
    UNITED STATES of America, Appellee, v. Solomon KAPLAN Defendant-Appellant.
    Docket No. 05-5531-cr.
    United States Court of Appeals, Second Circuit.
    Argued: Nov. 7, 2006.
    Decided: April 11, 2007.
    
      Zachary Margulis-Ohnuma, New York, New York, for Defendant-Appellant.
    Miriam E. Rocah, Assistant United States Attorney, (Michael J. Garcia, United States Attorney, Timothy Treanor and Jonathan S. Kolodner, Assistant United States Attorneys, on the brief), United States Attorney’s Office for the Southern District of New York, for Appellee.
    Before: FEINBERG, LEVAL, and CABRANES, Circuit Judges.
   FEINBERG, Circuit Judge:

Solomon Kaplan appeals from a judgment of conviction, entered following a jury trial in the United States District Court for the Southern District of New York (Batts, J.), on all seven counts of an indictment charging Kaplan’s participation in an insurance fraud scheme (Counts One through Five) and Kaplan’s interference with an investigation into that scheme (Counts Six and Seven). Specifically, the indictment charged Kaplan with one count of conspiracy in violation of 18 U.S.C. § 371 (Count One); two counts of mail fraud in violation of 18 U.S.C. §§ 1341 and 2 (Counts Two and Three); one count of making false statements in connection with health care matters in violation of 18 U.S.C. §§ 1035 and 2 (Count Four); one count of health care fraud in violation of 18 U.S.C. §§ 1347 and 2 (Count Five); one count of witness tampering in violation of 18 U.S.C. §§ 1512(b) and 2 (Count Six); and one count of making false statements to an agent of the Federal Bureau of Investigation (“FBI”) in violation of 18 U.S.C. § 1001 (Count Seven).

On appeal, Kaplan’s principal contentions are that (I) his conviction on the insurance fraud counts (Counts One through Five) must be vacated because the district court erred in admitting (A) lay opinion testimony regarding his knowledge of the fraud and (B) testimony concerning others’ knowledge of the fraud and (II) his conviction on the interference counts (Counts Six and Seven) must be vacated because (A) the district court’s jury instruction on Count Six was erroneous in light of the Supreme Court’s supervening decision in Arthur Andersen LLP v. United States, 544 U.S. 696, 125 S.Ct. 2129, 161 L.Ed.2d 1008 (2005); (B) the district court improperly gave a conscious avoidance jury instruction on Count Six; and (C) variance between a bill of particulars and proof at trial concerning Count Seven constituted a constructive amendment of the indictment or a prejudicial variance.

For the reasons set forth below, we agree that Kaplan’s conviction on Counts One through Five must be vacated because the district court erred in admitting, without adequate foundation, lay opinion testimony regarding Kaplan’s knowledge of the fraud and testimony regarding others’ knowledge of the fraud, and that at least the first of these errors was not harmless. However, we affirm his conviction on Counts Six and Seven because these evi-dentiary errors were harmless as to those counts, which relied on strong independent evidence of the crimes charged in those counts, and because we find no merit in Kaplan’s other arguments on appeal. The case is remanded for further proceedings consistent with this opinion.

BACKGROUND

Viewed in the light most favorable to the Government, see Jackson v. Virginia, 443 U.S. 307, 318-19, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979), the evidence showed the following.

Josef Sherman, a medical doctor, and his brother, Yevgeny Sherman, operated a medical clinic in Brooklyn, New York (the “Clinic”). The Clinic hired “runners” to recruit patients by staging automobile accidents and identifying individuals who had been in legitimate accidents but were willing to exaggerate their injuries. At the Clinic, these accident participants received unnecessary treatment for their feigned injuries and were compensated with a kickback. The Clinic then submitted fraudulent insurance claims for medical expenses to collect money under New York State’s no-fault insurance law.

The accident participants were also referred to a cooperating law office, which submitted on the participants’ behalf false or inflated insurance claims for bodily injury. From January 2000 until July 30, 2001, most of the Clinic’s cases were referred to a law office (the “Law Office”) operated in the name of Alexander Gaiko-vich, a lawyer hired by the Shermans and their associate Gennady “Gene” Medve-dovsky to serve as counsel of record in the referred cases. The accident participants signed a retainer agreement providing that the Law Office received one-third of any insurance settlement as well as expenses. Medvedovsky, although not an attorney, managed the Law Office on behalf of the Shermans through a management company called Starlin Executive Management. Gaikovich paid almost all of the proceeds he received from the insurance company settlements to Starlin Management, and received $1,000 per week as salary and occasional bonuses. The Shermans and Medvedovsky extracted the insurance proceeds from the law office principally by submitting to Starlin Management fraudulent bills from fictitious entities or by paying themselves salaries from Starlin Management.

By 2001, the Law Office had more than 3,000 active cases, and was receiving approximately 80 to 200 new cases per month, a significant portion of which came from the Sherman Clinic. Approximately five to 10 percent of the cases at the Law Office resulted from staged accidents, and 60 to 70 percent of the cases involved clients who exaggerated or faked the injuries.

In July 2001, Gaikovich was arrested by the FBI and charged with filing false and fraudulent claims and coaching clients to lie to the insurance companies. Because Gaikovich thus stood to lose his law license, the Shermans and Medvedovsky sought a replacement to serve as attorney-of-record in the fraudulent cases. They settled on Kaplan, with whom they were familiar because a few of the Clinic’s cases had previously been referred to him. An employee of the Clinic, Alexander Burman, testified that the eases referred to Kaplan were those that had been rejected by other lawyers because they were considered too obviously fraudulent.

In September 2001, Kaplan began representing 1,200 of Gaikovich’s 3,000 clients. The Shermans and Medvedovsky, worried that the transfer of all 3,000 cases from Gaikovich to Kaplan would look suspicious, arranged for Kaplan to formally purchase the law firm from Gaikovich. To make the deal appear legitimate, they hired a lawyer to draft a contract and conduct a closing, and Kaplan wrote several checks to Gaiko-vich totaling $50,000, including a $20,000 check at the closing. But at least some of the money came from the Shermans and Medvedovsky and was later returned to them by Gaikovich. Thus, the Shermans and Medvedovsky essentially bought the Law Office from themselves, but structured the transaction to look as if Kaplan had bought it from Gaikovich.

Gaikovich testified that on the way to the closing in October 2001, Kaplan and Gaikovich discussed the sale of the Law Office, and Kaplan stated that “he had handled cases like this before,” which Gai-kovich understood to mean that Kaplan had previously handled fake accident cases.

The Law Office continued to operate much as it had before, with Kaplan now formally representing almost all of Gaiko-vich’s 3000 clients. Medvedovsky and others, including Emik Aguronov, who was responsible for drafting false medical narratives to be submitted to insurance companies on behalf of clients seeking settlements, remained. Medvedovsky formed a new entity called Prostaff Support Services, Inc., which performed the same functions that Starlin Management had. From October 2001 through March 2002, the Law Office had revenues of approximately $892,000. The proceeds of the fraud were transferred to Medvedovsky and the Shermans through Prostaff, which was paid over $100,000 per month to “manage” the Law Office. Kaplan received approximately $74,000 in payments from the Law Office.

Kaplan, the Shermans, and Medvedov-sky all agreed that Kaplan would stay away from the Law Office and appear only to sign checks and for essential meetings. A photograph of Kaplan’s office at the Law Office, Kaplan’s appointment book, and the testimony of FBI Special Agent Rothe revealed that Kaplan was rarely present at the Law Office.

Following the sale of the Law Office to Kaplan, Gaikovich began cooperating with the FBI. Between approximately December 27, 2001 and February 22, 2002, Gaiko-vich recorded his conversations with the Shermans, Medvedovsky, and Kaplan on approximately 10 separate occasions. In addition to general discussions about the operations of the Law Office, the conversations recorded by Gaikovich detailed the efforts by the Shermans, Medvedovsky, and Kaplan to prevent Gaikovich from cooperating with law enforcement authorities in its investigation of the Sherman Clinic and the Law Office. Specifically, after Gaikovich informed his co-conspirators that he had been arrested and that the FBI had inquired about the Shermans, a meeting was arranged with Gaikovich on January 8, 2002. Medvedovsky told Gai-kovich that the purpose of the meeting was to “sit down and decide everything” about Gaikovich’s case. Josef Sherman and Gene Medvedovsky initially met with Gai-kovich alone and questioned him about the statement he had given to the FBI at the time of his arrest. Later, at Gaikovich’s request, Kaplan joined the meeting, and Gaikovich told Kaplan that they needed to discuss the “transition of my practice to yours” and “this thing that I’m going to need for ... possibly for court or possibly for the Disciplinary Committee.” Medve-dovsky, Sherman, and Kaplan suggested a variety of false stories to explain the Law Office transfer.

In March 2002, Kaplan, the Shermans, and Medvedovsky were arrested by the FBI. Immediately following his arrest, Kaplan agreed to be interviewed by the FBI. During that interview, Kaplan made a number of statements that the Government contends were false, including that in August 2001 he was introduced to Gaiko-vich and Medvedovsky by a Vladimir Scheckman; that, as part of his purchase of the Law Office, he was to receive only 1,000 files; that he did not recall who gave him the $20,000 check used to purchase the Law Office; that the Law Office was sold to him for $120,000; that he was assured by Medvedovsky and Gaikovich that the charges against Gaikovich “meant nothing because Gaikovich had done nothing wrong”; that he “just stopped by” the January 8 meeting between Gaikovich, Medvedovsky, and Sherman without planning in advance to participate; and that he did not take the conversation on January 8 “seriously” and they were just “joking around.”

In August 2002, Josef Sherman, Eugene Sherman, and Medvedovsky pled guilty to a three-count information charging them with conspiracy, mail fraud, and witness tampering. Kaplan was tried on the seven-count indictment against him described above. After an approximately two-week trial, the jury found Kaplan guilty on all counts. In August 2005, the district court sentenced Kaplan principally to 27 months of imprisonment, three years of supervised release, and $200,000 in restitution, but granted bail pending appeal pursuant to 18 U.S.C. § 3143(b).

This appeal followed.

DISCUSSION

I. Counts One through Five: The Insurance Fraud Scheme

Turning first to the insurance fraud counts (Counts One through Five), Kaplan argues on appeal, among other things, that his conviction must be vacated because the district court erred in admitting (A) lay opinion testimony regarding his knowledge of the fraud and (B) testimony concerning others’ knowledge of the fraud. We consider each of his arguments in turn.

A. Admission of Gaikovich’s lay opinion testimony

The first issue before us concerns the district court’s admission of lay opinion testimony regarding Kaplan’s knowledge of the fraud. Kaplan principally objects to two colloquies in which Gaikovich recounted a conversation he and Kaplan had as they drove together to the meeting in October 2001 to finalize the sale of the Law Office to Kaplan. After describing the conversation, Gaikovich was allowed to offer his lay opinion testimony regarding Kaplan’s knowledge of the fraud. First, on direct examination, Gaikovich testified as follows:

[Prosecutor]: Did you have any discussions in the car ride on the way to Davis’ office?
[Gaikovich]: Yes. It was actually the first time we really talked, me and Solomon Kaplan. And I asked him, What do you do? What kind of work do you do? Are you familiar with car accident cases, with the process of settlement and what it takes to settle? And he explained, yes he has handled cases like this before. Yes, he has settled cases before.
He explained that he has experience with these kinds of cases.
[Prosecutor]: What did you understand him to mean when he said “these kinds of cases”?
[Defense Counsel]: Objection.
The Court: I will allow it.
[Gaikovich]: That he understood that these were car accident cases where people exaggerated their injuries, where it was crucial to have a narrative report that exaggerated the injuries, that these reports were bought for the best of prices to get the best of reports and that you could settle these cases for very good money in a short period of time.

Joint Appendix (“JA”) 104-05. Then, on redirect, Gaikovich elaborated:

[Prosecutor]: What happened in this conversation?
[Gaikovich]: I asked him what experience he had with the car accident cases and generally what kind of experience he had, and he told me that he knew about these car accidents, he knew how to handle these cases, he knew how to maximize potential recoveries, and what is supposed to be in the files, how they are supposed to be worked up. [Prosecutor]: What was your purpose in asking Kaplan this question?
[Gaikovich]: I wanted to know how much he knew about the fraudulent office that he is participating in. [Prosecutor]: And after you got this answer from Mr. Kaplan, what did you think?
[Gaikovich]: I think he knew exactly what he was getting into.

JA 138-39.

We review a district court’s decision to admit evidence, including lay opinion testimony, for abuse of discretion. See United States v. Yuri Garcia, 413 F.3d 201, 210 (2d Cir.2005) (citing Old Chief v. United States, 519 U.S. 172, 174 n. 1, 117 S.Ct. 644, 136 L.Ed.2d 574 (1997)). “A district court ‘abuses’ or ‘exceeds’ the discretion accorded to it when (1) its decision rests on an error of law (such as application of the wrong legal principle) or a clearly erroneous factual finding, or (2) its decision — though not necessarily the product of a legal error or a clearly erroneous factual funding — cannot be located within the range of permissible decisions.” Zervos v. Verizon N.Y., Inc., 252 F.3d 163, 169 (2d Cir.2001) (footnotes omitted).

The Federal Rules of Evidence, in a sharp departure from the common law, see Asplundh Mfg. Div. v. Benton Harbor Eng’g, 57 F.3d 1190, 1195 (3d Cir.1995) (Becker, J.), permit lay witnesses to testify in the form of opinions to address a problem identified by Judge Learned Hand many years ago:

Every judge of experience in the trial of causes has again and again seen the whole story garbled, because of insistence upon a form with which the witness cannot comply, since, like most men, he is unaware of the extent to which inference enters into his perceptions. He is telling the ‘facts’ in the only way that he knows how, and the result of nagging and checking him is often to choke him altogether, which is, indeed, usually its purpose.

Central R.R. Co. of N.J. v. Monahan, 11 F.2d 212, 214 (2d Cir.1926); see also Yuri Garcia, 413 F.3d at 211 (“eyewitnesses sometimes find it difficult to describe the appearance or relationship of persons, the atmosphere of a place, or the value of an object by reference only to objective facts”). Accordingly, Rule 701 of the Federal Rules of Evidence was adopted “to accommodate and ameliorate these difficulties by permitting lay witnesses, in appropriate circumstances, to testify in language with which they are comfortable.” 4 Jack B. Weinstein & Margaret A. Berger, Wein-stein’s Federal Evidence § 701.02 (Joseph M. McLaughlin ed., 2d ed.2004).

But to ensure that lay opinion testimony is reliable and does not usurp the jury’s role as fact-finder, Rule 701 imposes certain foundation requirements that must be satisfied if such testimony is to be admitted:

If the witness is not testifying as an expert, the witness’ testimony in the form of opinions or inferences is limited to those opinions or inferences which are (a) rationally based on the perception of the witness, (b) helpful to a clear understanding of the witness’ testimony or the determination of a fact in issue, and (c) not based on scientific, technical, or other specialized knowledge within the scope of Rule 702.

Fed.R.Evid. 701 (2001). In interpreting these requirements, we have observed that (a) the rational-basis requirement “is the familiar requirement of first-hand knowledge or observation,” United States v. Rea, 958 F.2d 1206, 1215 (2d Cir.1992) (quoting Fed.R.Evid. 701 advisory committee’s note on 1972 Proposed Rules); (b) the helpfulness requirement is principally “designed to provide assurance!] against the admission of opinions which would merely tell the jury what result to reach,” id. (quoting Fed.R.Evid. 704 advisory committee’s note on 1972 Proposed Rules); and (c) the “not based on specialized knowledge” requirement requires that “a lay opinion must be the product of reasoning processes familiar to the average person in everyday life,” Yuri Garcia, 413 F.3d at 215. See also Fed.R.Evid. 701 advisory committee’s note on 2000 amendments.

The government’s evidence failed to demonstrate that Gaikovich’s lay opinion testimony was “rationally based on the perception of the witness.” Fed R. Evid. 701(a). We note that Rule 701(a) requires that lay opinion testimony be both (a) based on the witness’s first-hand perceptions and (b) rationally derived from those first-hand perceptions.

As to the first of these requirements, Rule 701(a) reflects, in part, the Rules’ more general requirement that “[a] witness may not testify to a matter unless evidence is introduced sufficient to support a finding that the witness has personal knowledge of the matter.” Fed.R.Evid. 602; see also United States v. Durham, 464 F.3d 976, 982 (9th Cir.2006) (“opinion testimony of lay witnesses must be predicated upon concrete facts within their own observation and recollection — that is facts perceived from their own senses, as distinguished from their opinions or conclusions drawn from such facts” (internal quotation marks omitted)). When Gaikovich was asked to articulate the basis for his opinion, he answered, “I based it on the only thing I could base it on, which is my experience there, what people said about [Kaplan], my conversation -with [Kaplan], everything that I[had] been involved in. That’s what my opinion could be based on.” JA 140. Although Gaikovich asserts that his testimony was based in part on first-hand experience — -principally his prior experiences at the Law Office and his conversation with Kaplan — his response was extremely vague. Thus, Gaikovich’s testimony failed to show that his opinion as to Kaplan’s knowledge was rationally based on facts he had observed.

We are therefore unable to conclude, as we must under Rule 701, that the opinion he offered was rationally based on his own perceptions. See Rea, 958 F.2d at 1216 (“When a witness has not identified the objective bases for his opinion, the proffered opinion obviously fails completely to meet the requirements of Rule 701 ... because there is no way for the court to assess whether it is rationally based on the witness’s perceptions .... ”). We applied this requirement to similar facts in Rea, and observed that lay opinion testimony regarding a defendant’s knowledge will, in most cases, only satisfy the rationally-based requirement if the witness has personal knowledge of one or more “objective factual bases from which it is possible to infer with some confidence that a person knows a given fact ... including] what the person was told directly, what he was in a position to see or hear, what statements he himself made to others, conduct in which he engaged, and what his background and experience were.” 958 F.2d at 1216. Because the Government did not lay an adequate foundation, Gaikovich’s testimony expressing his opinion as to Kaplan’s knowledge was not admissible.

Accordingly, having found that Gaiko-vich’s lay opinion testimony does not satisfy Rule 701, we conclude that the district court erred in admitting it.

B. Admission of Gaikovich’s additional testimony regarding his and others’ knowledge

Kaplan asserts that it was error for the district court to admit Gaikovich’s additional testimony regarding Gaikovich and others’ knowledge of the fraud as circumstantial evidence of Kaplan’s knowledge. Specifically, Gaikovich was permitted to testify that (1) when he first saw the building in which the Law Office was located, he thought, “[t]his is where I am going to get arrested,” JA 84; (2) everyone he spoke with told him not to buy the Law Office, JA 87; (3) the fraud was “done very subtly” because “this whole industry was a very big sham and it was big lies,” JA 90; (4) the fraud was not discussed explicitly because it was “kind of like ... hear no evil, see no evil,” and he was warned to “be very careful in what you say,” JA 94; (5) “[e]veryone knew what was going on, but you don’t say it,” JA 94; (6) at a party at the Law Office, Eugene Sherman read a poem about clients getting paid for sham injuries, JA 97; and (7) Steven Rosenberg, a lawyer at the Law Office, quit because he saw a file with “blatant” fraud, JA 116-19, and Michael Brummer, another lawyer at the Law Office, was “very sensitive and nervous about what was going on in the office” and was “always very nervous when we were skirting the issue of fraud,” JA 119-21.

As previously noted, we review a district court’s evidentiary rulings for abuse of discretion. See Yuri Garcia, 413 F.3d at 210 (citing Old Chief, 519 U.S. at 174 n. 1, 117 S.Ct. 644). Kaplan argues that the district court abused its discretion because the evidence was irrelevant and should have been excluded under Fed.R.Evid. 402 or, in the alternative, that its probative value was substantially outweighed by the danger of unfair prejudice and should have been excluded under Fed.R.Evid. 403. The Government argues that the testimony was relevant to the question of whether Kaplan was aware of the fraud because evidence that the fraud was obvious and widely-known tended to make it more probable that Kaplan also knew about it. As the prosecutor argued to the jury, “[Kaplan] had to know. Everybody else did.”

The parties vigorously dispute which of two precedents-United States v. Patrisso, 262 F.2d 194 (2d Cir.1958), and United States v. Schultz, 333 F.3d 393 (2d Cir. 2003)-controls this case. In Patrisso, a truckload of television tubes was hijacked; Patrisso, who knew they were stolen, sold them to Ellis, who also knew; Ellis, in turn, sold them to Postrel; finally, Postrel sold 1,000 of them to defendant Mankes. 262 F.2d at 196. We reversed Mankes’s conviction, finding that the district court had erred by, inter alia, allowing the Government to introduce Postrel’s knowledge of the theft without evidence that Postrel, or anyone else, had communicated that fact to Mankes. Id. at 197. Over four decades later, in Schultz, we distinguished Patrisso in upholding the admission of testimony regarding other individuals’ knowledge of a particular Egyptian law to prove the defendant’s knowledge of the law, finding that it was “relevant both to explain the practice of the industry in which this prosecution arose and to establish what someone with [defendant’s] extended background in the industry probably would know.” 333 F.3d at 416 (quoting United States v. Leo, 941 F.2d 181, 197 (3d Cir.1991)).

We believe that Patrisso and Schultz, though they reach different outcomes, stand for the same principle: evidence regarding the knowledge of individuals other than the defendant should be admitted only if there is some other evi-clence in the record-concerning, for example, the nature of the fraud or the relationship of the parties-from which to conclude that the defendant would have the same knowledge. Indeed, the Schultz court noted that the principal difference between the two cases was the nature of the knowledge involved: Schultz was likely to have the same knowledge, the defendant in Patrisso wasn't. 333 F.3d at 416. What of Kaplan?

We turn first to relevance. Relevant evidence includes any “evidence having any tendency to make the existence of any fact that is of consequence to the determination of the action more probable or less probable than it would be without the evidence.” Fed.R.Evid. 401; see also Fed.R.Evid. 402. “Implicit in that definition are two distinct requirements: (l)[t]he evidence must be probative of the proposition it is offered to prove, and (2) the proposition to be proved must be one that is of consequence to the determination of the action.” United States v. Diaz, 878 F.2d 608, 614 (2d Cir.1989). As the Supreme Court has observed, “[t]he Rules’ basic standard of relevance ... is a liberal one.” Daubert v. Merrell Dow Pharms., 509 U.S. 579, 587, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993). Despite this liberal standard, we think this evidence had little relevance in the circumstances of Kaplan’s case. Evidence of others’ knowledge would have been highly relevant had it been supplemented by evidence supporting the conclusion that such knowledge was communicated to Kaplan, or that Kaplan had been exposed to the same sources from which these others derived then-knowledge of the fraud. In the absence of such evidence, the relevance of others’ knowledge was at best minimal in proving Kaplan’s knowledge.

Nor does our inquiry end there — even as to evidence that is plainly relevant, the trial judge retains discretion to exclude the evidence “if its probative value is substantially outweighed by the danger of unfair prejudice.” Fed.R.Evid. 403. This evidence was of minimal probative value for two reasons. First, as noted, the Government failed to offer evidence that would connect the third parties’ knowledge of the fraud to Kaplan. Under the Government’s own theory of the case, Kaplan spent very little time at the law office and reviewed very few claims directly. Moreover, the Government proffered no evidence that anyone who allegedly was aware of the fraudulent scheme actually had communicated his knowledge to Kaplan. In fact, Gaikovich testified that the participants in the fraud were careful not to speak openly about the fraudulent nature of the injury claims: he testified that clients did not directly admit to the firm that their accidents were staged, but rather that “generally it was done very subtly,” and that he did not have explicit conversations about the fraud with clients or other people in the. office “because you don’t want to get in trouble ... It is kind of like, you know, hear no evil, see no evil.... Everyone knew what was going on, but you don’t say it.” This was not an office where the illegal nature of the business was necessarily visible and audible to everyone who worked there. In that sense, we think this case is more analogous to Patrisso than to Schultz: as in Patrisso, the Government failed to offer evidence that would explain how defendant Kaplan would have obtained the third parties’ knowledge of the criminal scheme.

Second, much of the testimony concerning knowledge of the fraud was so speculative or flawed in other respects that it had little or no probative value. For example, the Government did not lay a proper foundation for Gaikovich’s statement that it was “[his] understanding that this whole business was very very — this whole industry was a very big sham and it was big lies”; Gaikovich was not qualified as one having special knowledge of the personal injury “industry,” so it is difficult to see how Gaikovich had sufficient knowledge of the industry to testify competently on its criminal nature. See Fed.R.Evid. 602 (“A witness may not testify to a matter unless evidence is introduced sufficient to support a finding that the witness has personal knowledge of the matter.”); Woodman v. WWOR-TV, Inc., 411 F.3d 69, 86-87 (2d Cir.2005) (affirming district court’s decision to exclude plaintiffs testimony that her age was “well known throughout the industry” on ground that plaintiff was “hardly competent to testify to how others in the broadcast community perceived her age" (citing Fed.R.Evid. 602)). testimony concerning the alleged knowledge of Rosenberg, another lawyer in the office, was hearsay upon hearsay; Rosenberg allegedly made statements to a secretary in the office, who allegedly told Galkovich, who offered the jury the that Rosenberg had quit his job of fraud. Cf. Brown v. Keane, 355 F.3d 82, 90 (2d Cir.2004) ("An assertion of fact based on conjecture and surmise, to which the declarant would not be allowed to testify if called to the witness box, does not become admissible under an exception to the hearsay rule . .

We conclude, furthermore, that this limited probative value is substantially by the risk of unfair prejudice. Although relevant evidence is always to one side, we conclude that the risk of unfair prejudice here-in the likelihood that jurors would render a decision on an improper basis by giving this testimony undue weight or improperly holding Kaplan liable because they he should have known of the fraud-was great. The jury was required to draw a series of inferences, unsupported by other evidence, to connect Galkovich's testimony about his guilty knowledge (and that of others) to Kaplan's own knowledge, the ultimate issue in the case. Under the circumstances, the District Court should have concluded that whatever slight value the testimony might have had was outweighed by the risk that the jury would draw improper inferences from the testimony. Cf. United States v. Ravich, 421 F.2d 1196, 1204 n. 10 (2d Cir.1970) (Friendly, J.) ("The length of the chain of inferences necessary to connect the with the ultimate fact to be proved necessarily lessens the probative value of the evidence, and may therefore render it more susceptible to exclusion as unduly confusing, prejudicial, or time-consuming • . . ."). Moreover, Galkovich's testimony that the entire industry in which Kaplan operated was "a very big sham and it was big lies" was so inflammatory that it should have been excluded as prejudicial under the circumstances. Furthermore, the likelihood of prejudice was increased by the government's improper use of the evidence of others' knowledge. In the prosecutor argued, "[Kaplan] had to know. Everybody else did." As noted, the evidence did not support any such inference.

Accordingly, with respect to Galkovich's testimony regarding his and others' of the fraud, we conclude that the risk of unfair prejudice substantially the limited probative value, and hold that the district court erred in this evidence.

C. Harmless error analysis

We will reverse on account of these evidentiary errors only if they affect "substantial rights." See Fed.R.Crim.P. 52(a); Fed.R.Evid. 103(a); Yuri Garcia, 413 F.3d at 210. In Kotteakos v. United States, the Supreme Court set forth the analysis for determining whether a error is harmless:

If, when all is said and done, the is sure that the error did not the jury, or had but very slight effect, the verdict and the judgment should stand, except perhaps where the departure is from a constitutional norm or a specific command of Congress But if one cannot sa~y, with fair after pondering all that happened without stripping the erroneous action from the whole, that the judgment was not substantially swayed by the error, it is impossible to conclude that substantial rights were not affected. The inquiry cannot be merely whether there was enough to support the result, apart from the phase affected by the error. It is rather, even so, whether the error itself had substantial influence. If so, or if one is left in grave doubt, the conviction cannot stand.

328 U.S. 750, 764-65, 66 S.Ct. 1239, 90 L.Ed. 1557 (1946) (footnote omitted); see also United States v. Dukagjini, 326 F.3d 45, 62 (2d Cir.2003) (holding that non-constitutional error affects substantial rights if it had a “substantial and injurious effect or influence” on the jury’s verdict). But “we are not required to conclude that it could not have had any effect whatever; the error is harmless if we can conclude that that testimony was ‘unimportant in relation to everything else the jury considered on the issue in question, as revealed in the record.’ ” Rea, 958 F.2d at 1220 (quoting Yates v. Evatt, 500 U.S. 391, 403, 111 S.Ct. 1884, 114 L.Ed.2d 432 (1991)). In conducting this inquiry, we principally consider “(1) the overall strength of the prosecution’s case; (2) the prosecutor’s conduct with respect to the improperly admitted evidence; (3) the importance of the wrongly admitted testimony; and (4) whether such evidence was cumulative of other properly admitted evidence.” Zappulla v. New York, 391 F.3d 462, 468 (2d Cir.2004).

Applying these factors, it is clear that Gaikovich’s lay opinion testimony may have “substantially swayed,” Kotteakos, 328 U.S. at 765, 66 S.Ct. 1239, the jury’s verdict on Counts One through Five because it was not “unimportant in relation to everything else the jury considered on the issue in question.” The Government’s ease on those five counts rested principally on the following evidence: the recorded conversations during which Kap-lan, Sherman, and Medvedovsky told Gai-kovich to lie, Kaplan’s sham purchase of the Law Office, Kaplan’s disinterest in cases filed by the Law Office in his name, Kaplan’s absence from the Law Office, the arrangement between the Law Office and Prostaff and Medvedovsky, Kaplan’s receipt of over $70,000 for his limited services at the Law Office, and his false statements upon his arrest. Although the Government may, of course, prove its case exclusively with such circumstantial evidence, their case with respect to those counts was not strong.

As a result, and because Kaplan’s knowledge of the fraud was the central disputed issue in the case, Gaikovich’s lay opinion testimony was vitally important — just the sort of evidence that might well sway a jury confronted with a marginal circumstantial case. Our concern is heightened by the Government’s trial strategy with respect to the evidence; the Government repeatedly called the jury’s attention to Gaikovich’s lay opinion testimony. In its opening statement, the Government told the jury that “Gaikovich will recount for you conversations he had with Kaplan during the sale in which they discussed the fraudulent nature of the law practice.” In its closing, the Government reminded the jury that “Gaikovich explained to you that that conversation, and Kaplan’s comments, satisfied Gaikovich that Kaplan understood all about the fraud,” and that “[w]e know ... from the testimony of the witnesses, that Solomon Kaplan knew, undeniably knew, about the fraud at the law firm.” In rebuttal, the Government stated that “[Kaplan] essentially admitted to Gaiko-vich that he knew what was going on at this law office. That was how Gaikovich understood what he said. It is not his opinion. He was there. He had the conversation.” And, finally, we observe that this evidence was unique and thus was not cumulative of properly admitted evidence.

Because we cannot, in light of the say with fair assurance that lay opinion testimony did not sway[ ]" the jury's verdict as to Counts One, Two, Three, Four, and Five, we conclude that this error was not and therefore vacate Kaplan's on those counts.

II. Counts Six and Seven: Interfering with the Investigation

We turn next to Kaplan's challenging his conviction for with the investigation by tampering with a witness (Count Six) and giving false statements to the FBI (Count Seven). He argues on appea', among other things, that his conviction on those two counts must be vacated due to the district court's witness tampering and conscious avoidance jury instructions and a constructive amendment of, or prejudicial variance from, the as clarified by a bill of particulars. We address his arguments in turn.

A. Witness tampering jury

Kaplan contends that the district court's jury instruction on Count Six, alleging tampering, is deficient in light of the Supreme Court's decision in Arthur Andersen LLP v. United States, 544 U.S. 696, 125 S.Ct. 2129, 161 L.Ed.2d 1008 (2005), decided after his conviction.

Kaplan made no objection to this aspect of the jury instruction. Therefore, we his argument under the plain error standard of Fed.R.Crim.P. 52(b). We have ruled that when the claim of error derives from a supervening decision a settled rule of law in the Circuit, as it does here, the claimed error should be assessed under a standard of "modified plain-error." See United States v. Viola, 35 F.3d 37, 41-44 (2d Cir.1994). Ordinarily, the defendant asserting plain error bears the burden of persuasion as to but under our "modified plain-error" review, the Government bears that burden. See id. at 41-42. In this case it makes no difference whether the standard applied is the conventional or the modified "plain error" standard because any error in the charge was inconsequential and did not rise to the level of either standard.

Count Six alleges violations of 18 U.S.C. §~ 1512(b)(1) and (b)(3), which provide that:

(b) Whoever knowingly uses threatens or corruptly persuades another person, or attempts to do so, or engages in misleading conduct toward another person, with intent to-
(1) influence, delay or prevent the of any person in an official proceeding;
(3) hinder, delay, or prevent the communication to a law enforcement officer or judge of the United States of information relating to the commission or possible commission of a Federal offense or a violation of conditions of probation, supervised release, parole, or release pending judicial proceedings;

shall be fined under this title or imprisoned not more than ten years, or both.

In Arthur Andersen, the Supreme Court found the district court’s instructions were deficient in two respects — they failed to convey the requirements of 18 U.S.C. § 1512(b)(2) of mens rea and nexus to an official proceeding. As to the first of these deficiencies, the district court in Arthur Andersen instructed the jury that it could convict Arthur Andersen of witness tampering in relation to an official proceeding if it found that Arthur Andersen intended to “subvert, undermine, or impede” governmental factfinding by suggesting to its employees that they enforce the document retention policy, and that “even if [Arthur Andersen] honestly and sincerely believed that its conduct was lawful, you may find [Arthur Andersen] guilty.” 544 U.S. at 706, 125 S.Ct. 2129. These instructions, the Supreme Court held, did not properly convey the mens rea required for a violation of 18 U.S.C. § 1512(b) — they “diluted the meaning of ‘corruptly’ so that it covered innocent conduct,” id. — because “[o]nly persons conscious of wrongdoing can be said to ‘knowingly ... corruptly persuad[e],’ ” id. (emphasis supplied).

As to the second deficiency — regarding the so-called “nexus requirement” — the Supreme Court in Arthur Andersen found the instructions deficient because “[a] ‘knowingly ... corrupft] persuadefr]’ cannot be someone who persuades others to shred documents under a document retention policy when he does not have in contemplation any particular official proceeding in which those documents might be material,” id. at 708, 125 S.Ct. 2129, and the district court’s instructions “led the jury to believe that it did not have to find any nexus between the ‘persua[sion]’ to destroy documents and any particular proceeding,” id. at 707, 125 S.Ct. 2129. Cf. United States v. Arthur Andersen, 374 F.3d 281, 298 n. 32 (5th Cir.2004) (reciting district court’s charge in Arthur Andersen defining “official proceeding”). Instead, a “knowingly ... corrupt persuader” must believe that his actions are likely to affect a particular, existing or foreseeable official proceeding. See Arthur Andersen, 544 U.S. at 708, 125 S.Ct. 2129; see also United States v. Quattrone, 441 F.3d 153, 181 (2d Cir.2006) (holding that Arthur Andersen requires that there be “some nexus between the effort to tamper ... pertaining to the relevant proceeding and awareness that such conduct was likely to affect the proceeding”).

Kaplan contends that the district court’s instructions in this ease are similarly deficient. The district court instructed the jury on Count Six as follows:

The first element the government must prove is that the defendant corruptly persuaded a person, or attempted to do so.
The word “corruptly” simply means “having an improper purpose. An intent to subvert or undermine the fact-finding ability of an official proceeding is an improper purpose ...”
The second element that the government must prove is that the defendant acted knowingly and with the specific intent to influence the testimony of another person in an official federal proceeding.
An act is done “knowingly” if it is done voluntarily and intentionally and not because of mistake or accident.
By specific intent, I meant that the defendant must have acted knowingly and with the unlawful intent to influence the testimony of another person in an official federal proceeding; or to hinder, delay or prevent the communication to a federal law enforcement officer or judge information relating to the commission or possible commission of a federal offense.

JA 287-88.

Applying the lessons of Arthur Andersen, we find that the charge adequately conveyed the statute’s mens rea requirement. Viewing the charge as whole, the district court conveyed the substantial equivalent of Arthur Andersen’s holding that the defendant must be “conscious of wrongdoing” by instructing the jury to convict only if it found that the defendant acted with an “improper purpose” and “acted knowingly and with the unlawful intent to influence the testimony.” We note that the Supreme Court in Arthur Andersen expressly faulted the jury instructions in that case for (1) specifying that the jury could convict if it found that Arthur Andersen intended to “subvert, undermine, or impede ” (emphasis added) because “ ‘impede’ has broader connotations than ‘subvert’ or even ‘undermine,’ and many of these connotations do not incorporate any ‘corruptness’ at all,” 544 U.S. at 706-07, 125 S.Ct. 2129 (brackets omitted), and (2) instructing the jury that “even if [Arthur Andersen] honestly and sincerely believed that its conduct was lawful, you may find [Arthur Andersen] guilty,” id. at 706, 125 S.Ct. 2129. The district court here charged nothing of the sort. To convict, the jury had to find that Kaplan acted with an “improper purpose” and with “unlawful intent.”

Second, with regard to Kaplan’s argument that the district court’s instructions were erroneous in light of Arthur Andersen’s discussion of the statute’s nexus requirement, we note first that the charges in Arthur Andersen were brought under two clauses of § 1512(b)(2), both of which explicitly include as an element that the obstruction or tampering relate to an “official proceeding,” see id. at 702, 125 S.Ct. 2129. Kaplan, however, was charged under §§ 1512(b)(1) and (3). With respect to § 1512(b)(3), it is unclear whether Arthur Andersen’s nexus requirement is applicable because § 1512(b)(3) does not explicitly refer to an “official proceeding.” See United States v. Byrne, 435 F.3d 16, 23-25 (1st Cir.2006). We need not decide this issue, because even if the nexus requirement is applicable to prosecutions under § 1512(b)(3), and the district court’s instructions under § 1512(b)(3) were erroneous for failure to discuss nexus, any such error was harmless in the circumstances of this case for the reasons discussed below.

Kaplan was, as noted, also charged under § 1512(b)(1), which does contain an explicitly stated element of an “official proceeding.” The jury instructions on this charge undoubtedly needed to comply with the nexus requirement discussed in Arthur Andersen. The district court instructed the jury that the government “must prove” that the defendant acted “with the specific intent to influence the testimony of another person in an official federal proceeding.” The instructions did not identify the official proceeding. In view of the Supreme Court’s discussion in Arthur Andersen, 544 U.S. at 707, 125 S.Ct. 2129 (finding the instructions infirm because they led the jury to believe that it did not have to find any nexus between the “persuasion” to destroy documents and any “particular proceeding”), it would surely have been more prudent, even where the evidence only points to one federal proceeding, for the district judge to identify the “particular” federal proceeding that the defendant intended to obstruct. We need not decide whether the failure to do so in this case was error, because as we note below, it was harmless in any event.

Furthermore, in Quattrone, 441 F.3d at 153, the district court, following the Supreme Court’s formulation of the nexus requirement in United States v. Aguilar, 515 U.S. 593, 599, 115 S.Ct. 2357, 132 L.Ed.2d 520 (1995), described nexus to the jury as “some relationship in time, causation or logic, between the defendant’s actions and the grand jury proceeding so that his action or actions may be said to have the natural and probable effect of interfering with that proceeding.” Quattrone, 441 F.3d at 177 n. 24. We ruled that this instruction accurately described the nexus requirement. Id. at 178. The instructions given below did not contain this language, or its reasonable equivalent. In that regard, they were deficient.

Nevertheless, Kaplan is not entitled to reversal of his conviction on Count Six. Despite this error and other arguable deficiencies in the charge on nexus, any deficiencies were harmless in the particular circumstances of the case and did not amount to plain error. Unlike the defendants in Arthur Andersen and Quattrone, who, in accordance with a routine file purging policy, had urged destruction of documents that they may not have known were relevant to any “particular proceeding,” Arthur Andersen, 544 U.S. at 707, 125 S.Ct. 2129, Kaplan, according to the government’s evidence, directly participated in an effort to influence Gaikovich’s testimony. In his summation, Kaplan implicitly conceded that, if the jury found that he urged Gaikovich to testify falsely, it was with respect to either the state disciplinary proceeding or the federal criminal proceeding. Kaplan argued that the jury should find that any such effort related to the state disciplinary proceeding, and was therefore not covered by § 1512. His argument, however, implicitly conceded that, if the jury rejected his contention that his efforts related to the disciplinary proceeding, the efforts “relat[ed] in time, causation or logic,” Quattrone, 441 F.3d at 177 n. 24, to Gaikovich’s federal criminal proceeding. Accordingly, the court’s failure to explain in full an element that Kaplan had essentially conceded was harmless.

In sum, any error in the jury instructions was harmless and does not meet the standard of plain error. We reject Kap-lan’s argument that we must vacate his conviction on Count Six.

B. Conscious avoidance jury instruction

Kaplan contends that it was error for the district court to give a conscious avoidance charge on Count Six. We agree but conclude that the error was harmless.

An instruction on conscious avoidance is proper 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.” Quattrone, 441 F.3d at 181 (internal citations omitted). As for the second of these requirements, a factual predicate exists when “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.” Id. (internal quotation marks omitted).

Evidence sufficient to find actual knowledge does not necessarily constitute evidence sufficient to find conscious avoidance. See United States v. Ferrarini, 219 F.3d 145, 157 (2d Cir.2000) ("The evidence shows that [defendant] actually knew of the frauds; it is not sufficient to permit a finding that he consciously avoided them. The fact that a jury the evidence-find actual knowledge does not mean that it can also find avoidance."). Because the only evidence indicates that Kaplan had actual knowledge of the witness there was no factual predicate for a conscious avoidance charge on Count Six, and it was error for the district court to give it.

Nevertheless, we find that the error was harmless because there was overwhelming evidence of Kaplan's actual knowledge and direct involvement in the witness see ante at 116. See Quattrone, 441 F.3d at 181 (quoting Ferravini, 219 F.3d at 154) ("But an erroneously given conscious avoidance instruction constitutes harmless error if the jury was charged on actual knowledge and there was `overwhelming evidence' to support a finding that the defendant instead possessed actual of the fact at issue.").

C. Variance between bill of and proof

Finally, Kaplan argues that the Government's proof at trial constructively amended, or prejudicially varied from, Count Seven of the indictment, as clarified in a bill of particulars. We disagree.

Count Seven, which arises from the statements that Kaplan made to the FBI after his arrest, alleges that Kaplan "made materially false, fictitious, and fraudulent statements and representations, to wit, KAPLAN falsely informed an agent of the Federal Bureau of Investigation about the circumstances surrounding his purchase of a law practice from [Galkovich]." JA 17-18. These false statements were disclosed to Kaplan in an FBI report prior to trial. In response to Kaplan's request for a bill of particulars, the Government stated that "the statements contained in paragraphs three and four on page one of Kaplan's post-arrest statement form the basis of Count Seven."

At trial, the Government offered ampi proof of numerous false statements. I summation, the Government principall contended that seven statements were false: Kaplan told the FBI that (1) he was introduced to Galkovich and Medvedovsky by a Vladimir Scheckman; (2) "he received a thousand cases from Galkovich as a re~ sult of the sale;" (3) the Law Office was sold to him for $120,000; (4) he did not recall who gave him the $20,000 check used to purchase the Law Office; (5) he was assured by Medvedovsky and that the charges against Galkovich "were nothing [because] Galkovich had done nothing wrong"; (6) he "just stopped by" the January 8 meeting between lvledvedovsky, and Sherman without planning in advance to participate; and (7) he did not take the conversation on January 8 "very seriously" and that they were just "joking around." Only one of these statements-concerning how Kaplan had met Galkovich and Medvedovsky-was specified in the bill of particulars.

This does not constitute a constructive amendment of the indictment. "To prevail on a constructive amendment claim, a defendant must demonstrate that either the proof at trial or the trial court's jury instructions so altered an essential element of the charge that, upon review, it is uncertain whether the defendant was convicted of conduct that was the subject of the grand jury's indictment." United States v. Salmonese, 352 F.3d 608, 620 (2d Cir.2003). Here, because the Government proved the essential elements of the crime charged in Count Seven-albeit with dif~ ferent proof, as indicated above-the indictment was not constructively amended. See, e.g., United States v. Wallace, 59 F.3d 333, 337 (2d Cir.1995) (holding that no constructive amendment occurs "where a generally framed indictment encompasses the specific legal theory or evidence used at trial").

A variance, on the other hand, occurs when the charging terms remain unaltered but the facts proven at trial differ from those alleged in the indictment or bill of particulars. See United States v. Dupre, 462 F.3d 131, 140 (2d Cir.2006); United States v. Glaze, 313 F.2d 757, 759 (2d Cir.1963) (holding that once the Government has responded with a bill of particulars, it is "strictly limited to proving what it has set forth in it."). However, we will reverse on account of a variance only if it prejudices the defendant by infringing on the "substantial rights" that indictments exist to protect-"to inform an accused of the charges against him so that he may prepare his defense and to avoid double jeopardy." Dupre, 462 F.3d at 140; see also Glaze, 313 F.2d at 759 ("it is well settled that a variance between the proof and the bill of particulars is not grounds for reversal unless the [defendant] is prejudiced by the variance.").

Assuming arguendo there was a variance, we detect no prejudice to Kap-lan's ability to prepare his defense or interpose a defense of double jeopardy. MThere the defendant has notice of the "core of criminality" to be proven at trial, we have permitted "significant flexibility" in proof without finding prejudice. See United States v. LaSpina, 299 F.3d 165, 181-82 (2d Cir.2002); see also Salmonese, 352 F.3d at 621 ("A defendant cannot demonstrate that he has been prejudiced by a variance where the pleading and the proof substantially correspond .. . ."). For example, in Dnpre, we concluded that the proof of a wire transfer other than that specified in the indictment was a variance but did not prejudice the defendant because she was not surprised by the proof, 462 F.3d at 141-42, and suffered no risk of double jeopardy, id. at 143 n. 12. Additionally, we have routinely found that no prejudice results from a variance between overt acts charged in an indictment and those proved at trial. See, e.g., Frank, 156 F.3d at 337; LaSpina, 299 F.3d at 182-83; Salmonese, 352 F.3d at 622 ("In this case, because [the][i]ndictment gave [defendant] fair and adequate notice that the conspiratorial scheme achieved its ultimate economic purpose through the conspirators' multiple sales of stripped securities and their receipt of proceeds through June 1996, [defendant] cannot show that he was prejudiced by proof of a few uncharged proceed receipts after May 8, 1996.").

The alleged variance here in issue similarly does not justify reversal. Before trial, the government gave Kaplan the FBI report detailing all of his statements. Count Seven of the indictment was broadly-framed, giving Kaplan notice of the “core of criminality” to be proven at trial. And Kaplan’s counsel did not object to the admission of the false statements not specified in the bill of particulars, nor did he request a continuance when they were introduced. There is therefore no indication in the record that the evidence adduced at trial unfairly surprised Kaplan, exposed him to a risk of double jeopardy, or unfairly prejudiced him in any other-way. The variance did not create error, much less the plain error that arguably was required to be shown as a result of Kaplan’s failure to object.

We therefore reject Kaplan’s argument that the Government’s proof at trial constructively amended, or prejudicially varied from, Count Seven of the indictment.

CONCLUSION

We have considered all of Kaplan’s remaining arguments on appeal and find them to be without merit. For the reasons set forth above, the judgment of conviction is vacated as to' Counts One, Two, Three, Four, and Five, and affirmed as to Counts Six and Seven. The case is remanded for further proceedings consistent with this opinion. 
      
      . The indictment alleged that the objects of the conspiracy were mail fraud, wire fraud, making false statements relating to health care matters, health care fraud, and witness tampering.
     
      
      . The two colloquies quoted above are hereafter frequently referred to as "Gaikovich’s lay opinion testimony.”
     
      
      . In 2000, after Rea was decided, Rule 701 was amended to include 701(c).
     
      
      . As a result of this conclusion, we need not evaluate whether the error in admitting Gal-kovich’s testimony regarding others’ knowledge was harmless.
     
      
      . We note briefly that to the extent Kaplan~ argues that the evidentiary errors discussed in Section I compel us to vacate his conviction on Counts Six and Seven as well, we reject his argument. As to Counts Six and Seven, the evidentiary errors were harmless because the improper evidence was at most tangential to the theory of Counts Six and Seven. The Government's case rested on substantial evidence, including tape recordings of Kaplan's participation in witness Kaplan's post-arrest statement, and by the FBI agent who took Kaplan's post-arrest statement, to support his Moreover, the Government did not em~ phasize the testimony in issue in urging the jury to convict Kaplan of these two counts.
     
      
      . The Government argues that the Supreme Court's decision in Johnson v. United States, 520 U.S. 461, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997), requires that this Court abandon its "modified plain-error" test. We need not address the merits of this argument because we find that, even under the modified plain error test, Kaplan is not entitled to have his witness tampering conviction overturned.
     
      
      . We reject Kaplan's argument that it was error for the district court to give a conscious avoidance charge when the government actual knowledge in the alternative. we noted in Fenarini that evidence sufficient to find actual knowledge does not necessarily establish a factual predicate for conscious avoidance, 219 F.3d at 157, we have held that a conscious avoidance charge is "not inappropriate merely because the has primarily attempted to prove that the defendant had actual knowledge, while urging in the alternative that if the defendant lacked such knowledge it was only because he had studiously sought to avoid knowing what was plain," United States v. Hopkins, 53 F.3d 533, 542 (2d Cir.1995). So long as the Government can establish a factu-al predicate for conscious avoidance, it is free to argue alternative theories of conscious avoidance and actual knowledge.
     
      
      . On appeal, Kaplan does not challenge the sufficiency of this evidence to support his conviction on Count Seven.
     