A California jury found Petitioner Bruce Wayne Morris guilty of first-degree murder and robbery, and he was sentenced to death in 1987. We have considered his petition for a writ of habeas corpus, and a related mandamus petition, on three previous occasions and have already vacated Petitioner’s death sentence and ordered a new penalty-phase trial. In this fourth appeal, we consider the last two remaining guilt-phase issues: alleged failure of the prosecution to turn over material exculpatory evidence in violation of Brady v. Maryland, 373 U.S. 83 , 83 S.Ct. 1194 , 10 L.Ed.2d 215 (1963), and alleged presentation of perjured testimony in violation of Mooney v. Holohan, 294 U.S. 103 , 55 S.Ct. 340 , 79 L.Ed. 791 (1935) (per curiam), and Napue v. Illinois, 360 U.S. 264 , 79 S.Ct. 1173 , 3 L.Ed.2d 1217 (1959). For the reasons that we explain below, we affirm Petitioner’s convictions. The case is remanded with instructions to grant the writ as to the penalty subject to the state’s retrying the penalty phase within a reasonable time.

FACTUAL BACKGROUND

A. The Crimes and the Trial

In 1985, Petitioner, his girlfriend Avette Barrett, and Barrett’s sister Allison Eck-strom hitchhiked from Sacramento to Lake Tahoe, California. The victim, Rickey Van Zandt, picked them up in the Lake Tahoe area. They drove to a campsite some miles north of Lake Tahoe. Petitioner, Barrett, and Eckstrom discussed stealing Van Zandt’s van, and Barrett apparently suggested that Petitioner kill him. The prosecution’s theory was that while Van Zandt was sleeping, Petitioner hit him in the head with a rock approximately 13 times, then took his body from the van and dumped it down an embankment. Upon discovering that Van Zandt was still alive, Petitioner beat him several times with a stick. Petitioner, Barrett, and Eckstrom then cleaned the van and burned some bloody clothing and blankets. They drove through several states, making purchases with the victim’s credit cards. In Nebraska, they picked up a hitchhiker named Tom Logan. Petitioner confessed to Logan that he had killed Van Zandt. Logan fled and called the police, who arrested Petitioner, Barrett, and Eckstrom the next day. Among the items seized by police after the arrest were Petitioner’s blood-spattered jeans.

Petitioner confessed to police that he had murdered Van Zandt. Later, while Petitioner was in custody, he sent a letter to Barrett stating, “I’ve killed once for you, and if I have to I’ll do it again!!! And you know that I can, and I don’t need a rock to do it either.” Petitioner also confessed to fellow inmates that he had murdered Van Zandt.

At his trial, Petitioner claimed innocence. He testified that he did not see the killing; rather, he went fishing and re turned to the van to find Barrett and Eckstrom with blood on their clothing. According to Petitioner’s testimony, Barrett told him that Van Zandt had tried to rape her, and Eckstrom confessed to having killed him. Petitioner said that he then went to the van, found that Van Zandt was still alive, pulled him out, and moved him to the embankment. Later, Eckstrom saw that Van Zandt was moving and hit him with a stick. Petitioner testified that he falsely confessed to the murder in order to protect Barrett, who he believed was pregnant with his child, and Eckstrom.

Barrett and Eckstrom testified at trial that Petitioner had killed Van Zandt. Both Barrett and Eckstrom were thoroughly cross-examined about inconsistent statements to police and prosecutors; both contradicted themselves on the stand; and both admitted to having lied in the past.

Despite his testimony, the jury convicted Petitioner of first-degree murder and robbery.

B. The Roberts Letter

Barrett sent a letter to her mother, Michele Roberts, on November 1,1985, which was well before Petitioner’s 1987 trial. On November 6, 1985, Roberts forwarded the letter to Phil Lowe, the Sierra County District Attorney who was originally prosecuting the case against Petitioner. Her cover letter to Lowe contained the following statement: “I suppose Pete V. should see the letter also as I received a letter from him (Pete) saying Avette was saying Allison was as guilty as she and Bruce.” The letter from Barrett to Roberts was turned over to the defense before trial and is not at issue in this appeal, but the cover letter from Roberts to Lowe was not turned over until 2004.

At the evidentiary hearing before the district court on federal habeas, Petitioner’s trial counsel, Tom Condit, testified that he “didn’t think [Roberts’ trial testimony] was very effective for our side, because she asserted that the inconsistencies that her daughter had made went to an issue that was not important.” He explained that, if he had possessed the Roberts letter, he would have “confronted her with this information and I think possibly gotten her to admit that there was more.” Overall, Condit opined that the letter “would have added support to our contention that Avette and Allison were the guilty parties in the killing of Rickey Van Zandt,” and “it would have altered the course of our investigation” in that the defense would have interviewed Villareal. Condit further asserted that the letter could have been used to impeach Barrett because Barrett’s testimony indicated that she was protecting Eckstrom, while the letter said that Eckstrom was guilty.

C. The Gumz Status Report

At some point after Barrett testified against Petitioner at his trial, Diane Gumz, a legal assistant at the Attorney General’s Office, prepared a routine status report on Barrett’s case. In relevant part the report reads: “Defendant [Barrett] perjured herself at trial. Prelim set for 6/24/87 was postponed until court transcripts were received to determine exactly what defendant said.” In the “future action” section of the report, Gumz made a notation reading: “Pre-prelim 7/22/87 (for determination of Barrett’s perjury).” This report was not turned over to the defense until 2004.

Who saw the Gumz report at the time, and what response (if any) it prompted, is something of a mystery. The prosecution in Petitioner’s case was special prosecutor, Gary Rossi. Rossi died in 2002. Robert Marshall, who hired Rossi, testified on behalf of the state at the district court evi-dentiary hearing. Marshall served as acting District Attorney of Sierra County for approximately six months starting around November 1986, and then returned to the California Attorney General’s office, which had supervisory authority over Barrett’s case (because the new District Attorney was Barrett’s former defense lawyer). Gumz testified that she got the information for status reports from reading documents associated with the case and talking to the attorney assigned to it. She named Marshall as the assigned attorney in the Barrett case, but she did not remember his ever having used the word “perjury” in discussing the case.

Marshall testified that he recalled little about the Morris or Barrett eases and that he did not recall having seen the Gumz status report. He testified that he did not start an investigation into alleged perjury by Barrett at Petitioner’s trial, nor did he report it to his superiors. He did not inform Petitioner’s defense lawyers about the Gumz status report.

According to Condit, nobody on the prosecution team ever told him that they suspected Barrett had committed perjury. If the Gumz status report, and its notation indicating- that Barrett had perjured herself, had been disclosed to him, Condit said, he would have asked for a new trial.

DISCUSSION

A. The Roberts Letter

To establish a Brady violation, the defendant must show that exculpatory or impeaching evidence was suppressed by the state, either willfully or inadvertently, resulting in prejudice. Strickler v. Greene, 527 U.S. 263, 281-82 , 119 S.Ct. 1936 , 144 L.Ed.2d 286 (1999). With respect to the Roberts letter, we conclude that the government’s failure to disclose it did not constitute a Brady violation. The letter is only marginally exculpatory or impeaching, and Petitioner was not prejudiced by the government’s failure to turn it over.

The Roberts letter was, at best, minimally exculpatory. It did not say that Petitioner was not guilty, or that he was any less guilty than Barrett or Eckstrom. Instead, it suggested that they were all equally guilty, although it failed to explain why that might be so. The letter does not suggest that Barrett and Eckstrom, but not Petitioner, struck Van Zandt.

Moreover, the letter had limited impeachment value. It is unlikely that de fense counsel could have confronted Barrett and Eckstrom with the letter because it contains triple hearsay. Assuming that counsel could have overcome that hurdle, the letter, at most, might have lessened the credibility of Barrett’s and Eckstrom’s testimony, because they maintained that Petitioner was entirely responsible for the killing.

Even if the Roberts letter had some exculpatory value, Petitioner suffered insufficient prejudice from the government’s failure to produce it. In order to make out a Brady violation, the evidence must be material, which means there must be “a reasonable 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), quoted in United States v. Alvarez, 86 F.3d 901, 904 (9th Cir.1996). The Roberts letter fails to meet that standard for two reasons.

First, the defense already possessed, and presented at trial, significant impeachment evidence against Barrett and Eck-strom. Both admitted on the stand that they had lied to police and prosecutors. Even the prosecutor conceded in his closing argument that they were not reliable: “As to the testimony of Miss Barrett and Miss Eckstrom, I think[it] was very obvious that both girls were not totally honest. ... I have some problems with some of their testimony in this courtroom.” Thus, insofar as the Roberts letter would have shown that Barrett had made a statement inconsistent with her testimony at trial, it would have been cumulative. See United States v. Marashi, 913 F.2d 724, 732 (9th Cir.1990) (holding that where disclosure of impeachment evidence would not have enabled counsel to further discredit the witness, the evidence was “merely cumulative” and did not give rise to a Brady violation).

Second, there was compelling evidence of Petitioner’s guilt. He had admitted to several different people that he had murdered Van Zandt. See Arizona v. Fulminante, 499 U.S. 279, 296 , 111 S.Ct. 1246 , 113 L.Ed.2d 302 (1991) (noting that a defendant’s confession may be the most probative and damaging evidence); Morris II, 273 F.3d at 836-37 (holding that any possible error in admitting Barrett’s and Eck-strom’s testimony was harmless because Petitioner’s admissions were the “cornerstone” of the state’s case). Additionally, physical evidence, including Petitioner’s blood-spattered jeans, corroborated those confessions. In the circumstances, there simply is no reasonable likelihood that, had the defense known about the Roberts letter, the result of the trial would have been different.

Petitioner counters that he was prejudiced by the government’s failure to turn over the Roberts letter because it “revealed the existence of a credible, independent witness.” In other words, had Petitioner known about the letter, he argues, he would have interviewed Pete Villareal. But Petitioner’s lawyers and investigator already had an incentive to speak to Villa-real. Their theory was that Barrett and Eckstrom were the killers, so they had every reason to contact Barrett’s probation officer, with whom she might have discussed the crime or acted as if she had something to hide.

Petitioner’s lawyer also claims that the letter would have induced him to contact anyone to whom Barrett had talked, from the time of the crime through the time of the trial. But, as we will discuss later in this opinion, the defense team did interview several people to whom Barrett had talked in prison, and the defense offered testimony from them at trial about Barrett’s self-incriminating statements. Petitioner identifies no additional individuals who would have been contacted had the Roberts letter been disclosed.

In summary, we hold that the government’s failure to turn over the Roberts letter did not constitute a Brady violation.

B. The Gumz Status Report

1. Under Brady

Petitioner argues, first, that the government’s failure to turn over the Gumz status report gave rise to a Brady violation. As an initial matter, we consider whether the Gumz status report could constitute Brady material.

The Gumz status report contains a statement of the prosecutor’s (or, more accurately, an agent of the prosecutor’s) opinion about something that happened at Petitioner’s trial. The Supreme Court has not decided whether the government’s work product must be disclosed under Brady. See Goldberg v. United States, 425 U.S. 94 , 98 n. 3, 96 S.Ct. 1338 , 47 L.Ed.2d 603 (1976) (reserving this question). Nor has this court squarely addressed the issue. In Paradis v. Arave, 240 F.3d 1169, 1173 (9th Cir.2001), the petitioner raised a Brady claim with respect to the withholding of notes made by the prosecutor. The petitioner conceded that some notes were “taken by [the prosecutor] at [the petitioner’s] own trial and, therefore, are not Brady material.” We held that other notes — which recorded the opinion of an expert witness — were Brady material, but those notes contained underlying factual information. Id.

The Eleventh Circuit has considered this question and has concluded that Brady does not require a prosecutor to disclose to the defense most “opinion work product,” that is, material encompassing only an attorney’s mental impressions or legal theories. Williamson v. Moore, 221 F.3d 1177, 1182 (11th Cir.2000); cf. Mincey v. Head, 206 F.3d 1106 , 1133 n. 63 (11th Cir.2000) (citing district courts, a state court, and a commentator that have addressed this question, and noting that most have concluded that there is no automatic exemption from disclosure of work product under Brady). We agree with the Eleventh Circuit’s approach, which is consistent both with Paradis and with the purpose of Brady .

The animating purpose of Brady is to preserve the fairness of criminal trials. 373 U.S. at 87 , 83 S.Ct. 1194 . However, fairness does not encompass an obligation on the prosecutor’s part to reveal his or her strategies, legal theories, or impressions of the evidence. The Brady rule is not meant to “displace the adversary system”; “the prosecutor is not required to deliver his entire file to defense counsel, but only to disclose evidence favorable to the accused, that, if suppressed, would deprive the defendant of a fair trial.” Bagley, 473 U.S. at 675 , 105 S.Ct. 3375 (emphasis added; footnote omitted). Extending the Brady rule to opinion work product would greatly impair the government’s ability to prepare for trials. See Williamson, 221 F.3d at 1182 (noting that if work product were accessible, “ ‘much of what is now put down in writing would remain unwritten’ ” (quoting Hickman v. Taylor, 329 U.S. 495, 511 , 67 S.Ct. 385 , 91 L.Ed. 451 (1947))). Thus, in general, a prosecutor’s opinions and mental impressions of the case are not discoverable under Brady unless they contain underlying exculpatory facts.

The Gumz status report is best characterized as a statement of the prosecutor’s opinion or a recording of his thoughts about whether Barrett testified truthfully enough to receive the benefit of her plea bargain (given the inconsistencies in her testimony, which were apparent to defense and prosecution alike). So understood, it is not Brady material. Even if the Gumz status report referred to exculpatory facts unknown to the defense, and thus is Brady material, Petitioner was not prejudiced by the government’s failure to turn the status report over, for the reasons discussed in the next section.

2. Under Mooney-Napue

Petitioner’s final argument is that the Gumz status report establishes that the government knowingly presented perjured testimony and failed to disclose or correct it. The prosecution was obliged to, but did not, investigate the possibility that a government witness had perjured herself. We conclude, however, that Petitioner suffered no prejudice, because the witness’ testimony was thoroughly discredited at trial and there was independent, compelling evidence of Petitioner’s guilt.

Under the Mooney-Napue line of cases, a conviction will be reversed if two conditions are met: first, the prosecution knowingly presented false evidence or testimony at trial; and, second, it was material, that is, there is a reasonable likelihood that the false evidence or testimony could have affected the judgment of the jury. United States v. Agurs, 427 U.S. 97, 103 , 96 S.Ct. 2392 , 49 L.Ed.2d 342 (1976); Hayes v. Brown, 399 F.3d 972, 984 (9th Cir.2005) (en banc). Further, a prosecutor must correct false evidence whenever it appears. Napue, 360 U.S. at 269 , 79 S.Ct. 1173 .

Gumz testified at the evidentiary hearing that she wrote the status report in June or July of 1987. The jury in Petitioner’s trial reached a guilt-phase verdict on June 22, 1987, and Petitioner was sentenced to death on July 17, 1987. It is unclear whether the Gumz status report was written while Petitioner’s trial was still ongoing. The Gumz status report does not support an inference that the prosecutor knew ahead of time that Barrett would lie, but only an inference that the prosecutor concluded after the fact that she had.

In this respect, Petitioner presents an atypical Mooney-Napue claim. Often it is clear that false evidence has been presented. See, e.g., Giglio v. United States, 405 U.S. 150, 152-53 , 92 S.Ct. 763 , 31 L.Ed.2d 104 (1972); Hayes, 399 F.3d at 980 . Here, however, there is only an allegation or suspicion that a witness lied, but no followup investigation or explanation by the state. As a result, we must decide wheth er the prosecution had a duty to investigate an allegation or suspicion of perjury, separate from its duty to disclose perjury that has definitely taken place.

When a prosecutor suspects perjury, the prosecutor must at least investigate. The duty to act “is not discharged by attempting to finesse the problem by pressing ahead without a diligent and good faith attempt to resolve it. A prosecutor cannot avoid this obligation by refusing to search for the truth and remaining willfully ignorant of the facts.” Northern Mariana Islands v. Bowie, 243 F.3d 1109, 1118 (9th Cir.2001). This principle is supported by Mooney, Nwpue, and their progeny. In Mooney , the Supreme Court held that a conviction obtained through “deliberate deception of court and jury ... is ... inconsistent with the rudimentary demands of justice.” 294 U.S. at 112 , 55 S.Ct. 340 . In Napue , the Court elaborated on the prosecutor’s role, noting that “ ‘the district attorney has the responsibility and duty to correct what he knows to be false and elicit the truth.’ ” 360 U.S. at 270 , 79 S.Ct. 1173 (quoting People v. Savvides, 1 N.Y.2d 554 , 154 N.Y.S.2d 885 , 136 N.E.2d 853, 854 (1956)). The Court has emphasized that the presentation of false evidence involves “a corruption of the truth-seeking function of the trial process.” Agurs, 427 U.S. at 103 , 96 S.Ct. 2392 . This truth-seeking function cannot be fulfilled when the state, knowing that a witness may have perjured herself, proceeds without conducting an investigation to ensure that a new trial is not warranted. The duty to investigate flows from the “constitutional obligation of the State and its representatives to collect potentially exculpatory evidence, to prevent fraud upon the court, and to elicit the truth.” Bowie, 243 F.3d at 1117 .

Ward Campbell, who argued the case on behalf of the State of California, stated at oral argument that he discovered the Gumz status report in 1996 but did not conduct an investigation and did not disclose the status report to Petitioner’s counsel until 2004, during discovery in advance of the evidentiary hearing.

The prosecutor’s failure to act in a timely manner means that Rossi, who could have explained what the Gumz status report really meant, rather than leaving the parties (and us) to speculate, was unavailable by the time of the evidentiary hearing. He died in 2002, and the status report was not disclosed to Petitioner until two years later. Marshall, the only other government lawyer who could have recounted what Gumz was told that led her to write that Barrett committed perjury, testified that he had no relevant recollection of the case. In these circumstances, we must conclude that Barrett’s testimony at trial was false in some respect. But the nature and extent of the perjury is unknown and, at this point, unknowable.

If the Gumz status report referred simply to the fact that Barrett changed her story on the stand, as the state contends, then the defense knew it already. Barrett’s inconsistent stories provided fertile ground for cross-examination at trial. If the state had conducted an investigation and formed a good-faith belief that the Gumz status report was referring only to the already-known inconsistencies in Barrett’s testimony that were aired at trial, there would have been nothing for the prosecution to disclose and no duty under Mooney and Napue .

Had an investigation revealed that Barrett’s perjury went beyond that, however, then the state would have been bound to correct the falsehood. Even if the Gumz status report referred to something more, though, there is no evidence to suggest that Barrett-if completely truthful-would have corroborated Petitioner’s version of events, that is, would have said that Peti tioner did not participate actively in killing Van Zandt.

There was some evidence presented at trial supporting Petitioner’s story. One of Barrett’s fellow inmates, Crane, testified that Barrett told her that Petitioner was a “stupid son of a bitch” for taking a murder rap for her. According to Crane, Barrett said that she had hit the victim with a stick. Another jailhouse acquaintance of Barrett, named Baker, testified that Barrett had bragged about Petitioner’s taking the fall for her. But Baker also testified that Barrett told her that Petitioner had done the actual killing. A final jailhouse witness, Cicero, testified that Barrett told her that she had hit the victim with a stick.

That evidence, although favorable to Petitioner, does not corroborate his testimony. Rather, it conflicts with his theory that he was absent from the scene and that it was Eckstrom who said that she had beaten and killed Van Zandt. Also, nothing in Barrett’s statements suggests that anyone except Petitioner hit the victim with a rock. Barrett has never testified that Petitioner is innocent of the murder, despite several opportunities to do so; Petitioner did not call her as a witness at the evidentiary hearing. There is no evidence that Barrett ever confessed to any law enforcement officer or prosecutor that she had inflicted any injury on Van Zandt. We cannot speculate, contrary to anything in the record, that Barrett would have testified that Petitioner did not kill Van Zandt had the perjury been revealed in a timely fashion and had he received a new trial.

Thus, with respect to the first part of the Mooney-Napue analysis, we hold that false testimony was presented at Petitioner’s trial, but that there is insufficient evidence to permit a conclusion that correcting the falsehood (whatever it may have been) would have provided support for his claim of actual innocence. The second part of the Mooney-Napue analysis requires us to consider whether the prosecutor’s failure to investigate and disclose Barrett’s perjury resulted in prejudice. If not, we will not reverse the conviction. United States v. Zuno-Arce, 339 F.3d 886, 889 (9th Cir.2003). The test for prejudice for a Mooney-Napue claim is the same as that for materiality in a Brady claim. Silva v. Brown, 416 F.3d 980 , 986 n. 1 (9th Cir.2005). That is, we must decide whether, despite the use of perjured testimony, Petitioner received a “ ‘trial resulting in a verdict worthy of confidence.’ ” Hall v. Dir. of Corr., 343 F.3d 976, 984 (9th Cir.2003) (per curiam) (quoting Kyles v. Whitley, 514 U.S. 419, 434 , 115 S.Ct. 1555 , 131 L.Ed.2d 490 (1995)).

Petitioner argues for a different standard of review. He urges that when perjured testimony has been presented, “ ‘reversal is virtually automatic.’ ” Bowie, 243 F.3d at 1114 (quoting United States v. Wallach, 935 F.2d 445, 456 (2d Cir.1991)). Petitioner takes that phrase out of context; the Bowie court did state and apply the prejudice/materiality standard. Id. at 1116, 1122-23 . There is no rule of automatic reversal. See Hall, 343 F.3d at 983 (“A new trial is not automatically required when false evidence is discovered.”).

The state argues that the reference to Barrett’s perjury in the Gumz status report was an “unremarkable” reference to the well-aired contradictions and lies that were exposed by cross-examination at trial and, accordingly, that Barrett’s perjury was not material. But, as we have explained, the absence of a contemporaneous and full investigation by the state requires us to assume that the “perjury” referred to something more.

We are persuaded that Petitioner suffered insufficient prejudice from Barrett’s perjury, for much the same reasons that we conclude he suffered insufficient prejudice from the failure to turn over the Roberts letter. First, the jury already was shown that Barrett was completely inconsistent and dishonest, that is, that she was a liar in at least some respects. Second, even if the jury had been informed that Barrett’s testimony was perjured and should be disregarded entirely, or even if there had been a new trial at which Barrett did not testify, we are persuaded beyond any doubt that the outcome of the trial would have been the same. Petitioner testified in his own defense and implicated Barrett and Eckstrom in the crime, but the jury disbelieved him despite being told, by both defense counsel and the prosecutor, that Eckstrom and Barrett were unreliable. The physical evidence and Petitioner’s own, multiple confessions pointed to his guilt. Therefore, we reject Petitioner’s Mooney-Napue claim.

Convictions AFFIRMED; case REMANDED with instructions to grant the writ with respect to the penalty subject to the state’s retrying the penalty phase within a reasonable time.

FERGUSON, Circuit Judge, concurring: The Roberts letter at issue in this appeal is the latest piece of evidence that calls into question the administration of the death penalty in this case. I write separately to underscore the prosecutor’s abuse of his discretion in singling out the Petitioner for the death penalty, when it is the state’s position that the three defendants are equally guilty of the felony murder of Van Zandt. As long as a prosecutor’s discretion in seeking the ultimate penalty — death—remains thus unbridled, the administration of the death penalty in the United States will violate the guarantees of due process and freedom from cruel and unusual punishment enshrined in the Constitution.

In 2004, the government informed this Court of its position that the three defendants in this case, Morris, Barrett, and Eckstrom, are equally guilty of the felony murder of Van Zandt. Morris v. U.S. Dist. Court (In re Morris), 363 F.3d 891 , 895 n. 1 (9th Cir.2004) (per curiam) (Ferguson, J., concurring specially). The Roberts letter at issue in this appeal provides further evidence that Eckstrom is as guilty of Van Zandt’s murder as are Morris and Barrett. The Roberts letter is thus material to Morris’s penalty phase and should have been disclosed to the defense. Instead, the prosecution suppressed the letter and dropped all charges against Eck-strom, a minor, in return for her testimony against Morris. The prosecution also agreed to charge Barrett only for grand theft auto in exchange for her testimony against Morris. Of the three co-eonspira-tors, Morris alone was prosecuted for murder, and he alone was singled out for the death penalty.

Barrett was ultimately sentenced to three years’ imprisonment for violating California Vehicle Code § 10851. Eck-strom was never prosecuted. Morris was sentenced to death.

II.

Over thirty years ago, the Supreme Court declared that death is different. The death penalty must be imposed fairly, without prejudice or whim, or it may not be imposed at all. Furman v. Georgia, 408 U.S. 238 , 92 S.Ct. 2726 , 33 L.Ed.2d 346 (1972); see Gregg v. Georgia, 428 U.S. 153, 188 , 96 S.Ct. 2909 , 49 L.Ed.2d 859 (1976) (interpreting Furman). In the years since Furman , legislatures and courts have struggled to meet this daunting challenge, yet “the death penalty remains fraught with arbitrariness, discrimination, caprice, and mistake.” Callins v. Collins, 510 U.S. 1141, 1144 , 114 S.Ct. 1127 , 127 L.Ed.2d 435 (1994) (Blackmun, J., dissenting from denial of writ of certiorari). The problems today are not identical to those of thirty years ago. Rather, those problems that were originally “pursued down one hole with procedural rules and verbal formulas have come to the surface somewhere else, just as virulent and pernicious as they were in their original form.” Id. Even as the courts have tried to limit the jury’s discretion to impose the death penalty, “discrimination and arbitrariness at an earlier point in the selection process nullify the value of later controls on the jury.” DeGarmo v. Texas, 474 U.S. 973, 975 , 106 S.Ct. 337 , 88 L.Ed.2d 322 (1985) (Brennan, J., dissenting from denial of writ of certiorari).

Here, the prosecutor’s unbridled discretion to single out Morris for prosecution under the death penalty, when the guilt is equally spread among his co-defendants, is a rank example of “arbitrariness at an earlier point in the selection process.” Id. This sort of gross disparity in the treatment of equally guilty defendants “highlights the utter failure of the elaborate sentencing schemes approved by the [Supreme] Court in Gregg and its companion cases to meaningfully limit the arbitrary infliction of death by the States.” Id. at 974-75, 106 S.Ct. 337 . Such arbitrariness in the administration of the death penalty violates the Eighth Amendment and the Due Process Clause. See Gregg, 428 U.S. at 188-89 , 96 S.Ct. 2909 (arbitrary infliction of the death penalty violates the Eighth Amendment); United States v. Redondo-Lemos, 955 F.2d 1296, 1298-99 (9th Cir.1992), overruled on other grounds by United States v. Armstrong, 48 F.3d 1508 (9th Cir.1995), rev’d 517 U.S. 456 , 116 S.Ct. 1480 , 134 L.Ed.2d 687 (1996) (arbitrary charging decisions violate due process).

As a remedy to the constitutional violations in this case, I reiterate my position in In re Morris, 363 F.3d at 896 (Ferguson, J., concurring specially): Morris’s sentencing jury must be instructed that it may consider, as a mitigating factor, the fact that the prosecution pursued substantially more lenient punishment against Morris’s equally guilty co-defendants. Such instructions provide a mechanism for directing and limiting the prosecutor’s unbridled discretion to seek the death penalty. Through its regular functioning, the capital jury can account for and remedy an unconstitutionally arbitrary infliction of the death penalty' — by removing the threat of that ultimate penalty. See United States v. Bin Laden, 156 F.Supp.2d 359, 369 (S.D.N.Y.2001) (finding that in enacting the statute under which the defendant was prosecuted, Congress intended for juries to consider, as a mitigating factor, that “another defendant or defendants, equally culpable in the crime, will not be punished by death” so as to “provide[ ] jurors with a means of improving the likelihood that the death penalty would not be administered in an arbitrary or random manner”); see also 18 U.S.C. § 3592 (a)(4).

Cabining prosecutorial discretion by permitting capital juries to consider all manners of information in the form of mitigation evidence accords with Supreme Court precedent regarding the process that must be afforded capital defendants during their penalty phase. The Supreme Court has held that the Eighth and Fourteenth Amendments require that capital juries be permitted to consider, as mitigating factors, any aspects of a defendant’s character or of “the circumstances of the offense that the defendant proffers as a basis for a sentence less than death.” Lockett v. Ohio, 438 U.S. 586, 604 , 98 S.Ct. 2954 , 57 L.Ed.2d 973 (1978). In keeping with Lockett , the sentencing jury in this case must be permitted to consider the prosecutor’s grossly disparate treatment of Morris’s equally guilty co-defendants as a circumstance of the offense justifying a sentence less than death.

As the reasoning in my earlier concurrence makes clear, see In re Morris, 363 F.3d at 896 (Ferguson, J., concurring specially), providing additional mitigation instructions to the capital jury as a means of cabining prosecutorial discretion also avoids a separation-of-powers issue. This Court in Redondo-Lemos determined that although an arbitrary exercise of prosecu-torial discretion violates the Due Process Clause, there is no judicial remedy available because courts generally may not inquire into prosecutors’ decision-making processes. 955 F.2d at 1299 . The Redon-do-Lemos majority concluded that permitting courts to make such an inquiry would impermissibly entangle the judicial branch “in the core decisions of another branch of government.” Id. at 1300 . In this case, introducing evidence of the sentences received by Morris’s co-defendants would not require the courts to investigate the internal chai-ging decisions of the prosecutor. Instead, it would compel the prosecution to live with the charging decisions it made: if the jury found that the exercise of discretion in seeking the death penalty against Morris was arbitrary, it would be free to use that fact as a mitigating factor.

This Court has ordered a new penalty phase in this case. Morris v. Woodford, 273 F.3d 826, 843 (9th Cir.2001). During this second penalty phase, the jury must be permitted to consider, as a mitigating factor in its determination of whether to impose the death penalty, the government’s admission that it singled out Morris for capital punishment among three equally guilty perpetrators.

. The prior decisions are Morris v. Woodford, 229 F.3d 775 (9th Cir.2000) ("Morris I”); Morris v. Woodford, 273 F.3d 826 (9th Cir. 2001) ("Morris 17”); and Morris v. U.S. Dist. Court (In re Morris), 363 F.3d 891 (9th Cir.2004) (per curiam) ("Morris III"). . In Moms I, we granted a certificate of ap-pealability on three issues: whether Petitioner received ineffective assistance of counsel, "whether Petitioner was denied a fair trial by the state's erroneous introduction of excluded evidence,” and whether Petitioner was competent to assist in his own defense. Morris I, 229 F.3d at 781 . We remanded those issues to the district court. Monis II, 273 F.3d at 828, 843 . During the discovery process leading up to the evidentiary hearing, Petitioner received documents that, he contends, show a Brady violation and a Mooney-Napue violation. See Monis III. The district court permitted amendment of the petition for habeas corpus. When the district court conducted the hearing, Petitioner presented evidence concerning, and argued, only those two new claims. After rejecting the two new claims, the district court granted a certificate of ap-pealability on both. Petitioner presented no evidence on the three previously remanded claims, nor are they addressed in his brief to this court. Therefore, those issues have been abandoned. See Harik v. Cal. Teachers Ass’n, 326 F.3d 1042, 1052 (9th Cir.2003) (holding that issues abandoned in the district court will not be considered on appeal). . For more detailed summaries of the facts, see the California Supreme Court's decision on direct appeal, People v. Morris, 53 Cal.3d 152 , 279 Cal.Rptr. 720 , 807 P.2d 949 (1991), and this court's decisions in Monis I and Morris II. ' . “Pete V.” appears to refer to Pete Villareal, Barrett’s probation officer. He was interviewed by Petitioner’s counsel after the Roberts letter was discovered but had no recollection of the case. . Barrett's plea agreement provided that, in return for her complete and truthful testimo ny in all proceedings, the prosecution would dismiss all charges pending against her except "grand theft auto." Originally, the agreement also specified that Barrett’s representation that she had not personally inflicted any injuries on Van Zandt had to be truthful. This condition was deleted before Petitioner’s trial. The government honored its agreement with Barrett. . We review de novo the district court's denial of the petition for habeas corpus. Beardslee v. Woodford, 358 F.3d 560, 568 (9th Cir.2004). Because this petition was filed before the effective date of the Antiterrorism and Effective Death Penalty Act of 1996 ("AED-PA”), pre-AEDPA law applies. See Morris II, 273 F.3d at 830 ; Silva v. Brown, 416 F.3d 980, 985 (9th Cir.2005). Thus, we may set aside the state court conviction only if Petitioner proves that his "detention violates the fundamental liberties of the person.” Hayes v. Brown, 399 F.3d 972, 978 (9th Cir.2005) (en banc) (internal quotation marks omitted). . Mooney originated the rule that a conviction obtained through the use of perjured testimony violates due process. Mooney, 294 U.S. at 112 , 55 S.Ct. 340 ; Hayes, 399 F.3d at 983 . Napue expanded Mooney to encompass false testimony bearing only on the credibility of a witness. Napue, 360 U.S. at 269 , 79 S.Ct. 1173 ; Hayes, 399 F.3d at 983-84 . For convenience, we refer to Petitioner’s argument as a "Mooney-Napue claim.” See United States v. Zuno-Arce, 339 F.3d 886, 889 (9th Cir.2003) (discussing elements of a "Mooney-Na-pue claim”). . Although the Supreme Court has suggested that the presentation of perjured testimony gives rise to a type of Brady claim, see United States v. Agurs, 427 U.S. 97, 103 , 96 S.Ct. 2392 , 49 L.Ed.2d 342 (1976) (noting that the Brady rule applies in three different situations, one of which is "typified by Mooney ”), we view this Mooney-Napue claim as analytically distinct from the Brady claim. We deal not with whether the Gumz status report itself ought to have been turned over to the defense, but with whether there was perjured testimony by Barrett that ought to have been investigated and disclosed by the prosecution. See Banks v. Dretke, 540 U.S. 668 , 690 n. 11, 124 S.Ct. 1256 , 157 L.Ed.2d 1166 (2004) (declining to decide whether a Napue claim must be pleaded separately from a Brady claim). . Petitioner argues that the length of the jury’s deliberation shows that the case was close and that the outcome might have been different had the jury known that Barrett had perjured herself in some way beyond what the jury knew already. See Jennings v. Woodford, 290 F.3d 1006, 1019 (9th Cir.2002). We disagree. It is true that the jury deliberated for approximately a day and a half, but that does not mandate a finding of prejudice. See United States v. Velarde-Gomez, 269 F.3d 1023, 1036 (9th Cir.2001) (en banc) (suggesting that length of deliberations is but one factor among many in the prejudice analysis).