
676 F.3d 260 (2011)
UNITED STATES of America, Appellee,
v.
Ronald E. FERGUSON, Christopher P. Garand, Elizabeth A. Monrad, Robert D. Graham, Christian M. Milton, Defendants-Appellants.[*]
Docket Nos. 08-6211-cr(L), 09-0121-cr(Con), 09-0313-cr(XAP), 09-0507-cr(Con), 09-0881-cr(XAP), 09-1072-cr(Con), 09-1120-cr(XAP), 09-1677-cr(Con), 09-1723-cr(XAP), 09-2127-cr(Con), 09-2141-cr(XAP).
United States Court of Appeals, Second Circuit.
Argued: November 17, 2010.
Decided: August 1, 2011.
Amended: December 19, 2011.
*265 Seth P. Waxman (Jonathan E. Nuechterlein, Catherine M.A. Carroll, Washington, *266 DC, and Paul A. Engelmayer, Daniel C. Richenthal, Julia M. Lipez, New York, NY, on the brief), Wilmer Cutler Pickering Hale and Dorr LLP, Washington, DC, for Defendant-Appellant Ronald E. Ferguson.
Robert J. Cleary (Anthony Pacheco, Keith Butler, Los Angeles, CA, and Mark D. Harris, William C. Komaroff, New York, NY, on the brief), Proskauer Rose LLP, New York, NY, for Defendant-Appellant Christopher P. Garand.
Ira M. Feinberg (Katie M. Lachter, New York, NY, and Dominic F. Perella, Washington, DC, on the brief), Hogan Lovells U.S. LLP, New York, NY, for Defendant-Appellant Elizabeth A. Monrad.
Alan Vinegrad (Jonathan L. Marcus and Pamela A. Carter, on the brief), Covington & Burling LLP, New York, NY, for Defendant-Appellant Robert D. Graham.
Frederick P. Hafetz (Victor J. Rocco, Tracy E. Sivitz, Hafetz Necheles & Rocco, New York, NY, and Kannon K. Shanmugam, George W. Hicks, Jr., Williams & Connolly LLP, Washington, DC, on the brief), Hafetz Necheles & Rocco, New York, NY, for Defendant-Appellant Christian M. Milton.
Eric J. Glover and Raymond E. Patricco, Jr. (William J. Nardini and David B. Goodhand, on the brief), Assistant United States Attorneys, for Nora R. Dannehy, Acting United States Attorney for the District of Connecticut, New Haven, CT, and Neil H. MacBride, United States Attorney for the Eastern District of Virginia, Alexandria, VA, for Appellee.
Before: JACOBS, Chief Judge, KEARSE and STRAUB, Circuit Judges.


                                   TABLE OF CONTENTS
BACKGROUND ..............................................................................267
  I-  All Defendants' Claims ............................................................273
      A-  Stock Price Data ..............................................................273
      B-  Jury Charges ..................................................................275
          1-  "Willfully Caused" Instruction ............................................275
          2-  "Conscious Avoidance" Instruction .........................................277
          3-  "Specific Unanimity" Instruction ..........................................279
          4-  "No Ultimate Harm" Instruction ............................................280
      C-  Prosecutorial Misconduct ......................................................281
          1-  Napier's Potential Perjury ................................................281
          2-  Summation Remarks .........................................................283
 II-  Ferguson's Claims .................................................................284
      A-  Sufficiency Challenge to Scienter Evidence ....................................284
      B-  Graham's Email: "[Ferguson] et al[.] have been advised of, and have
            accepted, the potential reputational risk" ..................................285
          1-  Double-Hearsay ............................................................285
          2-  Severance from Graham .....................................................286
          3-  Summation .................................................................288
      C-  Finding that Conspiracy Began with First Call .................................288
III-  Graham's Claims ...................................................................289
      A-  Requested Jury Instructions ...................................................289
          1-  Professional Responsibility Rules .........................................289
          2-  Non-Contractual Understandings ............................................290
      B-  Treatment of Graham's Boss ....................................................290
 IV-  Milton's Claims ...................................................................291
      A-  Admissibility of Recordings Denigrating AIG ...................................291
      B-  Severance from Gen Re Defendants ..............................................292



*267
  V-  Monrad's Claims ...................................................................293
 VI-  Garand's Claims ...................................................................294
CONCLUSION ..............................................................................294

DENNIS JACOBS, Chief Judge:
This criminal appeal arose from a "finite reinsurance" transaction between American International Group, Inc. ("AIG") and General Reinsurance Corporation ("Gen Re"). That transaction (the "Loss Portfolio Transfer," or "LPT") reallocated risk in a way that shored up AIG's flagging loss reserves, which were feared to be dragging down its stock price. Finite reinsurance transactions, which entail some (usually low) risk, are acceptable accounting measures in the insurance industry, and have their uses; but in this instance it is charged that the transaction entailed no risk at all, and was a fraud. The defendants, four executives of Gen Re and one of AIG, appeal from judgments entered in 2008 and 2009 by the United States District Court for the District of Connecticut (Droney, J.), convicting them of conspiracy, mail fraud, securities fraud, and false statements made to the Securities and Exchange Commission ("SEC"). They were sentenced principally to prison terms ranging from one to four years, and are free on bail pending this appeal.
The government's case depended heavily on testimony from two cooperating witnesses  Richard Napier, a senior executive of Gen Re; and John Houldsworth, a senior executive of Cologne Re Dublin ("CRD"), an Irish subsidiary of Gen Re  who had pled guilty to similar charges. Their testimony was bolstered by contemporaneous recordings of calls involving Houldsworth (a normal business practice in Ireland for derivatives traders). The government also introduced AIG stock-price data to show the LPT's material effect on investors: The price declined steeply as details about regulatory scrutiny of the deal were released. After a six-week trial, the jury convicted the defendants on all counts.
The defendants appeal on a variety of grounds, some in common and others specific to each defendant, ranging from evidentiary challenges to serious allegations of widespread prosecutorial misconduct. Most of the arguments are without merit, but the defendants' convictions must be vacated because the district court abused its discretion by admitting the stock-price data.

BACKGROUND
AIG's announcement of its 3Q earnings in 2000 met analysts' expectations, but the stock price dropped significantly nevertheless. The cause was thought to be a $59 million decline in loss reserves that quarter.
Loss reserves are liabilities on an insurer's balance sheet that approximate expected claims on insurance contracts. Stock analysts and investors evaluate loss reserves in conjunction with new policies: If loss reserves do not rise when new policies are written (or worse, if they fall), the insurer's stock may drop notwithstanding better-than-expected income because a contract of insurance that is not covered by sufficient loss reserves inflates present income at the expense of future income. Thus, counterintuitively, a net decrease in a balance sheet liability may cause a stock price to drop.
Loss reserves can be transferred between companies through reinsurance arrangements. *268 In an ordinary reinsurance transaction, an insurer purchases coverage from a reinsurer for potential losses on policies it has issued, thus ceding substantial or unlimited risk to the reinsurer. In finite reinsurance, however, a company cedes a smaller, circumscribed (hence, finite) amount of risk to the reinsurer. To over-simplify, traditional reinsurance is primarily used by an insurer to lay off risk, whereas finite reinsurance lends itself to accounting goals because it can be strategically designed but also carefully bounded.
An insurer's creativity with finite reinsurance transactions is not unconstrained: Accounting rules require that each transaction transfer a threshold of risk. Under Financial Accounting Standards ("FAS") 113,[1] a reinsurance transaction must have "significant insurance risk," so that it is "reasonably possible" that the reinsurer may realize a "significant loss" from the deal. See FAS 113 ¶ 9. An industry rule of thumb provides clearer guidance: A transaction with more than a 10% chance of incurring more than a 10% loss (of the contractual limit) satisfies FAS 113, and can be booked as reinsurance.
Transactions that fall short of the risk threshold in FAS 113 cannot be treated as reinsurance; any premium paid must be deposit accounted, which has no effect on loss reserves. Each party makes its own determination as to whether a transaction has risk sufficient to qualify as reinsurance. Since risk can be hard to quantify, counterparties' good-faith determinations may conflict, with one booking the transaction as reinsurance and the other, as a deposit. Such asymmetric accounting may draw the attention of regulators, but is not a violation per se. See FAS 113 ¶ 47 (rejecting symmetrical-accounting requirement).

A
In view of the defendants' convictions, we summarize the facts in the light most favorable to the government. United States v. Riggi, 541 F.3d 94, 96 (2d Cir. 2008).
Maurice "Hank" Greenberg, CEO of AIG, was convinced that AIG's decreased loss reserves were depressing the stock. On October 31, 2000, he called Ronald Ferguson, the CEO of Gen Re, to discuss ways to shore up AIG's reserves. AIG was Gen Re's largest client, so Ferguson was eager to assist. (Greenberg was named as an unindicted coconspirator; Ferguson is a defendant.)
Greenberg requested a particular deal: AIG wished to "borrow" a specific range of loss reserves ($200 million to $500 million) over a six- to nine-month time period. This was unusual in several respects. Cooperating witness Napier, who had worked on hundreds of reinsurance deals, had never encountered a deal premised on a request for a specific amount of loss reserves. To the contrary, loss reserves are typically calculated through a detailed actuarial analysis, after a deal has been negotiated. It was also uncommon for AIG to act as the reinsurer; it typically sought reinsurance from Gen Re. The deal was to be largely funds-withheld, meaning that the ceding party would retain a large percentage of the premium it owes and only claim such losses as exceed the premium. A funds-withheld arrangement may not be irregular, but the insistence upon it is suggestive: AIG could register a substantial *269 change in loss reserves without Gen Re remitting a comparably large payment.
An important question for this case is whether the call between Ferguson and Greenberg initiated a conspiracy. It may have been a high-level brainstorming session about using accounting rules aggressively  but lawfully  to achieve an accounting objective; but it may (instead or also) have been an unlawful agreement to deceive AIG stockholders by booking a no-risk transaction (which by definition would not satisfy FAS 113) as reinsurance.

B
Shortly after the call from Greenberg, Ferguson created a deal team at Gen Re. He briefed Napier, a senior VP and the manager of Gen Re's relationship with AIG, and asked him to spearhead the effort. Ferguson suggested that Napier contact: (1) Christian Milton, the VP of reinsurance at AIG, to discuss specific requirements of the deal; and (2) Joe Brandon, the president of Gen Re. (Milton is a defendant; Brandon was named as an unindicted co-conspirator.)
Napier spoke with Brandon the same day, and Milton the next. Milton confirmed that AIG only wanted reserve impact to address criticism from stock analysts, and he and Napier began preliminary discussions about the particulars of the deal. Brandon suggested that defendant Chris Garand, a senior VP and Chief Underwriter of Gen Re's finite reinsurance operations, would be a good resource.
It is unclear when Napier first contacted Garand. Emails reflect that within a week, defendant Elizabeth Monrad, the Chief Financial Officer of Gen Re, became involved. Napier claims that he contacted Garand that same week, and that Garand pitched a "no risk" transaction in a November 13 meeting with Monrad and him.
For convenience, the role and affiliation of each player referenced in this opinion are listed in the margin.[2]
Napier's testimony concerning Garand is suspicious. Garand was added as a defendant in a superseding indictment, filed more than seven months after the other defendants were charged. On the same day the superseding indictment was filed, Napier confidently named Garand as the source of the no-risk idea. This identification contradicted Napier's previous identifications (recanted at trial) of Milton and Ferguson as the source. He made that identification of Milton while he was looking at the same undated page of notes that he later attributed to a meeting with Garand and Monrad (which he does not contend that Milton attended). Garand calls this allegation a perjurious attempt to curry favor with the government, and argues that the government was complicit in the perjury, or complacent.
The documentary evidence bearing on the November 13 meeting does not show *270 what happened. Late in the morning on November 13, Monrad emailed a warning to Ferguson about asymmetric accounting, which suggests that the no-risk idea had been hatched. Later that day, defendant Monrad called cooperating witness Houldsworth,[3] the CEO of CRD, to solicit his help with the transaction. She cautioned that AIG would not be charged any losses on the deal, and that Ferguson requested strict confidentiality. The next day, Houldsworth called Garand about the transaction, broaching the subject delicately because of the confidentiality warning. But Garand showed no familiarity with it, presuming instead that Houldsworth was referring to another AIG deal:

HOULDSWORTH: AIG, uh, you may have heard about this, I, I presume it's highly confidential  well, it's definitely high  [Monrad] told me not to tell anyone.... Do you know anything about this  or not?

GARAND: No, only to the extent that Milan mentioned it and 

HOULDSWORTH: Okay.

GARAND:  Tad had a meeting with AIG.

HOULDSWORTH: Okay. Well, it's nothing to do with [your other AIG deal].

GARAND: Okay.

HOULDSWORTH: Um, the  the issue is, and I, for I, don't know why you don't know in that case. I mean, maybe it's,  I don't know how these things work. Anyway, I'm gonna tell you anyway. If I get in trouble, heigh ho, uh, we have to work together, so it's stupid otherwise.
Joint Appendix at 1959-60. Garand claims that this call on November 14 was the first he heard of the LPT.

C
The Gen Re team continued designing the transaction. On November 15, Houldsworth circulated a draft slip contract of the LPT to Monrad, Garand, Napier, and two others. (Ferguson did not receive the email, but he reviewed a hard copy of the email and slip.) The draft contract contemplated Gen Re paying AIG $10 million for assuming $100 million of risk. The premium was $500 million on a 98% funds-withheld basis, meaning that Gen Re would pay only $10 million but could charge AIG only for losses beyond the $500 million premium (up to a $600 million cap on losses, yielding $100 million of risk).
The slip omitted two key terms of the transaction, however, which were discussed frankly in the cover email:
First, in selecting contracts for AIG to reinsure, Houldsworth designated over $300 million in contracts that had already been reinsured, leaving "no possibility" (or making it "virtually impossible") for the remaining contracts to have claimable losses (i.e., over the $500 million premium). Trial Tr. at 2286. This accommodated AIG's request, which Houldsworth characterized as seeking loss reserves "with the intention that no real risk is transferred." Joint Appendix at 1978.
Second, Gen Re was to receive a fee for the deal as well as reimbursement for the portion of the premium it would pay:
Contract we provide must give A[IG] a potential upside in entering the transaction. *271 Given that we will not transfer any losses under this deal it will be necessary for A[IG] to repay any fee plus the margin they give us for entering this deal.
Joint Appendix at 1978 (emphasis added) (Houldsworth's cover email). Houldsworth confirmed at trial that the exclusion of these fees from the slip was intentional, because AIG wanted a piece of paper that would allow the contract to be booked as a reinsurance deal. At one point, Napier clumsily suggested that fees be written into the contract. Houldsworth replied:
But I think to give them a deal with no risk in it, and just charge them a fee, I, you know, I mean, you can assume their auditors are, you know, are being, you know, pushed in one direction, but I think that's just going too far.... I'd be staggered if they would get away with that.
Joint Appendix at 2003. Similarly, Monrad rejected the idea of memorializing the fees in a separate written agreement: "Those always get a little tricky because sometimes firms ... feel obliged to show their auditors them." Joint Appendix at 2015.[4]
By November 17, Ferguson had secured Hank Greenberg's agreement to the rough contours of the deal, including the no-risk aspect, the repayment of the $10 million premium, and Gen Re's $5 million net fee. Greenberg tapped Milton as a point person for the transaction at AIG. Napier then forwarded the slip contract to Milton, describing in his cover email the fee and premium repayment (which remained conspicuously absent from the contract).

D
With a preliminary agreement in place, Gen Re began internal discussions about the accounting treatment of the deal. At some point during these discussions, defendant Robert Graham, an in-house lawyer at Gen Re, joined the team. The Gen Re side understood that AIG wished to book the transaction as reinsurance to invigorate its loss reserves. But recognizing that the deal lacked the necessary risk, they wanted to protect Gen Re by booking the transaction using deposit accounting.
Ferguson asked his team to alert AIG that Gen Re was contemplating asymmetric accounting. On November 20, defendants Monrad, Graham, and Garand (and co-conspirator Napier) of Gen Re called defendant Milton of AIG to advise that Gen Re would be booking the LPT as a deposit transaction. Milton confirmed that Gen Re's deposit accounting would not be an issue for AIG. Napier relayed the good news to Ferguson.

E
Milton accepted the deal on AIG's behalf on December 7, but he asked Gen Re to create a paper trail to disguise the transaction's origin. On December 18, Houldsworth circulated an offer letter to AIG suggesting that the deal had first been solicited by Gen Re. Milton at AIG circulated the offer letter and draft contract to AIG accountants and informed them that it would be booked as reinsurance, thus ensuring that the usual underwriting and actuarial due diligence on such a large transaction would not be performed.
Gen Re's in-house counsel Graham drafted the final contracts for the deal. They omitted the $5 million fee and $10 *272 million premium repayment. As he drafted, Graham expressed some discomfort with the accounting of the deal to his boss, Tim McCaffrey, the General Counsel of Gen Re:
Tim 
The AIG project continues....
Our group will book the transaction as a deposit. How AIG books it is between them, their accountants and God; there is no undertaking by them to have the transaction reviewed by their regulators.
[Ferguson] et al[.] have been advised of, and have accepted, the potential reputational risk that US regulators (insurance and securities) may attack the transaction and our part in it.
Rob
Joint Appendix at 2192. The discomfort must have been fleeting, however, because the contracts were shortly thereafter finished and sent off to Milton.
In January 2000, the offer letter (annotated with written instructions from Milton) and draft slip contract were routed to Lawrence Golodner, an assistant comptroller at AIG. (The documents were sent by Golodner's boss, John Blumenstock, but their route from Milton to Blumenstock is not entirely clear.) Golodner followed Blumenstock's instructions, booking $250 million in loss reserves for each of 4Q 2000 and 1Q 2001.[5]

F
The deal worked, up to a point. AIG announced increased loss reserves in 4Q 2000 and IQ 2001 that, without the LPT, would have been declines.
Meanwhile, Gen Re sought to enforce the unwritten fee agreements. It refused to deliver the $10 million premium (that it was contractually obliged to pay) until it had collected the $15 million that it was owed under the secret side deal. Garand orchestrated a scheme to effect the payment without directly transferring funds (which could have attracted regulatory scrutiny).[6] Milton agreed. On December 28, the final steps of Garand's scheme were executed; the same day, Gen Re wired the $10 million premium to an AIG subsidiary.
The matter was dormant for several years. In a typical reinsurance arrangement, the ceding company files claims with the reinsurer, which pays the claims and, over time, reduces loss reserves commensurately. But Gen Re made no claims, AIG paid no claims, and there were no adjustments to AIG's loss reserves (excluding a $250 million reduction in AIG's loss reserves in late 2004, when the parties commuted half of the deal).
Beginning in February 2005, the SEC and the Office of the New York Attorney General began investigating the transaction. News articles about the investigations trickled out for several months, while AIG's stock price declined steadily. On May 31, 2005, AIG concluded that the LPT did not transfer sufficient risk for reinsurance accounting, and restated its financials for the duration of the LPT's existence.

G
The defendants were charged with conspiracy, mail fraud, securities fraud, and making false statements to the SEC.[7] The *273 trial began in January 2008. The government's two cooperating witnesses, Napier and Houldsworth, provided critical testimony that narrated the events of the deal and was used to argue the parties' fraudulent intent. The particulars of much of their testimony were corroborated by contemporaneous emails and Houldsworth's recorded phone conversations. (Much of this corroboration was admitted into evidence as co-conspirator statements, the court having found that the conspiracy began with the Ferguson-Greenberg call.[8]) The testimony was not uncontroverted, however. The cross-examination of Napier was especially fierce: He acknowledged some mistaken recollection, but refused to recant his allegedly perjurious claim that Garand conceived the idea of doing a no-risk deal.
After four days of deliberations, the jury convicted the defendants of all charges. The court denied the defendants' perfunctory Fed.R.Crim.P. 33 motions for a new trial.[9]
In hundreds of pages of briefing, the defendants raise numerous issues on appeal, ranging from evidentiary challenges to serious allegations of widespread prosecutorial misconduct.

I
Several arguments affect all five defendants. We consider each in turn.

A
Materiality is an element of most of the charged offenses. There must have been a "substantial likelihood" that the LPT-related misstatements would be important to a reasonable investor. See Basic Inc. v. Levinson, 485 U.S. 224, 231, 108 S.Ct. 978, 99 L.Ed.2d 194 (1988). As evidence of materiality, the government introduced (inter alia) articles about the LPT's impropriety, which it connected to contemporaneously declining stock prices. Excluded as overly prejudicial was a line graph tracing AIG's stock price from February to March 2005 (as it declined by 12%). However, the court permitted the government to show a functionally identical chart to the jury during opening statements, *274 and it admitted into evidence three bar-charts showing single-day stock prices for the days following each publication.
The charts were prejudicial because the LPT was one of several problems besetting AIG at that time. Unrelated allegations of bid-rigging, improper self-dealing, earnings manipulations, and more, had to be redacted from the articles about the LPT, to avoid prejudicing the defendants. The stock-price evidence presented the defendants with a dilemma: [i] allow the jury to attribute the full stock-price decline to the LPT, or [ii] introduce prejudicial evidence of the other besetting scandals, wrongdoing, and potentially illegal actions at AIG. The defendants sought to sidestep by stipulating to materiality, but the government refused.
We conclude that the district court abused its discretion in admitting the three bar-charts and that the defendants' substantial rights were affected. Marcic v. Reinauer Transp. Cos., 397 F.3d 120, 124 (2d Cir.2005).
The district court's rulings on the stock-price charts were inconsistent. The chart showing the full decline in stock price was excluded as overly prejudicial, but it was functionally identical to the chart shown during the government's opening argument. In any event, the court's solution, to allow only isolated ranges of stock-price data, did not mitigate the prejudice: Instead of a downward line, there were three dropping sets of dots; it is inevitable that jurors would connect them. So the risk that jurors would attribute the full 12% decline to the LPT was unabated by the court's precaution.
The government may of course reject a proposed stipulation in order to present a "coherent narrative" of its case. Old Chief v. United States, 519 U.S. 172, 191-92, 117 S.Ct. 644, 136 L.Ed.2d 574 (1997). But the charged offenses here do not require a showing of loss causation ("a causal connection between the material misrepresentation and the loss"). Dura Pharms., Inc. v. Broudo, 544 U.S. 336, 342, 125 S.Ct. 1627, 161 L.Ed.2d 577 (2005). The stock-price evidence therefore fell "outside the natural sequence of what the defendant[s] [were] charged with thinking and doing." Old Chief, 519 U.S. at 191, 117 S.Ct. 644. Although the evidence was admitted only to show materiality, the government exploited it to emphasize the losses caused by the transaction. For example, the government reminded the jury during rebuttal summation that:
[B]ehind every share of [AIG] stock is a living and breathing person who plunked down his or her hard-earned money and bought a share of stock, maybe [to] put it in their retirement[ ] accounts, maybe to put it in their kids' college funds, or maybe to make a little extra money for the family.
Trial Tr. at 4584. The prosecution's use of the evidence, while aggressive, was not "egregious misconduct" that "so infect[ed] the trial with unfairness as to make the resulting conviction a denial of due process." United States v. Shareef, 190 F.3d 71, 78 (2d Cir.1999) (internal quotation marks omitted). Still, the government used this evidence to humanize its prosecution, not to complete the narrative of its case.
If no offer to stipulate were forthcoming, the government could have relied upon the sufficiency of its other materiality evidence[10] or offered expert testimony about *275 the LPT's effect on the stock price.[11] The charts suggested that this transaction caused the price of AIG shares to plummet 12% during the relevant time period (at a time when AIG's market capitalization was in excess of $150 billion). This is without foundation. Nevertheless, the government bookended its case with the theme of the financial loss suffered by the average investor. Its opening statement told the jury that, beginning on the day the government issued subpoenas to AIG about the LPT deal, "AIG's stock price began a dovraward spiral. On February 14th, AIG shares were worth $70.33. By March 14th, March 15th, AIG's shares were worth only $61.03. This was approximately a 12 percent decline in AIG stock price over that one month period alone." Trial Tr. at 90. And during its rebuttal summation, the government envisioned that "behind every share of [AIG] stock is a living and breathing person who plunked down his or her hard-earned money and bought a share of stock." Trial Tr. at 4584.
The fraudulent aspect of the transaction at issue was that it entailed zero risk as opposed to an ordinary  and permissible  finite reinsurance transaction, in which the risk would be minimal. As between a permissible transaction with minimal risk, and an illegal one with none, it would be difficult for the jury to appreciate what adverse effect the transaction would have had on anyone. Moreover, since the permissible transaction with minimal risk would have had the same advantageous effect on AIG's accounts as the impermissible transaction with no risk, there would seem in fact to have been no effect whatsoever in financial terms  except that stockholders might have developed qualms about the honesty of the company's leadership. Other than the charts, nothing in the record conveyed the impression that the transaction had a palpable financial impact and that stockholders were hurt  and hurt seriously.

B
The defendants challenge the particulars of the "willfully caused" jury instruction, as well as the district court's refusal to give certain instructions and insistence upon giving others. We review the jury charge de novo, examining "the entire charge to see if the instructions as a whole correctly comported with the law." United States v. Jones, 30 F.3d 276, 283 (2d Cir.1994). A defendant challenging jury instructions must show that he was prejudiced by a charge that misstated the law. See United States v. Goldstein, 442 F.3d 777, 781 (2d Cir.2006).

1
A defendant commits an offense if he "willfully causes an act to be done which if directly performed by him or another would be an offense against the United States." 18 U.S.C. § 2(b). In seeking to accommodate the reasonable phrasings offered by the various parties, the court ended up with a charge that allowed the jury to convict without finding causation. The court instructed the jury about "willfully causing" liability through a similar pair of questions for each offense:
The meaning of the term "willfully caused" can be found in the answers to the following questions:
With regard to securities fraud:

*276 First, did the defendant act knowingly, willfully, and with an intent to defraud as I defined those terms for you in my instructions about securities fraud?

Second, did the defendant intend that this crime, as explained to you in my earlier instructions, would actually be committed by others?
. . .
If you are persuaded beyond a reasonable doubt that the answer to both of these questions is "yes," then the defendant is guilty of the crime charged just as if he or she had actually committed it.
Trial Tr. at 4760-61.
It appears the judge was led into error. The original instruction submitted by the government contained a proper causation standard;[12] the defendants challenged a vague phrase ("take some action") in the government's instruction and proposed a lengthier instruction that tracked the actual elements of each offense (but that also properly charged on causation).[13] The court fashioned a compromise from the parties' submissions, but neglected to include either side's causation instruction: The court's first question instructs about both the requisite act ("did the defendant act") and the requisite mental state ("knowingly, willfully, and with an intent to defraud"); the second question merely refines the mental state requirement ("did the defendant intend that this crime ... would actually be committed by others?").
The instruction is not saved by the plain meaning of "willfully caused," which is the term the court undertook to define. The word "cause" should convey a causation requirement. But the jury was not invited or permitted to rely on the phrase's plain meaning, given the superseding definition provided in the charge: "The meaning of the term `willfully caused' can be found in the answers to the following questions...." Trial Tr. at 4760.
Although the defendants objected to the instruction, they did not "specific[ally] object[ ]" about causation; the objection on that ground was thus not preserved, and we review for plain error. See Fed.R.Crim.P. 30(d). But the error is plain enough. When a jury is instructed on multiple theories of liability, one of which is defective, a court must ascertain whether a flawed instruction had a "`substantial and injurious effect or influence in determining the jury's verdict.'"[14]Hedgpeth *277 v. Pulido, 555 U.S. 57, 58, 129 S.Ct. 530, 172 L.Ed.2d 388 (2008) (per curiam) (quoting Brecht v. Abrahamson, 507 U.S. 619, 623, 113 S.Ct. 1710, 123 L.Ed.2d 353 (1993)); see also Skilling v. United States, ___ U.S. ___, 130 S.Ct. 2896, 2934 n. 46, 177 L.Ed.2d 619 (2010) (applying Hedgpeth on direct review). The defendants here would not be prejudiced by the infirm instruction if either: (1) the jury would have necessarily found the causation element satisfied beyond a reasonable doubt even with a proper instruction, see Neder v. United States, 527 U.S. 1, 15-16, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999); or (2) the jury would have necessarily found the defendants guilty on one of the properly instructed theories of liability, see Hedgpeth, 555 U.S. at 61, 129 S.Ct. 530; United States v. Skilling, 638 F.3d 480, 482 (5th Cir.2011); United States v. Riley, 621 F.3d 312, 322-24 (3d Cir.2010).
The government argued a "willfully causing" theory of liability to the jury, rendering it a likely basis for the jury's conviction. See, e.g., Trial Tr. at 4203 ("[Did defendants] document a false [transaction] in order to deceive AIG's internal auditors and their external auditors and accountants[?]"); Trial Tr. at 4620 ("These five defendants helped actively create false documents to deceive AIG's accountants. So they cannot now try to say that they expected those very same accountants to do their job properly."). But the record contained sufficient evidence for the jury to find in the defendants' favor on the omitted causation element  more precisely, the record contained insufficient evidence of why AIG booked the transaction as it did to render a finding of causation a foregone conclusion  and the defendants vigorously argued that the government failed to demonstrate that their conduct caused AIG to book the transaction as no risk. Trial Tr. at 4458-59 ("[T]he black hole in this case is the prosecution's failure to prove beyond a reasonable doubt anything the defendants allegedly did had any effect on AIG's accounting decision.").
We doubt, but need not decide, that the jury would have necessarily returned a verdict of guilty on any of the other theories of liability.[15] The erroneous admission of the stock price data sufficiently tainted the defendants' trial to require that the verdict be set aside.

2
The district court instructed the jury that the government could prove that a defendant acted knowingly if he "was aware of a high probability that [a] statement was false" but "deliberately and consciously avoided confirming that fact, unless the evidence show[s] that [he] actually believed the statement was true." Trial Tr. at 4730. Such a conscious avoidance *278 instruction[16] may be given only
(i) "when a defendant asserts the lack of some specific aspect of knowledge required for conviction," and
(ii) "the appropriate factual predicate for the charge exists, i.e., the evidence is such that a rational juror may reach the conclusion beyond a reasonable doubt that the defendant was aware of a high probability of the fact in dispute and consciously avoided confirming that fact."
United States v. Quattrone, 441 F.3d 153, 181 (2d Cir.2006) (internal citations and quotation marks omitted). The government need not choose between an "actual knowledge" and a "conscious avoidance" theory. United States v. Kaplan, 490 F.3d 110, 128 n. 7 (2d Cir.2007).
The defendants claim not to have known (1) that the LPT contained insufficient risk transfer or (2) how AIG would account for the LPT. The government argues that the factual predicate for the charge is the same evidence that establishes scienter. (The government emphasizes the November 20 call ordered by Ferguson in which Monrad, Napier, Graham, and Garand told Milton that Gen Re would use deposit accounting for the LPT.)
Red flags about the legitimacy of a transaction can be used to show both actual knowledge and conscious avoidance. See United States v. Nektalov, 461 F.3d 309, 312, 317 (2d Cir.2006). In Nektalov, a jeweler was convicted of money laundering for repeatedly selling gold to a government informant posing as a narcotics dealer. Id. at 312. We upheld a conscious avoidance instruction because the prior dealings between the parties (cash payments using small bills) and the statements about the transactions ("moving gold" to Colombia; money from selling "product" "in the streets") provided the factual predicate for the charge. Id. at 317. Similarly, several red flags are waving here, including: the secret side agreements, the fake offer letter, the accounting pretext for the reinsurance transaction, and the insistence on strict confidentiality.
The defendants claim they could not have consciously avoided present knowledge of how AIG would book the LPT on some future date. It is true that, "in general, conscious avoidance instructions are only appropriate where knowledge of an existing fact, and not knowledge of the result of defendant's conduct, is in question." United States v. Gurary, 860 F.2d 521, 526 (2d Cir.1988). In Gurary, the defendants sold fake invoices that were commonly used by purchasers to fraudulently reduce taxable income. Id. at 523. The defendants challenged the conscious avoidance instruction on the ground that they could not know the nefarious ends of the purchasers. We upheld the instruction because the repeated (subsequent) frauds provided sufficient "`proof of notice of high probability'" of purchasers' tax fraud. Id. at 527. But we also noted that a "future conduct" challenge to a conscious avoidance instruction "might hold water if th[e] case involved the sale of invoices on a single occasion." Id. at 526.
*279 Although the LPT was a single transaction, it is dissimilar to the "single occasion" theory in Gurary. The parameters of the deal were developed over a number of months, and there were numerous forward-looking meetings, emails, and negotiations. Moreover, AIG's accounting decisions informed Gen Re's accounting decisions to some extent, which brought AIG's accounting into the transaction's purview (even if asymmetric accounting in general is unobjectionable).
The conscious avoidance instruction was not error.

3
The jurors were presented with four theories of liability: principal, aiding and abetting, willfully causing, and Pinkerton.[17] The district court denied the defendants' request for a "specific unanimity" instruction, which would have ensured that, as to each defendant, the jurors unanimously agreed on the theory for conviction. "A general instruction on unanimity is sufficient to insure that such a unanimous verdict is reached, except in cases where the complexity of the evidence or other factors create a genuine danger of jury confusion." United States v. Schiff, 801 F.2d 108, 114-15 (2d Cir.1986) (internal citations omitted).
In dicta, we have suggested that a jury is unanimous even if some jurors convicted on a theory of principal liability and others on aiding and abetting. United States v. Peterson, 768 F.2d 64, 67 (2d Cir.1985); accord, e.g., United States v. Garcia, 400 F.3d 816, 820 (9th Cir.2005) ("It does not matter whether some jurors found that [the defendant] performed these acts himself, and others that he intended to help someone else who did, because either way, [his] liability is the same...."). Just as there is "no general requirement that the jury reach agreement on the preliminary factual issues which underlie the verdict," neither must it agree on "alternative mental states." Schad v. Arizona, 501 U.S. 624, 631-32, 111 S.Ct. 2491, 115 L.Ed.2d 555 (1991) (internal quotation marks omitted) (holding that specific unanimity not required for theories of Arizona first-degree murder  premeditated and felony murder).
Nothing limits the Peterson analysis to principal versus aiding-and-abetting liability. The four theories are compatible  they are zones on a continuum of awareness, all of which support criminal liability.[18] This view is consistent with case law *280 maintaining distinctions among mental states where different mental states form elements of different offenses. Compare, e.g., Schad, 501 U.S. at 630-31, 111 S.Ct. 2491 ("[P]etitioner's real challenge is to Arizona's characterization of first-degree murder as a single crime" that encompasses "premeditated murder and felony murder"), with, e.g., People v. Gonzalez, 1 N.Y.3d 464, 467, 775 N.Y.S.2d 224, 807 N.E.2d 273 (2004) (affirming the reversal of depraved indifference murder conviction for defendant acquitted of intentional murder count, because "only reasonable view of the evidence here was that defendant intentionally killed the victim").
Even assuming that the jury had to agree on the theory of liability, the general unanimity instruction  "it is necessary that each juror agrees to [the verdict]," Trial Tr. at 4788  was sufficient to remove any genuine danger that the jury would convict on disparate theories. The accounting and insurance concepts in the case may have been complicated, but they did not add significant complexity to the theories of liability. At the same time, the assurance of a just result would have been reinforced if the instruction were given.

4
The court instructed the jury that "[n]o amount of honest belief on the part of a defendant that the scheme will ultimately make a profit for the investors, or not cause anyone harm, will excuse fraudulent actions or false representations by him or her." Trial Tr. at 4730. Graham claims that this "no ultimate harm" instruction lacked a factual basis and undermined his defense of good faith.
Our leading precedent on the "no ultimate harm" instruction is United States v. Rossomando, 144 F.3d 197, 200-03 (2d Cir. 1998), which rejected the instruction in a case in which a former firefighter underreported his post-retirement income on pension forms. Rossomando believed that he was causing no harm to the pension fund, which distinguished him from a person for whom the instruction is proper:
[W]here some immediate loss to the victim is contemplated by a defendant, the fact that the defendant believes (rightly or wrongly) that he will "ultimately" be able to work things out so that the victim suffers no loss is no excuse for the real and immediate loss contemplated to result from defendant's fraudulent conduct.
Id. at 201.
Rossomando is "limited to the quite peculiar facts that compelled [its] result," United States v. Gole, 158 F.3d 166, 169 (2d Cir.1998) (Jacobs, J., concurring), so Graham's analogy is not persuasive. The "no ultimate harm" instruction given in the present case ensured that jurors would not acquit if they found that the defendants knew the LPT was a sham but thought it beneficial for the stock price in the long run. It may well have been proven beneficial to AIG stockholders, but the immediate harm in such a scenario is the denial of an investor's right "to control [her] assets by depriving [her] of the information necessary to make discretionary economic decisions." Rossomando, 144 F.3d at 201 n. 5 (citing United States v. Dinome, 86 F.3d 277, 280, 284 (2d Cir. 1996)). Moreover, the jury charge given here could not have undermined Graham's good-faith defense; the instructions made clear that "[a] defendant who acted in good faith cannot be found to have acted knowingly, willfully, and with the unlawful intent required for the charge you are considering," Trial Tr. at 4711, and that "[h]owever misleading or deceptive a plan may be, it is not fraudulent if it was devised or carried out in good faith," id. at 4729.


*281 C
The defendants argue that prosecutorial misconduct  ranging from intentional grammatical errors to eliciting perjury  warrants reversal because the ensuing "substantial prejudice" "so infect[ed] the trial with unfairness as to make the resulting conviction a denial of due process." United States v. Shareef, 190 F.3d 71, 78 (2d Cir.1999) (internal quotation marks omitted). Certain factual inconsistencies in Napier's testimony are sufficiently obvious to raise an eyebrow, but most of the arguments are meritless.

1
Compelling inconsistencies suggest that Napier may well have testified falsely. Napier provided important testimony (i) that he attended a meeting with Monrad at which AIG's CFO was warned that Gen Re would book the LPT on a deposit basis; and (ii) that Garand first proposed a no-risk deal.
(i) Napier testified that Monrad, at Ferguson's behest, led a meeting at AIG in late November or early December 2000, in which she informed Howie Smith and Mike Castelli (AIG's CFO and Controller, respectively) that Gen Re would book the LPT as a deposit. The disclosure ensured that AIG could not later claim to be surprised by Gen Re's accounting. The testimony was thus strong evidence of Monrad's scienter.
Neither Napier nor the government could produce "one scrap of paper" showing that the meeting actually took place (Trial Tr. at 1274): no preparatory documents or emails, no AIG sign-in or security records confirming that Monrad and Napier had visited the office at that time; no records of the Gen Re car (and drivers) that Napier claimed provided their transportation. Napier's calendar entries could not confirm the meeting, because none of his historic calendar data was recoverable. No one sent an email summarizing the discussion for those not in attendance or memorializing it for those who were.
Monrad's counsel cross-examined Napier about an email describing an earlier meeting he had with Howie Smith at AIG on an unrelated matter. The earlier meeting  which Monrad did not attend  contradicted Napier's testimony that the purported LPT meeting was the first time that he had met Smith. Napier admitted that he may have confused the LPT meeting with this meeting.
(ii) Garand challenges as perjury (and relatedly, as government misconduct) Napier's belated recollection (with "certain[ty]," Trial Tr. at 1670) that it was Garand who originated the idea of a no-risk transaction. Among the circumstances he cites as suspicious are: Napier did not recollect Garand's role as originator until the day that the government filed the superseding indictment in which Garand was first named as a defendant; Napier's certainty is incompatible with his concession on cross-examination that he was "having a hard time remembering the events of [that day]" and was "drawing a blank on the entire date"; Napier had earlier been uncertain about Garand's first involvement (he had suggested that Garand may not have been involved until Gen Re collected on the side deal in 2001); the identification contradicted Napier's previous identification (recanted at trial) of Milton as the source; and his identification of Milton was made while looking at the same undated page of notes that he attributed at trial to a meeting with Garand and Monrad (which he does not contend that Milton attended). Moreover, when Houldsworth delicately broached the LPT in a call with Garand the next day, Garand evinced no recognition of the transaction. Houldsworth formed the impression that Garand *282 "didn't appear to know anything about it." (Garand claims to have first learned about the transaction during this call with Houldsworth.)
The government argues that we should not review these arguments at all because the defendants waived them; but where a defendant does not "intentional[ly] relinquish[ ] or abandon[ ]" a known right, but simply "fail[s] to make the timely assertion of [it]," the result is not waiver but forfeiture. United States v. Olano, 507 U.S. 725, 733, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993) (internal quotation marks omitted). We review such forfeited arguments for plain error. If these arguments had been presented to the trial court, a factual record about Napier's potential perjury (and the extent of the government's awareness and diligence) could have been made. The district court requested substantive briefing and argument on the issue, but was not taken up. The defendants may have had their reasons for sidestepping the issue of Napier's possible perjury and the government's alleged responsibility for it; but "our review for plain error [is] more rigorous" where the failure to object was a "strategic decision" that "resulted in an incomplete record or inadequate findings." United States v. Brown, 352 F.3d 654, 665 (2d Cir.2003).
There are ambiguities in our case law regarding the proper standard to use, which could not have helped the district judge in sorting this out. The parties appear to agree that the two-part test from United States v. Wallach applies:
(1) Whether the perjury was material to the jury's verdict;
(2) The extent to which the prosecution knew or should have known about the perjury;[19]
935 F.2d 445, 456 (2d Cir.1991). But that test is in tension with the four-part test from United States v. Zichettello, which supplements the Wallach factors with two factors from precedent[20] (italicized):
(i) "the witness actually committed perjury;"[21]
(ii) "the alleged perjury was material";
(iii) "the government knew or should have known of the alleged perjury at the time of trial"; and
(iv) "the perjured testimony remained undisclosed during trial."

208 F.3d 72, 102 (2d Cir.2000) (internal quotation marks omitted). Later cases add to the confusion by applying Wallach without referencing Zichettello. See, e.g., United States v. Stewart, 433 F.3d 273, 297 (2d Cir.2006). The tests are not necessarily incompatible, however.[22]
*283 Since we are vacating the judgments on the basis discussed above, we need not reconcile these cases or decide whether the prosecution's actions amounted to misconduct. (Our decision would have been hindered by the defendants' gamesmanship; and their fact-intensive arguments[23] are blunted by the underdeveloped record.) No doubt it is dangerous for prosecutors to ignore serious red flags that a witness is lying, and the government will doubtless approach Napier's revised recollections with a more skeptical eye on remand. At the same time, Napier's inconsistent statements concern facts that could not have been conclusively verified by the government, and the potential that Napier had lied in these respects was fully presented in cross-examination and summation to the jury, which resolved the credibility issue against the defendants.

2
The remainder of the misconduct claims involve the government's comments at opening statement, in summation, and on rebuttal. "It is a `rare case' in which improper comments ... are so prejudicial that a new trial is required." United States v. Rodriguez, 968 F.2d 130, 142 (2d Cir.1992) (quoting Floyd v. Meachum, 907 F.2d 347, 348 (2d Cir.1990)). Such comments do not amount to a denial of due process unless they constitute "egregious misconduct." Shareef, 190 F.3d at 78 (internal quotation marks omitted). In assessing a claim, we consider: (1) "the severity of the misconduct"; (2) "the measures adopted to cure it"; and (3) "the certainty of conviction in the absence of the misconduct." Id. (internal quotation marks omitted).
The defendants did not contemporaneously object to the statements they now claim constitute misconduct. (The one objection was made a day after the challenged statement was made.) They were thus able to pore at leisure over the transcript, hunting for any plausible (or nearly plausible) claims. The remarks do not amount to misconduct, separately or in the aggregate.[24]
First, Graham challenges two mistakes that the prosecution made when quoting his email:
In quoting the line that "regulators (insurance and securities) may attack the transaction," the prosecutor repeatedly used "would" rather than "may." However, the distinctions among "may," "might," "will" and "would" are among the slipperiest in the English language. The distinction should have been preserved, but it cannot be said that a slip  even a recurring slip  was misconduct. It is easy to make such mistakes, but there is reason to think that there will be heightened vigilance on retrial.
The prosecution also misquoted "potential reputational risk" as "potential risk" in two slides shown to the jury (and referenced by the prosecutor twice). Graham *284 failed to object to the omitted word; he chose instead to correct the government's mistake in his closing argument, which dovetailed nicely with an argument that the email's use of "reputational" was pregnant. Having tried that (and having called the miscue a "good faith" mistake in closing argument, Trial Tr. at 4508), he now argues that was reversible misconduct. We conclude that the omissions were honest mistakes, and any harm was cured by Graham's tactical discussion of the error.
These misstatements were "minor aberrations in a prolonged trial," rather than misconduct. United States v. Modica, 663 F.2d 1173, 1181 (2d Cir.1981) (internal quotation marks omitted).
Second, the defendants contend that the government oversimplified the case by emphasizing Gen Re's lack of need for reinsurance and AIG's acquiescence to paying a $5 million fee despite accepting risk. These features of the transaction, it is claimed, are irrelevant because they may also inhere in any lawful finite reinsurance transaction. It is true that these facts alone are insufficient to support a conviction, but neither are they irrelevant: It was within the province of the jury to determine whether these were incidents of fraud or incidental to a lawful transaction of that specialized kind.
Third, the defendants claim that the government knowingly invited a false inference. One defense theory was that the LPT could not have been fraudulent because Warren Buffet  the CEO of Berkshire Hathaway, Gen Re's parent company  vetted aspects of the deal. Defendants attack the prosecution statement that there was no evidence Warren Buffet knew anything significant about the deal. No false inference was invited: Although the government led with its inference that Buffett knew nothing of significance, it then described in detail all of the Buffett evidence. The jury could assess the evidence as it saw fit; the prosecution was free to offer its assessment as well.
Finally, the defendants challenge the government's rebuttal assertion that Milton's delivery of the fake slip and offer letter successfully deceived two AIG employees, who therefore booked the transaction as reinsurance. They claim the statement was intended to paper over the government's missing proof of causation. But this single sentence had an insignificant (if any) effect on the prosecution's causation evidence, especially in view of the court's later reminder to the jury that statements from summations are not evidence. The statement fell far short of misconduct.

II
Ronald Ferguson, the CEO of Gen Re, and Hank Greenberg envisioned a creative and unusual deal, but Ferguson claims that he was unaware that the deal would be fraudulent. He cautions that the jury may have disregarded the flimsy evidence of his scienter (some of which he claims was admitted in error) and convicted him merely because he was in charge as CEO.

A
Ferguson argues that the jury finding of his scienter was supported by insufficient evidence. He has the "heavy burden" of showing that "no rational juror could have found [his scienter] beyond a reasonable doubt." United States v. Bryce, 208 F.3d 346, 352 (2d Cir.1999) (internal quotation marks omitted). For a sufficiency challenge, we view "all of the evidence in the light most favorable to the government and draw all reasonable inferences in its favor." Id.
*285 Napier testified extensively about Ferguson's knowledge of the no-risk aspect of the deal and his insistence on disclosing Gen Re's deposit accounting to AIG; that testimony alone was likely sufficient to support the jury's finding. See United States v. Parker, 903 F.2d 91, 97 (2d Cir. 1990) ("The fact that a conviction may be supported only by the uncorroborated testimony of a single accomplice is not a basis for reversal if that testimony is not incredible on its face and is capable of establishing guilt beyond a reasonable doubt.").
In any event, that testimony, taken together with other evidence  Ferguson's unusual request for internal confidentiality, his review of the Houldsworth email noting that "no real risk"[25] would be transferred and that "[CRD] w[ould] not transfer any losses under this deal" (Joint Appendix at 1978), and his proactivity with the tortuous fee recovery (which was not in the LPT documents)  were sufficient for a rational juror to have found his scienter beyond a reasonable doubt.

B
Ferguson challenges the admissibility of a December 2000 email from Graham, which assured Gen Re's General Counsel, Timothy McCaffrey, that:
[Ferguson] et al[.] have been advised of, and have accepted, the potential reputational risk that US regulators (insurance and securities) may attack the transaction and our part in it.
Joint Appendix at 2192. The email was admitted as a coconspirator statement under Rule 801(d)(2)(E). Ferguson argues that the email: (1) was inadmissible double-hearsay; (2) mandated severance because it created tension between his lack-of-scienter defense and Graham's good-faith defense; (3) and led the government to invite the inference that Graham himself told Ferguson about the potential reputational risk, which it knew to be false.

1
Double-hearsay[26] is a potential issue because the December email is written in the passive voice: Ferguson and others have been advised about the potential reputational risk by some unidentified person. Whether this statement constitutes double-hearsay is a legal issue, which we review de novo. See, e.g., Biegas v. Quickway Carriers, Inc., 573 F.3d 365, 378 (6th Cir. 2009) ("Whether a statement is hearsay is a question of law, which we review de novo."); United States v. Collicott, 92 F.3d 973, 978 (9th Cir.1996) ("Whether the district court correctly construed the hearsay rule is a question of law reviewable de novo."). (However, a district court's hearsay rulings based upon factual findings or the exercise of its discretion warrant additional deference.[27])
*286 The phrase "have been advised of" is used to convey the idea that they "know"; if the email said "Ferguson et al. know the potential reputational risk" there would be no double-hearsay issue.[28] Without indicia of evasiveness, it is not necessary to look for the speaker behind every sentence written in the passive voice. It is unlikely that the email was carefully drafted for hearsay subterfuge, especially in view of the incautious discussion about AIG answering to God about its accounting practices. This is not an instance in which a sentence is carefully manipulated to smuggle hearsay evidence into pending litigation. See, e.g., Official Comm. of Unsecured Creditors v. Hendricks, No. 1:04-cv-066, 2008 WL 3007989, at *4, 2008 U.S. Dist. LEXIS 116318, at *12 (S.D.Ohio Aug. 1, 2008) (striking passive-voice sentence in affidavit submitted with motion for summary judgment).
In any event, the unnamed speaker need not be identified to conclude that the statement is nonhearsay. First, the statement was not offered for its truth; it was offered solely for the purpose of showing that the statement was made to Ferguson. See, e.g., George v. Celotex Corp., 914 F.2d 26, 30 (2d Cir.1990) ("[A]n out of court statement offered not for the truth of the matter asserted, but merely to show that the defendant was on notice of a danger, is not hearsay."). Second, no nonmember of the conspiracy could have given Ferguson the advice about potential reputational risk, because only co-conspirators would have been aware of the particular reputational risk that the conspiracy's object entailed (especially in view of Ferguson's order for an unusual level of internal secrecy about the deal). The statement, made in furtherance of the conspiracy, is thus also a nonhearsay co-conspirator statement under Rule 801(d)(2)(E).

2
Ferguson wished to keep Graham's email out of evidence, but Graham wanted it in  as evidence that he acted in good faith by soliciting his supervisor's imprimatur on a legally questionable transaction. Ferguson and Graham claim that the email created unavoidable tension between Ferguson's lack-of-scienter defense and the good-faith defense mounted by Graham, and that severance was therefore warranted. Ferguson claims additional prejudice from his inability  without infringing Graham's rights  to place before the jury an exculpatory statement from Graham's proffer session.
"Motions to sever under Rule 14 are committed to the sound discretion of the trial judge"; to compel reversal, the *287 defendant has the "heavy burden" to "show prejudice so severe that his conviction constituted a miscarriage of justice." United States v. Rittweger, 524 F.3d 171, 179 (2d Cir.2008) (internal quotation marks omitted).
It was well within the district court's discretion to conclude that any tension between defenses was insufficient to warrant severance. If the jury concluded that both Graham and Ferguson feared reputational risk because the LPT was objectionable but non-fraudulent  like, for example, an aggressive (but defensible) offshore tax position  it could have credited both defense theories.
Nor was severance necessary to permit Ferguson to introduce evidence from Graham's proffer session, in which Graham denied personally informing Ferguson about the reputational risk of the transaction. Ferguson argues that in a joint trial, he was hamstrung: He could not introduce the government's notes containing the statement, because Graham was given limited-use immunity; nor could he compel Graham to testify, because of Graham's Fifth Amendment right against self-incrimination.
We have identified several factors for determining whether to grant severance based on a defendant's need to call a co-defendant as a witness:
(1) the sufficiency of the showing that the co-defendant would testify at a severed trial and waive his Fifth Amendment privilege;
(2) the degree to which the exculpatory testimony would be cumulative;
(3) the counter arguments of judicial economy; and
(4) the likelihood that the testimony would be subject to substantial, damaging impeachment.
United States v. Finkelstein, 526 F.2d 517, 523-24 (2d Cir. 1975) (internal citations omitted). Although the district court did not recite or explicitly apply these factors,[29] its discretion is not conditioned upon reciting or considering them. Rather, the factors are what the district court "properly could have considered"; they were announced "[w]ithout purporting to delimit the trial court's field of inquiry." Id. at 523 (emphasis added).
In any event, the factors weigh in favor of the district court's ruling. Only the second factor favors Ferguson: Graham's testimony about the email would not have been cumulative, because there is no adequate substitute for further clarification from the drafter himself. But the other factors favor the government: first, Ferguson merely assumes Graham would testify, and it is unclear whether the conflicting statement from the proffer session would even be admissible if he did not;[30]*288 second, the exculpatory value of the statement would be hugely outweighed by staging another multi-week trial (with another potential appeal) for Ferguson alone; third, the cross-examination of Graham would elicit testimony damaging to Ferguson about Graham's qualms concerning the deal, and the basis for his statement that Ferguson knew of the reputational risk.

* * *
At bottom, "Rule 14 does not require severance even if prejudice is shown"; instead, "the tailoring of the relief to be granted, if any, [is in] the district court's sound discretion." Zafiro v. United States, 506 U.S. 534, 538-39, 113 S.Ct. 933, 122 L.Ed.2d 317 (1993). Ferguson and Graham have not made a showing that justifies upending the district court's exercise of its discretion.

3
The government's closing statement juxtaposes Napier's comment that the LPT is Ferguson's deal with a description from Graham's email that Ferguson was advised of the potential reputational risk. Ferguson argues that this sequence (and a similar one in the government's opening statement) amounted to prosecutorial misconduct, because it advanced the inference that Graham had personally discussed reputational risk with Ferguson, which the government knew to be false.
Prosecutors are well advised to tread carefully when it comes to arguing for inferences that are fair in terms of evidence but are doubtful (if not foreclosed) based on what they were told in proffer sessions. However, any inference arose from the structure of the government's argument, rather than its substance. Although it is possible that the misleading structure of a prosecutor's argument could amount to misconduct, the misconduct here (if any) was insufficient to create the "substantial prejudice" necessary to warrant vacatur. United States v. Valentine, 820 F.2d 565, 570 (2d Cir.1987).

C
The government insists that the deal was tainted from the very first call between Ferguson and Greenberg, when AIG asked to rent a specific amount of reserves for a defined period. Yet it also theorized that the idea of a no-risk deal did not surface until Garand suggested it in mid-November. Ferguson claims that this is a contradiction that renders untenable the district court's finding that the conspiracy began with the Greenberg-Ferguson call on October 31, a ruling that allowed the government to introduce co-conspirator statements made starting on October 31 (rather than starting from mid-November).[31] The district court's decision to admit the co-conspirator statements under Fed. R. 801(d)(2)(E) is reviewed for clear *289 error. See United States v. Farhane, 634 F.3d 127, 160-61 (2d Cir.2011).
The government's theories are not irreconcilable. Although the details of the plan were not settled during the October 31 call, Greenberg and Ferguson agreed to a highly unusual deal: The transaction was prompted predominately by stock market concerns; it inverted their customary commercial roles as cedant and reinsurer, even though there was no evidence that Gen Re wanted reinsurance; and AIG requested a specific dollar range of loss reserves for a specific term.
Even if Greenberg and Ferguson had hoped to accomplish their objectives legally, execution of a no-risk transaction was not unforeseeable. These very senior executives agreed to pursue specific parameters. And their objective predictably exerted pressure on their subordinates on the deal team to get the transaction done that way no matter what.[32] Under these circumstances, we cannot say that it was clearly erroneous for the district court to find that the conspiracy began on October 31.

III
The defense of Robert Graham, in-house lawyer for Gen Re and the only attorney among the defendants, is that he acted in good faith throughout, was unaware of the true nature of the deal, and vetted his ethical concern with Gen Re's General Counsel, Timothy McCaffrey. Graham argues that (1) his requests for certain jury instructions were improperly denied, and (2) government conduct rendering McCaffrey unavailable was a hindrance to Graham's good-faith defense.

A
Graham claims that the district court improperly denied his request for jury instructions about professional responsibility rules for attorneys, and non-contractual understandings (i.e., handshake deals). Although a criminal defendant is entitled to a jury instruction when there is "any foundation in the evidence" for it, an instruction that "divert[s] the attention of the jury to questions of little significance" and that would "have confused, if not misled, the jury" are properly rejected. United States v. Russo, 74 F.3d 1383, 1393-94 (2d Cir.1996) (internal quotation marks omitted).

1
Graham sought a jury instruction explaining that an attorney confronted with "an arguable question of professional duty" may discharge his ethical responsibilities by consulting with a supervisory lawyer and relying on his resolution of the matter. Joint Appendix at 2844-45. He argues that his email to his boss,[33] which the government says is a basis for liability, was in fact and effect compliance with his ethical responsibilities:
Tim 
The AIG project continues. It is now a two step loss portfolio deal between Cologne Re Dublin and National Union of Pittsburgh, with $250 million booked in the 4th quarter of 2000 and $250 million more to be booked in 2001 (probably 1st quarter). While it will be booked in the third quarter, it is retroactive to 1/12/2000.

*290 Our group will book the transaction as a deposit. How AIG books it is between them, their accountants and God; there is no undertaking by them to have the transaction reviewed by their regulators.

[Ferguson] et al[.] have been advised of, and have accepted, the potential reputational risk that US regulators (insurance and securities) may attack the transaction and our part in it.
Rob
Joint Appendix at 2192 (emphasis added).
The professional responsibility rules shed no light, however: A memo reporting on what is being done, without more, is not the kind of consultation contemplated by the rules. See Conn. Rules of Prof'l Conduct 5.2. Nor was there a foundation for the instruction in the record, because Graham presented no evidence implicating the rules. (He evidently decided not to call the ethics expert witness he had considered. See United States v. Ferguson, No. 06-cr-137, 2007 WL 4539646, at *2 (D.Conn. Dec. 14, 2007).)

2
The government emphasized that Graham drafted the contracts to omit the $15 million in payments to Gen Re. Graham countered that the payments were non-contractual "handshake" deals and thus needed not be included in a written contract. He sought a jury instruction about handshake deals to bolster this argument.
There was no foundation for the instruction; and the instruction only would have created confusion about an unimportant collateral issue. The testimony on handshake deals was that they are "agreement[s] over many years, often decades" in which "one party may or may not make up losses to the other party if they do particularly well," Trial Tr. at 3435, and that such arrangements are common in the insurance industry. But numerous comments in the record confirm that the $15 million payment was not a handshake. (The only comment to the contrary was a stray remark by Houldsworth that appears to have been a misstatement). Moreover, the money trail does not suggest that the parties were dealing on the trustful basis of a handshake: Gen Re withheld the payment it was contractually obligated to make until AIG had routed the $15 million from the side deal (premium repayment and fee) to Gen Re.

B
In a government interview, Graham's boss, Tim McCaffrey, recalled that when he received Graham's email, he saw nothing that warranted questioning or follow up, and presumed that Graham would have been more explicit if truly concerned about legal or ethical issues. Graham subpoenaed McCaffrey to testify about these matters (and Graham's good character), but McCaffrey, who was changed to an "unindicted co-conspirator" in the superseding indictment,[34] declined to testify unless immunized. The government refused. Graham claims that McCaffrey should have been immunized so that he could testify on Graham's behalf or, failing that, he should have had the benefit of a missing witness instruction.
*291 "The situations in which the United States is required to grant statutory immunity to a defense witness are few and exceptional." United States v. Praetorius, 622 F.2d 1054, 1064 (2d Cir.1979). So few and exceptional are they that, in the nearly thirty years since establishing a test for when immunity must be granted, we have yet to reverse a failure to immunize. The test requires three findings:
(1) "[T]he government has engaged in discriminatory use of immunity to gain a tactical advantage or, through its own overreaching, has forced the witness to invoke the Fifth Amendment";
(2) "[T]he witness' testimony will be material, exculpatory and not cumulative"; and
(3) The testimony "is not obtainable from any other source."
United States v. Burns, 684 F.2d 1066, 1077 (2d Cir.1982). We review the court's factual findings about government actions and motive for clear error, but its ultimate balancing for abuse of discretion. United States v. Ebbers, 458 F.3d 110, 118 (2d Cir.2006).
Graham's claim fails on two grounds. As to the first part, a prosecutor does not overreach by refusing to immunize a legitimate target of an ongoing investigation, and the district court's finding that McCaffrey was a target was not clearly erroneous: He failed to act following Graham's remark that "[h]ow AIG books [the LPT] is between them, their accountants and God," and he annotated a list of transactions in his personal files to indicate a hidden letter for the LPT.
As for the second part, McCaffrey's interpretation of Graham's email was not material because it was non-contemporaneous and self-serving: It would have been disadvantageous for McCaffrey to concede that the email had raised red flags that he subsequently ignored. Moreover, McCaffrey's assumption, that Graham would have been more explicit if he had a qualm, was conjectural.
Similarly, the district court did not abuse its discretion by omitting a missing witness instruction. Such an instruction need not be given "in the absence of circumstances that indicate the government has failed to immunize an exculpatory witness." United States v. Myerson, 18 F.3d 153, 160 (2d Cir.1994). For the reasons discussed above, it was within the court's discretion to conclude that McCaffrey's potential testimony was insufficiently exculpatory to warrant the instruction.

IV
Christian Milton, a Vice President of Reinsurance at AIG, was the only AIG employee who was indicted in this case. His primary defense is that the trial was a vilification of AIG, and he was convicted by association.

A
Milton argues that the court abused its discretion by admitting comments from the recordings of the Gen Re defendants that impugned AIG generally.[35]*292 The statements are undeniably prejudicial to AIG, but they are also highly probative of the scienter of the Gen Re defendants. The district court conscientiously conducted extensive balancing, evaluating and redacting recordings on a line-by-line basis. See United States v. Ferguson, 246 F.R.D. 107, 121-23 (D.Conn.2007) (excluding comments that sarcastically referenced the "lovely people" and "nice people" at AIG, that there were good reasons to avoid doing business with AIG, and that "Ferguson doesn't like it because it's AIG"). Moreover, the court gave a charge that guilt could not be conferred
based solely on [the defendants'] senior position or positions that he or she held within their respective companies.
... [Y]ou may not infer that any of the defendants, based solely on his or her position at Gen Re ... or AIG, had any knowledge of the alleged fraud.
Trial Tr. at 4766-67. The instruction is easy to follow, and "juries are presumed to follow their instructions."[36]Richardson v. Marsh, 481 U.S. 200, 211, 107 S.Ct. 1702, 95 L.Ed.2d 176 (1987). Although fortified limiting instructions for each recording could have provided additional (perhaps redundant) protection, Milton's strategy was to forgo additional instructions to avoid drawing unnecessary attention to the recordings.[37]
Under these circumstances, we cannot say that the district court abused its discretion in admitting the statements.[38] At the same time, some of the phrases (from the recordings) that denigrated AIG were exploited rhetorically by the government. Although "cook the books" is a cliche that comes easily to the lips, the government pushed its luck by harping on it twenty-three times (by Milton's count)  after a recording containing that phrase was excluded. Such conduct draws appellate scrutiny, and could neutralize the effect of an otherwise sufficient limiting instruction. We need not consider this further; but the government would be well served to avoid gratuitous prejudice of that kind at retrial.

B
The recordings denigrating AIG did not require that Milton's trial be severed.[39] Joint trials "play a vital role in the criminal justice system" by promoting efficiency and avoiding the "scandal and inequity of inconsistent verdicts." Zafiro v. *293 United States, 506 U.S. 534, 537, 113 S.Ct. 933, 122 L.Ed.2d 317 (1993) (internal quotation marks omitted). As discussed, the decision whether to sever is confided to the "sound discretion" of the district court and is "virtually unreviewable" unless the conviction "constituted a miscarriage of justice," United States v. Yousef, 327 F.3d 56, 149-50 (2d Cir.2003) (internal quotation marks omitted).
There was substantial other evidence to convict Milton besides the recordings denigrating AIG. He spoke frequently with Napier  sometimes a couple of times a day, and other times "almost daily"  to "keep the pressure" on Gen Re to get the deal done. Joint Appendix at 789. He also spoke candidly with Napier about the no-risk nature of the deal, and their discussions are memorialized in at least one email. He orchestrated the circulation of deal documents within AIG to avoid the customary actuarial and underwriting due diligence. He signed the final contracts on behalf of AIG, which omitted the $15 million fee, and then helped effect (and disguise) the payment of the fee, including by signing the relevant paperwork on AIG's behalf.
Even if the recordings were likely inadmissible had Milton been tried alone (they were admitted for the limited purpose of showing scienter of the Gen Re defendants), we cannot say  in view of the line-by-line redaction of the recordings, the court's steps to minimize each recording's effect on Milton,[40] and the jury charge rejecting conviction based upon corporate title  that the district court abused its discretion in denying severance.

V
Elizabeth Monrad, the Chief Financial Officer of Gen Re, argues principally the prosecutorial misconduct and other issues discussed above. In addition, she argues that testimony by Houldsworth and Napier about what others (sometimes she herself) meant by certain statements was equivalent to asking what she knew, which improperly prejudiced the jury as to her scienter.[41] Such testimony, she claims, was neither "rationally based" on the witnesses' perception nor helpful to the jury. See Fed.R.Evid. 701.
Monrad relies heavily upon United States v. Kaplan, 490 F.3d 110 (2d Cir. 2007). In Kaplan, conspirators from a medical office and a law firm set up auto *294 accidents to collect insurance. Id. at 114-16. A cooperating witness, a lawyer involved in the ring, testified that when a successor lawyer in the ring claimed experience with "these kinds of cases," the witness "understood" the cases as auto insurance scams. Id. at 117. That was held to be improper lay opinion testimony because the government did not establish that it was "rationally based" on the perception of the witness (who was extremely vague when explaining the basis for his testimony). Id. at 119.
Those concerns are absent here. Napier and Houldsworth testified only about calls or emails they were involved with, and their testimony was rationally based on the perceptions that they formed from those communications and as key players in the LPT deal. The testimony was helpful to the jury because of the jargon,[42] the heavy involvement by Napier and Houldsworth in the LPT, and their experience in the reinsurance industry.
Although this Court has "suspect[ed] that in most instances a proffered lay opinion will not meet the requirements of Rule 701" when the issue is a party's knowledge, we have advised that, like here, "[l]ay opinion testimony will probably be more helpful when the inference of knowledge is ... from such factors as the defendant's history or job experience." United States v. Rea, 958 F.2d 1206, 1216 (2d Cir.1992). The district court therefore did not abuse its discretion in admitting the testimony about these statements.

VI
All of the arguments raised by Chris Garand, a Senior VP and Chief Underwriter of Gen Re's finite reinsurance operation in the U.S., are raised by other defendants and are discussed above.

CONCLUSION
For the foregoing reasons, the defendants' convictions are vacated and the case is remanded to the district court for a retrial.
NOTES
[*]  The Clerk of the Court is directed to amend the official caption to conform to the names listed above.
[1]  The Financial Accounting Standards are accounting rules that help define Generally Accepted Accounting Principles ("GAAP"), which companies must use for public filings in the United States.
[2]  Defendants:

 Ronald Ferguson: CEO of Gen Re
 Christopher Garand: Senior VP and the Head and Chief Underwriter of Gen Re's finite reinsurance operations
 Robert Graham: Legal counsel and Senior VP at Gen Re
 Christian Milton: VP of Reinsurance at AIG
 Elizabeth Monrad: CFO of Gen Re
Co-operating Witnesses:
 Richard Napier: Senior VP at Gen Re, who managed its relationship with AIG
 John Houldsworth: CEO of Cologne Re Dublin, a Gen Re subsidiary
Unindicted Co-Conspirators:
 Maurice "Hank" Greenberg: CEO of AIG
 Timothy McCaffrey: General Counsel of Gen Re
 Joseph Brandon: President of North American Operations at Gen Re
[3]  Houldsworth was not in his office when he fielded this after-business-hours call. It therefore was not recorded like most of his other calls. The description of the call comes from Houldsworth's testimony. (Phone records confirm that the call was made, however.)
[4]  Houldsworth also rejected the idea of treating the fee as a non-contractual "handshake." We discuss handshake deals below in connection with Graham's argument that a jury instruction on handshakes should have been given.
[5]  The parties decided to split the deal into two $250 million tranches.
[6]  The scheme entailed (1) offsetting $15 million from the $30 million that a Gen Re subsidiary held for an AIG entity under an existing contract, and (2) concealing the offset with a sham reinsurance contract.
[7]  Garand was first charged in the superseding indictment filed in September 2006; the rest of the defendants were indicted back in February 2006.

All defendants were charged with one count of conspiracy under 18 U.S.C. § 371 and three counts of mail fraud under 18 U.S.C. § 1341. All defendants except Garand were also charged with seven counts of securities fraud, 15 U.S.C. §§ 78j(b) & 78ff, and five counts of making false statements to the SEC, 15 U.S.C. §§ 78m(a) & 78ff; Garand was also charged with three counts of securities fraud and three counts of making false statements.
[8]  Certain findings are necessary to admit co conspirator statements under Fed.R.Evid. 801(d)(2)(E). See United States v. Geaney, 417 F.2d 1116, 1120 (2d Cir.1969). The court conditionally admitted co-conspirator statements during the government's presentation of its case; after the government rested, the court made the necessary Geaney findings and admitted the statements.
[9]  Prior to submission to the jury, the defendants had moved for acquittal pursuant to Fed.R.Crim.P. 29(a) and renewed their motions for severance under Fed.R.Crim.P. 14(a). (Only Ferguson, joined by Garand, submitted motion papers.) The court reserved decision.

After the verdict, the defendants moved for a new trial under Rule 33 only if the court granted their Rule 29(a) motions for any of the counts. They submitted skeletal memos without substantive argument, declined to file Rule 29(c) motions, and declined oral argument despite the court's request. (Ferguson initially requested oral argument on his motions, but then withdrew his request.)
The district court denied the defendants' pre-verdict motions for severance (Rule 14) and acquittal (Rule 29(a)), thus mooting their conditional motions for a new trial (Rule 33). United States v. Ferguson, 553 F.Supp.2d 145, 163-64 (D.Conn.2008).
[10]  The government's other materiality evidence was substantial: Two stock analysts and an AIG investor-relations manager testified about the importance of loss reserve information to investors and analysts.
[11]  If expert testimony were used, the probative value of the evidence would be reinforced because confounding factors could be excluded. Cf., e.g., United States v. Schiff, 538 F.Supp.2d 818, 836 (D.N.J.2008) (deeming stock-price data irrelevant for materiality in the absence of expert testimony). The expert could, for example, estimate the extent of the 12% drop attributable to the LPT.
[12]  The first question in the government's proposed instruction enunciated the causation requirement:

The meaning of the term "willfully caused" can be found in the answers to the following questions:
First, did the defendant take some action without which the crime would not have occurred?
Second, did the defendant intend that the crime would be actually committed by others?
Joint Appendix at 300 (emphasis added).
[13]  The second question from the defendants' proposed charge instructed on causation:

Second, as to each count and each Defendant, did the Defendant (a) intentionally cause other people to use a deceptive device in connection with the purchase or sale of AIG stock, that is to say, did he or she, in connection with the purchase or sale of AIG stock, intentionally cause other people to make either a deliberate affirmative misstatement of material fact or a deliberate omission of material fact by one who had a legal duty to disclose that fact, and, (b) intentionally cause some other person to knowingly use, or cause to be used, an instrumentality of communication in interstate commerce (i.e., the mails) in furtherance of such fraudulent scheme or conduct?
Joint Appendix at 556 (emphases added).
[14]  In United States v. Joseph, 542 F.3d 13 (2d Cir.2008), we held that we must reverse in such a circumstance "unless it can [be] determine[d] with absolute certainty that the jury based its verdict on the ground on which it was correctly instructed." Id. at 18. This standard is inconsistent with the Supreme Court's intervening decision in Hedgpeth, and therefore no longer controls. See In re Zarnel, 619 F.3d 156, 168 (2d Cir.2010).
[15]  If the jury did base its conviction on the willfully caused instruction, it was well on its way to finding the defendants guilty of aiding and abetting. Based on the court's willfully causing instruction, the jury would have found that the "defendant[s] act[ed] knowingly, willfully, and with an intent to defraud" and that the defendants "intend[ed] that this crime ... would actually be committed by others." Trial Tr. at 4760. However, given the conflicting evidence about who did the deceiving and who was deceived, it cannot be ascertained that the jury would have necessarily found a guilty principal whom the defendants could have aided and abetted as required by the aiding and abetting instruction. See Trial Tr. at 4757.
[16]  The Supreme Court appears to now prefer the appellation "willful blindness." Global-Tech Appliances, Inc. v. SEB S.A., ___ U.S. ___, 131 S.Ct. 2060, 2070 & n. 9, 179 L.Ed.2d 1167 (2011) (citing United States v. Svoboda, 347 F.3d 471, 477-78 (2d Cir.2003), which uses the term "conscious avoidance," as an example of this Court's "articulat[ion of] the doctrine of willful blindness"); see also United States v. Reyes, 302 F.3d 48, 54 (2d Cir.2002) ("The doctrine of conscious avoidance, also known as deliberate ignorance or willful blindness...."). We retain the designation "conscious avoidance" in order to conform to the briefs and the district court opinion.
[17]  See Pinkerton v. United States, 328 U.S. 640, 646-48, 66 S.Ct. 1180, 90 L.Ed. 1489 (1946) (ruling that liability for reasonably foreseeable acts within the scope and in furtherance of a conspiracy is attributable to all conspirators).
[18]  The defendants argue that Peterson cannot be extended because the four theories of liability have "clearly different elements that the jury must find." Garand Br. at 56 n. 14. But even Pinkerton liability  which requires the jury to find certain facts such as participation in the conspiracy  is premised on a mental state. See Pinkerton, 328 U.S. at 647, 66 S.Ct. 1180 ("The criminal intent to do the act is established by the formation of the conspiracy."); United States v. Thirion, 813 F.2d 146, 153 (8th Cir.1987) ("In the Pinkerton analysis.... [t]he mens rea necessary to transform the act into a criminal offense is evidenced by the defendant's participation in the conspiracy."). All four theories are thus various mental states in which the same crime may be committed; they may differ in "brute facts" underlying the mental state element, but none requires proof of other "factual elements" of the crime (which must be found unanimously by the jury). Richardson v. United States, 526 U.S. 813, 817, 119 S.Ct. 1707, 143 L.Ed.2d 985 (1999); cf. United States v. Sanchez, 917 F.2d 607, 612 (1st Cir.1990) ("As with the `aiding and abetting' theory, vicarious co-conspirator liability under Pinkerton is not in the nature of a separate offense.").
[19]  Two standards of review are set, based upon the prosecution's knowledge. If the prosecution knew or should have known of the perjury, the conviction must be set aside "if there is any reasonable likelihood that the false testimony could have affected the judgment of the jury." Id. (internal quotation marks omitted). But where the government was unaware of the perjury, a new trial "is warranted only if the testimony was material and the court [is left] with a firm belief that but for the perjured testimony, the defendant would most likely not have been convicted." Id. (internal quotation marks omitted).
[20]  See United States v. Helmsley, 985 F.2d 1202, 1205 (2d Cir.1993); United States v. Blair, 958 F.2d 26, 29 (2d Cir.1992).
[21]  In Wallach, the government conceded that the witness had committed perjury. See 935 F.2d at 455.
[22]  The government in essence collapses the Zichettello factors into the two Wallach factors, arguing that the perjury (if any) was immaterial because it was disclosed at trial and fully corrected by the defendants' forceful attack on Napier's credibility during cross-examination and summation, yet the jury nevertheless convicted Monrad and Garand.
[23]  For example, Garand argues that a subset of Napier's notes produced by the government was in rough chronological order, suggesting that the undated notes page was from between November 15 and 17, rather than from November 13. The government's use at trial of an identical copy of the notes from elsewhere in the production (rather than the version from the chronological subset), he argues, shows intent to obscure the correct date for the notes.
[24]  Several of the misconduct arguments are discussed elsewhere, including the alleged misuse of (1) stock-price data, (2) the recordings with derogatory comments about AIG, and (3) Graham's email to McCaffrey. The argument about characterizing the deal as "no risk" from the start duplicates Ferguson's argument about co-conspirator statements discussed below.
[25]  Ferguson attempts to distinguish "no real risk" from "no risk," arguing that the former is simply a reference to a legitimate finite reinsurance transaction with low risk. He introduced CRD materials that arguably use "no real risk" in this manner. However, the evidence is inconclusive, and in any event the potential distinction was before the jury. A rational juror could have, but need not have, credited the distinction.
[26]  As Ferguson notes, statements admitted under Rule 801(d)(2)(E) are technically nonhearsay, rather than hearsay exceptions. Ferguson's argument is thus a first-order hearsay issue (which happens to be embedded in a nonhearsay co-conspirator statement). The distinction is irrelevant for present purposes. We therefore use "double-hearsay" for ease of reference and to conform to the framing of the arguments in the district court.
[27]  See, e.g., United States v. Fell, 531 F.3d 197, 231 (2d Cir.2008) (reviewing statement admitted as excited utterance under Rule 803(2) for abuse of discretion); United States v. Padilla, 203 F.3d 156, 161 (2d Cir.2000) (reviewing district court's findings for admitting co-conspirator statements under Rule 801(d)(2)(E) for clear error).
[28]  Such a formulation would raise a problem as to the speaker's competence to say what is in the mind of another person, however. It is unclear whether Graham knew that Ferguson had been informed or whether some degree of conjecture was involved. We have never explicitly held that co-conspirator statements admitted under Rule 801(d)(2)(E) need not satisfy Rule 602's personal knowledge requirements. The government argues that personal knowledge is not required, noting that several other Circuits have so held, see, e.g., United States v. Lindemann, 85 F.3d 1232, 1237-38 (7th Cir.1996), and that we have rejected such a requirement for a similar provision (for admissions by a party's agents under Rule 801(d)(2)(D), see United States v. Lauersen, 348 F.3d 329, 340 (2d Cir.2003), vacated and remanded on other grounds, 543 U.S. 1097, 125 S.Ct. 1109, 160 L.Ed.2d 988 (2005)).

The potential personal knowledge issue was waived, however, because double-hearsay is what was argued and there was no ruling on personal knowledge in the first place. See Norton v. Sam's Club, 145 F.3d 114, 117 (2d Cir.1998).
[29]  Ferguson's motion to sever based on this email was made on the eve of trial, because it was prompted by the government's eleventh-hour Brady/Jencks disclosures. The district court thus had to decide the motion on an expedited basis, and did so orally (before the government was even able to file a written response). The court later provided a written explanation for the denial of Ferguson's renewed motion to sever (raised in connection with his Rule 29 motions), but that too omitted any reference to the Finkelstein factors. (Ferguson had once previously renewed the motion to sever, which was also denied orally.)
[30]  Ferguson argues that the proffer session statement would be admissible under Rule 804(b)(3) as a statement against interest or under the Rule 807 residual exception. But "only those declarations ... that are individually self-inculpatory" are admissible under Rule 804(b)(3), and a court could exclude the statement for "d[oing] little to subject [Graham] himself to criminal liability." See Williamson v. United States, 512 U.S. 594, 599, 604, 114 S.Ct. 2431, 129 L.Ed.2d 476 (1994) (emphasis added). As for the residual exception, it is unclear that the notes have the necessary "equivalent circumstantial guarantees of trustworthiness," Fed.R.Evid. 807, because Graham was not cross-examined and there was no transcript from the hearing.
[31]  Extrajudicial statements made among co-conspirators during and in furtherance of a conspiracy (whose existence is established by a preponderance of the evidence) are admissible against co-conspirators. See Fed.R.Evid. 801(d)(2)(E); United States v. Tellier, 83 F.3d 578, 580 (2d Cir.1996).

The court conditionally admitted the statements during the government's case, but admitted the statements only after conducting a hearing after the government rested; the court found that the government satisfied its burden of proving the necessary Rule 801(d)(2)(E) elements by a preponderance. See United States v. Geaney, 417 F.2d 1116, 1120 (2d Cir.1969).
[32]  The government presented evidence that Ferguson and Greenberg ratified the transaction even after it became no-risk.
[33]  This is the same email that Ferguson challenged (discussed above) for asserting that he had been advised of the potential reputational risk of the transaction.
[34]  Graham argues that the reference to McCaffrey as a co-conspirator in the indictment should have been stricken. Motions to strike surplusage from an indictment are granted only when the challenged phrases are "not relevant to the crime charged and are inflammatory and prejudicial." United States v. Hernandez, 85 F.3d 1023, 1030 (2d Cir. 1996) (internal quotation marks omitted). The phrase was relevant because, as discussed below, McCaffrey was legitimately being investigated; he had to be referenced because he was the only recipient of Graham's infamous email.
[35]  Milton challenges the admission of four passages:

1) HOULDSWORTH: [I]f there's enough pressure on at [AIG's] end, they'll, they'll find ways to cook the books, won't they?
2) MONRAD: I'm not sure [AIG] use[s] all the same rules we use.
3) HOULDSWORTH: ... I mean, how much cooking goes on in, in there [at AIG]?
. . . .
. . .
GARAND: They'll do whatever they need to make their numbers look right.
4) GRAHAM: [AIG's] organizational approach to compliance issues has always been, pay the speeding ticket.
[36]  Milton argues that no limiting instruction could cure the prejudice, by analogy to cases where plea allocutions were introduced, see, e.g., United States v. Riggi, 541 F.3d 94 (2d Cir.2008). The analogy is not apt.
[37]  It is easy to see why the defendant did not seek further limiting instructions. Confusion abounded when the first limiting instruction was given, causing the jury to hear the prejudicial statement three times in short succession. Nevertheless, the option to seek further instructions was available.
[38]  Two aspects of the district court's treatment of Houldsworth's "cook the books" comment give pause. First, the court excluded the comment from evidence after previously allowing the jury to hear it during the government's opening. But the statement was not so explosively prejudicial as to taint the jury after one hearing; nor was it irrational for the district court to revisit its Rule 403 balancing and exclude the comment. Second, excluding the comment seems inconsistent with admitting Houldsworth's other comment that asked how much cooking goes on at AIG. But the "cooking the books" metaphor was not the only objectionable part of the excluded comment; Houldsworth also explained that "We won't help them do that too much. We won't, we'll do nothing illegal."
[39]  Milton moved for a severance both before and after trial. The pretrial motion was based on the above recordings and evidence of a similar transaction, which was excluded before trial. The post-verdict motion for severance lacked supporting legal arguments.
[40]  (The content of the recordings is discussed above, see supra note 32.)

Although the first recording was permitted to be played in the government's opening, the court did not allow the government to bring it out in Houldsworth's testimony.
The second passage is not inflammatory, but in any event, the court instructed the jury that it "may not consider this statement at all as to any of the other defendants" besides Monrad, the speaker.
The court offered to give a limiting instructions to the jury for the third recording "if requested by Milton," 246 F.R.D. at 120, but Milton did not request one.
Milton expressly requested that the court not give a limiting instruction for the fourth recording.
[41]  Monrad argues in general about improper lay opinions by Napier and Houldsworth, but focuses on four interpretations:

1) Testimony by Houldsworth and Napier concerning her statement that "We told AIG that there would not be symmetrical accounting here." Joint Appendix at 2094.
2) Houldsworth's testimony about Napier's statement that "The accounting does not appear to be an issue for AIG." Joint Appendix at 2067.
3) Houldsworth's testimony about his comment that "to me it sounds like [Monrad's] got something in mind." Joint Appendix at 1957.
4) Testimony by Houldsworth and Napier concerning her statement that AIG "may have a tough time getting the accounting they want." Joint Appendix at 1997.
[42]  Monrad's analogy to "drug code" cases is unpersuasive. See, e.g., United States v. Grinage, 390 F.3d 746 (2d Cir.2004). Attributing highly inculpatory meaning to otherwise innocuous phrases (e.g., "I need something bad, bad, bad," and "I need about nearly four" interpreted as needing four ounces of PCP) differs from providing context for obscure accounting terminology.
