
612 F.3d 1107 (2010)
UNITED STATES of America, Plaintiff-Appellee,
v.
Tammy A. THOMAS, Defendant-Appellant.
No. 08-10450.
United States Court of Appeals, Ninth Circuit.
Argued and Submitted December 7, 2009.
Filed July 22, 2010.
*1110 Ethan A. Balogh, Coleman & Balogh LLP, San Francisco, CA, for the defendant-appellant.
Laurie Kloster Gray, Assistant U.S. Attorney, San Francisco, CA, for the plaintiff-appellee.
Before A. WALLACE TASHIMA, SUSAN P. GRABER, and JAY S. BYBEE, Circuit Judges.

OPINION
BYBEE, Circuit Judge:
Defendant-Appellant Tammy Thomas, a former professional cyclist, challenges her convictions, after a jury trial, of three counts of perjury under 18 U.S.C. § 1623(a) and one count of obstruction of justice under 18 U.S.C. § 1503. We affirm.

I
Thomas was prosecuted for statements she made at a November 6, 2003, appearance before a Northern District of California grand jury that was investigating the distribution of anabolic steroids to professional athletes and money laundering by persons affiliated with Burlingame, California-based BALCO Laboratories. The investigation into BALCO began in the summer of 2002, when the IRS Criminal Investigation Division began looking into allegations that BALCO and its principal, Victor Conte, were distributing illegal performance-enhancing drugs and were laundering profits from those sales. During the first part of the investigation, the IRS obtained and analyzed financial records. Later, investigators, including IRS Special Agent Jeff Novitzky, seized items such as needle wrappers and used syringes from BALCO's trash. These were later found to contain steroids.
In the spring of 2003, investigators uncovered an email exchange between Conte and a track-and-field coach in Greece referencing an individual named Patrick Arnold as "the clear man." In August 2003, the United States Anti-Doping Agency (USADA) sent a fluid-filled syringe to Dr. Don Catlin, the director of a UCLA laboratory that tested urine specimens obtained by USADA. The syringe, which had been sent to USADA by a track coach, was found to contain tetrahydrogestrinone (THG), a previously unidentified substance. USADA documented five athletes who had tested positive for THG and prepared to publicize these positive tests. At the behest of Dr. Catlin, USADA contacted Novitzky, and criminal investigators obtained search warrants ahead of public disclosure of these positive tests.
In September 2003, law enforcement officers executed search warrants at several locations, including BALCO and Conte's home. At the time the warrants were served, both Conte and James Valente, BALCO's vice president, voluntarily answered investigators' questions, telling agents that BALCO distributed two substances: "the clear," a substance with anabolic properties, so named because it had been undetectable by anti-doping testers, and "the cream," a testosterone-based substance. Conte and Valente independently told investigators that "the clear" came from Patrick Arnold. After the search, both Conte and Valente obtained counsel and refused to cooperate further.
The BALCO searches yielded documentary evidence indicating the distribution of illegal performance-enhancing drugs, including doping calendars, steroid tests, invoices with athletes' names, and other documents suggesting a relationship among BALCO, Conte, and numerous athletes in *1111 a variety of sports. Among the documents seized in the BALCO search was a May 1, 2002 fax from Arnold's business to BALCO. This fax included a document from the United States Olympic Committee Athletic Center referencing a urine sample collected from Tammy Thomas on March 14, 2002, which had tested positive for norbolethone, an anabolic steroid.[1]
The BALCO search also turned up numerous emails, including the following message sent from Arnold to Conte on May 1, 2002:
I know the girl who they just snagged for norbolethone. It is not the same girl I was helping in the Olympics, but a cyclist girl. I saw her tests and everything. She is trying to fight it, and I am advising her technically on how to do it. Needless to say, if you know anyone taking the stuff who is taking subject to testing, then tell them to stop.
The May 1, 2002 email also included an earlier exchange between the two men regarding Dr. Catlin's discovery of norbolethone.
After the BALCO search, approximately thirty athletes, including Tammy Thomas, were subpoenaed to testify before a Northern District of California grand jury that was investigating the illegal distribution of steroids. Most of these athletes had been linked to BALCO in some way. Thomas, on the other hand, appeared to have a direct link to Arnold, the manufacturer of norbolethone and the apparent source for THG, and was subpoenaed because investigators believed she would have material information linking Arnold to Conte.
Thomas appeared before the grand jury on November 6, 2003. By the time of this appearance, Thomas had tested positive for norbolethone in urine samples collected on August 30, 2001, March 14, 2002, and April 10, 2002. The August 2001 and April 2002 urine samples also tested positive for THG. At the grand jury hearing, Thomas testified pursuant to an immunity agreement in which she acknowledged that she understood that her statements at the grand jury could not be used against her in a subsequent criminal proceeding as long as they were truthful, but that she could be subject to criminal or civil liability for answering untruthfully. Thomas was placed under oath and told she was not a subject or target of the grand jury investigation and that she could consult with counsel outside the grand jury room at any point in her testimony if she wished to do so.
In response to questions, Thomas testified that she received the legal supplement 1-AD, but nothing else, from Patrick Arnold. She denied ever getting any other "products" from Arnold, ever "tak[ing] anything that Arnold gave [her]," and "ever tak[ing] anabolic steroids."
In February 2004, the first BALCO grand jury handed down a forty-two count indictment against Conte, Valente, Greg Anderson (a BALCO-linked trainer), and Remi Korchemny (a BALCO-linked track *1112 and field coach). Among other things, the indictment alleged a conspiracy to distribute anabolic steroids. In July 2005, all four defendants pleaded guilty. According to the government, the failure to indict Arnold along with Conte, Valente, Anderson, and Korchemny in 2004 was attributable at least in part to Thomas's grand jury testimony, because Thomas's denials before the grand jury led investigating agents to think that they had "lost... the opportunity to have the one witness with direct knowledge and direct contact with Patrick Arnold in the early stages of th[eir] investigation."
In August 2005, more than a year after the initial BALCO defendants were indicted, a second grand jury in the Northern District of California began investigating Arnold and his activities related to BALCO. In November 2005, this grand jury handed down a three-count indictment in the Northern District of California against Arnold, charging him with, among other things, conspiring with Conte to distribute norbolethone and THG. On May 1, 2006, Arnold pleaded guilty.
On December 14, 2006, a third Northern District of California grand jury indicted Thomas on six counts relating to Thomas's November 6, 2003 grand jury testimony. Counts one through five of the indictment against Thomas alleged material false declarations in violation of 18 U.S.C. § 1623, while count six of the indictment alleged that Thomas obstructed justice in violation of 18 U.S.C. § 1503 "by knowingly giving Grand Jury testimony that was intentionally evasive, false, and misleading, including but not limited to the false statements made by the defendant as charged in Counts One through Five of this indictment."
Arnold, Novitzky, Catlin, and several other witnesses testified for the government at Thomas's trial. Viewing the evidence in the light most favorable to the jury's verdict, see United States v. Nevils, 598 F.3d 1158, 1161 (9th Cir.2010) (en banc), the following facts were established at Thomas's trial: Arnold successfully manufactured norbolethone starting in 1998 and sent norbolethone to Conte on five to seven occasions from approximately 2000 to July 2001. Later, Arnold also provided Conte with THG, a "designer steroid" that Arnold created and that was unavailable from any source but Arnold.[2] Arnold's goal in developing THG was to create a substance with steroid-like effects that would be undetectable by drug-testing authorities. His efforts were successful, at least for a whileTHG was distributed by Conte to numerous elite athletes and nicknamed "the clear" because it was, for a time, undetectable to drug testers. However, while Conte, through BALCO, provided THG to many athletes, Thomas received THG and norbolethone from Arnold directly. Like THG, norbolethone was intended to be undetectable and, according to Arnold, that was why Thomas used it.
Thomas first contacted Arnold in 2000 in an email in which she introduced herself as a competitive cyclist who was interested in any products that might help her performance. Arnold replied that he could help, and the two subsequently talked on the telephone approximately five times. During these conversations, Arnold discussed with Thomas his products, her performance, and his recommendations for using his products. Initially, Arnold provided Thomas with nutritional supplements and his best-selling legal supplement, 1A-D. Based on his discussions with Thomas, it soon became clear to Arnold that Thomas wanted steroids. Arnold's practice was to "never" send THG or norbolethone to anyone "without an inquiry or discussion and *1113 an agreement" as to what was being received. Thus, according to Arnold, when he personally sent THG to Thomas at least once, and perhaps two to three more times, in the spring of 2002, "Thomas understood full well that [THG] was undetectable and that was its intention."
In addition to Arnold's personally sending THG to Thomas, Arnold's then-live-in girlfriend Kelcey Dalton sent norbolethone to Thomas several times between 2000 and 2002. Dalton spoke on the phone with Thomas dozens of times during a three-to-four month period, and she "facilitated" shipping drugs to Thomas. When Thomas needed more of something, Dalton relayed that request to Arnold. Dalton never sent either norbolethone or THG without consulting Arnold or without Arnold's knowledge.
During their conversations, Dalton and Thomas compared notes on their strength training. Dalton described Thomas as having a "steroid voice." Based on her conversations with Thomas, Dalton understood that "Tammy wanted to win medals in her sport, sprint cycling. She wanted the norbolethone and the THG in order to gain the athletic edge necessary to win those medals." Dalton and Thomas used the word norbolethone during their conversations, and Dalton testified that Thomas understood that norbolethone was not listed on any of the doping agencies' testing lists and thus would allow her to use steroids without getting caught. Dalton testified: "[A]t no time was ... Tammy Thomas not aware that she was taking undetectable steroids. These weren't pet names or game names. It was very clear in our conversations between Tammy and I what she was taking."
After Thomas's positive drug test for norbolethone in March 2002, Arnold spoke with Thomas and expressed his disappointment with her because he had previously told her on several occasions to stop taking norbolethone, which was a "hot potato" for testers. To deal with Thomas's situation, Arnold and Thomas "brainstormed" a "cover story""a false story to cover up the truth." This scheme was reflected in the text of the email Arnold sent to Conte stating that Arnold "kn[e]w the girl who they just snagged for norbolethone" and that "[s]he is trying to fight it, and I am advising her technically on how to do it."
Shortly after the execution of the BALCO search warrants, Thomas and Arnold again exchanged emails. Thomas used the name "Ann Frank" in corresponding with Arnold and wrote:
I have no idea what this is in reference to, but a federal agent tracked me down today and subpoenaed me to testify in front of a grand jury in San Francisco. This is totally bizarre because I don't really know anyone in San Francisco.... This is really weird since I can't think of anyone except maybe V.C. out in San Fran. Have you heard anything else on his case?
Arnold responded, "[i]f you have no connections to the guy in S.F., then it makes no sense why they would subpoena you," to which Thomas replied:
I spoke with feds briefly today. They said they found a record with my name at V.C. and they said they thought V.C. and P.A. were giving steroids to athletes. I was never one of V.C.'s athletes that he helped. I only contacted him after I was notified of my hearing for the ban.... He was supposed to clean house after it was all over, but, apparently, did not.... It would be helpful if I knew what documents V.C. still had that the feds got their hands on, but I really don't know anything about anything.
At the close of Thomas's perjury trial, both parties submitted proposed jury instructions. Thomas sought an explicit instruction *1114 regarding her so-called "literal truth" theory of defense, but the district court refused to give the jury Thomas's proposed literal truth instruction or any separate instruction on "literal truth."
On April 4, 2008, the jury acquitted Thomas on counts two and five of her indictment and convicted Thomas on counts one, three, four, and six. In a special verdict form, the jury indicated that its verdict on count sixthe obstruction of justice countrested on the statements alleged in counts one and three, plus two additional statements appended to the indictment. On October 10, 2008, Thomas was sentenced to five years' probation with six months' monitored home detention, 500 hours of community service, and a mandatory $400 special assessment. Thomas timely appealed.[3]

II
Section 1623(a) of Title 18 makes it unlawful to knowingly make a false material declaration under oath before a grand jury. See United States v. McKenna, 327 F.3d 830, 838 (9th Cir.2003). Thomas challenges her convictions on counts one, three, and four of the superseding indictment, asserting that her answers were "literally truthful." Thomas's convictions rested on the following answers given under oath before the grand jury:
Q: Did you everbesides this one instance of getting the 1-AD from Mr. Arnold, did you ever get any other services from Mr. Arnold or products?
A: No, no other products.

(Count one of superseding indictment).
Q: Did you take anything that Patrick Arnold gave you?
A: No.

(Count three of superseding indictment).
Q: Now, let me ask, as you sit here now, and before this grand jury "today, have you ever taken anabolic steroids"?
A: No.

(Count four of superseding indictment).
Thomas argues that her statements before the grand jury were in fact "literally true" and that, therefore, no reasonable jury could have convicted Thomas of making material false statements. In the alternative, Thomas argues that the district court erred in refusing to submit her proposed "literal truth" instruction to the jury and that she is entitled to a new trial on this basis. We reject both of Thomas's "literal truth" arguments.

A
Thomas's contention that she is entitled to a judgment of acquittal because her statements to the grand jury allegedly were "literally true" stems from the Supreme Court's decision in Bronston v. United States, 409 U.S. 352, 93 S.Ct. 595, 34 L.Ed.2d 568 (1973).[4]See United States v. Cowley, 720 F.2d 1037, 1041-42 (9th Cir.1983) (applying Bronston to a prosecution under 18 U.S.C. § 1623). In Bronston, the Court "consider[ed] a narrow but important question in the application of *1115 the federal perjury statute, 18 U.S.C. § 1621: whether a witness may be convicted of perjury for an answer, under oath, that is literally true but not responsive to the question asked and arguably misleading by negative implication." 409 U.S. at 352-53, 93 S.Ct. 595 (footnote omitted). Defendant Samuel Bronston was indicted for perjury based on the following exchange that occurred under oath at a bankruptcy hearing:
Q: Do you have any bank accounts in Swiss banks, Mr. Bronston?
A: No, sir.
Q: Have you ever?
A: The company had an account there for about six months, in Zurich.
Q: Have you any nominees who have bank accounts in Swiss banks?
A: No, sir.
Q: Have you ever?
A: No, sir.
Id. at 354, 93 S.Ct. 595. It was "undisputed that for a period of nearly five years... [Bronston] had a personal bank account at the International Credit Bank in Geneva, Switzerland, into which he made deposits and upon which he drew checks." Id. It was "likewise undisputed that [Bronston's] answers were literally truthful." Id. "The Government's prosecution for perjury went forward on the theory that in order to mislead his questioner, [Bronston] answered the second question with literal truthfulness but unresponsively addressed his answer to the company's assets and not to his ownthereby implying that he had no personal Swiss bank account at the relevant time." Id. at 355, 93 S.Ct. 595.
At Bronston's trial, the district court instructed the jury that Bronston "could... be convicted if he gave an answer `not literally false but when considered in the context in which it was given, nevertheless constitute[d] a false statement.'" Id. at 355, 93 S.Ct. 595 (alteration in original). Bronston was convicted of one count of perjury, and the Second Circuit affirmed, holding, among other things, that "an answer containing half of the truth which also constitutes a lie by negative implication, when the answer is intentionally given in place of the responsive answer called for by a proper question, is perjury." Id. at 356, 93 S.Ct. 595 (internal quotation marks omitted).
The Supreme Court reversed Bronston's conviction, reasoning that "[t]he words of the [perjury] statute confine the offense to the witness who `willfully ... states ... any material matter which he does not believe to be true,'" and that "the statute does not make it a criminal act for a witness to willfully state any material matter that implies any material matter that he does not believe to be true." Id. at 357-58, 93 S.Ct. 595 (alterations in original). The Court emphasized the "responsibility of the lawyer to probe.... If a witness evades, it is the lawyer's responsibility to recognize the evasion and to bring the witness back to the mark, to flush out the whole truth with the tools of adversary examination." Id. at 358-59, 93 S.Ct. 595. The Court also noted that it was "no answer to say that here the jury found that petitioner intended to mislead his examiner," as "[a] jury should not be permitted to engage in conjecture whether an unresponsive answer, true and complete on its face, was intended to mislead or divert the examiner." Id. at 359, 93 S.Ct. 595.
Although the Supreme Court's decision in Bronston underpins Thomas's "literal truth" defense, the case is not directly on point here. Bronston's allegedly perjurious statements were undisputedly literally true, id. at 354, 93 S.Ct. 595, which is not the case here. Fortunately, in the thirty-odd years since the Supreme Court's decision in Bronston, we have elaborated on and interpreted that decision, providing us with guidance on how Bronston applies to *1116 a case such as Thomas's, in which a defendant asserts that her answers were "literally true" but the government contests the assertion.
In United States v. Matthews, 589 F.2d 442 (9th Cir.1978), and United States v. Sainz, 772 F.2d 559 (9th Cir.1985), we addressed the relevance of Bronston to testimony as to which "literal truth" depended on the defendant's understanding of the questions he was asked. In both cases, we emphasized the importance of context to the Bronston inquiry. Thus, in Sainz, where the defendant raised the issue of literal truth, we explained that, "[i]n reviewing a perjury conviction, ... [o]ur central task is to determine whether the jury could conclude beyond a reasonable doubt that the defendant understood the question as did the government and that, so understood, the defendant's answer was false." 772 F.2d at 562 (citation and internal quotation marks omitted). We stressed that "[o]ur inquiry into the defendant's allegedly perjurious statement must begin with an appreciation of the context in which the statement was offered," because "[t]he practice of lifting statements uttered by a witness out of context can serve no useful purpose in advancing the truth-seeking role of the perjury statutes." Id. "Consequently, we must look to the context of the defendant's statement to determine whether the defendant and his questioner joined issue on a matter of material fact to which the defendant knowingly uttered a false declaration." Id.
Our decision in Matthews is factually on point with the case at bar. In Matthews, the defendant was convicted under 18 U.S.C. § 1623 for providing the following grand jury testimony:
Q: Do you know where Mr. Rasmussen got that money?
A: No.
589 F.2d at 443. The government contended that Matthews testified falsely because he did, in fact, know the source of Rasmussen's money. Matthews, however, argued that he was entitled to acquittal under Bronston because "the question that he [wa]s charged with answering falsely c[ould] be construed as an inquiry as to the person from whom Rasmussen physically received the money," and "the government... presented no evidence that the question so construed was answered falsely." Id. at 444. In response, "[t]he government assert[ed] that in the light of all circumstances it must have been clear to appellant that the question in issue sought to ascertain the source of the money," and that "[o]ther portions of appellant's testimony before the grand jury support[ed] the government's position." Id.
In our decision in Matthews, we began by distinguishing Bronston, stating:
In Bronston the answer given was perfectly true, but it was unresponsive to the question asked ....
This holding, however, is of no assistance here. The answer here was not unresponsive. It was a forthright "no." It was not, as in Bronston, a statement of fact the truth of which could be ascertained without reference to the question that elicited it. Here the answer cannot be separated from the question if the truth of the answer is to be determined. If the "no" is to have meaning it must be read to echo and negate the language of the question itself.... The problem presented here, which was not presented in Bronston, is that the question asked of appellant is asserted to be subject to two different meanings.
Id. (footnote omitted).
We then turned to the specifics of Matthews's literal truth argument: "If language in which a question is couched is plausibly subject to two interpretations, that language contains within it two different questions." Id. at 444. Matthews arguedas *1117 Thomas arguesthat "if the answer is to be proved false it must be proved false as to both questions." Id. Because Matthew's answer "was not proved false under the interpretation he advance[d]," he had not (in his view) "been proved guilty of perjury." Id.
We ultimately rejected Matthews's argument. We first noted that the argument "assume[d] that the interpretation advanced by [Matthews] [wa]s a plausible one[,] ... a question upon which we entertain[ed] great doubt." Id. at 444 n. 2. We then explained why Matthews could not prevail on his literal truth challenge to his perjury conviction:
Here ... the government's construction of the question was plausible and a jury issue was presented as to the defendant's understanding of the question. There is ample evidence from which the jury could conclude beyond a reasonable doubt that [Matthews] understood the question asked of him as pursuing the same inquiry ... that was explicitly made the subject of other questions asked in the course of the same interrogation. Indeed, we do not see how the question could rationally be given any other meaning. We conclude that appellant was not entitled to an acquittal as [a] matter of law; that the question of guilt remained one for the jury.
Id. at 445.
The "literal truth" framework outlined in Sainz and Matthews controls here. When a defendant claims that her allegedly perjured testimony was literally true based on her own purported understanding of the government's questions, the issue is "whether the jury could conclude beyond a reasonable doubt that the defendant understood the question as did the government and that, so understood, the defendant's answer was false." Sainz, 772 F.2d at 562 (internal quotation marks omitted); see also Matthews, 589 F.2d at 445. We now apply this framework to the three statements for which Thomas was convicted under 18 U.S.C. § 1623.

1
As to Thomas's conviction on count onebased on Thomas's response that she never received any "products" other than 1-AD from Patrick ArnoldThomas argues that her answer was literally truthful because she believed she received the products THG and norbolethone only from Dalton, Arnold's then-live-in-girlfriend, because Dalton was Thomas's primary phone contact in the Arnold household. This argument lacks merit, as there was ample testimony at Thomas's trial for a reasonable jury to conclude that Thomas had, indeed, "g[otten] any other [products besides 1-AD] from Mr. Arnold." Most importantly, Arnold testified that he "for sure" sent Thomas THGa "product" after Arnold and Dalton had split up. This direct testimony was bolstered by circumstantial evidence at Thomas's trial. For example, Arnold testified that Thomas first contacted him in 2000 and that the two spoke on the phone approximately five times after the initial contact. After Thomas tested positive for norbolethone, she and Arnold "brainstormed" a "cover story""a false story to cover up the truth"to deal with the positive test. Arnold and Thomas also exchanged emails regarding Thomas's receipt of a grand jury subpoena. The jury could reasonably have concluded from all this testimony that Thomas's grand jury testimony that she never received any "other products" from Arnold was not literally true.

2
Thomas was convicted on count three of the indictment for answering "No" to the question, "Did you take anything that Patrick Arnold gave you?" Thomas *1118 argues that her answer to this question was literally true because trial testimony established that all the products she received from Arnold were either bought or bargained-for, and thus nothing was "g[iven]" to Thomas in the sense that "give" can mean "to confer the ownership of without receiving a return." WEBSTER'S THIRD NEW INTERNATIONAL DICTIONARY 959 (1993).
"Give," however, is commonly used and understood in a variety of ways, many if not most of which do not require that an object be exchanged without consideration. Dictionary definitions of "give" include "to put into the possession of another for his use," id., "to yield possession of by way of exchange," id. at 960, and "to dispose of for a price," id. Thus, we commonly speak of "giving" gifts on certain holidays, although we may also describe this as an "exchange." And it is common usage to say that a doctor "gives" us a shot, even though we have paid the appropriate co-pay.
The Matthews case, in which we rejected a perjury defendant's literal truth claim where "the question asked of [the defendant wa]s asserted to be subject to two different meanings," 589 F.2d at 444, squarely controls here. In Matthews, we emphasized that where "the government's construction of the question [i]s plausible[,] a jury issue [i]s presented as to the defendant's understanding of the question," and where "[t]here is ample evidence from which the jury could conclude beyond a reasonable doubt that appellant understood the question asked of him as pursuing the same inquiry ... that was explicitly made the subject of other questions asked in the course of the same interrogation[,]... the question of guilt remain[s] one for the jury." Id. at 445; see also Sainz, 772 F.2d at 562 ("Our central task is to determine whether the jury could conclude beyond a reasonable doubt that the defendant understood the question as did the government and that, so understood, the defendant's answer was false." (internal quotation marks omitted)).
Nothing in Thomas's grand jury testimony suggests that either the government questioner or Thomas herself carefully distinguished between "gave" in the sense of "made a present" and "gave" in the sense of "put into the possession of another for h[er] use" or "yield[ed] possession of by way of exchange." Indeed, the grand jury transcript reveals that the government attorney questioning Thomas repeatedly used general verbs such as "get" and "give" to indicate acquisition through unspecified means. For example, directly before being asked whether she had ever taken anything Arnold "gave" her, Thomas was asked whether she ever "g[o]t an anabolic steroid from anybody in connection with[her] training up to the time of March 2002" and whether she "ever t[ook] one, at least knowingly, anyway." Thomas responded that in 1998 and 1999 she "had been given some testosterone without [her] knowledge." Thomas was then asked, "did Patrick Arnold ever give you anything that was a steroid," and then immediately asked whether she "t[ook] anything that Patrick Arnold gave [her]." There is simply no indication from the grand jury transcript that, in light of the tenor and context of the government attorney's questions and his use of the general verbs "get" and "give," Thomas understood the attorney to be asking only if Arnold had "made a gift" of substances such as THG and norbolethone to Thomas. Perhaps the government's questions could have been phrased to avoid any ambiguity, but the phrasing actually used was not only plausible, but well within the bounds of ordinary construction. A reasonable jury could conclude beyond a reasonable doubt that Thomas understood the question in count three of the indictment as the *1119 government understood it and that she answered falsely based on this understanding. Thus, Thomas is not entitled to a judgment of acquittal on count three. See Matthews, 589 F.2d at 445.

3
Thomas's final sufficiency challenge targets her conviction on count four for answering "No" to the question, "as you sit here now, and before this grand jury today, have you ever taken anabolic steroids?" Thomas claims that her answer to this question was literally true because, although Thomas unquestionably "took" THG and norbolethone, neither of those substances was specifically listed under the definition of "anabolic steroids" in 21 U.S.C. § 802(41)(A) (2003) ("The term `anabolic steroid' means any drug or hormonal substance, chemically and pharmacologically related to testosterone (other than estrogens, progestins, and corticosteroids) that promotes muscle growth, and includes [certain enumerated substances].") at the time of Thomas's grand jury testimony,[5] and the government failed to present any evidence that THG and norbolethone were not excluded "progestins."
Thomas's argument on this issue is unavailing. The government was not required to prove that THG and/or norbolethone were statutorily prohibited under 21 U.S.C. §§ 802(41)(A) and 841 at the time of Thomas's testimony, and no one had suggested during Thomas's questioning before the grand jury that "anabolic steroids" referred only to the statutory definition. Rather, the government was required to prove only that Thomas lied in stating that she had never "taken anabolic steroids." Under Sainz and Matthews, this means that the government had to prove that when Thomas testified that she had never "taken anabolic steroids," Thomas understood "anabolic steroids" to include THG and/or norbolethone. There was ample evidence at Thomas's trial for a jury reasonably to conclude that Thomas understood "anabolic steroids" to include THG and/or norbolethone during her November 2003 grand jury testimony. Three witnessesDr. Catlin, Arnold, and Daltonall testified, without contradiction, that both THG and norbolethone were anabolic steroids by their chemical structure, their pharmacological effects, and as a matter of common understanding. Arnold testified that Thomas contacted him because she wanted steroids, which is why Arnold sent her THG. Arnold also testified that "Thomas understood full well that this was undetectable and that was its intention." Dalton testified that both drugs Arnold supplied to ThomasTHG and norbolethonewere undetectable anabolic steroids and that Thomas was well aware of this fact.
In sum, there was sufficient evidence for the jury reasonably to conclude that Thomas did not offer literally true answers in the exchange charged in count four of the indictment.

B
Thomas also contends that the district court erred by refusing to provide to the jury a proposed jury instruction regarding literal truth. "Defendant's Proposed Jury Instruction Number 10" stated:
You are not permitted to convict the defendant of the charges in the indictment based on any testimony that was literally true. In other words, testimony that is literally true, even if it has an especially strong tendency to be misleading, cannot support a conviction for *1120 making material false statements to the grand jury or for obstructing justice. If you find that any answer to any question charged in the indictment is literally true, you must acquit the defendant of the charge presented by her literally truthful answer.
The district court declined to issue the above instruction, instead instructing the jurors as follows as to counts one, three, and four:
In order for the defendant to be found guilty of Count 1 [or 2, or 3, respectively], the government must prove each of the following elements beyond a reasonable doubt:
1. The defendant testified under oath before a grand jury;
2. The testimony described above was false;
3. The testimony described above was material to the grand jury before which she testified; and
4. The defendant knew that the testimony described above was false and was material to the grand jury before which she testified.
A statement was material if it had a natural tendency to influence, or was capable of influencing, the decision of the decisionmaking body to which it is addressed.
This instruction followed the applicable Ninth Circuit Model Instruction. See Ninth Circuit Model Criminal Jury Instruction 8.112 (False Declaration Before Grand Jury or Court, 18 U.S.C. § 1623).
"A defendant is entitled to have the judge instruct the jury on h[er] theory of defense, provided that it is supported by law and has some foundation in the evidence." United States v. Mason, 902 F.2d 1434, 1438 (9th Cir.1990). "A failure to give such instruction is reversible error; but it is not reversible error to reject a defendant's proposed instruction on h[er] theory of the case if other instructions, in their entirety, adequately cover that defense theory." Id. Thus, Thomas must satisfy three hurdles in order to prevail on her motion for a new trial based on the district court's refusal to give a separate jury instruction on literal truth.[6] First, Thomas must show that a literal truth jury instruction "ha[d] some foundation in evidence." United States v. Johnson, 459 F.3d 990, 992 (9th Cir.2006) (internal quotation marks omitted). Second, Thomas must show that her literal truth theory of defense "[wa]s supported by law." Id. at 995 (internal quotation marks omitted). Finally, even if Thomas's literal truth theory had an evidentiary foundation and was supported by law, Thomas must show that the district court committed reversible error by refusing to give a separate instruction on her theory where "other instructions, in their entirety, adequately cover that defense theory." Mason, 902 F.2d at 1438. We address each question in turn.

1
Before an instruction is required on Thomas's theory of literal truth, the theory must have had some foundation in the evidence presented at trial. We review for abuse of discretion whether there is a factual foundation for a proposed instruction. See Johnson, 459 F.3d at 992 n. 3.
*1121 We have described as "generous" the legal standard for whether a defendant's proposed instruction has evidentiary foundation. United States v. Kayser, 488 F.3d 1070, 1076 (9th Cir.2007). A defendant "need only show that there is evidence upon which the jury could rationally find for the defendant." Johnson, 459 F.3d at 993 (internal quotation marks omitted). "We have cautioned that a mere scintilla of evidence supporting the defendant's theory is not sufficient to warrant a defense instruction," but "we have also repeatedly stated that the defendant is entitled to h[er] proposed instruction even if h[er] evidence is weak, insufficient, inconsistent, or of doubtful credibility." Id. (internal quotation marks omitted). "[T]his standard protects the defendant's right to have questions of evidentiary weight and credibility resolved by the jury." Kayser, 488 F.3d at 1076.
Here, there was "some foundation in evidence" for Thomas to claim that her allegedly perjured statements were literally truthful, but misleading. Thomas's literal truth arguments with respect to counts one, three, and four of her indictment (recounted in some detail in Part II-A of this opinion) were not particularly persuasive, but neither were they completely implausible. Moreover, although the only evidence supporting Thomas's literal truth arguments was arguably "weak, insufficient, inconsistent," and "of doubtful credibility," the evidentiary foundation for Thomas's literal truth defense was still more than "a mere scintilla" and supported an instruction on literal truth under our precedent. Johnson, 459 F.3d at 993 (internal quotation marks omitted); see also id. at 994 (even though defendant's "story[wa]s weakly supported and suffer[ed] from various problems," defendant was entitled to jury instruction on his theory of defense "[i]n light of the low evidentiary threshold [defendant] must clear"). If the jury rejected, on credibility grounds or otherwise, large swaths of testimony by trial witnesses such as Novitzky, Arnold, Dalton, and the USADA testers, it rationally could have agreed with Thomas's literal truth defense.
"In light of the low evidentiary threshold [Thomas] must clear" in this inquiry, id. at 994, we reject the government's argument that there was no evidentiary foundation for the district court to issue a literal truth instruction.

2
We next consider whether Thomas's literal truth theory was "supported by law," an issue we review de novo. Id. at 995 & n. 8 (internal quotation marks omitted). Thomas's literal truth theoryand, indeed, her proposed jury instruction on literal truth (insofar as the instruction dealt with Thomas's perjury charges)closely resembled language from the Supreme Court's decision in Bronston. Compare Bronston, 409 U.S. at 352-53, 93 S.Ct. 595 ("We ... consider ... [an] important question in the application of the federal perjury statute: whether a witness may be convicted of perjury for an answer, under oath, that is literally true but not responsive to the question asked and arguably misleading by negative implication." (citation and footnote omitted)), with Defendant's Proposed Jury Instruction Number 10, supra at 10500 ("You are not permitted to convict the defendant of the charges in the indictment based on any testimony that was literally true. In other words, testimony that is literally true, even if it has an especially strong tendency to be misleading, cannot support a conviction for making material false statements to the grand jury ...."). We note that Thomas's proposed instruction deviated from Bronston in a potentially important respect: Bronston spoke of "arguably misleading" testimony, while Thomas's proposed instruction *1122 described testimony with "an especially strong tendency to be misleading." However, without approving the phrasing of Thomas's proposed instruction itself,[7] we are willing to assume that her literal truth theory of defense flows from Bronston and is therefore "supported by law."

3
Even though Thomas's literal truth theory had "some evidentiary foundation" and was "supported by law," the district court was not required to give a separate jury instruction on this theory "if other instructions, in their entirety, adequately cover[ed] that defense theory." Mason, 902 F.2d at 1438. "We review de novo the question of whether the district court's instructions adequately cover the defense theory." Id. We hold that the district court did not err by refusing to issue a separate instruction on literal truth.
Although we noted in Mason that "[e]xpressing the theory of the defense in an instruction that precisely defines that theory is far superior to reliance on the jury's ability to piece the theory together from various general instructions," we immediately qualified this statement by explaining that "[w]e ... look to the instructions as a whole and a refusal to give a proper specific instruction can be remedied by other instructions that cover the subject." Id. at 1441. "[I]n reviewing the instructions as a whole, we must consider how they will reasonably be understood by the jury in the context of the whole trial." Id.
The model jury instruction used by the district court explicitly required the jury to find, beyond a reasonable doubt, that "[t]he testimony described above was false" and further required the jury to find (again beyond a reasonable doubt) that "defendant knew that the testimony described above was false." (Emphasis added.) When a statement is proved to be false, and the witness making that statement knows it to be false, there is no logical possibility that this same statement could be "literally true." This is in contrast to a statement that is, instead of false, merely "misleading," "evasive," or "deceptive." Had the jury instructions described "misleading," "evasive," or "deceptive" testimony rather than knowingly false testimony, Thomas might well have been entitled to a separate instruction based on her literal truth theory. See Bronston, 409 U.S. at 353, 93 S.Ct. 595 (describing the testimony at issue as "literally true but not responsive to the question asked and arguably misleading by negative implication"). But the jury instructions at Thomas's trial did not refer to "misleading," "evasive," or "deceptive" testimony; they used the term "false," and thus "adequately cover[ed] th[e] defense theory" of literal truth. Mason, 902 F.2d at 1438.
We also note that defense counsel specifically argued in his closing argument that Thomas's allegedly perjurious statements were "literally true" and therefore were not false. This simply underscores our conclusion that "in the context of the whole trial" the jury "reasonably ... understood" *1123 the instructions on falsity to exclude statements that were literally true. Id. at 1441.
In short, it was therefore "not reversible error to reject ... defendant's proposed instruction on [t]his theory." Id. at 1438.

III
In addition to challenging her § 1623(a) convictions on the ground that her statements before the grand jury were "literally true," Thomas also argues that her statements were not material because they lacked a jurisdictional nexus to the Northern District of California. In the alternative, she argues that the district court erred in refusing to separately instruct the jury on her jurisdictional nexus defense. We hold that there was sufficient evidence for a rational jury to find the element of materiality and that the district court did not err in declining to instruct the jury on Thomas's jurisdictional nexus defense.

A
Thomas seeks a judgment of acquittal on her § 1623(a) convictions on counts one, three, and four on the ground that there was insufficient evidence for a reasonable jury to find that the grand jury statements underlying these counts were material. Our "case law has established very broad parameters for the definition of materiality." United States v. Dipp, 581 F.2d 1323, 1328 (9th Cir.1978). To be material, a false statement need only be "relevant to any subsidiary issue under consideration," and "[t]he government need not prove that the perjured testimony actually influenced the relevant decision-making body." United States v. McKenna, 327 F.3d 830, 839 (9th Cir.2003) (internal quotation marks omitted). In short, we may direct a verdict for Thomas only if, viewing the evidence in the light most favorable to the prosecution, no rational trier of fact could have found her statements to the grand jury to be "relevant to any subsidiary issue under [the grand jury's] consideration." Id.; see Nevils, 598 F.3d at 1170. We conclude that Thomas is not entitled to a judgment of acquittal under this standard.

1
Count one of the superseding indictment alleged that Thomas made a material false statement when she told the grand jury that she never received any products from Arnold. Likely recognizing that Thomas's receipt of products from Arnold would be directly relevant to a potential indictment against Arnold in some judicial district, Thomas argues only that this testimony lacked any jurisdictional nexus to the Northern District of California. We reject this argument.
Had Thomas truthfully testified that she had received products from Arnold, the grand jury could have inquired into the specifics of these products, particularly whether Arnold distributed THG and norbolethone, designer steroids that were found in only two groups of athletes around 2000-2002athletes with a connection to Arnold and those with a connection to BALCO. Had the grand jury heard truthful testimony from Thomas, it might have been able to connect Arnold to BALCO, and making this connection would have been "relevant to [an] issue under [the grand jury's] consideration." McKenna, 327 F.3d at 831.
Critically, there was evidence at Thomas's trial regarding the connections between Arnold, BALCO, and THG and norbolethone. Based on this evidence, the trial jury could reasonably have concluded, for example, that Thomas's truthful response to the question underlying count one could have helped to establish a link between BALCO and Arnold, which could have led to Arnold's indictment in the Northern District of California for conspiracy *1124 to distribute steroids along with BALCO-linked defendants Victor Conte, James Valente, Greg Anderson, and Remi Korchemny. There was no error.

2
Counts three and four of the superseding indictment alleged that Thomas made material false statements when she testified that she never took anything Arnold gave her (count three) and that she never took anabolic steroids (count four). Thomas argues that this testimony was immaterial because whether or not Thomas actually ingested steroids or other substances she received from Arnold was only relevant to Thomas's potential criminality, and she could not be prosecuted because of her immunity agreement. We disagree with the premise of Thomas's argument. Whether Thomas actually ingested Arnold's products, and whether these products were steroids, were relevant to any potential indictment of Arnold. As the BALCO prosecutions show, evidence relating to an athlete's ingestion of steroids (e.g., a doping calendar or medical records indicating allergies and blood type) can be highly probative of a defendant's distribution of steroids. Here, the trial jury could reasonably have concluded that Thomas's statements as to whether she ingested products sent by Arnold and/or steroids were material to whether Arnold could ultimately be indicted in the Northern District of California as part of the BALCO investigation.

B
Next, Thomas argues that the district court erred in refusing to submit to the jury her proposed instruction on immateriality due to lack of jurisdictional nexus. Thomas's "Proposed Jury Instruction Number 12" stated:
A federal grand jury only has the authority to return an indictment against an individual if the facts show probable cause to believe that the individual committed a federal crime that occurred, in whole or in part, within its jurisdiction. The jurisdiction of the federal grand jury before which the defendant is alleged to have testified was the Northern District of California, which is limited to [enumerated counties in California].
Instead of Thomas's proposed materiality instruction, the district court provided the jury with the Ninth Circuit's model jury instruction on materiality, which states that "[a] statement was material if it had a natural tendency to influence, or was capable of influencing, the decision of the decisionmaking body to which it is addressed."
As explained earlier, see supra at 10501-02, we employ a three-prong inquiry to determine whether a new trial is warranted based on the district court's refusal to instruct the jury on a defense theory: (1) whether the theory underlying the jury instruction has some foundation in evidence, Johnson, 459 F.3d at 992; (2) whether the theory is supported by law, id. at 995; and (3) whether other instructions adequately cover the defense theory, Mason, 902 F.2d at 1438. The theory underlying Thomas's Proposed Instruction Number 12 fails on the first prong of this analysis, as it lacked evidentiary foundation.
The evidence at Thomas's trial indicated that the grand jury before which she testified was investigating a possible drug and money laundering conspiracy involving Victor Conte and other individuals linked to BALCO Laboratories in the Northern District of California. The trial evidence made clear that Thomas was called to testify about her dealings with Arnold because of Arnold's potential involvement in the BALCO conspiracy in the Northern District of California, not because of any investigation into potential crimes committed elsewhere. Importantly, there is nothing *1125 in the recordnot even "evidence [that] is weak, insufficient, inconsistent, or of doubtful credibility," Johnson, 459 F.3d at 993 (internal quotation marks omitted), to contradict this view of the evidence. There is, therefore, no "evidence upon which the jury could rationally find" that Thomas's false statements to the grand jury were only relevant to crimes committed wholly outside of the Northern District of California. Id. Accordingly, the district court did not abuse its discretion in refusing to give Thomas's proposed instruction.

IV
Thomas challenges her conviction under count six of the superseding indictment alleging obstruction of justice under 18 U.S.C. § 1503on three grounds, arguing that: (1) the immunity order pursuant to which Thomas testified before the 2003 grand jury precluded the government from using her testimony to prosecute her for anything other than perjury, making material false declarations, or refusing to testify; (2) the indictment failed to allege, and the grand jury failed to find, all of the elements of 18 U.S.C. § 1503, namely the materiality of the statements through which Thomas was alleged to have obstructed justice, and the district court failed to instruct the jury as to the materiality requirement; and (3) the district court erroneously permitted the government to amend the obstruction count following the close of evidence by presenting for the first time four new alleged statements in support of an obstruction finding. None of these asserted errors warrants reversal of Thomas's conviction on count six.

A
Thomas testified at the 2003 grand jury hearing pursuant to a grant of immunity under 18 U.S.C. § 6002. This immunity order stated that her "testimony and other information compelled from" her "may not be used against her in any criminal case, except for a prosecution for perjury, false declaration, or otherwise failing to comply with this order." [20] Thomas was ultimately indicted for obstruction of justice under 18 U.S.C. § 1503 for
corruptly influenc[ing], obstruct[ing], and imped[ing], and endeavor[ing] to corruptly influence, obstruct, and impede, the due administration of justice, by knowingly giving Grand Jury testimony that was intentionally evasive, false, and misleading, including but not limited to the false statements made by the defendants as charged in Counts One through Five of this indictment.
Thomas's immunity grant was governed by 18 U.S.C. § 6002, which provides:
[N]o testimony or other information compelled under the order [granting immunity] (or any information directly or indirectly derived from such testimony or other information) may be used against the witness in any criminal case, except a prosecution for perjury, giving a false statement, or otherwise failing to comply with the order.
Thomas argues that 18 U.S.C. § 6002 bars prosecution for obstruction of justice under 18 U.S.C. § 1503 for false and/or misleading grand jury testimony. We disagree.
Section 6002 does not strictly confine permissible prosecutions for immunized testimony to perjury and false declarations charges: the statute states that testimony pursuant to a grant of immunity can be used in "a prosecution for perjury, giving a false statement, or otherwise failing to comply with the [immunity] order." Id. (emphasis added). The question, then, becomes whether a prosecution for obstruction of justice falls within the ambit of "otherwise failing to comply with the [immunity] order." Thomas urges that the "otherwise" clause of § 6002 applies only *1126 to prosecutions for contempt. See United States v. Paxson, 861 F.2d 730, 738 (D.C.Cir.1988) (noting defendant's argument that the "otherwise" clause of § 6002 allows only prosecutions of contempt, and not prosecutions for obstruction of justice, but declining to reach the issue).
We have rejected the contempt-only view of the "otherwise" in § 6002. In United States v. Duran, we "h[e]ld that the final clause of § 6002 encompasses a prosecution for conspiracy to commit perjury." 41 F.3d 540, 545 (9th Cir.1994). We explained:
A conspiracy to commit perjury is simply an agreement between defendants not to comply with the order. Such an agreement to commit perjury frustrates the purpose of the grant of immunity. Pillsbury Co. v. Conboy, 459 U.S. 248, 253, 103 S.Ct. 608, 74 L.Ed.2d 430 (1983) (finding the purpose of the immunity statute, which includes § 6002, is to provide the government with an evidence gathering tool). Thus, when a person falsely testifies under a grant of immunity, the government may use that testimony as evidence of a conspiracy to commit perjury.
Duran, 41 F.3d at 545.
In light of Duran, we are compelled to reject a "contempt-only" view of § 6002's "otherwise" clause. Moreover, our reasoning in the Duran case is helpful in the case at bar. In Duran, we reasoned that conduct that "frustrates the purpose of the grant of immunity" can fall within the "otherwise" clause of § 6002. Here, the immunity order compelling Thomas's testimony before the grand jury explained, in relevant part:
1. Tammy Thomas may be called to testify before the grand jury; and
2. In the judgment of the United States Attorney, Tammy Thomas is likely to refuse to testify on the basis of her Fifth Amendment privilege against selfincrimination; and
3. In the judgment of the United States Attorney the testimony and other information to be obtained from Tammy Thomas is necessary to the public interest...
IT IS THEREFORE ORDERED that TAMMY THOMAS soon as she may be called, shall testify under oath and provide other information including documents in this case and in any further ancillary proceedings....
This immunity order embodies what the Supreme Court has described as the Constitution's "rational accommodation between the imperatives of the privilege[against selfincrimination] and the legitimate demands of government to compel citizens to testify." Kastigar v. United States, 406 U.S. 441, 446, 92 S.Ct. 1653, 32 L.Ed.2d 212 (1972); see also United States v. Tramunti, 500 F.2d 1334, 1342 (2d Cir.1974) ("[B]y perjuring himself the witness commits a new crime beyond the scope of the immunity which was intended to protect him against his past indiscretions."). The purpose of the immunity order in this case was to compel Thomas to testify truthfully and in good faith before the grand jury to assist it in its investigation into the BALCO case. Indeed, the Supreme Court has explained that 18 U.S.C. § 6002 was enacted to assist law enforcement, and was therefore intended to provide as little immunity as the Constitution will allow. See Pillsbury Co., 459 U.S. at 253, 103 S.Ct. 608 ("The major purpose of the Organized Crime Control Act of 1970, of which § 6002 was a key provision, was to provide the criminal justice system with the necessary legal tools to strengthen the evidence gathering process and insure that the evidence will then be available and admissible *1127 at trial. Congress sought to make the grant of immunity more useful for law enforcement officers ..." (internal quotation marks, citations, ellipses, and brackets omitted)); id. at 253 n. 8, 103 S.Ct. 608 ("[T]he House explained that § 6002 was not to provide an `immunity bath'...."). To the extent that Thomas "knowingly g[ave] Grand Jury testimony that was intentionally evasive, false, and misleading," this act "frustrate[d] the purpose of the grant of immunity," and thus would appear to be a "fail[ure] to comply with the [immunity] order" under our decision in Duran.
The Supreme Court has repeatedly explained that 18 U.S.C. § 6002 is grounded in the Fifth Amendment privilege against compelled self-incrimination and "is intended to be as broad as, but no broader than, the privilege against self-incrimination." United States v. Apfelbaum, 445 U.S. 115, 123, 100 S.Ct. 948, 63 L.Ed.2d 250 (1980) (quoting S.Rep. No. 91-167, at 145 (1969); H.R.Rep. No. 91-1549, at 42 (1970), U.S.Code cong. & Admin.News 1970, p. 4007); see also Apfelbaum, 445 U.S. at 122, 100 S.Ct. 948 ("Congress intended the perjury and false-declarations exception [in § 6002] to be interpreted as broadly as constitutionally permissible."). The immunity order compelling Thomas's testimony made clear that the basis for its grant of § 6002 immunity was that "Tammy Thomas is likely to refuse to testify on the basis of her Fifth Amendment privilege against self-incrimination." Under Apfelbaum and Pillsbury Co., the prosecution of Thomas under 18 U.S.C. § 1503 for her grand jury testimony is barred by 18 U.S.C. § 6002 only if the Fifth Amendment bars such prosecution.
The Fifth Amendment provides that "[n]o person ... shall be compelled in any criminal case to be a witness against himself." U.S. Const. amend. V. The Fifth Amendment privilege against self-incrimination "has never been construed to mean that one who invokes it cannot subsequently be prosecuted." Kastigar, 406 U.S. at 453, 92 S.Ct. 1653. Rather, the "sole concern" of the privilege "is to afford protection against being forced to give testimony leading to the infliction of penalties affixed to criminal acts." Id. (internal quotation marks and ellipses omitted) (emphasis added). Here, the criminal charges against Thomas, including the obstruction of justice charge, were not directed at compelled or forced testimony. Thomas was subpoenaed to testify truthfully about past, potentially criminal, activity relating to her alleged relationship with Patrick Arnold and use of his designer performance-enhancing drugs, and her subsequent prosecution on perjury and obstruction of justice charges did not relate to any prior actions about which she was compelled to testify truthfully, but instead related only to Thomas's unforced decision to testify in an untruthful and misleading manner.
"The principle that the Fifth Amendment privilege against compulsory self-incrimination provides no protection for the commission of perjury ... [is] firmly established constitutional law." Apfelbaum, 445 U.S. at 127, 100 S.Ct. 948. This principle is grounded in the fact that a grand jury witness is not compelled to lie or obstruct justice. Thus, in Apfelbaum, "the Fifth Amendment d[id] not prevent the use of [the defendant's] immunized testimony at his trial for false swearing because, at the time he was granted immunity, the privilege would not have protected him against false testimony that he later might decide to give." Id. at 130, 100 S.Ct. 948. As the Court had explained in United States v. Freed, 401 U.S. 601, 606-07, 91 S.Ct. 1112, 28 L.Ed.2d 356 (1971):
Appellees' argument assumes the existence of a periphery of the Self-Incrimination Clause which protects a person against incrimination not only against past or present transgressions but which *1128 supplies insulation for a career of crime about to be launched. We cannot give the Self-Incrimination Clause such an expansive interpretation.
The Sixth Circuit has squarely addressed the issue we face here, and we agree with its analysis. In United States v. Black, the court stated:
18 U.S.C. § 6002[ ] specifically exempts from the grant of immunity a prosecution "for perjury, giving a false statement, or otherwise failing to comply with the order." ... Though the statute does not include prosecutions for obstructions of justice among its exceptions, we agree with the court in United States v. Caron, 551 F.Supp. 662 (E.D.Va.1982), aff'd mem., 722 F.2d 739 (4th Cir.1983), cert. denied, 465 U.S. 1103, 104 S.Ct. 1602, 80 L.Ed.2d 132 (1984), ... that § 6002 does not proscribe the use of immunized testimony in a prosecution for obstruction of justice.... [A] grant of immunity relates to the past, not to future conduct, and the obstruction of justice must have occurred after immunity had been granted.... [T]he exceptions in § 6002 are broad enough to permit the use of immunized testimony against one who has allegedly committed perjury or otherwise subverted the functioning of a tribunal.

776 F.2d 1321, 1327 (6th Cir.1985) (internal quotation marks and citations omitted); accord Caron, 551 F.Supp. at 671-72 (holding that a § 1503 prosecution fell within the "otherwise" clause of § 6002).
If Thomas, having been compelled to testify to the grand jury, had testified that she had knowingly and intentionally used anabolic steroids without a valid prescription, the Fifth Amendment would bar a prosecution under 21 U.S.C. § 844(a) for such possession. But Thomas was not in any way compelled to "knowingly giv[e] Grand Jury testimony that was intentionally evasive, false, and misleading" by virtue of her grand jury subpoena.
The prosecution of Thomas under 18 U.S.C. § 1503 was "constitutionally permissible" under the Fifth Amendment, and "Congress intended the perjury and falsedeclarations exception [in 18 U.S.C. § 6002] to be interpreted as broadly as constitutionally permissible." Apfelbaum, 445 U.S. at 122, 100 S.Ct. 948. Because the obstruction of justice charge against Thomas satisfied the immunity exception for "otherwise failing to comply with the [immunity] order," 18 U.S.C. § 6002, and was "constitutionally permissible," Apfelbaum, 445 U.S. at 122, 100 S.Ct. 948, Thomas's conviction on count six was not barred by her grant of immunity.

B
Thomas next argues that the government failed to plead and prove materiality as an element of its obstruction charge. Unlike 18 U.S.C. § 1623(a), which explicitly proscribes "knowingly mak[ing] any false material declaration" (emphasis added), 18 U.S.C. § 1503(a) contains no express materiality element. Recognizing that § 1503, by its terms, does not include an express materiality element, Thomas argues that two of our cases, United States v. Rasheed, 663 F.2d 843 (9th Cir.1981), and United States v. Ryan, 455 F.2d 728 (9th Cir.1972), hold that materiality is an implicit element of § 1503(a).
These casesand the Ryan case in particularsupport Thomas's argument that § 1503 implicitly requires materiality. In Ryan, we held that "[t]he acts complained of" as the basis for an obstruction charge under 18 U.S.C. § 1503 "must bear a reasonable relationship to the subject of the grand jury inquiry," and we reversed the defendant's § 1503 conviction for destroying documents where "there was no evidence that the Grand Jury was investigating *1129 any of the[] matters" to which the documents pertained. 455 F.2d at 734-35. In Rasheed, we described materiality as the basis for our decision in Ryan, explaining that "[i]n Ryan, we reversed the defendant's conviction because the subpoenaed documents were immaterial to the grand jury proceedings." 663 F.2d at 851. In Rasheed, we actually rejected the defendant's challenge to her § 1503 conviction based on jury instructions that did not include an express materiality element, but in doing so we made clear that the requisite materiality element was included in the jury instructions as a whole, because the instructions required the jury to "find that ... [the defendant] knew the grand jury was conducting an investigation, that [the defendant] knew what documents were covered by the subpoena, and that, knowing that the particular documents were covered by the subpoena, she willfully concealed or endeavored to conceal them from the grand jury." Id. at 852.
In light of Ryan and Rasheed, we conclude that although not expressly included in the text of § 1503, materiality is a requisite element of a conviction under that statute. Our conclusion does not, however, mandate a reversal of Thomas's obstruction conviction, because it is clear that the jury found the requisite element of materiality in convicting Thomas on count six. The jury unanimously returned a special verdict on Thomas's § 1503(a) charge indicating that the false statements alleged in counts one and three of Thomas's indictment obstructed justice, and the jury in turn had found Thomas guilty of making material false statements with respect to counts one and three. By convicting Thomas of perjury on counts one and three, the jury necessarily found the statements in those counts to be material. And by indicating in a special verdict form that these statements obstructed justice, the jury necessarily found that Thomas's obstruction conviction was based on two material statements.[8]

C
Thomas's final argument as to why her conviction on count six should be reversed relates to a purported "amendment" of the obstruction count of the superseding indictment at the close of trial. Count six of the superseding indictment charged that
On or about November 6, 2003, in the Northern District of California, and elsewhere, the defendant TAMMY A. THOMAS did corruptly influence, obstruct, and impede, and endeavor to corruptly influence, obstruct, and impede, the due administration of justice, by knowingly giving the Grand Jury testimony that was intentionally evasive, false, and misleading, including but not limited to the false statements made by the defendant as charged in Counts One through Five of this indictment.
All in violation of Title 18, United States Code, Section 1503.
(Emphasis added.)
At trial, Thomas objected to the charging language of count six as overbroad, and the district court asked the government to explain its theory of obstruction. The government responded that, in addition to the five alleged false statements expressed in counts one through five, there "was a pattern of evasive and misleading conduct throughout [defendant's] testimony." However, to "cure some of the court's concerns," the government agreed to "specify ... a few specific instances in *1130 which the defendant testified evasively, or false, or in a misleading way above and beyond the false statements." After Thomas rested, the government submitted new instructions and a new verdict form narrowing Thomas's alleged obstructive conduct from her entire grand jury testimony to the five specific allegations already pleaded in the indictment and four specific instances of "evasive, false, misleading conduct."
The district court made one amendment to the government's proposed jury instruction on count six, and ultimately instructed the jury that
[i]n order for the defendant to be found guilty of Count 6, the government must prove each of the following elements beyond a reasonable doubt:
1. The defendant corruptly, that is, for the purpose of obstructing justice,
2. Obstructed, influenced or impeded, or endeavored to influence, obstruct or impede, through one of the below listed statements,
3. The due administration of justice.
In order for the defendant to be found guilty of Count 6, you must all agree that one or more of the following statements obstructed, influenced or impeded the due administration of justice, or was made for the purpose of obstructing, influencing or impeding the due administration of justice. (All of you must agree as to which statement or statements so qualify):
1. The statement contained in Count 1;
2. The statement contained in Count 2;
3. The statement contained in Count 3;
4. The statement contained in Count 4;
5. The statement contained in Count 5;
6. Statement A: [A portion of Thomas's grand jury testimony that did not appear in the superseding indictment.]
7. Statement B: [A portion of Thomas's grand jury testimony that did not appear in the superseding indictment.]
(Emphasis added.) The jury ultimately found Thomas guilty as to count six, and "unanimously agree[d] that" four specific statements obstructed justice: the statement contained in count one, the statement contained in count three, and statements A and B in count six.
Thomas argues that the district court "improperly amended" count six of the indictment at the last moment by narrowing its scope from "including but not limited to the false statements made by the defendant as charged in Counts One through Five of this indictment" (emphasis added), to the nine enumerated statements in the jury instructions, four of which did not specifically appear in the indictment. Even if Thomas were correct, she would not be entitled to reversal of her conviction on count six. The jury instructions on this count expressly stated that Thomas would be guilty of obstruction of justice if "one or more of the following statements obstructed, influenced or impeded the due administration of justice," and cautioned the jurors that "[a]ll of you must agree as to which statement or statements so qualify." (Emphasis added.) The jury ultimately unanimously concluded that the statements contained in count one and count three both "obstructed, influenced or impeded the due administration of justice, or were made for the purpose of obstructing, influencing or impeding the due administration of justice," thus supporting a guilty verdict on the obstruction of justice count even if the district court erred in allowing the jury to consider statement A and statement B. The jury's unanimous findings that statements one and three independently supported its obstruction of justice verdict render any error on the instructions ultimately submitted to the jury harmless.

V
Finding no basis for reversing Thomas's convictions, we affirm the judgment of the *1131 district court.[9]
NOTES
[1]  Thomas's March 14, 2002 urine sample was the first time that norbolethone, an anabolic steroid related to testosterone, had been detected in an athlete's urine. Dr. Catlin explained his identification of norbolethone at Thomas's trial, testifying that initial test results on Thomas's urine contained "many suspicious features" that "looked like an anabolic steroid" but "didn't match with anything we knew," so Catlin's lab had to "figure out its chemical structure." Once this was done, the chemical structure was inputted into a database and the name "norbolethone" appeared as a match. Further research revealed that norbolethone was developed in the 1960s and patented by Wyeth, a large pharmaceutical company, but the substance was never sold commercially. A sample of norbolethone obtained from Wyeth proved an identical match for the unknown substance found in Thomas's urine.
[2]  Even though he did not invent it, Arnold was also the primary source for norbolethone.
[3]  The government filed a notice of cross-appeal, but this cross-appeal was later dismissed on the government's own motion.
[4]  We review a district court's denial of a motion for judgment of acquittal de novo, United States v. Stewart, 420 F.3d 1007, 1014 (9th Cir.2005), but this review is "highly deferential to the jury's findings," United States v. Bancalari, 110 F.3d 1425, 1428 (9th Cir. 1997) (internal quotation marks omitted). A sufficiency challenge can succeed only if, viewing the evidence in the light most favorable to the prosecution, no rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt. United States v. Lo, 231 F.3d 471, 475 (9th Cir.2000).
[5]  In 2004, Congress amended 21 U.S.C. § 802(41)(A) to include THG and norbolethone as enumerated "anabolic steroids."
[6]  Although Thomas submitted a specific proposed instruction on the issue of literal truth, our case law makes clear that the question here is not whether the district court should have given Defendant's Proposed Jury Instruction Number 10 to the jury, but whether the district court should have given the jury some separate instruction on defendant's literal truth defense. See United States v. Escobar de Bright, 742 F.2d 1196, 1198 & n. 5 (9th Cir.1984) ("Of course, the precise phrasing of the instruction is within the district court's discretion.").
[7]  The district court rejected as "argumentative" the specific literal truth instruction proposed by Thomas. See generally United States v. Felix-Gutierrez, 940 F.2d 1200, 1211 (9th Cir.1991) ("[A] court is not required to accept a proposed instruction which is manifestly intended to influence the jury towards accepting the evidence of the defendant as against that of the prosecution." (quotation marks omitted)); United States v. Sarno, 73 F.3d 1470, 1485 (9th Cir.1995) ("A defendant may not draw upon the right to present a `theory of the case' to compel a certain resolution to a disputed question of fact."). Because we hold that the jury instructions adequately covered Thomas's literal truth theory of defense, we need not address the district court's rejection of Thomas's proposed instruction on the theory that it was argumentative.
[8]  Because the special verdict forms establish that the jury necessarily found the element of materiality in voting to convict Thomas on count six, we need not address Thomas's related argument that the jury instructions for that count were erroneous in failing to include a materiality element.
[9]  We dispose of other issues raised by Thomas in her appeal in an unpublished memorandum disposition filed today.
