II, 2006 WL 2595574, at *6. This characterization ignores a critical difference between Agent Urso's notes and Silvester's later statements. The proffer notes support an alternative version of the Silvester-Spadoni conversation about finder's fees, one entirely at odds with the government's theory of the case at trial. In Silvester's later statements, including his testimony at trial, the only requests Spadoni "decline[d]," id., were the requests to characterize the payments as finders' fees and to make the payments before Silvester left office. These later statements do not include an indication that Spadoni declined to agree to the substance of Silvester's request. Indeed, Silvester claimed that Spadoni agreed to pursue an arrangement economically identical to Silvester's finder's fee proposal, but he would characterize it differently and perform it when Silvester left office. By contrast, when Silvester first proffered (through his attorney) to the government, his statements strongly suggested that Spadoni had declined to make payments that would amount to a bribe. Agent Urso's notes, and particularly the notation that Spadoni responded that the arrangement with Triumph had to be at "arms length" and "needed to make sense," suggest that Spadoni refused to enter into an arrangement economically identical to Silvester's proposal. By far the most natural interpretation of these phrases in Agent Urso's notes is that Spadoni indicated that a deal would have to make economic sense and be at arm's length; that is, that it would be anything but economically identical to Silvester's proposal.
The difference between the government's final version of events and the version supported by Silvester's initial proffer, which was suppressed, is directly relevant to the intent element of the consulting contract bribe charges. In the final version, Spadoni agreed in substance to pay Silvester's associates for work they did not perform, which provided strong support for the jury's finding that Spadoni intended to influence an official act and to defraud the people of the State of Connecticut of their right to Silvester's honest services. Silvester's initial version of that conversation, however, provided scant if any support for the inference that Spadoni possessed the requisite intent to bribe or defraud. By suppressing Urso's notes of that proffer, the government deprived Spadoni of exculpatory evidence going to the core of its bribery case against him.
Spadoni could have used the proffer notes not merely to support his version of his conversation with Silvester, but also to impeach Silvester's credibility. While the notes did not record Silvester's words, Spadoni could have attributed them to him through cross-examining Silvester, questioning Agent Urso, and if necessary calling Silvester's attorney. See, e.g., United States v. Gil, 297 F.3d 93, 104 (2d Cir.2002) (noting that Brady material need not be admissible if it "could lead to admissible evidence" or "would be an effective tool in disciplining witnesses during cross-examination *163 by refreshment of recollection or otherwise," and citing cases).
The notes were taken at a meeting where Silvester's attorney approached the government on Silvester's behalf to relate Silvester's account of his criminal activity in an attempt to convince the government to offer him a cooperation agreement. Spadoni could have argued that Silvester initially authorized his attorney to tell the truth, which inculpated others and exculpated Spadoni, but that once he began to cooperate with the government he fabricated a new, inculpatory version of his dealings with Spadoni to enhance the value of his cooperation and his expected reward. It is by no means certain that this argument would have swayed the jury, but it is a real enough possibility to undermine confidence in the verdict.
In contending otherwise, and in maintaining that the phrases "arms length" and "needed to make sense" in Agent Urso's proffer notes do not cast the case in such a different light, the government overestimates the strength of its case. Other than Silvester's testimony, the evidence regarding Spadoni's intent was far from overwhelming. See Gil, 297 F.3d at 103 ("Where the evidence against the defendant is ample or overwhelming, the withheld Brady material is less likely to be material than if the evidence of guilt is thin."). It is true that suspiciously one-sided consulting contracts with Stack and Thiesfield were eventually signed, but the timing of these contracts was hotly disputed. If the consulting contracts were signed after the investment contract was finalized, as Spadoni contends, the inference that they were intended to influence Silvester's decision to set the final amount of the investment contract is weak absent evidence of some prior agreement or understanding. Silvester's testimony provided evidence of such an agreement. Indeed, based on the strength of Silvester's testimony, the government told the jury in summation that it could convict even if it found that the consulting contracts were signed after the investment contract. However, in the absence of Silvester's testimony, it would be more difficult for the jury to have found that Spadoni possessed the requisite intent to influence Silvester's actions regarding the investment contract unless it found that the consulting contracts themselves were signed before the investment contract was finalized.[12]
If the jury credited the government's evidence that Stack's consulting contract was executed on November 11, 1998, the day before the investment contract, it could certainly have found that Spadoni possessed the requisite intent. However, the evidence that Stack's contract was signed on that date is thin. While Triumph's records reflect that Triumph sent him a fax on November 9, 1998 and visitor logs confirm that Stack met with Spadoni and McCarthy on November 11, 1998, Stack's testimony was the only evidence that the fax was a draft contract or that a *164 contract was signed at the November 11, 1998 meeting. Moreover, Silvester corroborated Stack's earlier statements indicating that when the two of them were in Tortola, after the signing of the investment contract, they were still discussing the future prospect of a consulting contract for Stack.
In light of this factual dispute, the jury could easily have found that the consulting contract was signed after the investment contract but still convicted Spadoni based on Silvester's account of their conversation. There is thus no reason to believe that the jury would still have concluded that Spadoni possessed the requisite intent if its confidence in Silvester's testimony had been undermined. In short, the government suppressed evidence in its possession which was both exculpatory and impeaching, and there is a reasonable probability that if the evidence had been disclosed, the outcome of the proceeding would have been different.
The government argues that we must defer to the district court's determination that the remarks recorded in Agent Urso's notes did not materially differ from Silvester's other statements, noting that the district court is in the best position to assess the impact of an omitted piece of evidence on the proceedings. It is true that we review the denial of a motion for a new trial for abuse of discretion in recognition of the district court's greater familiarity with the proceedings. E.g., Rivas, 377 F.3d at 199. However, we have reversed convictions based on Brady violations where the district court concluded that the omitted evidence was not material in circumstances at least as favorable to the government as these.
In United States v. Gil, we reversed a district court's finding that a document was not material even though, if credited, the document would not directly establish a defense. 297 F.3d at 104-05. In that case, a general contractor accused of fraudulently inflating invoices from his subcontractors argued that he had been authorized by his clients to do so in a side deal in consideration for his performance of some additional duties. Id. at 97-98. The government suppressed a memorandum describing a meeting where the defendant agreed to perform extra-contractual work for the client and the client authorized the defendant to charge a premium on subcontractor bills. Id. at 97. The district court found that this memorandum was not material because it did not state that the defendant was authorized to collect this premium by falsifying the amounts recorded on his subcontractor invoices. Id. at 101. However, we found that in providing evidence for a side deal similar to that described by the defendant, the memorandum provided enough circumstantial support for the defense case that it was material. Id. at 104-05. The proffer notes here, which directly tend to negate the intent element in Spadoni's bribe charges, are at least as central to the defense case as the memorandum in Gil, which did not record any indication that the defendant was authorized to falsify the amounts of his subcontractors' invoices.
In United States v. Rivas, we reversed the district court and granted a new trial where the suppressed evidence was inculpatory as well as exculpatory, because its exculpatory character harmonized with the theory of the defense case. 377 F.3d at 199-200. That case concerned the appeal of Edgar Rivas, a seaman convicted of smuggling cocaine in his ship cabin. Id. at 196. Rivas's cabinmate Pulgar, the key prosecution witness, testified at trial that he saw Rivas store cocaine in their cabin and that Rivas had told him of Rivas's drug trafficking activity, but no evidence was produced regarding how the drugs *165 were brought aboard the ship. Id. at 197-98. The government suppressed a statement in which Pulgar claimed that he had brought the cocaine onto the ship for Rivas and delivered it to him, believing it to be alcohol until Rivas admitted its true nature. Id. at 198. The district judge denied Rivas's Brady motion, finding the suppressed testimony was not material because it was consistent with Pulgar's trial testimony indicating that Rivas owned the drugs. Id. at 198-99. Although we acknowledged that the suppressed statement was inculpatory as well as exculpatory, we nonetheless vacated the district court's denial of Rivas's Brady motion and remanded for a new trial, as the statements supported the defense's theory that Pulgar had possessed the drugs all along and blamed Rivas for them. Id. at 199-200.
The proffer notes here were at least as crucial to Spadoni's defense as Pulgar's statement was to the defense in Rivas. The government contended at oral argument that Agent Urso's proffer notes were inculpatory, arguing that Spadoni's refusal indicated his consciousness of wrongdoing. We find this argument to be strained. In any event, any inculpatory effect of Spadoni's refusal to accept Silvester's proposal is dwarfed by its tendency to exculpate Spadoni. Based on Agent Urso's notes, Spadoni could have contended at trial that the first time Silvester recounted the critical conversation with Spadoni, he told the government that Spadoni would engage only in an "arm's length" deal that would "make sense" for Triumph. The fact that Agent Urso was the case agent during the investigation and at trial renders the nondisclosure of his notes of the attorney proffer especially difficult to comprehend. The government could not explain at oral argument why the notes were withheld. We must conclude that there is a reasonable probability that if the government had not inexplicably withheld Agent Urso's proffer notes, the jury would have harbored a reasonable doubt about Spadoni's guilt. Accordingly, Spadoni is entitled to a new trial on the racketeering, racketeering conspiracy, bribery, and wire fraud charges.[13]

D. The Sufficiency of the Evidence of Obstruction of Justice

Spadoni claims that he is entitled to a judgment of acquittal on the charge of violating 18 U.S.C. § 1503 because there was insufficient evidence to support the jury's finding that he knew his actions were likely to affect the grand jury proceedings. He bases this argument on United States v. Aguilar, 515 U.S. 593, 115 S.Ct. 2357, 132 L.Ed.2d 520 (1995), where the Supreme Court held that § 1503's intent element incorporates a requirement that the defendant have "knowledge that his actions are likely to affect the judicial proceeding" being obstructed. Id. at 599, *166 115 S.Ct. 2357.[14]
The Court in Aguilar derived this requirement from several decisions limiting the facially broad language of the obstruction statute. Id. at 598-99, 115 S.Ct. 2357. The provision at issue is § 1503's "omnibus clause," which follows several clauses criminalizing endeavors to injure, intimidate or influence grand or petit jurors or court officers. The omnibus clause contains a more general and open-ended prohibition, punishing anyone who "corruptly or by threats or force, or by any threatening letter or communication, influences, obstructs, or impedes, or endeavors to influence, obstruct, or impede, the due administration of justice." § 1503. Wary of the breadth of this language in a criminal prohibition, courts have limited its scope in several ways.
In Pettibone v. United States, 148 U.S. 197, 13 S.Ct. 542, 37 L.Ed. 419 (1893), the Supreme Court imposed a requirement that a judicial proceeding actually exist, and that the defendant know or have notice of its existence. Id. at 206-07, 13 S.Ct. 542. The Court in Pettibone deemed it impossible to intend to obstruct a proceeding of which one was unaware or had no notice, and thus inferred the requirement of knowledge from the statute's corrupt intent element. Id.
Additionally, several courts of appeals, apparently in order to confine § 1503's application to sufficiently harmful conduct, imposed a "nexus" requirement, under which the defendant's conduct must be such "`that its natural and probable effect would be the interference with the due administration of justice.'" United States v. Wood, 6 F.3d 692, 695 (10th Cir.1993) (quoting United States v. Thomas, 916 F.2d 647, 651 (11th Cir.1990)). Of course, the government was not required to prove that conduct with the natural and probable effect of obstructing justice actually succeeded in doing so in order to prove an endeavor to obstruct justice under § 1503. Wood, 6 F.3d at 695-96 & n. 4; Thomas, 916 F.2d at 651.
In Aguilar, the Supreme Court adopted this "nexus" requirement with a slight modification. 515 U.S. at 600, 115 S.Ct. 2357. It applied Pettibone's requirement of knowledge[15] not just to the existence of a proceeding, but to the likelihood that the defendant's actions would affect it, holding that "as in Pettibone, if the defendant lacks knowledge that his actions are likely to affect the judicial proceeding, he lacks the requisite intent to obstruct." Aguilar, 515 U.S. at 599, 115 S.Ct. 2357.[16] The *167 Court apparently used the word "likely" to stand in for the requirement that the action have the natural and probable effect of obstructing the judicial proceeding. See Aguilar, 515 U.S. at 613, 115 S.Ct. 2357 (Scalia, J., concurring in part and dissenting in part) (adopting this interpretation of the majority opinion).
However, the facts of Aguilar appeared to complicate its holding that a defendant must have knowledge that his or her action is likely to affect the proceeding. The government presented evidence that Robert Aguilar made false statements to an FBI agent regarding an unauthorized disclosure of a wiretap, and that he did so while aware that a grand jury was investigating the same matter. Id. at 596-97, 115 S.Ct. 2357. The government's theory was that the defendant endeavored to obstruct the grand jury proceeding by lying to the agent with the intent that the agent communicate his false statements to the grand jury. Id. at 600-01, 115 S.Ct. 2357. The Supreme Court found the evidence insufficient to show that Aguilar "knew that his false statement would be provided to the grand jury" and thus reversed his § 1503 conviction. Id. at 601, 115 S.Ct. 2357. On its face, this statement seems in tension with the standard that the Court announced, which requires only that a defendant know that her actions would be likely to affect the proceeding, not that she know that her actions would have that effect.
Later in the opinion, however, the Court indicated that the operative question was whether Aguilar knew that his false statement was likely to be provided to the grand jury, and that the Court's reference to whether Aguilar knew that his statement "would be provided" was an imprecise shorthand. The Court posited several situations where a defendant would violate § 1503 by knowing that his false statements were likely to be provided to the grand jury despite the fact that the statements were not actually provided to the grand jury, such as where a defendant lies to "a subpoenaed witness who is ultimately not called to testify, or who testifies but does not transmit the defendant's version of the story." Aguilar, 515 U.S. at 602, 115 S.Ct. 2357. In indicating that a defendant could be culpable where false statements were not in fact provided to the grand jury, the Court made clear that the inquiry is whether the defendant knew his statements were likely to be presented to the grand jury, not whether he knew they would be presented.[17]
*168 Spadoni argues that his conduct was directly analogous to Aguilar's. Destroying a document does not in fact affect a grand jury proceeding if the grand jury never requests the document. While Spadoni deleted several documents from his company laptop, at no time did he delete a document for which there was an outstanding subpoena. Just as there was insufficient evidence to prove that Aguilar knew his false statements would later be communicated to the grand jury (or, as indicated above, were likely to be communicated to the grand jury), so, goes the argument, there is insufficient evidence to prove that Spadoni knew the documents he deleted would later be, or were likely later to be, requested by the grand jury.
Spadoni's argument ignores a key difference between the issuance of a grand jury subpoena duces tecum seeking the production of documents and the questioning of a subject by an investigating agent. Grand jury subpoenas duces tecum are customarily employed to gather information and make it available to the investigative team of agents and prosecutors so that it can be digested and sifted for pertinent matter. Before the subpoenas are issued, the government often does not have at its disposal enough information to determine precisely what information will be relevant. Cf. United States v. R. Enters., Inc., 498 U.S. 292, 297, 111 S.Ct. 722, 112 L.Ed.2d 795 (1991) ("`[T]he identity of the offender, and the precise nature of the offense, if there be one, normally are developed at the conclusion of the grand jury's labors, not at the beginning.'" (quoting Blair v. United States, 250 U.S. 273, 282, 39 S.Ct. 468, 63 L.Ed. 979 (1919))). Indeed, "[t]he function of the grand jury is to inquire into all information that might possibly bear on its investigation until it has identified an offense or has satisfied itself that none has occurred. As a necessary consequence of its investigatory function, the grand jury paints with a broad brush." Id. at 297, 111 S.Ct. 722 (internal citation and quotation marks omitted).
Accordingly, subpoenas duces tecum are often drawn broadly, sweeping up both documents that may prove decisive and documents that turn out not to be. This practice is designed to make it unlikely that a relevant document will escape the grand jury's notice, and it is generally effective. Destruction of a relevant document is therefore likely to impact the grand jury's deliberations. Cf. Aguilar, 515 U.S. at 601, 115 S.Ct. 2357 ("delivery of false documents" to the grand jury would be obstruction of justice).
*169 By contrast, an investigating agent collecting statements from witnesses (or even, as in Aguilar, from a suspect) does not always act as "an arm of the grand jury," and "what use will be made of false testimony given to an investigating agent who has not been subpoenaed or otherwise directed to appear before the grand jury is... speculative." Id. 515 U.S. at 600-01, 115 S.Ct. 2357.
This context provides a crucial distinction between Aguilar's conduct and Spadoni's. The inference that Aguilar's statements to the agent would be presented to the grand jury was not strong. The statements were not obtained by grand jury subpoena, and statements made to investigating agents are not communicated to grand juries as a matter of course. By contrast, the inference that the grand jury would issue a subpoena for the Thiesfield and Stack contracts was quite strong, perhaps inescapable. The government produced evidence suggesting Spadoni's awareness of the comprehensive nature of the subpoenas duces tecum typically issued in federal grand jury investigations. The jury heard evidence that Triumph's attorneys anticipated further subpoenas, Trial Tr. vol. 5, 179-80; that Spadoni had received advice from a former prosecutor indicating that the grand jury would be likely to inspect the data contained on his laptop, id. at 180-81; that Spadoni stated his belief that federal investigations are "very comprehensive and thorough," id. at 194; and that Spadoni asked Silvester to destroy copies of a different contract, id. at 191-93.
The Stack and Thiesfield contracts were not peripheral documents; they were at the very core of the transaction the government was investigating. The jury could have concluded that Spadoni was aware that further subpoenas covering a broad range of documents would issue, and that he knew that it was likely that the Stack and Thiesfield contracts would be requested. Accordingly, Spadoni's conviction for obstruction of justice, based on his destruction of those documents in his computer files, was supported by sufficient evidence.

E. The Jury Instructions Regarding Obstruction of Justice

Spadoni argues that even if the evidence was sufficient to support his conviction, the district court's instructions allowed the jury to convict him on a theory inconsistent with Aguilar's mens rea requirement. He challenges the instruction that in order to convict, the jury had to "find that in the defendant's mind, his or its conduct had the natural and probable effect of obstructing or interfering with the grand jury proceeding." Trial Tr. vol. 16, 123. The district court rejected the defendant's request to insert the words "he knew that" after "in the defendant's mind." Def.'s Exceptions to Jury Instructions 7. Spadoni argues that the instruction as given allowed the jury to convict him if it merely found that he had a belief or impression that his conduct was likely to affect the grand jury investigation, a mental state that falls short of § 1503's requirements. We are not persuaded.
Courts have long struggled to determine, in the § 1503 context and elsewhere, precisely what level of certainty is necessary to constitute knowledge. The issue has not been fully resolved,[18] but Aguilar *170 makes clear that § 1503 is not violated if the defendant merely has the impression that his conduct will have the natural and probable effect of obstructing justice. See Aguilar, 515 U.S. at 601-02, 115 S.Ct. 2357. The jury in that case heard evidence that the agent had told Aguilar that a grand jury would be hearing evidence about the investigation, 515 U.S. at 600, 115 S.Ct. 2357, and Aguilar himself testified that, at the end of the interview, it was his "impression" that his remarks would be conveyed to the grand jury. Id. at 614, 115 S.Ct. 2357 (Scalia, J., concurring in part and dissenting in part). The jury could reasonably have inferred that Aguilar, at the time he made false statements, had the impression that those statements would be conveyed to the grand jury. Yet the Supreme Court found the evidence insufficient to support a finding that he had the requisite mental state for a § 1503 conviction. That element necessarily requires more than a showing that the defendant merely had the impression that his conduct would have the natural and probable effect of obstructing justice.
In this case, the district court instructed the jury that the government must prove that "in the defendant's mind, his or its conduct had the natural and probable effect of obstructing or interfering with the grand jury proceeding." Trial Tr. vol. 16, 123. The court's language was taken from our opinion in United States v. Schwarz, 283 F.3d 76 (2d Cir.2002), where we explained that under Aguilar, the conduct offered to prove § 1503's intent element "must be conduct that is directed at the court or grand jury and that, in the defendant's mind, has the `natural and probable effect' of obstructing or interfering with that entity." Schwarz, 283 F.3d at 109 (quoting Aguilar, 515 U.S. at 599, 115 S.Ct. 2357). We conclude that it sufficiently conveyed the high degree of certainty § 1503 requires. The district court did not tell the jury that § 1503 required a finding that in Spadoni's mind, his conduct might have had the natural and probable effect of obstructing justice, but rather a finding that in his mind, his conduct did have the natural and probable effect of obstructing justice. That is, the instruction adequately conveyed that Spadoni must be certain enough of his conduct's natural and probable obstructive effect that, to him, the effect was a fact, not a mere possibility. The district court's charge thus imposed on the government a burden substantially more demanding than merely proving that Spadoni had the impression that his conduct would have the natural and probable effect of obstructing justice, a formulation that connotes a highly tentative belief. See, e.g., American Heritage Dictionary of the English Language (4th ed.2000), available at http:// www.bartleby.com/61/57/I0065700.html (defining "impression" as, inter alia, "2. A vague notion, remembrance, or belief"). Without resolving the difficult question of precisely what degree of certainty amounts to knowledge under § 1503, we can say that the formulation at issue here sufficed to convey the requisite certainty.
It is true that our reference in Schwarz to what was "in the defendant's mind," *171 which was reproduced by the district court in the challenged instruction, could have made it clearer that the defendant's subjective belief has to amount to knowledge. Compare Schwarz, 283 F.3d at 109 (stating that the conduct evincing the defendant's specific intent "must be conduct that is directed at the court or grand jury and that, in the defendant's mind, has the natural and probable effect of interfering with that entity") (internal quotation marks omitted), with id. ("`[I]f the defendant lacks knowledge that his actions are likely to affect the judicial proceeding, he lacks the requisite intent to obstruct.'" (emphasis added) (quoting Aguilar, 515 U.S. at 599, 115 S.Ct. 2357)). If the case had involved a claimed error in the jury charge, as opposed to a sufficiency challenge to Schwarz's § 1503 conviction, we might have addressed more squarely the issue before us here. In any event, though the district court's use of our language from Schwarz was understandable, we believe the better course in future cases would be for courts to make it explicit that to violate § 1503's omnibus clause, a defendant must know that her conduct has the natural and probable effect of obstructing the judicial or grand jury proceeding in question (or, what amounts to the same thing, that the defendant must know that her conduct is likely to obstruct the proceeding).
However, a jury instruction need only adequatelynot perfectlyinstruct the jury on the applicable legal standard. See, e.g., United States v. Naiman, 211 F.3d 40, 51 (2d Cir.2000) ("`A jury instruction is erroneous if it misleads the jury as to the correct legal standard or does not adequately inform the jury on the law.'") (quoting United States v. Walsh, 194 F.3d 37, 52 (2d Cir.1999)). The fact that a jury could conceivably misinterpret an instruction as to the requisite state of mind does not render the instruction erroneous. See, e.g., Victor v. Nebraska, 511 U.S. 1, 6, 114 S.Ct. 1239, 127 L.Ed.2d 583 (1994) ("[T]he proper inquiry is not whether the instruction `could have' been applied in an unconstitutional manner, but whether there is a reasonable likelihood that the jury did so apply it.") (citing Estelle v. McGuire, 502 U.S. 62, 72 & n. 4, 112 S.Ct. 475, 116 L.Ed.2d 385 (1991)); see also Jones v. United States, 527 U.S. 373, 390 & n. 9, 119 S.Ct. 2090, 144 L.Ed.2d 370 (1999) (similar). Thus, while we encourage district courts in the future to make clear that the required mental state is knowledge that the defendant's conduct had the natural and probable effect of obstructing justice, we do not find that the instruction as given was erroneous.[19]
Accordingly, Spadoni is not entitled to a new trial on the obstruction of justice charge.

CONCLUSION
For the foregoing reasons, we affirm Spadoni's conviction on Count 24, but reverse the judgment of the district court and remand for a new trial on Counts 1, 2, *172 and 19-23. Because we cannot be certain that the 36-month concurrent sentence on Count 24 was not affected by the convictions that we have reversed, the sentence on Count 24 is vacated and the case is remanded for resentencing on that count.
NOTES
[1]  The Honorable John Gleeson of the United States District Court for the Eastern District of New York, sitting by designation.
[2]  Unbeknownst to Spadoni, Stack had previously given Silvester illegal kickbacks of portions of his finder's fees.
[3]  As discussed below, Silvester initially provided a different account of this interaction to investigators through his attorney, and the failure to disclose that initial version underlies Spadoni's Brady claim on this appeal.
[4]  At trial, Stack also testified that while he was in Tortola, the day after he allegedly signed the $1 million contract, he was in a bad mood because of his company's financial difficulties.
[5]  In addition, Silvester testified that Thiesfield's contract was signed in January 1999, after she returned from a December 1998 trip to California.
[6]  Thiesfield and McCarthy pled guilty to bribery charges. Andrews was tried and convicted separately.
[7]  This objection is recorded in Spadoni's undated written statement of his exceptions to the jury instructions. This statement indicates that the objections were made during the preliminary and final charge conferences, which were both held off the record in chambers. Def.'s Exceptions to Jury Instructions 1. The government does not object to the accuracy of this document.
[8]  This was the third written memorandum opinion in Spadoni's case because it followed the district court's denial of Spadoni's motion for a new trial pursuant to Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), and Giglio v. United States, 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), discussed in more detail infra. See United States v. Spadoni (Spadoni II), No. 00-CR-217 (EBB), 2006 WL 2595574, at *7 (D.Conn. Sept.7, 2006) (denying Brady/Giglio motion).
[9]  Although the note omits the word "not," the government agrees that Agent Urso intended to write "could not pay related to a deal" and left out the word "not" merely as an oversight.
[10]  Spadoni remains on bond pending the outcome of this appeal.
[11]  Spadoni takes issue with the district court's statement, in its decision denying his insufficiency challenge, that the jury could reasonably have inferred that the consulting contracts were intended to induce Silvester to enter into the investment contract in the first place. See Spadoni I, 2005 WL 2275938, at *6 ("Regardless of whether Spadoni and McCarthy knew of Silvester's intent to increase the investment, they agreed to pay Stack and Thiesfield the equivalent of finder's fees in an attempt to influence Silvester's decision to invest in Triumph Capital."). The government's theory in the indictment and bill of particulars, and at trial, was that the campaign contribution bribesof which Spadoni was acquittedwere the sole incentive for Silvester to enter into the investment contract. Thus, Spadoni argues that his conviction on the consulting contracts bribe cannot be upheld by evidence suggesting that it was intended to induce the creation of the contract. Given that the evidence is sufficient for a reasonable jury to find that Spadoni intended the consulting contracts to induce Silvester to increase the size of the investment, we have no occasion to address this contention.
[12]  The district court's instructions on the bribery and wire fraud counts charged the jury solely on a theory of bribery, not on an illegal gratuity theory. In instructing the jury on the bribery count, the district court stated that the offer of a bribe had to precede the official act being influenced. Trial Tr. vol. 16, 181. The district court's instructions on the wire fraud count defined the charged scheme or artifice to defraud as simply the same charged bribe, referred to her previous instructions on the bribery, and noted that the bribe had to be understood to be offered "in exchange for an official act to be performed." Id. at 190-91 (emphasis added). Even had the district court not instructed the jury that the offer of a bribe had to precede the official act, the inference of corrupt intent would still have been weaker had the jury found that Spadoni agreed to the consulting contracts only after the investment contract was finalized.
[13]  Spadoni argues that the impeaching effect of the proffer notes might have led the jury to discredit Silvester's testimony in its entirety, including his statements regarding the obstruction of justice charge, and thus that the suppressed evidence raises a reasonable probability that the jury would not have convicted Spadoni of obstruction of justice. We disagree. Even if the suppressed notes had an impeaching effect so strong as to call into question Silvester's testimony on other matters, the government's evidence of Spadoni's obstruction of justice was overwhelming. In light of the forensic examiner's detailed testimony regarding the suspicious timing of the deletion of relevant files from Spadoni's laptop using Destroy-It! software, and its corroboration with Trevisani's testimony about Spadoni's mention of Destroy-It! as software to be used in order to "hide something," Trial Tr. vol. 7, 217, we do not think that the suppression of Agent Urso's notes raises a reasonable probability that the verdict on the obstruction of justice count would have been different.
[14]  Prior to Aguilar, we had noted that "destroying documents in anticipation of a subpoena can constitute obstruction." United States v. Ruggiero, 934 F.2d 440, 450 (2d Cir.1991). Spadoni argues that we must reevaluate this precedent in light of Aguilar.
[15]  It did not, however, allow a conviction based on mere notice, as opposed to knowledge.
[16]  The Court also noted, puzzlingly, that the "natural and probable effect" requirement was equivalent both to the requirement that the defendant intend to influence judicial or grand jury proceedings as opposed to ancillary proceedings, Aguilar, 515 U.S. at 599, 115 S.Ct. 2357 (citing United States v. Brown, 688 F.2d 596, 598 (9th Cir.1982)), and also to the more general requirement that the charged act "have a relationship in time, causation, or logic with the judicial proceedings." Aguilar, 515 U.S. at 599, 115 S.Ct. 2357 (citing Wood 6 F.3d at 696, United States v. Walasek, 527 F.2d 676, 679 & n. 12 (3d Cir.1975)).

The equivalences the Court perceived do not have a great deal of practical significance. An instruction solely on the requirement that the defendant intend to influence judicial or grand jury proceedings as opposed to ancillary proceedings, without stating that the defendant must know that his actions are likely to obstruct the proceedings, is insufficient to support a § 1503 convictionthe jury in Aguilar certainly could have found that the defendant intended to influence the grand jury proceeding, 515 U.S. at 613-14, 115 S.Ct. 2357 (Scalia, J., concurring in part and dissenting in part). Similarly, an instruction stating only that the jury must find that the defendant's act had "a relationship in time, causation or logic with the judicial proceedings," id. at 599, 115 S.Ct. 2357, without specifying that the defendant must know that his act is likely to obstruct the proceeding, is also insufficient to support a § 1503 conviction. Any action has some relationship in time, causation or logic with judicial proceedings, and without specific guidance on what that relationship must be, a jury may vote to convict without finding that a defendant knew that his action was likely to obstruct a judicial or grand jury proceeding.
Conversely, if the jury is instructed that a defendant, in order to violate § 1503's omnibus clause, must know that his action was likely to obstruct a judicial or grand jury proceeding, there is no need to instruct the jury that the defendant's act must have a relationship in time, causation or logic with the judicial proceedings. The jury must, however, be instructed that the defendant cannot be convicted if he does not intend to obstruct the judicial or grand jury proceedings.
[17]  We note that Aguilar seems to allow the term "likely," perhaps because it is used interchangeably with the natural and probable consequence formulation, to encompass plans where success reasonably appears to be likely but is in fact impossible. A witness who is known to be subpoenaed to testify before the grand jury but who secretly has no intention of relating the defendant's version of the story appears to still qualify as "likely" to pass along the defendant's false statements for the purposes of the defendant's mental state. See id. ("Were a defendant with the requisite intent to lie to a subpoenaed witness ... who testifies but does not transmit the defendant's version of the story, the defendant has endeavored to obstruct, but has not actually obstructed, justice. Under our approach, a jury could find such defendant guilty."). In using the term "likely" or "natural and probable consequence" in this way, the Court simply follows the longstanding rule that apart from the nonexistence of a judicial proceeding, see Pettibone, 148 U.S. at 207, 13 S.Ct. 542 (finding that predecessor to § 1503 could not be violated if no judicial proceedings existed to be obstructed); United States v. Reed, 773 F.2d 477, 485 (2d Cir.1985) ("[T]he existence of an ongoing proceeding is an element of a § 1503 violation...."), impossibility is not a defense to obstruction of justice, see Osborn v. United States, 385 U.S. 323, 333, 87 S.Ct. 429, 17 L.Ed.2d 394 (1966) (affirming § 1503 conviction where the defendant passed bribe money to a government cooperator who had no intention of delivering it to the target juror).
[18]  See United States v. Solow, 138 F.Supp. 812, 816 & n. 14 (S.D.N.Y.1956) (finding that belief and reasonable grounds to believe that destroyed documents would be requested by a grand jury sufficed for § 1503 conviction); compare United States v. Golomb, 811 F.2d 787, 792 (2d Cir.1987) ("Knowledge and belief are very different mental states; knowledge implies a much higher degree of certainty."), with United States v. Coté, 504 F.3d 682, 688 & n. 7 (7th Cir.2007) (distinguishing Golomb as a case dealing with completed offenses and concluding that the defendant in a prosecution for a criminal attempt articulated "no discernable difference between `belief' and `knowledge'" in his case); cf. Model Penal Code § 2.02(7) (Proposed Official Draft 1962) ("When knowledge of the existence of a particular fact is an element of an offense, such knowledge is established if a person is aware of a high probability of its existence, unless he actually believes that it does not exist.").
[19]  Our conclusion in this regard is further supported by our examination of how the case was actually presented to the jury at trial. In summation, the prosecutor explicitly stated that Spadoni could not be convicted unless he knew that his conduct had the natural and probable effect of obstructing the proceeding. Trial Tr. vol. 14, 49 ("[D]id he corruptly endeavor to obstruct the Grand Jury? That is, did he know that his contactconduct, excuse me, would have the natural and probable effect of interfering with the Grand Jury?"). Spadoni's counsel conveyed the same standard. Id. at 195 ("He has to know that what he's doing has the natural and probable effect of obstructing the Grand Jury."). Under the circumstances, the risk that the jury would misunderstand the district court's instruction was negligible.
