OPINION

Dixie Wilson was murdered on April 6, 1981, while she was working at the Baird Oil gas station in Garden City, Idaho. Approximately $385 was taken from the station’s cash drawer and safe. Both Petitioner Lacey Sivak and his co-defendant Randall Bainbridge (who was tried separately) admitted that they were present when the crime occurred, but each insisted that the other was responsible for the murder and robbery. The jury convicted Sivak of felony murder and/or aiding and abetting felony murder, but acquitted him of pre-meditated murder. The trial judge then imposed the death penalty. After pursuing other remedies, Sivak filed a federal habeas petition before the effective date of the Antiterrorism and Effective Death Penalty Act of 1996 (AEDPA). The district court denied him relief.

Although we reject many of Sivak’s contentions on appeal, we conclude that absent the State’s knowing presentation of perjured inmate testimony, the result of Sivak’s penalty-phase hearing could have been different. The only direct evidence establishing that Sivak, not Bainbridge, committed the murder came from Bainbridge’s unsworn statement during a police interrogation, and a pair of jailhouse informants. One of the informants admitted on the witness stand that he was a habitual liar; the other committed perjury regarding his motives for testifying and his expectations of receiving preferential treatment from the State. Accordingly, we hold that the State violated Sivak’s due process rights under Napue v. Illinois, 360 U.S. 264 , 79 S.Ct. 1173 , 3 L.Ed.2d 1217 (1959), and we reverse the district court’s denial of the writ with respect to Sivak’s death sentence.

FACTUAL AND PROCEDURAL BACKGROUND

A. Guilt Phase

The State charged Sivak with robbery, two counts of murder in the first degree (premeditated murder and felony murder), and possession of a firearm during the commission of the robbery and murder.

1. The Crime

According to her husband, Dixie Wilson left her home around 6:20 on the morning of April 6, 1981, to go to her job at the Baird Oil gas station.

By 7:00, she had been stabbed and shot repeatedly. Numerous witnesses testified that, when they found her on the gas station floor, she was breathing faintly and appeared to be unconscious, her “face was all bloody,” and blood was coming out of her mouth. One person noted that “her blouse was kind of up above her breasts,” and another said that her shirt was “pulled way up” so that “she was naked from ... [the] top of her breast down to her pants line.” There was a pool of blood on the floor around her, and a knife blade was lying on the ground. The gas station’s money drawer was open, and contained only loose change, and no bills. An empty money bag was sitting out on the counter-top.

Wilson was unconscious when she arrived at the hospital, and was declared dead less than an hour later. An autopsy revealed that she had been shot at least five times in the head and face, and the coroner recovered seven separate bullet fragments from her skull. She was stabbed approximately twenty times around the head, neck, and shoulder, as well as on her left hand, which a physician described as a defensive wound. An x-ray appeared to show the tip of a pocketknife blade lodged in her skull, and the tip of a knife blade was recovered from her hair.

The State sought to establish that Sivak had two motives for the crime: to rob the gas station and to resolve a grudge against Wilson. In addition to the two motives proffered by the State, Sivak’s former supervisor suggested an alternative motive for the robbery and killing: Bainbridge’s attraction to Wilson. According to the supervisor, Bainbridge appeared at the station two days after the murder and “started talking about Dixie,” saying “two or three times ... how she turned him on.” Bainbridge appeared “very nervous,” his lips quivered, his voice broke, and his hands were shaky. (In fact, in the separate prosecution of Bainbridge, “the prosecutor ... pursue[d] a sexual motivation theory.” State v. Bainbridge, 108 Idaho 273 , 698 P.2d 335, 337 (1985).)

2. Jailhouse Informants

During the trial, the State introduced testimony from a number of inmate witnesses. The first, Jimmy Leytham, said that when he and Sivak were in jail together, Sivak confessed to murdering Wilson. Leytham claimed that he was interested in learning about the crime because he had heard about the murder on the television news, and had heard from other inmates that the victim “always helped convicts and stuff.” He asked Sivak “why he shot her and stabbed her so many times,” and Sivak responded, “because she kept on moving.” He then asked Sivak “what happened to the knife handle,” and Sivak said that he “threw it in the river over by the fairgrounds.” He asked about what type of gun was used, and Sivak said that “they used a .22.” He asked about Sivak’s motive, and Sivak said that “he holds grudges against people,” and that “he used to work at the place” and Wilson had “fired him.”

Leytham admitted to the jury that he had been in jail on charges of burglary and escape. When asked about his motive for testifying, Leytham said that he had “a wife and kids out on the streets,” and he did not “want anything to happen to them.” Asked whether he was seeking “any particular favoritism from State authorities” in exchange for his testimony, Leytham said “[n]o.” Asked whether the prosecutor’s “office or any other State agency” took any actions “with regard to your incarceration [in] the Ada County jail,” Leytham said only that his escape charge was dismissed after the preliminary hearing, and a charge pending in another city was also dismissed. Leytham said that he did not know whether the prosecutor’s office was involved in the dismissals.

On cross-examination, the defense asked Leytham about his lengthy criminal history, which included convictions for burglary and insufficient funds. The defense pointed out that Leytham’s sentencing “[jjudge probably wasn’t real happy about seeing [him] the second time” after being lenient during his first conviction. Leytham acknowledged having a third burglary charge against him, and admitted that, in the words of Sivak’s counsel, he had been a “prime candidate for the penitentiary” before his charges were dismissed following the preliminary hearing. The defense also asked whether Leytham talked to the other inmates, which elicited a discussion about an inmate named Nathan Crispin. The defense asked Leytham whether he had ever left the jail to travel to Kansas, and Leytham said he had done so for “[pjersonal reasons.” The defense then asked Leytham whether he had testified against Nathan Crispin in a murder trial in Kansas, and Leytham responded cryptically and without elaboration, “You’ve got the information.”

The defense closed its cross-examination of Leytham by asking, “is it fair to say that you are a free man today because you testified here today and because you testified in Nathan Crispin’s case?” Leytham responded unequivocally, “No, sir.”

The State called a second inmate, Duane Grierson, to testify about Sivak’s jailhouse admissions. According to Grierson, Sivak was pleased that Grierson had testified against Bainbridge, and Sivak wanted to tell his version of events to Grierson. Sivak then said: “I didn’t rob anybody.... Bainbridge did.... I killed her, ... and so did he.” Sivak also said that “he reached a sexual climax from pushing on the body,” and that Bainbridge “reached one from playing with her boobs.”

Asked about why he was testifying, Grierson said, “Because I believe that they are guilty of it,” and “I believe that what I am doing is right.” Asked whether “any ... government entity” had made a deal in exchange for his testimony, he said, “No, sir,” and that he did not “expect” any leniency in his pending sentencing pro-' ceedings. On cross-examination, Grierson admitted that he had testified in two other cases in addition to Sivak’s and Bainbridge’s. Grierson initially denied giving information in the interest of obtaining favorable treatment, but later admitted to writing an affidavit for his .sentencing judge, which stated: “I have been promised deals by the prosecuting attorney’s office, the main gist of which were that if I testified in court in certain murder cases, I would not be sent to any prison, but would receive, at best, a county jail term.” He also wrote to the sentencing judge that he was “a chronic liar,” and “lying was a way of life” for him.

3.' The Verdict

The jury found Sivak guilty of robbery, first-degree felony murder, possession of a firearm during the commission of a robbery, and possession of a firearm during the commission of a felony murder. The jury acquitted Sivak of first-degree premeditated murder and possession of a firearm during the commission of first-degree premeditated murder.

B. Penalty Phase

Consistent with Idaho and federal law at the time, the trial judge was responsible for making the findings necessary to impose the death sentence. See State v. Lovelace, 140 Idaho 73 , 90 P.3d 298, 300-01 (2004) (discussing impact of Ring v. Arizona, 536 U.S. 584 , 122 S.Ct. 2428 , 153 L.Ed.2d 556 (2002), on Idaho capital-sentencing procedures). Because of the complicated procedural history of the case, we summarize here the evidence presented in all three of Sivak’s sentencing hearings.

The defense emphasized two points in favor of a life sentence instead of a death sentence. First, Sivak was not “in that last irredeemable, hopeless category of evil, ugly, horrible people” for whom the death penalty is warranted. Second, Bainbridge was primarily responsible for the robbery and murder, or at the very least, Sivak and Bainbridge were equally culpable in the robbery and murder, and Bainbridge had been spared the death penalty.

In the initial sentencing proceeding, the defense called four witnesses who testified about Sivak’s activities in his local church and in his former job as an Avon cosmetics salesman. The defense also presented mitigating evidence about Sivak’s childhood. Sivak’s sister and mother testified that Sivak’s alcoholic father “beat him,” and that the beatings occurred “a lot.” Sivak’s mother also testified that Sivak was “hyperactive” as a child, and an expert witness discussed the association between attention deficit hyperactivity disorder and criminal behavior. The evidence of Sivak’s hyperactivity was meant not to exculpate him of responsibility, but to raise doubts about whether it was an impulsive attack rather than a “planned [and] rational” one. In addition to highlighting Sivak’s character and upbringing, the defense also emphasized the magnitude of Bainbridge’s participation in the murder.

The judge found that “all of the mitigating circumstances do not outweigh the gravity of each aggravating circumstance so as to make unjust the imposition of the death penalty.” (Citing State v. Charboneau, 116 Idaho 129 , 774 P.2d 299 (1989), overruled, on other grounds by State v. Card, 121 Idaho 425 , 825 P.2d 1081, 1088 (1991).) The judge concluded that, because Sivak “actively participated in the brutal savage slaying and sexually molesting [sic] of a woman while at the same time butchering her alive,” he had “forfeited his rights to life,” and “society must be protected.” Accordingly, he imposed the death penalty.

JURISDICTION AND STANDARD OF REVIEW

We have jurisdiction under 28 U.S.C. §§ 1291 and 2253. Our review is governed by pre-AEDPA standards. Lindh v. Murphy, 521 U.S. 320, 326-27 , 117 S.Ct. 2059 , 138 L.Ed.2d 481 (1997). As the State implicitly concedes, this rule applies even though Sivak filed amended petitions after AEDPA was enacted. See Robinson v. Schriro, 595 F.3d 1086, 1098-99 (9th Cir.) (applying pre-AEDPA legal standards where initial petition was filed prior to AEDPA’s enactment and amended petition was filed after AEDPA’s enactment), cert. denied, — U.S. -, 131 S.Ct. 566 , 178 L.Ed.2d 427 (2010); Hamilton v. Ayers, 583 F.3d 1100, 1105 (9th Cir.2009) (same); Jackson v. Brown, 513 F.3d 1057, 1068-69 (9th Cir.2008) (same); accord Smith v. Mahoney, 611 F.3d 978, 993-95 (9th Cir.) (discussing this issue with respect to AED-PA statute of limitations), cert. denied, — U.S. -, 131 S.Ct. 461 , 178 L.Ed.2d 293 (2010).

Under pre-AEDPA law:

Jackson, 513 F.3d at 1069 (citations omitted) (quoting Hovey v. Ayers, 458 F.3d 892, 900 (9th Cir.2006)).

DISCUSSION

A. Brady and Napue Claims

Sivak contends that the State violated his due process rights under Brady v. Maryland, 373 U.S. 83 , 83 S.Ct. 1194 , 10 L.Ed.2d 215 (1963), and Napue v. Illinois, 360 U.S. 264 , 79 S.Ct. 1173 , 3 L.Ed.2d 1217 (1959), by failing to disclose impeachment evidence regarding jailhouse informant Jimmy Leytham, by allowing Leytham’s false testimony regarding his motives for testifying to stand uncorrected, and by failing to disclose statements made by another inmate, Louis Fazio, regarding Bainbridge’s actions during the murder.

1. Procedural Challenges

Before addressing the merits of Sivak’s contentions, we first address the State’s arguments that the Napue claim is procedurally defaulted and unexhausted.

a. Procedural Default

The district court assumed without deciding that the claim was not procedurally defaulted, and concluded that it failed on the merits. We review questions regarding procedural default de novo. Robinson, 595 F.3d at 1099 .

The State argues that the Napue claim is procedurally defaulted because the Idaho Supreme Court held that “Sivak has waived this issue under the provisions of I[daho] C[ode] § 19-2719(5).” Sivak v. State (Sivak V), 134 Idaho 641 , 8 P.3d 636, 645 (2000). That statutory provision requires defendants to “file any legal or factual challenge to the sentence or conviction that is known or reasonably should be known” within forty-two days after a death sentence is imposed. Idaho Code § 19-2719 (3). If a defendant fails to comply, he is “deemed to have waived such claims for relief as were known, or reasonably should have been known” within that forty-two day period. Id. § 19-2719(5). The only exception is “for issues that were not known or could not reasonably have been known,” and which “cast doubt on the reliability of the conviction or sentence.” Id. § 19-2719(5)(a)-(b). Applying these provisions, the Idaho Supreme Court concluded that Sivak’s Napue claim “reasonably should have been known at the time of Sivak’s first petition” for post-conviction relief in 1984. Sivak V, 8 P.3d at 645 .

We have previously held that Idaho’s forty-two day filing deadline was inadequate in cases involving allegations of ineffective assistance of counsel where the petitioner “continued to be represented by his original trial counsel during the forty-two day period.” Hoffman v. Arave, 236 F.3d 523, 534 (9th Cir.2001).. The reason for our conclusion was self-evident: “it is the rare attorney who can be expected to contend on appeal that his representation was so poor that he deprived his client of a fair trial.” Id. (alteration and internal quotation marks omitted). As a result, we concluded that “Idaho’s forty-two day filing deadline, as applied to Hoffman, is uniquely harsh.” Id. Because the rule “frustrated the exercise of a federal right,” it was “inadequate to preclude federal courts from reviewing the merits of the federal claim.” Id. at 531 (internal quotation marks omitted).

Once again, we conclude that “Idaho’s forty-two day filing deadline, as applied to [the petitioner], is uniquely harsh,” and is therefore inadequate. Id. at 534 . The Idaho Supreme Court’s invocation of the § 19-2719(5) time bar in Sivak’s case is perplexing. The court’s conclusion rested on the apparent premise that Sivak had not raised his Napue claim in his direct appeal or his first petition for post-convie tion relief. See Sivak V, 8 P.3d at 645 . But Sivak did present the Napue claim in his first post-conviction petition. Sivak’s Second Amended Petition for Post-Conviction Relief, filed in November 1984, included a Fourteenth Amendment argument premised on the prosecutor’s “failure ... to reveal the substance of plea bargain agreements with two of his witnesses.” After the trial court denied the petition, Sivak’s opening brief to the Idaho Supreme Court discussed Napue at length under the subheading, “Presentation of false testimony.” The Idaho Supreme Court then considered and rejected Sivak’s Napue claim on the merits. Sivak v. State (Sivak II), 112 Idaho 197 , 731 P.2d 192, 203 (1986) (citing Napue, 360 U.S. 264 , 79 S.Ct. 1173 ); see also id. at 219-22 (Bistline, J., dissenting in relevant part) (discussing Napue claim at length, explaining that “the majority’s conclusion against Sivak [on this claim] is without doubt the most unsubstantiated and distorted holding in the case”). In light of these undisputed facts, it is unsurprising that two justices dissented from the subsequent invocation of the § 19-2719(5) time bar. The justices wrote, correctly, that “Sivak’s claim concerning the alleged deal between the prosecutor and Leytham was raised, addressed on its merits, and denied in his prior petition for post-conviction relief.” Sivak V, 8 P.3d at 646 (Trout & Silak, JJ., dissenting).

The Idaho Supreme Court’s application of § 19-2719(5) is premised on an erroneous factual determination, and Sivak has met his AEDPA burden of introducing clear and convincing evidence to rebut the presumption that the state court’s determination is correct. See 28 U.S.C. § 2254 (e)(1) (2006). While it is unusual to reject a state court’s use of a procedural bar on the ground that it was erroneously applied, “[t]he procedural default doctrine self-evidently is limited to cases in which a ‘default’ actually occurred ie., cases in which the prisoner actually violated the applicable state procedural rule.” 2 Randy Hertz & James S. Liebman, Federal Habeas Corpus Practice and Procedure § 26.2[c] (6th ed. 2011). Here, the state court applied the state’s procedural rule to Sivak’s case in an erroneous and arbitrary manner. Thus, we follow the Supreme Court and our sister circuits in holding that an erroneously applied procedural rule does not bar federal habeas review.

b. Exhaustion

The State also argues that, even if the Napue claim is not procedurally barred, it is unexhausted because Sivak did not present the factual basis of the Napue claim to the Idaho courts.

In order to exhaust a claim, the petitioner must fully and fairly “present both the factual and legal basis for the claim to the state court.” Robinson, 595 F.3d at 1101 ; see also 28 U.S.C. § 2254 (b) (1994). “Full and fair presentation ... requires a petitioner to present the substance of his claim to the state courts, including a reference to a federal constitutional guarantee and a statement of facts that entitle the petitioner to relief.” Scott v. Schriro, 567 F.3d 573, 582 (9th Cir.) (per curiam), cert. denied, — U.S. -, 130 S.Ct. 1014 , 175 L.Ed.2d 618 (2009).

Sivak specifically argued to the Idaho Supreme Court that his federal due process rights were violated under Napue because “[t]he jury was never advised that there existed the possibility of prospective benefits to Leytham depending upon the quality of his testimony against petitioner.” Although Sivak subsequently discovered documentary evidence to bolster this argument, he undeniably presented both “the legal and factual basis of his federal constitutional claim” to the state court. Robinson, 595 F.3d at 1102 . He identified evidence showing that Leytham knew that the prosecution provided him benefits in exchange for his cooperation, that the prosecution told him that it would consider providing him with benefits, and that he testified at trial that he “d[id]n’t know” whether or not the prosecution was involved in providing him benefits. This evidence was more than sufficient to satisfy the exhaustion requirement. “[A]s long as the ‘ultimate question for disposition’ has remained the same in state and federal court, ... ‘variations in the legal theory or factual allegations urged in its support’ are entirely legitimate.” Id. at 1102 n. 14 (quoting Picard v. Connor, 404 U.S. 270, 277 , 92 S.Ct. 509 , 30 L.Ed.2d 438 (1971)).

Because Sivak’s Napue claim is neither procedurally barred nor unexhausted, we now turn to the merits of the Napue and Brady claims.

2. Legal Framework

In Brady v. Maryland, the Supreme Court held that “the suppression by the prosecution of evidence favorable to an accused upon request violates due process where the evidence is material either to guilt or to punishment, irrespective of the good faith or bad faith of the prosecution.” Brady, 373 U.S. at 87 , 83 S.Ct. 1194 . The Court has subsequently explained that “[t]here are three components of a true Brady violation: [1] The evidence at issue must be favorable to the accused, either because it is exculpatory, or because it is impeaching; [2] that evidence must have been suppressed by the State, either willfully or inadvertently; and [3] prejudice must have ensued.” Strickler v. Greene, 527 U.S. 263, 281-82 , 119 S.Ct. 1936 , 144 L.Ed.2d 286 (1999). Brady applies not only to information known to the prosecutor, but also to “evidence ‘known only to police investigators and not to the prosecutor.’ ” Id. at 280-81, 119 S.Ct. 1936 (quoting Kyles v. Whitley, 514 U.S. 419, 438 , 115 S.Ct. 1555 , 131 L.Ed.2d 490 (1995)).

In Napue v. Illinois, the Court held that “a conviction obtained through use of false evidence, known to be such by representatives of the State, must fall under the Fourteenth Amendment.” Napue, 360 U.S. at 269 , 79 S.Ct. 1173 . “A claim under Napue will succeed when ‘(1) the testimony (or evidence) was actually false, (2) the prosecution knew or should have known that the testimony was actually false, and (3) the false testimony was material’ ” Jackson, 513 F.3d at 1071 -72 (quoting Hayes v. Brown, 399 F.3d 972, 984 (9th Cir.2005) (en banc)). It is “irrelevant” whether the defense knew about the false testimony and failed to object or cross-examine the witness, because defendants “c[an]not waive the freestanding ethical and constitutional obligation of the prosecutor as a representative of the government to protect the integrity of the court and the criminal justice system.” N. Mariana Islands v. Bowie, 243 F.3d 1109, 1122 (9th Cir.2001); see also Belmontes v. Brown, 414 F.3d 1094, 1115 (9th Cir.2005) (“Whether defense counsel is aware of the falsity of the statement is beside the point.”), rev’d on other grounds sub nom., Ayers v. Belmontes, 549 U.S. 7 , 127 S.Ct. 469 , 166 L.Ed.2d 334 (2006).

3. Analysis

a. Leytham’s Letters

i. Factual Background

During discovery in the district court, Sivak discovered four letters in the prosecutor’s files regarding Leytham.

The first letter was written on May 7, 1981, by Ada County prosecutor Jim Harris to Dennis Albers, the prosecutor of a neighboring county in which a charge was pending against Leytham for escaping jail. Harris explained that Leytham had “come to law enforcement in Ada County with very damaging evidence] against three inmates in the Ada County Jail presently being held on murder charges.” Harris acknowledged that “[t]he escape charge that you have presently pending against Mr. Leytham is ... of serious concern to you and your office!,]” but he nevertheless “request[ed] that charges against Mr. Leytham in Idaho County, relating to the escape above mentioned, be dismissed by your office.” Harris explained that his request was “[b]ased on [Leytham’s] cooperation, as well as the fact that Mr. Leytham is presently serving a sentence in the Ada County Jail resulting from” a prior offense. Harris concluded by adding, “I do believe that based on the fact that Mr. Leytham will obviously not be sentenced to the State Penitentiary (based on his willingness to testify against these individuals), justice will be served adequately without proceeding further on the escape charge pending in your jurisdiction.”

The second letter was written a few days later by Harris to the chairman of the state Commission for Pardons and Parole. Harris “recommend[ed]” that Leytham “be given additional consideration for parole from the Idaho State Correctional Institute” during his upcoming parole hearing the following week. Harris’s “recommendation [was] based upon Mr. Leytham’s cooperation” with respect to the two trials in the Wilson murder, the murder trial in Kansas, and an additional local murder investigation.

The third letter was written on July 13, 1981, by Leytham to Jerry Brown, who was “evidently a Kansas prosecutor.” Sivak V, 8 P.3d at 640. Leytham wrote that Idaho investigator “Vaughn [Killeen] told me you are in the same position Idaho is in. They say they will help me when I get out but don’t do any thing [sic] about it.” Leytham added that he wanted $6,000 “cash” as “wit[]ness fees as soon as you can.” A few weeks later, Brown forwarded this letter to Killeen.

The fourth letter was written on July 27, 1981, by Killeen to Leytham. The letter began: “Your witness fee should arrive shortly.... I also wanted to communicate to you about your status and what this office has.done in regards to your current status. If you recall you had information regarding the Bainbridge-Sivak case and you approached me about a ‘deal.’ I informed you we couldn’t make ‘deals’ but if you desired to testify I would attempt to do something for you but there would be no guarantees.” Killeen then summarized the “arrangements” that» were made in anticipation of Leytham’s testimony.in the Sivak, Bainbridge, and Crispin cases; “the dismissing of criminal charges against you in two jurisdictions, a reduction in sentence and a parole from the Idaho State Correctional Institute.” Killeen closed by stating that “[w]ithout our intervention you would still be in prison with other pending criminal charges against you. I would suggest you stop trying to make a living off the system.... I will see you at the deposition” in Sivak’s case.

ii. Preliminary Brady/Napue analysis

These letters clearly satisfy the first two requirements of Brady : they are “favorable to the accused ... because [they are] ... impeaching,” and they “have been suppressed by the State, either willfully or inadvertently.” Strickler, 527 U.S. at 281-82 , 119 S.Ct. 1936 .

With respect to the first Brady requirement, the letters are classic examples of impeachment evidence. It is well established that “[ejvidence of a deal or promise of lenient treatment in exchange for a witness’s testimony against a defendant may constitute evidence that must be disclosed under Brady and Napue.” Hovey, 458 F.3d at 916 (citing Giglio v. United States, 405 U.S. 150, 154-55 , 92 S.Ct. 763 , 31 L.Ed.2d 104 (1972)); see also Banks v. Dretke, 540 U.S. 668, 685, 700-02 , 124 S.Ct. 1256 , 157 -L.Ed.2d 1166 (2004) (finding a Brady violation where prosecution failed to disclose that key witness “was an informant and that he had been paid $200 for his involvement in the case”). We have also explained that “[t]he deal or promise need not be express; failure to disclose an agreement or guarantee of leniency ‘indicated without making a bald promise’ also may violate Brady.” Hovey, 458 F.3d at 917 (quoting United States v. Butler, 567 F.2d 885 , 888 n. 4 (9th Cir.1978) (per curiam)). We have even found a Brady violation where the evidence “implie[d] that a tacit agreement was reached between [a witness] and the government ... in exchange for his cooperation.” United States v. Shaffer, 789 F.2d 682, 689 (9th Cir.1986). Here, although there is no evidence of an explicit meeting of the minds between Leytham and the prosecutor’s office, there is ample evidence of an implicit quid pro quo between the prosecutor’s office and Leytham, and that this agreement was known to Harris, Killeen, and Leytham. We therefore agree with the Idaho Supreme Court’s factual conclusion “that Leytham and the prosecutor had reached some type of deal in exchange for his cooperation.” Sivak V, 8 P.3d at 643.

With respect to the second Brady requirement, the state, undeniably suppressed the letters within the meaning of Brady . The district court correctly concluded “that the State suppressed the letters because they were not disclosed until the Court granted discovery in this habeas matter.” This finding is sufficient to satisfy the second element of Brady. See Benn v. Lambert, 283 F.3d 1040, 1053 (9th Cir.2002) (“[T]he terms ‘suppression,’ ‘withholding,’ and ‘failure to disclose’ have the same meaning for Brady purposes.”).

Having concluded that the letters were impeachment evidence and were withheld by the State, we next examine the materiality of the letters collectively in combination with the other Brady and Napue violations. See Jackson, 513 F.3d at 1076 .

b.Leytham’s False Testimony

During the guilt phase of the trial, Leytham testified that he had “a wife and kids out on the streets,” and he did not “want anything to happen to them.” Asked directly whether he “ask[ed] for any particular thing — any particular favoritism from State authorities” in exchange for his testimony, Leytham said “[n]o.” Asked whether the prosecutor’s “office or any other State agency” took any actions “with regard to your incarceration [in] the Ada County jail,” Leytham said only that his escape charge was dismissed after the preliminary hearing, and a charge pending in another city was also dismissed. Leytham said that he did not know whether the prosecutor’s office was involved in the dismissals.

The Idaho Supreme Court concluded that:

Sivak V, 8 P.3d at 645 . These factual findings are entitled to a presumption of correctness, and the State has not rebutted this presumption. 28 U.S.C. § 2254 (d) (1994). In addition to the false statements identified by the Idaho Supreme Court, Leytham also falsely testified that he had traveled to Kansas for personal reasons. The prosecutor knew or reasonably should have know that Leytham had traveled to Kansas in order to testify in the Crispin murder trial. Killeen’s July 27, 1981, letter to Leytham notes that Leytham “requested ... $6,000 for [his] continued testimony against Crisp[i]n,” and Harris’s letter to the parole commission stated that Leytham had “agreed to testify regarding a murder ... in the State of Kansas.”

These findings about Leytham’s testimony establish the first two elements of a Napue violation: “‘(1) the testimony (or evidence) was actually false [and] (2) the prosecution knew or should have known that the testimony was actually false.’” Jackson, 513 F.3d at 1071 (quoting Hayes, 399 F.3d at 984 ).

c. Fazio’s Statement to Police

Sivak also contends that the prosecution withheld a statement from inmate Louis Fazio, who told investigators that Bainbridge had admitted to fondling the victim’s breasts as she was dying. However, the statement was not suppressed during the penalty phase. As the district court correctly found, “Sivak’s counsel possessed this evidence before the 1988 sentencing hearing” and even “referred to it as part of Sivak’s mitigation case.” Because the prosecution disclosed the material to Sivak, it was not suppressed for purposes of Brady during the sentencing phase. In addition, had the Brady material regarding Fazio been disclosed during the guilt phase and been admissible, the total mix of evidence would have been essentially unchanged.

d. Materiality and Prejudice

Under Brady , withheld “evidence is material only if there is a reason able probability that, had the evidence been disclosed to the defense, the result of the proceeding would have been different. A ‘reasonable probability’ is a probability sufficient to undermine confidence in the outcome.” United States v. Bagley, 473 U.S. 667, 682 , 105 S.Ct. 3375 , 87 L.Ed.2d 481 (1985). In determining materiality, the “suppressed evidence” must be “considered collectively, not item by item.” Kyles, 514 U.S. at 436 , 115 S.Ct. 1555 . “[0]nce a reviewing court applying Bagley has found constitutional error there is no need for further harmless-error review.” Id. at 435, 115 S.Ct. 1555 .

In contrast, under Napue, a conviction (or capital sentence) is “set aside whenever there is ‘any reasonable likelihood that the false testimony could have affected the judgment of the jury.’ ” Jackson, 513 F.3d at 1076 (quoting Hayes, 399 F.3d at 985 ). Although “Napue does not create a ‘per se rule of reversal,]’ ” “[w]e have gone so far as to say that ‘if it is established that the government knowingly permitted the introduction of false testimony reversal is virtually automatic.’ ” Id. (quoting Hayes, 399 F.3d at 978, 984 ).

In Jackson , we explained the proper method of analyzing Brady and Napue errors in tandem:

Id. (ellipsis and citations omitted).

If Brady and Napue violations were prejudicial with respect to either the guilt or penalty phase of the trial, Sivak is entitled to relief. See, e.g., Cone v. Bell, 556 U.S. 449 , 129 S.Ct. 1769, 1784 , 173 L.Ed.2d 701 (2009) (“Evidence that is material to guilt will often be material for sentencing purposes as well; the converse is not always true, however....”); see also Jackson, 513 F.3d at 1076-77 (analyzing materiality of Napue violation with respect to sentencing phase).

i. Guilt Phase

Sivak has failed to show that the State’s Napue and Brady violations prejudiced him during the guilt phase of proceedings. Although the evidence of Sivak’s guilt was not quite as “overwhelming” as the State repeatedly contends in its brief, we are confident that the State’s misconduct could not reasonably have affected the jury’s verdict.

In his testimony to the jury, Sivak did not deny the fact that he was present at the crime scene when Wilson was shot five times and stabbed twenty times. Sivak’s theory of the case was that Bainbridge was the primary actor in the robbery and murder. Sivak contended he was merely present at the offense and was incapable of intervening because of his own fear and Bainbridge’s implicit and explicit threats against Sivak and Sivak’s family.

Ample evidence rebutted Sivak’s contentions. One eyewitness testified that prior to the murder, “it was [a] real uneasy atmosphere” in the gas station and Wilson appeared uncomfortable in Sivak and Bainbridge’s presence. This testimony rebutted Sivak’s claim that the murder was spontaneous. Another eyewitness testified that Bainbridge “was going through the drawer” that contained the cash, which rebutted Sivak’s claim that Bainbridge kept the derringer pointed at him following the murder. Witnesses testified that Bainbridge and Sivak seemed normal after the murder, which suggested that Sivak was not scared of Bainbridge. In fact, Bainbridge’s wife and a friend each testified that Bainbridge seemed more nervous than usual that morning after the murder, but no one (other than Sivak himself) testified that Sivak was similarly affected by the crime.

In addition, a number of employees from the gas station testified that Sivak disliked Wilson and thought she had been changing his deposits. They also testified about the station’s practice of keeping a limited amount of cash in the upper part of the safe and the remainder of the cash in the lower locked portion of the safe, and that Sivak thought Wilson had a key to the lower portion. Another employee testified that someone who sounded like Sivak called a few days before the murder to ask which employee would be working on Monday morning. Accordingly, there was ample evidence supporting the State’s theory of the case, as summarized in the State’s closing argument: “Lacey Sivak thought that Dixie still had the key to the lower part of that safe. He knew that on Monday the receipts from the previous weekend were in the lower part of that safe.”

Finally, and most importantly, Sivak himself admitted (after first denying it on the stand) that he stole the .22 pistol, derringer, and ammunition from the store where his mother worked. Sivak’s fingerprint was on the barrel of the .22, and he could not explain why there were no other fingerprints on the gun.

Thus, even if the State had not violated Napue and Brady , we have full confidence that the jury would still have convicted Sivak of first-degree murder. It is important to recall that the jury found Sivak guilty of first-degree felony murder, and that the jury was properly instructed that anyone who “aid[s] and abet[s]” a crime is “equally guilty” as the person who “directly commit[s] ... the offense.”

Under Idaho law, Sivak could have been convicted of either aiding and abetting Bainbridge’s felony. murder or directly committing the felony murder. Thus, the Napue violations could not reasonably have affected the guilt-phase outcome. There is no doubt that a robbery occurred and that Wilson was killed during the course of that robbery, thus supporting a first-degree felony-murder conviction. The only real dispute was, if Bainbridge committed both the murder and the rob^ bery, whether Sivak aided and abetted either of those acts. The jury heard and rejected Sivak’s contentions that he knew nothing about the plan to commit robbery and/or murder, that he did not participate in the robbery or murder, and that he was threatened into assisting Bainbridge cover up the crimes.

In light of this strong evidence of guilt under either a direct felony-murder theory or an aiding-and-abetting felony murder theory, the Napue violations could not have changed the jury’s guilt determina tion. Leytham’s testimony was wholly cumulative to the other evidence of Sivak’s guilt. Leytham’s testimony tended to establish that Sivak, not Bainbridge, was the actual killer, and tended to bolster Grierson’s testimony to the same effect. But even if the jury disbelieved Leytham entirely and, in turn, disbelieved Grierson (an admitted liar whose testimony was corroborated only by Leytham), there still is no “reasonable likelihood that the false testimony could have affected the judgment of the jury.” Jackson, 513 F.3d at 1076 (emphasis and internal quotation mark omitted). As in Morris v. Ylst, 447 F.3d 735, 746 (9th Cir.2006), Sivak “testified in his own defense and implicated[others] in the crime, but the jury disbelieved him,” and “the physical evidence” and various other pieces of evidence “pointed to his guilt.” Thus, we follow Morris and hold that Sivak has not shown prejudice on account of the guilt-phase Napue violations. Cf. Maxwell v. Roe, 628 F.3d 486, 507-08 (9th Cir.2010) (finding prejudice where perjured witness “was the ‘make-or-break’ witness for the State” (citing Hall v. Dir. of Corrs., 343 F.3d 976, 978, 984-85 (9th Cir.2003) (per curiam); Killian v. Poole, 282 F.3d 1204, 1209 (9th Cir.2002))), petition for cert. filed, 80 U.S.L.W. 3004 (U.S. June 21, 2011).

We reach the same result under Brady and our collective Napue-Brady analysis. Had the Brady material regarding Leytham been disclosed, Leytham would have been thoroughly discredited, and Grierson’s testimony would likewise have suffered. But Leytham’s substantive testimony was duplicative to the other evidence presented at trial with respect to Sivak’s guilt, and, as a result, we have full confidence in the verdict.

Because the errors are not prejudicial standing alone, we must “consider all of the Napue and Brady violations collectively and ask whether ‘there is a reasonable probability that ... the result of the proceeding would have been different.’ ” Jackson, 513 F.3d at 1076 (emphasis omitted). We conclude that the result would not have been different. Even if the jury had been properly advised that Leytham was committing perjury and heard from Fazio that Bainbridge admitted to fondling the victim, the guilt-phase verdict would have been the same. There was simply too much evidence placing Sivak at the scene of the crime while it occurred, and the jury rejected Sivak’s exculpatory testimony. Accordingly, there was no prejudice during the guilt phase on account of the Napue and Brady violations.

ii. Penalty Phase

We reach a contrary conclusion regarding penalty-phase prejudice. The penalty phase differed from the guilt phase in two important respects. First, under Idaho law, the prosecution was required to prove beyond a reasonable doubt that the murder satisfied at least one of Idaho’s statutory aggravating circumstances. Idaho Code § 19-2515 (g) (West 1992). Second, the prosecution was required to prove that one of these aggravating circumstances outweighed all of the mitigating circumstances, and that the death penalty was warranted. Idaho Code § 19-2515 (c) (West 1992); Charboneau, 774 P.2d at 323-24 ; see also Fetterly v. Paskett, 997 F.2d 1295, 1301 (9th Cir.1993) (“For whatever reason, Idaho has chosen to prohibit the grouping of aggravating circumstances in connection with their weighing against whatever elements of mitigation that appear in a given case.”); accord Hayes, 399 F.3d at 988 (noting that Napue violation “would have affected not only the special circumstance verdict, but also the jury’s ultimate decision to impose the penalty of death”).

Thus, in determining whether Sivak suffered prejudice as a result of the State’s misconduct, we must be confident in the trial judge’s conclusion that Sivak caused or intended to cause Wilson’s death, and in the judge’s balancing of aggravating and mitigating circumstances. The Idaho Supreme Court has observed that it is difficult to conduct harmless error review in capital-sentencing cases because “[ajggravating and mitigating factors are of a more subjective nature than evidence of guilt or innocence that can be objectively examined to determine whether a jury’s deliberations would have come out the same way as to the underlying offense.” State v. Lovelace, 140 Idaho 73 , 90 P.3d 298, 304 (2004) (internal quotation marks omitted). In conducting this “subjective” balancing process, the sentencer must be carefully attuned to “[t]he relative weights of the various types of’ available evidence. Sivak II, 731 P.2d at 205 . In light of these guiding principles of state sentencing law, it is important also to keep in mind the appropriate prejudice/harmless error standards: if there is any “reasonable probability that ... the result ... would have been different” without the Brady error, Bagley, 473 U.S. at 682 , 105 S.Ct. 3375 , or if “there is any reasonable likelihood that” the sentencer’s weighing process “could have” been different without the Napue error, Jackson, 513 F.3d at 1076 (internal quotation marks omitted), then we must grant Sivak’s petition.

The State first contends that Leytham’s perjury and the suppressed letters were not prejudicial because the sentencing judge had ample reasons to disbelieve Leytham. We disagree. In Jackson , the defense thoroughly cross-examined one of the witnesses “about his own attempts to benefit from his cooperation.” Id. at 1077 . We held that this cross-examination was an inadequate substitute for hard evidence of the witness’s cooperation: “evidence of an explicit promise of assistance by the trial prosecutor likely would have carried far greater weight than any speculative benefit [the witness] might have thought he could achieve on his own.” Id. Similarly, Jackson’s jury was aware that robbery charges were pending against the state’s other key witness, and could have “speculated” he was cooperating to try and get a better deal. Id. We held that “this speculation pales in comparison to the reality that law enforcement officers had actually promised” favors to the witness. Id. We concluded that the two witnesses’ “willingness to lie under oath to keep the promises secret would cast doubt on [their] entire testimony.” Id.; accord Silva v. Brown, 416 F.3d 980, 989 (9th Cir.2005) (holding that withheld impeachment evidence was material under Brady because “the undisclosed evidence was not duplicative of the impeachment evidence actually presented, but rather was of a different kind”).

Like the witnesses in Jackson , Leytham refused to admit that he was seeking (and indeed, had received) money and favorable treatment in exchange for his testimony. This testimony invited the factfinder to “speculate” about the possibility that he was testifying out of self-interest rather than public-minded generosity. Leytham even lied about his motivations for testifying and the benefits he had received from the prosecution. Had the sentencing judge known about Leytham’s true motivations for testifying, and had the judge known that Leytham had lied on the witness stand about those motivations, the judge very likely would have rejected Leytham’s testimony about Sivak’s purported confession.

There are a number of reasons why the sentencing judge’s assessment of Leytham’s testimony might have been different if the State had corrected his falsehoods or disclosed the four impeaching letters to Sivak. We have explained the importance of disclosing that a witness has received “prosecution provided benefits”: “ ‘Disclosure of an agreement to provide such benefits, as well as evidence of the benefits themselves, could have allowed the jury to reasonably conclude that the informant had a motive other than altruism for testifying on behalf of the State. Such a finding could have substantially impeached the informant’s credibility as a witness.’ ” Benn, 283 F.3d at 1057 (alterations omitted) (quoting Singh v. Prunty, 142 F.3d 1157, 1162 (9th Cir.1998)). The Supreme Court has suggested that such information is even more significant if the witness has not received a firm commitment from the prosecution, because witnesses have greater incentives to lie if the potential benefits are “not guaranteed through a promise or binding contract.” Bagley, 473 U.S. at 683 , 105 S.Ct. 3375 . And as we explained on remand from the Court, “ ‘[t]he more uncertain the agreement, the greater the incentive to make the testimony pleasing to the promisor.’ ” Bagley v. Lumpkin, 798 F.2d 1297, 1302 (9th Cir.1986) (quoting Boone v. Paderick, 541 F.2d 447, 451 (4th Cir.1976)).

In addition, if a witness’s false testimony is corrected by the prosecution, his “willingness to lie under oath” is exposed and his credibility is irreparably damaged. Jackson, 513 F.3d at 1077 . There is a substantial difference between “general evidence of untrustworthiness and specific evidence that a witness has lied.” Benn, 283 F.3d at 1056-57 . “ ‘All the other evidence used by the defense to punch holes in the informant’s credibility amount[s] only to circumstantial reasons why the informant might alter the truth to continue to feather his own nest. A lie would be direct proof of this concern, eliminating the need for inferences.’ ” Id. at 1057 (alterations omitted) (quoting United States v. Bernal-Obeso, 989 F.2d 331, 336 (9th Cir.1993)).

Furthermore, these rules are of particular significance when the witness is incarcerated. We have repeatedly observed that inmate testimony is inherently unreliable. “Defendants or suspects with nothing to sell sometimes embark on a methodical journey to manufacture evidence and to create something of value, setting up and betraying friends, relatives, and cellmates alike.” Bowie, 243 F.3d at 1124 .

All of these considerations are relevant in this case. Had Leytham’s lies been exposed, the factfinder very likely would have rejected Leytham’s testimony about Sivak’s confession. If the factfinder had rejected Leytham’s testimony, it would have been forced to reconsider Grierson’s testimony as well. See id. (“Frequently, and because they are aware of the low value of their credibility, criminals will even go so far as to create corroboration for their lies by recruiting others into the plot.”). Absent Leytham’s corroborating testimony regarding Sivak’s confession, Grierson’s version of Sivak’s confession would have been called into doubt. Because Grierson’s own lies were uncovered at trial and he even admitted to being a “chronic liar,” the corroborating power of Leytham’s testimony vis-a-vis Grierson’s is clear. Absent the overlap between Leytham’s and Grierson’s testimony about Sivak’s confession, the judge reasonably would have reconsidered his firm conviction that Sivak personally committed the murder.

Once Sivak’s purported confessions to Leytham and Grierson are removed from the mix of evidence, the sentencing judge’s aggravating findings are significantly weakened. We have repeatedly held that Brady and Napue violations were prejudicial where they impacted testimony regarding the defendant’s purported confession, particularly when the remaining case against the defendant was entirely circumstantial. Maxwell, 628 F.3d at 508 ; Jackson, 513 F.3d at 1077-79 ; Silva, 416 F.3d at 987 ; Horton v. Mayle, 408 F.3d 570, 579-81 (9th Cir.2005); Hayes, 399 F.3d at 985-87 ; Hall, 343 F.3d at 984 . Such a conclusion derives from common sense: a confession is typically “powerful evidence,” Premo v. Moore, — U.S. -, 131 S.Ct. 733, 744 , 178 L.Ed.2d 649 (2011), but is substantially less powerful if heard secondhand from a witness with a motive to lie and a demonstrated pattern of lying, cf. Barker v. Fleming, 423 F.3d 1085, 1097 (9th Cir.2005) (rejecting Brady claim where the record contained ample evidence in which the jury heard that the witness was “a serial liar whose story that [the defendant] confessed was highly doubtful”).

In light of these considerations, we are compelled to conclude that there is a reasonable likelihood that the sentencing outcome could have been different if the State had corrected Leytham’s perjury. Sivak simply did not receive a trial that “‘resulted] in a [sentence] worthy of confidence.’ ” Jackson, 513 F.3d at 1076 (quoting Kyles, 514 U.S. at 434 , 115 S.Ct. 1555 ). Although the trial judge heard ample evidence establishing Sivak’s'presence at the scene of the crime, the only direct evidence of Sivak’s personal participation in the crime consisted of testimony from inmates (Leytham and Grierson) and Sivak’s co-perpetrator Bainbridge. Given that these three pieces of evidence are inherently unreliable, we cannot be confident that the trial judge would have reached the same conclusion, had he known that Leytham (in addition to Grierson) lied on the witness stand.

Contrary to the State’s arguments, the Napue violations in the sentencing phase are not harmless errors that are merely cumulative to other evidence known to the factfinder. Unlike the cases cited by the State, this is not a case in which the witness at issue had already been exposed to the factfinder as a liar, e.g., Heishman v. Ayers, 621 F.3d 1030, 1035 (9th Cir.2010) (per curiam), petition for cert. filed, No. 10-10953 (U.S. June 1, 2011); Morris, 447 F.3d at 741, 746 , the witness already testified that he received immunity, Heishman, 621 F.3d at 1035 , the witness “explained the conditions of his plea agreement in open court,” Hamilton, 583 F.3d at 1111 , or the witness admitted that he “ ‘hope[d]’ ” to “receiv[e] lenient treatment in exchange for his testimony,” Hovey, 458 F.3d at 921 . Furthermore, unlike our harmless error analysis of the guilt phase of this case, there is not “overwhelming evidence of [Sivak’s] guilt” with respect to the aggravating factors supporting the death sentence, Hamilton, 583 F.3d at 1112 , and there is not reliable independent evidence that Sivak personally caused or intended to cause Wilson’s murder, such as clear physical evidence and a reliable confession from the defendant, e.g., Morris, 447 F.3d at 738-39, 741, 746 .

Accordingly, we conclude that the State’s failure to correct Leytham’s false testimony violated Sivak’s due process rights with respect to his capital sentence.

B. Double Jeopardy Clause

Sivak also contends that the trial court violated the Double Jeopardy Clause because its sentencing findings conflicted with the jury’s acquittal on the premeditated murder count. In his briefs, Sivak argues that this acquittal “necessarily determined that he lacked the specific intent to commit murder.” At oral argument, Sivak offered the more expansive argument that the acquittal established “that he was not the actual killer.” We disagree with both versions of Sivak’s Double Jeopardy claim.

1. Legal Framework

In Ashe v. Swenson, 397 U.S. 436 , 90 S.Ct. 1189 , 25 L.Ed.2d 469 (1970), the Supreme Court “held that the Double Jeopardy Clause precludes the Government from relitigating any issue that was necessarily decided by a jury’s acquittal in a prior trial.” Yeager v. United States, 557 U.S. 110 , 129 S.Ct. 2360, 2366 , 174 L.Ed.2d 78 (2009). “In evaluating collateral estoppel claims, we follow a three-step process. First, we identify the issues in the two actions to determine whether they are sufficiently material and similar to justify invoking the doctrine; second, we examine the record in the prior ease to determine whether the similar issue was litigated; third, we examine the record of the prior proceeding to determine whether the issue was necessarily decided in the first case.” Wilson v. Belleque, 554 F.3d 816, 830 (9th Cir.), cert. denied, — U.S. -, 130 S.Ct. 75 , 175 L.Ed.2d 53 (2009). Analysis of “[t]he preclusive effect of the jury’s verdict ... is a question of federal law which we ... review de novo.” Schiro v. Farley, 510 U.S. 222, 232 , 114 S.Ct. 783 , 127 L.Ed.2d 47 (1994). “The burden is ‘on the defendant to demonstrate that the issue whose relitigation he seeks to foreclose was actually decided in the first proceeding.’ ” Id. at 233 , 114 S.Ct. 783 (quoting Dowling v. United States, 493 U.S. 342, 350 , 110 S.Ct. 668 , 107 L.Ed.2d 708 (1990)).

When determining the preclusive effect of a jury verdict, we must “examine the record of [the] prior proceeding, taking into account the pleadings, evidence, charge, and other relevant matter, and conclude whether a rational jury could have grounded its verdict upon an issue other than that which the defendant seeks to foreclose from consideration.” Ashe, 397 U.S. at 444 , 90 S.Ct. 1189 (internal quotation marks omitted). If “there is more than one rational conclusion that can be drawn from the first jury’s verdict,” then collateral estoppel cannot apply because the issue was not necessarily decided by the jury’s verdict. Santamaria v. Horsley, 133 F.3d 1242, 1246 (9th Cir.1998) (en banc). “The inquiry ‘must be set in a practical frame and viewed with an eye to all the circumstances of the proceedings.’ ” Ashe, 397 U.S. at 444 , 90 S.Ct. 1189 (quoting Sealfon v. United States, 332 U.S. 575, 579 , 68 S.Ct. 237 , 92 L.Ed. 180 (1948)).

2. Analysis

Sivak was charged with two separate counts of first-degree murder. Count two of the Information charged that he murdered Wilson “willfully, unlawfully, deliberately, with premeditation and with malice aforethought.” ■ Count three charged him with murdering Wilson “in the perpetration of a robbery.” The jury found Sivak guilty on count three (felony murdei) but not count two (premeditated murder).

We reject Sivak’s argument that these verdicts conclusively establish that he did not personally stab and shoot Wilson. Sivak is correct that it is possible that the jury concluded that Bainbridge used the murder weapons. However, we cannot agree that the jury “necessarily decided” whether Sivak or Bainbridge was the killer. Yeager, 129 S.Ct. at 2366 . We have previously rejected a similar argument, using language directly applicable to this case: although the “two verdicts may be harmonized by concluding that the jury found that [Sivak] was guilty as an aider and abettor[,] ... it is also conceivable that all twelve jurors were convinced beyond a reasonable doubt that [Sivak] played some role, either as the shooter or as an aider and abettor, without ascertaining exactly which role. We have no way of knowing, because these alternatives ... are rationally consistent with the jury’s verdict in th[is] case.” Santamaria, 133 F.3d at 1246 . By convicting Sivak on count three, the jury determined that either Sivak or Bainbridge committed a murder “in the perpetration of, or attempt to perpetrate robbery,” and, to the extent that it found that Bainbridge committed the murder, that Sivak “aid[ed] and abet[ted] in its commission.” By acquitting Sivak on count two, the jury found reasonable doubt as to whether Sivak or Bainbridge committed a “willful, deliberate and premeditated killing,” or, to the extent it found that Bainbridge committed a “willful, deliberate and premeditated killing,” that there was reasonable doubt whether Sivak aided and abetted that crime. In other words, there are numerous “rational conclusion^] that can be drawn from the ... jury’s verdict,” and we are therefore unable to conclude that the jury found that Bainbridge, not Sivak, committed the murder. Id. Sivak’s argument rests entirely on impermissible “guesswork,” “conjecture,” and “speculation.” Yeager, 129 S.Ct. at 2368 .

Sivak also argues that his acquittal on count two (premeditated murder) should have, precluded the sentencing judge from finding beyond a reasonable doubt that- he acted with the “specific intent” to kill Wilson. He contends that Idaho’s “specific intent to kill” aggravating factor requires the State to establish all the elements of first-degree premeditated murder: namely, that “there must be wilfulness, deliberation and premeditation in addition to malice aforethought.” State v. Porter, 142 Idaho 371 , 128 P.3d 908, 911 (2005) (quot ing State v. Aragon, 107 Idaho 358 , 690 P.2d 293, 297 (1984)).

We are unpersuaded by Sivak’s construction of Idaho law. At the time of the final sentencing proceeding, the Idaho Code included the following statutory aggravating factor: “[1] The murder was one defined as murder of the first degree by section 18-4003, Idaho Code, subsections (b), (c), (d), (e) or (f) [which establish various types of felony murder], and [2] it was accompanied with the specific intent to cause the death of a human being.” Idaho Code § 19-2515 (g)(7) (West 1992). Sivak does not dispute that his particular felony-murder conviction fits within the first part of this statute, as it was a “murder committed in the perpetration of, or attempt to perpetrate, ... robbery.” Id. § 18-4003(d). Instead, he argues that the jury’s acquittal on count two conclusively established that he did not act “with the specific intent to cause the death of a human being,” id. § 19 — 2515(g)(7), and that he therefore may not be subjected to the death penalty on this ground.

Sivak’s argument is unavailing. The Idaho courts have held that the “specific intent to kill” aggravating factor is identical to the intent-to-kill element of first and second-degree murder, and we are unable to determine conclusively that the jury acquitted Sivak of premeditated murder because he lacked an intent to kill. Instead, the jury could have acquitted Sivak on numerous grounds other than his intent to kill — for example, if there was an absence of premeditation or deliberation, or if Bainbridge committed the murder. By acquitting Sivak of first-degree premeditated murder, the jury determined (in the language of the jury instruction) that Sivak did not commit a “willful, deliberate and premeditated killing.” The jury could have reached one of the following conclusions under the jury instructions: (1) Sivak did not intend to kill Wilson (and thus did not act willfully, deliberately, or with premeditation); (2) even if Sivak intended to kill Wilson, he did not form this intent to kill prior to killing her (and thus did not act deliberately or with premeditation); or (3) even if Sivak formed an intent to kill prior to killing Wilson, he did not reflect on this intention (and thus did not act with premeditation). If we take into account the court’s aiding and abetting instruction, the number of . possible conclusions is multiplied: if the jury concluded that Bainbridge committed the murder, it could have acquitted Sivak of aiding and abetting because (4) Bainbridge did not act intentionally, (5) Bainbridge did not form an intent prior to acting, (6) Bainbridge did not reflect on his intent, or (7), even if Bainbridge did commit a premeditated murder, Sivak did not aid and abet the murder.

Viewing the record with “realism and rationality,” Ashe, 397 U.S. at 444 , 90 S.Ct. 1189 , we must conclude that there is “more than one rational conclusion that can be drawn from the ... jury’s verdict” of acquittal. Sant amaria, 133 F.3d at 1246 . We have no way of knowing which .of these seven conclusions the jury actually reached. Accordingly, we agree with the district court’s conclusion that “Sivak has not established that the [sentencing judge’s] finding of specific intent ‘was actually decided in his favor’ ” and “[t]he trial court was free to conclude during the subsequent penalty phase, based on all of the evidence before it, that Sivak intended to kill Wilson.” (Quoting Schiro, 510 U.S. at 236 , 114 S.Ct. 783 .)

C. Ex Parte Information

Sivak argues that Judge New-house violated due process by basing his sentencing decision on extrajudicial information, including: (1) telephone calls and letters the judge received before (and possibly after) the initial sentencing; (2) an encounter with Sivak’s father where the father exclaimed, “Why the hell don’t you kill the son of a bitch”; (3) a telephone conversation with a person from the company that owned the Baird Oil gas station, in which the person indicated he had opposed the death -penalty prior to Sivak’s case but changed his mind afterwards; (4) a conversation with Judge Newhoúse’s clerk, in which she indicated that she had changed her attitude toward the death penalty because of Sivak’s case; and (5) exposure to “stories” and “rumors” about Sivak having an “unnatural relationship” with his mother.

In Gardner v. Florida, 430 U.S. 349 , 97 S.Ct. 1197 , 51 L.Ed.2d 393 (1977), a defendant challenged his death sentence because the sentencing judge announced that he was relying, in part, on a confidential presentence investigation report that had not been provided to the defense. Id. at 351, 353 , 97 S.Ct. 1197 . The Gardner plurality observed that when a sentencing judge-relies on information not disclosed to the defendant, there is an inherent risk that the information “may be erroneous, or may be misinterpreted.” Id. at 359 , 97 S.Ct. 1197 . The plurality stated that the “truth-seeking function of trials” requires courts to afford defense counsel the opportunity to comment on facts that might influence the judge’s sentence. Id. at 360 , 97 S.Ct. 1197 . Ultimately, the plurality concluded that the defendant “was denied due process of law when the death sentence was imposed, at least in part, on the basis of information which he had no opportunity to deny or explain.” Id. at 362 , 97 S.Ct. 1197 .

A majority of the justices in Gardner agreed with the plurality that, because the confidential information in the presentence report had never been provided to the defendant, the defendant’s conviction must be vacated. Id.; see also id. at 363-64, 97 S.Ct. 1197 (White, J., concurring); id. at 364 , 97 S.Ct. 1197 (Brennan, J., concurring). But because of the fragmented nature of the Gardner decision, the Court has subsequently clarified the scope of Gardner’s holding. In O’Dell v. Netherland, 521 U.S. 151, 162 , 117 S.Ct. 1969 , 138 L.Ed.2d 351 (1997), the Court concluded that Justice White’s concurrence, not the plurality opinion, was the controlling decision in Gardner . The O’Dell Court explained that Gardner stands only for the “narrow” proposition that “ ‘a procedure for selecting people for the death penalty which permits consideration of secret information relevant to the character and record of the individual offender1 violates the Eighth Amendment’s requirement of ‘reliability in the determination that death is the appropriate punishment.’ ” O’Dell, 521 U.S. at 162, 117 S.Ct. 1969 (ellipsis and alteration omitted) (quoting Gardner, 430 U.S. at 364 , 97 S.Ct. 1197 (White, J., concurring) (emphasis added in O’Dell)).

As our sister circuits have recognized, O’Dell’s interpretation of Gardner is now the law of the land. See Vining v. Sec’y, Dep’t of Corr., 610 F.3d 568 , 571 n. 1 (11th Cir.2010), cert. denied, - U.S. -, 131 S.Ct. 2898 , 179 L.Ed.2d 1193 (2011); Holland v. Anderson, 583 F.3d 267, 280 (5th Cir.2009), cert. denied, — U.S. -, 130 S.Ct. 2100 , 176 L.Ed.2d 731 (2010); see also O’Dell v. Netherlands 95 F.3d 1214 , 1224 (4th Cir.1996) (relying on Justice White’s Gardner concurrence), aff'd, 521 U.S. 151 , 117 S.Ct. 1969 . Although we have previously relied on the Gardner plurality opinion, e.g., McKenzie v. McCormick, 27 F.3d 1415, 1419-20 (9th Cir.1994), we are bound to follow the intervening interpretation set forth in O’Dell, see Miller v. Gammie, 335 F.3d 889, 900 (9th Cir.2003) (en banc) (“[Wjhere intervening Supreme Court authority is dearly irreconcilable with our prior circuit authority[,] ... a three-judge panel of this court ... should consider [it]sel[f] bound by the intervening higher authority and reject the prior opinion of this court as having been effectively overruled.”).

Applying the narrow standard articulated in Justice White’s Gardner concurrence, along with our case law establishing that the petitioner bears the burden of proving a Gardner violation, see McKenzie, 27 F.3d at 1420 , Sivak’s Gardner claims plainly lack merit. The sentencing judge’s phone calls, letters, conversation with Sivak’s father, and conversation with the court clerk did not contain “ ‘information relevant to the character and record of the individual offender.’ ” O’Dell, 521 U.S. at 162, 117 S.Ct. 1969 (internal quotation mark omitted). As the Idaho Supreme Court correctly observed, the “calls and letters received by Judge Newhouse ... apparently contained no more than an expression of personal opinions ” rather than “factual information.” Sivak II, 731 P.2d at 200 . Although Judge Newhouse’s subsequent deposition has provided additional details regarding his ex parte contacts, Sivak has not satisfied his burden of demonstrating that the trial judge received any “information” about Sivak’s “character and record” as a result of the phone calls, conversations, and written correspondence.

Sivak identifies only one item of ex parte information that contained facts about Sivak’s character and record: rumors regarding Sivak’s “unnatural relationship” with his mother. But even though these rumors were “ ‘relevant to [Sivak’s] character,’ ” O’Dell, 521 U.S. at 162, 117 S.Ct. 1969 (emphasis omitted), they did not constitute “secret information” for purposes of Gardner . Unlike the confidential, undisclosed information at issue in Gardner , Sivak knew that the trial judge had received information suggesting that Sivak had an unusual relationship with his mother. The Presentence Report (which was disclosed to Sivak) stated that “[s]everal” individuals had informed investigators about the “uniquely close” relationship between Sivak and his mother, and some individuals even “mistakenly referred to the defendant’s mother as his wife on several occasions.” Because Gardner requires only that the defendant be given an “opportunity to respond” to the information at issue, Gardner, 430 U.S. at 364 , 97 S.Ct. 1197 (White, J., concurring), and here Sivak had ample opportunity to explain the nature of his relationship with his mother, Sivak’s due process rights were not violated.

Accordingly, we hold that there was no Gardner violation during Sivak’s sentencing proceedings.

D. Judicial Bias

Sivak also contends that he was denied due process because the sentencing judge exhibited bias toward him and his counsel. Sivak raises two related arguments. First, Sivak argues that because Judge Newhouse made up his mind following the initial sentencing proceeding in 1981, he viewed subsequent resentencing hearings with “contempt” and “felt an overwhelming need to vindicate his initial ruling.” Second, Judge Newhouse “became ‘embroiled in a running, bitter controversy’ with [Sivak] and his counsel,” “such that a detached observer must conclude that a fair and impartial hearing was unlikely.” (Quoting Mayberry v. Pennsylvania, 400 U.S. 455, 465 , 91 S.Ct. 499 , 27 L.Ed.2d 532 (1971).)

“The Due Process clause ‘requires a fair trial in a fair tribunal before a judge with no actual bias against the defendant or interest in the outcome of his particular case.’ ” Smith, 611 F.3d at 997 (quoting Bracy v. Gramley, 520 U.S. 899, 904-05 , 117 S.Ct. 1793 , 138 L.Ed.2d 97 (1997)). However, “ ‘[o]nly in the most extreme of cases would disqualification on the basis of judge bias be constitutionally required.’ ” Id. (alteration omitted) (quoting Aetna Life Ins. Co. v. Lavoie, 475 U.S. 813, 821 , 106 S.Ct. 1580 , 89 L.Ed.2d 823 (1986)). In addition to cases involving actual judicial bias,

Crater v. Galaza, 491 F.3d 1119, 1131 (9th Cir.2007) (alterations and citations omitted).

Although the standard of review governing judicial bias claims in pre-AEDPA cases “is not entirely clear,” Poland v. Stewart, 117 F.3d 1094, 1103 (9th Cir.1997), we have repeatedly concluded that a “state court’s finding that the trial judge was impartial ... is a finding of historical fact that is entitled to a presumption of correctness” under 28 U.S.C. § 2254 (d) (1994), Gretzler v. Stewart, 112 F.3d 992, 1009 (9th Cir.1997); see also Ortiz v. Stewart, 149 F.3d 923, 938 (9th Cir.1998); Villafuerte v. Stewart, 111 F.3d 616 , 632 (9th Cir.1997) (per curiam). Petitioners must also overcome an additional presumption against their claim — the “strong presumption that a judge is not biased or prejudiced.” Rhoades v. Henry, 598 F.3d 511, 519 (9th Cir.2010) (citing Bracy, 520 U.S. at 909 , 117 S.Ct. 1793 ).

Here, Sivak has failed to overcome the twin presumptions against finding actual bias: the state courts determined that Judge Newhouse was impartial, and, in any event, we presume that Judge New-house was impartial because he was a judicial officer. The state courts’ conclusions are instructive. When Sivak filed a motion to disqualify Judge Newhouse during the final round of sentencing in 1992, the motion was referred to another trial judge, Gerald Schroeder (who later became Chief Justice of the Idaho Supreme Court). Judge Schroeder determined that there was “nothing before this Court that indicates Judge Newhouse cannot decide the merits of this case based upon facts that may properly be considered.” Judge Schroeder also determined that there was “no current record as to the judge’s attitude towards either the defendant or the defendant’s attorney” — i.e., Sivak had failed to introduce evidence of bias. Judge Schroeder accordingly denied Sivak’s motion to recuse Judge Newhouse, subject to Judge Newhouse’s “final review” to determine actual bias. Judge Newhouse then stated on the record that “I feel no basis for disqualification.” In a similar motion filed a few months later during post-conviction proceedings, Judge Newhouse concluded that he could “render a fair and impartial determination based upon the evidence placed before [him].” On Sivak’s appeal, the Idaho Supreme Court unanimously affirmed Judge Newhouse’s determination “that he could sit fairly and im partially and perform the proper legal analysis which the law requires to be performed.” State v. Sivak (Sivak IV), 127 Idaho 387 , 901 P.2d 494, 496 (1995).

Despite having the opportunity to depose Judge Newhouse during federal habeas proceedings, Sivak has failed to identify evidence of actual bias sufficient to overcome the presumptions against his claim. Judge Newhouse’s deposition contains various statements expressing frustration with the Idaho Supreme Court’s repeated reversals of his sentencing decisions. But even if a case has been reversed on appeal, “ ‘[i]t has long been regarded as normal and proper for a judge to sit in the same case upon its remand and to sit in successive trials involving the same defendant.’ ” Poland, 117 F.3d at 1103 (quoting Liteky v. United States, 510 U.S. 540, 551 , 114 S.Ct. 1147 , 127 L.Ed.2d 474 (1994)). While “[sjome may argue that a judge will feel the ‘motivation to vindicate a prior conclusion’ when confronted with a question for the second or third time, for instance, upon trial after a remand, ... we accept the notion that the ‘conscientious judge will, as far as possible, make himself aware of his biases of this character, and, by that very self-knowledge, nullify their effect.’ ” Liteky, 510 U.S. at 562 , 114 S.Ct. 1147 (Kennedy, J., concurring) (citations omitted). Although Judge Newhouse said that the Idaho Supreme Court’s remands were “ridiculous,” he also explained that after each remand, he “did it over, looked it over with the additional matters and came to the same conclusion” as he had in the original proceeding.

Sivak’s evidence of purported bias pales in comparison to the cases he relies upon. Unlike in Porter v. Singletary, 49 F.3d 1483, 1489 (11th Cir.1995) (per curiam), Sivak has not identified “specific and ostensibly reliable evidence that the judge had a fixed predisposition to sentence this particular defendant to death if he were convicted by the jury.” In Porter, the court clerk submitted a declaration that “before or during Porter’s trial,” the trial judge said that “ ‘he would send Porter to the chair.’ ” Id. at 1487 (quoting declaration); see also Franklin v. McCaughtry, 398 F.3d 955, 962 (7th Cir.2005) (finding due process violation where trial judge “decided that Franklin was guilty before he conducted Franklin’s trial”). Here, in contrast, Judge Newhouse made his sentencing decision after hearing the parties’ evidence and arguments during both the guilt and sentencing phases of trial.

Nor is this case like Harrison v. McBride, 428 F.3d 652 (7th Cir.2005), another case cited by Sivak. In Harrison , a witness told police detectives that the trial judge was connected with the local drug community. Id. at 655 . The trial judge essentially “transformed” a pretrial hearing “into a proceeding to vindicate ‘the credibility of this Court[,]’ ” id. at 669 (quoting trial transcript), and engaged in conduct that “revealed a personal interest in protecting his name and the judiciary in Posey County, an interest he specifically admitted,” id. at 670 (internal quotation marks omitted). We cannot agree with Sivak’s argument that Harrison is “much like,” or even anything like, this case. There is no evidence in the record establishing that Judge Newhouse conducted Sivak’s proceedings with the goal of vindicating himself or his court.

Sivak also suggests that Judge New-house harbored personal bias against Sivak and his counsel, pointing to the judge’s statements expressing frustration with Sivak and his counsel for filing multiple disqualification motions. Judge Newhouse complained that the motions were creating a “carnival atmosphere,” that was turning the proceedings into a “Roman circus.” These statements do not establish judicial bias. In Ortiz , we addressed a similar set of facts. The postconviction judge had complained about the seemingly endless cycle of ineffective assistance of counsel claims, calling the petitions part of a “game” aimed at extending proceedings into the indefinite future. Ortiz, 149 F.3d at 939 (emphasis omitted). We rejected the petitioner’s due process claim, explaining that the judge’s statements “simply reflected] the judge’s frustration with the number of petitioners who attempt to manipulate the criminal justice system to serve their own ends,” and did not reveal “deep-seated favoritism or antagonism that would make fair judgment impossible.” Id. at 940 (internal quotation marks omitted). The same conclusions apply here.

Although Judge Newhouse at one point suggested that Sivak’s “over-zealous counsel” had “fabricated” “innuendos of a partisan and personal nature,” the judge’s statements were fully consistent with the general rule that “[r]umor, speculation, beliefs, conclusions, innuendo, suspicion, opinion, and similar non-factual matters” do not form the basis of a successful recusal motion. Clemens v. U.S. Dist. Ct. for the Cent. Dist. of Cal, 428 F.3d 1175, 1178 (9th Cir.2005) (per curiam) (internal quotation marks omitted). Judge Newhouse had previously allowed Sivak to present evidence to substantiate his argument that the judge received ex parte information from letters and phone calls. Sivak’s witness (a reporter from a local newspaper who had written an article about the judge) failed to provide probative information supporting the claim. Judge New-house’s single reference to Sivak’s counsel’s “fabricat[ion]” does not establish that he was “personally embroiled ... in a running, bitter controversy” with Sivak or his counsel. See Mayberry, 400 U.S. at 465 , 91 S.Ct. 499 (internal quotation marks omitted); see also Little v. Kern Cnty. Super. Ct., 294 F.3d 1075, 1083 (9th Cir.2002) (finding due process violation where trial judge presided over contempt proceeding against attorney after, in judge’s words, attorney “ ‘smirkingly’ ” submitted a drawing containing an “ ‘obscene characterization’ ” of the judge that was “ ‘offensive in the extreme’ ”). Sivak relies upon Anderson v. Sheppard, 856 F.2d 741, 747 (6th Cir.1988), but in that case the “judge exhibited] ... open hostility and bias [from] the beginning of [the] ... proceeding,” essentially telling the pro se plaintiff he would lose and then “assum[ing] the posture of an advocate” to ensure as much. Nothing like that occurred here; Judge Newhouse simply concluded that Sivak had not met his burden of substantiating his allegations that the judge had received ex parte information.

Sivak also suggests that Judge New-house violated his due process rights “when he decided to sit for an interview with a news reporter to discuss [Sivak’s] case and his initial sentencing decision.” While various state and federal canons of judicial ethics proscribe such conduct, it is important to remember that “ ‘[t]he Due Process Clause demarks only the outer boundaries of judicial disqualifications. Congress and the states ... remain free to impose more rigorous standards for judicial disqualification than those’ ” imposed by the federal Constitution. Caperton v. A.T. Massey Coal Co., 556 U.S. 868 , 129 S.Ct. 2252, 2267 , 173 L.Ed.2d 1208 (2009) (quoting Aetna Life Ins., 475 U.S. at 828 , 106 S.Ct. 1580 ). Although Sivak cites numerous cases in which trial judges have been ordered recused, e.g., United States v. Cooley, 1 F.3d 985, 995 (10th Cir.1993), or in which different judges have been assigned to the case upon remand, e.g., United States v. Microsoft Corp., 253 F.3d 34, 115 (D.C.Cir.2001) (en banc) (per curiam), he has not identified a single case holding that a judge violated a litigant’s constitutional rights by talking with the media.

Sivak has failed to overcome the presumption that the Idaho courts correctly determined that Judge Newhouse lacked actual bias, as well as the presumption that judicial officers act impartially. Sivak’s case does not involve the type of “extreme facts” establishing a violation of his due process rights to an impartial judge. Caperton, 129 S.Ct. at 2265-66 .

E. Confrontation Clause

Sivak contends that the sentencing court violated his Confrontation Clause rights by considering Bainbridge’s out-of-court statement. This argument is foreclosed by Williams v. New York, 337 U.S. 241 , 69 S.Ct. 1079 , 93 L.Ed. 1337 (1949), which held that the Confrontation Clause does not bar courts from considering unconfronted statements during sentencing proceedings. Sivak acknowledges that Williams is dispositive, and raises this argument solely to preserve it for review. The district court properly denied relief on this claim.

F. Evidentiary Hearing and Further Discovery

Habeas petitioners are “entitled to an evidentiary hearing only if they allege[ ] facts that, if proved, would entitle them to relief and if they did not receive a full and fair evidentiary hearing in the state court.” Swan v. Peterson, 6 F.3d 1373, 1384 (9th Cir.1993). “[I]f the record refutes the applicant’s factual allegations or otherwise precludes habeas relief, a district court is not required to hold an evidentiary hearing.” Schriro v. Landrigan, 550 U.S. 465, 474 , 127 S.Ct. 1933 , 167 L.Ed.2d 836 (2007).

“[D]iseovery is available only in the discretion of the court and for good cause shown.” Rich v. Calderon, 187 F.3d 1064, 1068 (9th Cir.1999). “Good cause exists ‘where specific allegations before the court show reason to believe that the petitioner may, if the facts are fully developed, be able to demonstrate that he is ... entitled to relief.’ ” Smith, 611 F.3d at 996 (quoting Bracy, 520 U.S. at 908-09 , 117 S.Ct. 1793 ). “We review the district court’s denial of discovery and an evidentiary hearing for abuse of discretion.” Id. at 997 .

The district court considered Sivak’s discovery requests and granted Sivak limited discovery. Sivak has not identified any colorable claims that were not developed in federal or state court. Nor has Sivak identified any “disputed facts” that would entitled him to relief if resolved in his favor after an evidentiary hearing. Hendricks v. Vasquez, 974 F.2d 1099, 1103 (9th Cir.1992) (citing Harris v. Pulley, 885 F.2d 1354, 1378 (9th Cir.1988)). Accordingly, the district court did not abuse its discretion by denying further discovery and an evidentiary hearing.

G.Uncertified Issues

We decline to address the issues not included in the district court’s certificate of appealability. See Cooper-Smith v. Palmateer, 397 F.3d 1236, 1245 (9th Cir.2005) (citing 9th Cir. R. 22—1(e)).

CONCLUSION

The State violated Sivak’s federal due process rights under Napue when it allowed jailhouse informant Jimmy Leytham to testify falsely. Absent this due process violation, Sivak’s penalty-phase proceedings could have resulted in a different outcome. We accordingly remand for the district court to vacate Sivak’s death sentence. Of course, “the State is not precluded from seeking to impose a death sentence upon petitioner, ‘provided that it does so through a new sentencing hearing at which petitioner is permitted to present any and all relevant mitigating evidence that is available.’” Hitchcock v. Dugger, 481 U.S. 393, 399 , 107 S.Ct. 1821 , 95 L.Ed.2d 347 (1987) (quoting Skipper v. South Carolina, 476 U.S. 1, 8 , 106 S.Ct. 1669 , 90 L.Ed.2d 1 (1986)).

The judgment of the district court denying the petition as to Sivak’s conviction is affirmed, the judgment of the district court denying the petition as to Sivak’s death sentence is reversed, and the case is remanded to the district court with instructions that the court enter an appropriate order for a penalty-phase retrial, if the State elects to seek such a retrial. The parties shall bear their own costs.

AFFIRMED IN PART, REVERSED IN PART, AND REMANDED.

. See James v. Kentucky, 466 U.S. 341, 351 , 104 S.Ct. 1830 , 80 L.Ed.2d 346 (1984) (deeming state procedural bar inadequate where defendant "sought to invoke the substance of his federal right”); Mapes v. Coyle, 171 F.3d 408, 429 (6th Cir.1999) (deeming state procedural bar inadequate where petitioner "properly raised [the issue] on post-conviction review and Ohio courts erroneously refused to consider it”); Forgy v. Norris, 64 F.3d 399, 402 (8th Cir.1995) (deeming state procedural bar inadequate where state supreme court’s procedural ruling was contrary to the record and applicable state rules); Kubat v. Thieret, 867 F.2d 351 , 366 n. 11 (7th Cir.1989) (deeming state procedural bar inadequate where state court "ruled that [petitioner] had waived the claim by failing to raise it on direct appeal[,]” but "[t]he record clearly shows ... that [petitioner] did in fact raise and argue the issue in his brief on direct appeal"). . We reject the State’s contention that Sivak conceded in the district court that the Napue claim was not exhausted. It is not clear that Sivak made such a concession, and the district court declined to rely on this ground. Absent the district court’s reliance on Sivak’s concession, Sivak is not bound by his prior assertions. See Reed Elsevier, Inc. v. Muchnick, - U.S. -, 130 S.Ct. 1237, 1249 , 176 L.Ed.2d 18 (2010) ("[W]e decline to apply judicial estoppel. ... [T]hat doctrine typically applies when, among other things, a 'party has succeeded in persuading a court to accept that party's earlier position, so that judicial acceptance of an inconsistent position in a later proceeding would create the perception that either the first or the second court was misled.' ” (quoting New Hampshire v. Maine, 532 U.S. 742, 750 , 121 S.Ct. 1808 , 149 L.Ed.2d 968 (2001))). Even if Sivak’s purported concession had been accepted by the district court, "this court can independently review the state court record to determine whether the issue has actually been exhausted.” Lopez v. Schriro, 491 F.3d 1029 , 1041 n. 7 (9th Cir.2007). . "The terms 'material' and 'prejudicial' are used interchangeably in Brady cases. Evidence is not 'material' unless it is 'prejudicial,' and not 'prejudicial' unless it is 'material.' " Benn, 283 F.3d at 1053 n. 9. . Because Idaho "treats aiding and abetting as a theory and not as a separate offense with distinct elements, ... there is no basis for a specific unanimity instruction” that would "requir[e] the basis for the jury’s verdict (aider and abettor or principal) be a unanimous decision.” State v. Johnson, 145 Idaho 970 , 188 P.3d 912, 919-20 (2008); see also Schad v. Arizona, 501 U.S. 624, 632 , 111 S.Ct. 2491 , 115 L.Ed.2d 555 (1991) (plurality opinion) (" '[T]here is no general requirement that the jury reach agreement on the preliminary factual issues which underlie the verdict.’ ”); Schad, 501 U.S. at 649 , 111 S.Ct. 2491 (Scalia, J., concurring) (“[I]t has long been the general rule that when a single crime can be committed in various ways, jurors need not agree upon the mode of commission.”). . The sentencing judge relied on the following provisions: (5) The murder was especially heinous, atrocious or cruel, manifesting exceptional depravity. (6) By the murder, or circumstances surrounding its commission, the defendant exhibited utter disregard for human life. (7) The murder was one defined as murder of the first degree by section 18-4003, Idaho Code, subsections (b), (c), (d), (e) or (£), and it was accompanied with the specific intent to cause the death of a human being. (8)The defendant, by prior conduct or conduct in the commission of the murder at hand, has exhibited a propensity to commit murder which will probably constitute a continuing threat to society. Idaho Code § 19-2515 (g) (West 1992). . "Bainbridge's statement had been obtained under questionable circumstances shortly after he was arrested and in custody when he was first confronted with the full impact of the state’s accusation that he was guilty of murder. Few aspects of the criminal law are more familiar than the phenomenon of co-defendants who implicate each other.... Bainbridge had every reason to lie.” State v. Sivak (Sivak III), 119 Idaho 320 , 806 P.2d 413, 424 (1990) (Bistline, J., dissenting); see also State v. Sivak (Sivak I), 105 Idaho 900 , 674 P.2d 396, 414 (1983) (Bistline & Huntley, JJ., dissenting) ("How a district judge could divine from the two sources of evidence, to wit, the sworn and unsworn statements of Sivak as against the unsworn statements of Bainbridge, who was telling the truth, if either, is beyond my powers of comprehension.”). . Having found Napue error, we need not address Sivak’s Brady claims. See Jackson, 513 F.3d at 1076 (setting forth two-step procedure for considering related Napue and Brady claims). . We assume, as the Supreme Court did in Schiro , that in capital-sentencing proceedings, "collateral estoppel could bar the use of [an] ... aggravating circumstance” that is inconsistent with the jury's verdicts. Id. at 232 , 114 S.Ct. 783 . The Eleventh Circuit appears to be the only circuit court to reach this issue. Delap v. Dugger, 890 F.2d 285, 316 (11th Cir.1989) (holding that Ashe's collateral estoppel "principle properly extends to the sentencing phase of a capital trial”). . Neither Idaho law nor .the federal constitution require jury unanimity regarding whether Sivak was guilty as a principal or as an aider and abettor. See supra at 913 n.4. . The Code has subsequently been amended to state: "The murder was committed in the perpetration of, or attempt to perpetrate, arson, rape, robbery, burglary, kidnapping or mayhem and the defendant killed, intended a killing, or acted with reckless indifference to human life.” Idaho Code § 19-2515 (9)(g). . In State v. Windsor, 110 Idaho 410 , 716 P.2d 1182 (1985), the court upheld the death penalty due to the jury’s finding that the defendant "intentionally murdered [the victim] while perpetrating or attempting to perpetrate a burglary.” Id. at 1191 (quoting jury instructions; emphasis added by Idaho Supreme Court). Despite the lack of a premeditation finding, the court held that the jury’s finding of intent was sufficient to satisfy both Idaho law’s statutory "specific intent to kill” aggravating factor and the Eighth Amendment ”inten[t] to kill” requirement set forth in Enmund v. Florida, 458 U.S. 782, 798 , 102 S.Ct. 3368 , 73 L.Ed.2d 1140 (1982). Id. at 1191-92. Similarly, in State v. Dunlap, 125 Idaho 530 , 873 P.2d 784 (1993), the court held that a guilt-phase finding of "express malice” is sufficient to establish a penalty-phase finding of specific intent to kill, because express malice exists "when there is manifested a deliberate intention to unlawfully take away a life.” Id. at 788 (citing Idaho Code § 18-4002 ). The Dunlap court added that "implied malice” was not sufficient to establish intent to kill, as implied malice exists "when the circumstances attending the killing show an abandoned and malignant heart.” Id. (citing Idaho Code § 18-4002 ). Dunlap thus establishes that something more than implied malice (i.e., “abandoned and malignant heart” murder) is required to satisfy the "specific intent to kill” aggravating factor, and (hat express malice (i.e., a deliberate and intentional killing) is sufficient. The Idaho courts’ approach is consistent with the traditional common law distinction between first- and second-degree murder: an intentional killing will support a second-degree murder conviction, and if the intentional killing is done with premeditation and deliberation, a first-degree murder conviction is appropriate. See 2 Charles E. Torcía, Wharton's Criminal Law § 142 (15 ed. 2010 Supp.) ("Although an intent to kill, without more, may support a prosecution for common law murder, such a murder ordinarily constitutes murder in the first degree only if the intent to kill is accompanied by premeditation and deliberation.”); 2 Wayne R. LaFave et al., Substantive Criminal Law § 14.7(e) (2d ed. 2008) (”[I]ntent-to-kill murder without the added ingredients of premeditation and deliberation is second degree murder.”); accord State v. Pina, 149 Idaho 140 , 233 P.3d 71, 74-78 (2010) (interpreting Idaho criminal law in light of statutory text and common law). . Because the judge did not define the terms "willful,'' "deliberate,” and "premeditated,” these terms "must be given their plain, nontechnical meanings.” State v. Henry, 138 Idaho 364 , 63 P.3d 490, 493 (App.2003). Here, the jury’s understanding of these undefined terms is best understood in light of counsel's closing arguments. See Schiro, 510 U.S. at 233-35 , 114 S.Ct. 783 (relying on arguments of counsel to determine collateral estoppel effect of jury verdict). Defense counsel explained that "willful, premeditated, [and] deliberate murder ... doesn’t require that you deliberate about it for any particular amount of time, but you have to meditate about it and think about it.” Counsel also explained that premeditation and deliberation required "that [Sivak] thought about it and decided to do it before it happened." In rebuttal, the State observed that "[i]f a woman who has been stabbed 20 times or so and shot five times is not evidence of premeditation and malice, then premeditation and malice don’t exist.” The jury presumably followed the attorneys’ explanations, which were consistent with the plain meaning of the relevant terms, with Idaho law, and with the facts in the record. The plain meaning of "willful” is "done deliberately^] not accidentally] or without purpose.” Webster's Third New International Dictionary 2617 (2002). The plain meaning of “deliberate” is "consideration of effects and consequences” and “awareness of the implications or consequences of one’s actions.” Id. at 596. The plain meaning of "premeditation” is "previous deliberation.” Id. at 1789. These definitions are consistent with the Idaho” Supreme Court's explanation that willfulness is "the intent to take life,” premeditation is "conceived beforehand,” and deliberation is "done with reflection.” Aragon, 690 P.2d at 298 . Consistent with defense counsel’s statements to the jury, the Idaho Supreme Court has also explained that "[p]remeditation does not require an appreciable space of time between the intention to kill and the killing; rather, it may be as instantaneous as two successive thoughts of the mind.” State v. Olin, 103 Idaho 391 , 648 P.2d 203, 210 (1982). . After the Court handed down O’Dell, we discussed Gardner at some length in Correll v. Stewart, 137 F.3d 1404 (9th Cir.1998), without relying on the Court's intervening discussion in O’Dell. Correll does not constitute binding circuit precedent regarding the proper application of Gardner . Our holding in Correll rested on the petitioner’s failure to show cause and prejudice to overcome his procedural default in state court; we did not address the merits of the Gardner claim. Correll, 137 F.3d at 1415-16 .