PATRICK E. HIGGINBOTHAM, Circuit Judge:

Colton Pitonyak was convicted of first-degree murder for the death of Jennifer Cave, who was killed in his apartment by a gunshot in the early morning hours of August 17, 2005. Pitonyak filed a federal habeas petition alleging that the prosecution violated Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), by failing to disclose an alleged jailhouse confession to the murder by Laura Hall. The district court denied relief and we granted a certificate of appealability ("COA"). With the benefit of oral argument, we deny the State's motion for reconsideration of the COA and affirm the judgment of the district court.

I.

A.

Jennifer Cave's dismembered body was discovered the evening of August 18, 2005, in the bathtub of Colton Pitonyak's apartment. She had been killed in the early morning hours of the previous day by a single gunshot that passed through her right arm, right lung, and aorta, causing her death within fifteen seconds. Her head and both hands had been severed from her body and were found in a black plastic garbage bag that lay on the floor next to the bathtub. A second bullet was fired post-mortem through the severed neck up into the skull, and there were twenty-nine post-mortem stab wounds on the face, chest, and hand.

Several days later, Mexican police apprehended Pitonyak and Laura Hall at a hotel in Piedras Negras, Mexico, and turned them over to U.S. Marshals on the international bridge. In 2007, a jury convicted Pitonyak of first-degree murder and sentenced him to fifty-five years of imprisonment. At a separate trial later that year, a jury convicted Hall of hindering apprehension and tampering with physical evidence by mutilating Cave's body. She received a ten-year sentence.

Pitonyak and Cave met in the summer of 2004 and became good friends. Pitonyak and Laura Hall met in the spring of 2005. Hall was "in love with" or "obsessed" with Pitonyak. Although he did not reciprocate those feelings, they had sexual relations about once per week through the spring and summer of 2005. Pitonyak, initially a stellar scholarship student at the University of Texas, had been struggling academically since he began seriously abusing drugs and alcohol in 2003. He also occasionally sold drugs, mostly to University of Texas students. As payment for a drug debt, Pitonyak had acquired a Smith & Wesson SW380 pistol. He later sold the pistol before reacquiring it as collateral for loaning his car to a friend of Hall's on August 13, 2005.

Around 6:00 p.m. on August 16, 2005, Pitonyak called Cave because he had not seen her in a while, and he wanted to know what she was doing. Cave returned his call around 9:00 p.m. She was excited because she had been offered a job at a law firm that was to begin the next day. She wanted to go downtown to get a drink and something to eat. Pitonyak had been drinking vodka steadily since 5:00 p.m. and had also taken several Xanax pills. Pitonyak testified that he recalled going to a restaurant with Cave and ordering drinks, but that he remembers nothing else until the following morning.

Later that night, several of Cave's friends talked to Cave and Pitonyak at a bar on Sixth Street in downtown Austin. Close to midnight, the group left to go to a different bar across the street. Pitonyak and Cave trailed behind as they closed their tab. They were last seen at the door of a second bar before they abruptly left without entering and walked down the street shortly after midnight.

No one who encountered Pitonyak and Cave that night gave any hint of a problem or argument between them, and several witnesses testified it was clear the two were not out on a date. Michael Rodriguez, a friend of Cave's, testified that he called her at 10:30 p.m. Cave told him she was going to hang out with Pitonyak, who was "having some issues." Cave and Rodriguez exchanged two text messages at 11:47 p.m. and 11:54 p.m. He also spoke with her twice more that night, once around midnight and once at 1:05 a.m. Rodriguez testified that on the call at midnight, Cave "basically said that [Pitonyak] was a little upset, basically mentioned something about the only people that could help him were people that were in jail." Cave "didn't sound like she was in any kind of problem or issue." Rodriguez last spoke with Cave by phone at 1:05 a.m. as Cave walked with Pitonyak towards her car. On that call, Cave indicated that Pitonyak was upset because he lost his phone and she was going to help him find it. She told Rodriguez that Pitonyak had just tried to break the window of a car and that he was urinating on a vehicle. Rodriguez testified that Cave "didn't sound like she was in trouble or anything," and that "she said she was fine" and did not seem intoxicated.

Nora Sullivan, a good friend of Pitonyak's whose apartment was approximately 100 feet from his, testified that Pitonyak came to her apartment for between 15 and 30 minutes shortly after 3:00 a.m. on Wednesday, August 17. Pitonyak said he had lost his cell phone and asked to borrow Sullivan's phone to place two calls, at 3:27 and 3:28 a.m., to a friend named Evan Ray. Ray did not answer the calls. Pitonyak then told Sullivan that he had been in a gunfight with "some Mexican guys" at his apartment. According to Pitonyak's story, the Mexicans had fired five shots, and Pitonyak had returned two shots. Pitonyak said one shot hit his sofa and implied one of the Mexicans had been hit. Pitonyak asked if he had any blood on him, and Sullivan for the first time noticed a small smear of what appeared to be blood on Pitonyak's forearm. However, Sullivan testified that she thought Pitonyak had made the story up because she had not heard gunshots and could tell he had been drinking. She described Pitonyak as intoxicated but "functioning fine, talking fine." Pitonyak then took a handgun out of his waistline and said he was unloading it as he took the magazine out and put the gun back in his waistline. The two then smoked a cigarette and had a beer together before Pitonyak returned to his apartment.

Cell phone records show that Pitonyak called Hall at 5:59 a.m. and they spoke for 13 minutes. Call records show they spoke again at 6:57 a.m. and 7:24 a.m.1 Pitonyak testified that some time on Wednesday morning, he "went back to the bathroom to use the restroom and [Cave] was laying in the bathtub." He was "scared" and "panicked." He "tried to call [Hall] back and tell her not to come by," but Hall "insisted." Pitonyak testified that he recalled this phone call, Hall's arrival, and the events thereafter.

When Hall arrived, she saw Cave's purse on the ground and asked Pitonyak "what was going on," and Pitonyak took her to the bathroom and showed her Cave's body. Pitonyak and Hall discussed cutting up the body to get rid of it. At 3:00 p.m., Pitonyak went to a hardware store with a shopping list of supplies that Hall had written for him. Telling the shop owner that he needed to cut up a turkey, Pitonyak purchased a hacksaw, shop towels, bathroom tissue, latex gloves, dust masks, 55-gallon drum liners, ammonia, carpet cleaner, and odor eliminator.

That same afternoon, an attorney at the law firm that had hired Cave called Cave's mother, Sharon, to report that Cave had not shown up to work that morning. When Sharon, too, was unable to reach Cave, she obtained from her cell phone provider a list of calls made to and from Cave's cell phone and began calling the numbers on that list. Sharon talked to Rodriguez and learned that Cave had last been seen with Pitonyak. Sharon also talked to Pitonyak, who acknowledged that he had seen Cave the previous night, but claimed they had parted ways downtown and that he did not know where she was. When Sharon called again around 8:30 p.m., she told Pitonyak that police were on their way to Pitonyak's apartment. Pitonyak and Hall threw some clothes in a bag and left the apartment. They stopped by Hall's apartment to pick up a bottle of rum and Hall's clothes before driving south to Del Rio, Texas, where they crossed the international border into Ciudad Acuna, Mexico, around 2:00 a.m. They checked into a hotel there before traveling to another hotel in Piedras Negras, Mexico.

The next day, Thursday, August 18, Sharon and her boyfriend Jim Sedwick drove to Austin. The Austin Police Department informed them that Cave's car had been found parked at the apartment complex where Pitonyak lived. That evening, Sharon, Sedwick, and Cave's sister Vanessa drove to Pitonyak's apartment. After police told them they were unable to help, Sharon and Sedwick unlocked a window to Pitonyak's apartment through a chip in the glass. Sedwick opened it and climbed into the apartment. When he went into the bathroom, he discovered Cave's dismembered body in the bathtub. The family immediately called the police, who arrived at the scene shortly thereafter.

After four days, Mexican police apprehended Hall and Pitonyak in Piedras Negras, Mexico, where they had absconded after fleeing the United States. Authorities turned them over to U.S. Marshalls at the border. Deputy United States Marshall Vincent Bellino testified that when he told Pitonyak he was being arrested, Pitonyak said, "If this is a murder charge, then I know exactly what this is about."

B.

There was no testimony at trial from witnesses who could describe the actual events that took place in Pitonyak's apartment on the night of the murder. Pitonyak denied firing the shot into Cave's decapitated head and stabbing and dismembering the body; he implied that Hall alone had done so. But Pitonyak accepted sole responsibility for Cave's death, while painting an extensive picture of drug and alcohol abuse to show he could not have had the requisite intent for first-degree murder. Pitonyak testified:

Forensic evidence showed the two bullets found in Cave's body were from the Smith & Wesson SW380 pistol that belonged to Pitonyak and was later found in his car. Three cartridge casings fired from that pistol were found in Pitonyak's apartment: two on the coffee table and one in the bathtub under Cave's body. Besides the two bullets found in Cave, the third bullet was never found.

DNA tests showed that neither Pitonyak nor Hall could be excluded as the source of DNA evidence on the grip, slide, and magazine of the gun. DNA on the handle of a machete found in the dishwasher and on two dust masks found on the sink counter in the bathroom was consistent with that of Cave and Pitonyak, but not Hall. A blood stain on a green washcloth was consistent with the DNA profile of Pitonyak but neither Hall nor Cave. A blue towel found on the coffee table in Pitonyak's apartment was not stained with blood but contained both Pitonyak's and Hall's DNA. DNA testing of a faucet knob from Pitonyak's apartment was consistent with Cave and Pitonyak, but not Hall. Blood stains on a pair of men's blue jeans found in Pitonyak's apartment were consistent with Pitonyak's DNA, and Cave's blood was found on a pair of grey men's briefs. A pair of red sweat pants tested positive for blood and contained trace evidence consistent with Hall's DNA. Only Cave's DNA was found on the blade of a hacksaw found lying on her chest; the handle was not tested.

Hall did not testify as a witness for either the defense or the prosecution. Ryan Martindill, an acquaintance of Hall's, testified that there was one night that summer when Hall stayed the night at Martindill's apartment and then asked him in the morning to drive her to her car. Martindill's recollection of the date of that night was uncertain. In his affidavit given nine days after the murder, Martindill stated that a Detective Walker had asked him whether Hall stayed at Martindill's apartment on August 16. Martindill said he did not remember her doing so, and that while Hall did stay on his couch one night, he did not recall what day it was. At Pitonyak's trial, Martindill testified that he did not remember the date, but he "believe[s]" it was the night of the 16th and morning of the 17th. He testified that on the night Hall stayed over, he and Hall were drinking and watching TV before Martindill fell asleep around midnight to 1:00 a.m. He slept on one couch while Hall slept on the other. Martindill slept through the night before Hall awoke him around 7:00 a.m. and asked him to drive her to her car because Pitonyak "had called her and she wanted to go hang out with him." Martindill did so.

Javier Rosales, who worked at a restaurant with Hall in 2006, testified that Hall had told him she had helped cut up a human body, and that she "masterminded" the escape to Mexico. Rosales testified that Hall did not provide any specific details regarding the murder itself.

A jury found Pitonyak guilty of first-degree murder and sentenced him to fifty-five years in prison. Pitonyak appealed. The state appellate court affirmed Pitonyak's conviction and sentence. Pitonyak v. State, 253 S.W.3d 834 (Tex.App.-Austin [3d Dist.] 2008, pet. ref'd). The Texas Court of Criminal Appeals ("TCCA") denied his subsequent petition for discretionary review.

C.

Pitonyak then filed an application for post-conviction review in state court. Ex parte Pitonyak, Application No. WR-73,-118-01 (Nov. 25, 2009). As relevant here, he asserted a claim pursuant to Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), based on a notation in Hall's jail file. Pitonyak argued that the prosecution had violated Brady when it failed to disclose that notation to defense counsel. The notation stemmed from Hall's imprisonment in the Travis County Jail after police arrested her in late August 2005. A jail counselor, Carrie Hoffman, made the following notation as "Remarks" in Hall's jail file on September 16, 2005:

Additionally, under "Prescription," Hoffman noted "Call IM's lawyer." Both appeared on the same page in Hall's jail file under a tab labeled "Medical" and a sub-tab labeled "Treatment."

The state trial court recommended that the TCCA reject Pitonyak's Brady claim. It made findings of fact that evidence at Pitonyak's trial "strongly supported the conclusion that he had killed Cave." It also concluded that as a matter of law, (1) Pitonyak failed to establish that "a mental health professional employed by a county jail has a duty to disclose potentially exculpatory information learned within the scope of a mental health relationship;" and (2) "[i]f Hall made the alleged statement that she committed this offense, and it had been disclosed to the defense, its admission would have had no reasonable impact on this trial given the statement's context, lack of detail or corroboration, and the weight of the other evidence."

Pitonyak filed objections to the trial court's recommended findings. The TCCA adopted the trial court's findings and denied relief "without [a] written order on [the] findings of [the] trial court without [a] hearing." Ex parte Pitonyak, Application No. WR-73,118-01, at cover (Jan. 13, 2010).

Pitonyak then filed a timely petition under 28 U.S.C. § 2254 for federal habeas relief in the district court. The district court denied Pitonyak's petition with prejudice and denied Pitonyak a COA. Pitonyak v. Thaler, Final Judgement, No. A-10-CV-081-LY (W.D. Tex. July 10, 2012). It adopted in full the Report and Recommendation of Magistrate Judge Pitman, who concluded that the undisclosed notation was not material under Brady, and it denied relief "for the reasons stated [in the Report]." Pitonyak v. Thaler, Order on Report and Recommendation, No. A-10-CV-081-LY (W.D. Tex. July 10, 2012). Specifically, the district court rejected Pitonyak's objections to the Magistrate Judge's factual findings. It denied the petition for a COA because it concluded reasonable jurists could not debate the denial of Pitonyak's petition on the merits.

Pitonyak timely appealed and moved for a COA. This Court granted Pitonyak a COA with respect to his Brady claim. The State's motion for reconsideration was carried with the case.

II.

At the threshold, we deny the State's motion to reconsider our grant of a COA in this case. To appeal the district court's denial of his habeas petition, Pitonyak must obtain a COA pursuant to the Anti-Terrorism and Effective Death Penalty Act of 1996 ("AEDPA"), 28 U.S.C. § 2253(c)(1)(A). A COA may only issue if the petitioner demonstrates that "jurists of reason could disagree with the district court's resolution of his constitutional claims or that jurists could conclude the issues presented are adequate to deserve encouragement to proceed further." Miller-El v. Cockrell, 537 U.S. 322, 327, 123 S.Ct. 1029, 154 L.Ed.2d 931 (2003). Pitonyak's Brady claim meets this threshold standard. Accordingly, we deny the State's motion for reconsideration and reach the merits of Pitonyak's appeal.

III.

To prevail on a Brady claim, the appellant must show Brady's three components: "The evidence at issue must be favorable to the accused, either because it is exculpatory, or because it is impeaching; that evidence must have been suppressed by the State, either willfully or inadvertently; and prejudice must have ensued." Strickler v. Greene, 527 U.S. 263, 281-82, 119 S.Ct. 1936, 144 L.Ed.2d 286 (1999). We are persuaded that on the facts here the state court was not unreasonable in concluding that the evidence was not suppressed within the meaning of Brady because the prosecution's Brady obligation did not extend to Counselor Hoffman. Accordingly, we affirm the district court and deny relief.

As relevant here, 28 U.S.C. § 2254(d) bars this Court from granting an application for a writ of habeas corpus with respect to any claim that was adjudicated on the merits by a state court unless the state court adjudication "resulted in a decision that was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States." 28 U.S.C. § 2254(d). "A federal court may not overrule a state court for simply holding a view different from its own, when the precedent from [the Supreme] Court is, at best, ambiguous." Mitchell v. Esparza, 540 U.S. 12, 17, 124 S.Ct. 7, 157 L.Ed.2d 263 (2003). In conducting its inquiry, "a habeas court must determine what arguments or theories supported or ... could have supported[] the state court's decision; and then it must ask whether it is possible fairminded jurists could disagree that those arguments or theories are inconsistent with the holding in a prior decision of [the Supreme] Court." Harrington v. Richter, ___ U.S. ___, 131 S.Ct. 770, 786, 178 L.Ed.2d 624 (2011). In other words, to obtain relief, Pitonyak must show that the state court's ruling was "so lacking in justification that there was an error well understood and comprehended in existing law beyond any possibility for fairminded disagreement." Id. at 786-87. This review, moreover, is "limited to the record that was before the state court that adjusticated the claim on the merits." Cullen v. Pinholster, ___ U.S. ___, 131 S.Ct. 1388, 1398, 179 L.Ed.2d 557 (2011).

With respect to Pitonyak's Brady claim, Kyles v. Whitley, 514 U.S. 419, 437, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995), held that Brady requires the "individual prosecutor [] to learn of any favorable evidence known to the others acting on the government's behalf in the case, including the police." Although the Supreme Court acknowledged that covered parties sometimes fail to inform a prosecutor of all they know, it explained that "procedures and regulations" must be established to effectuate the prosecutor's burden to learn. Id. at 438, 115 S.Ct. 1555 (quotations omitted). Under the facts of that case, the Court held that the prosecutor's Brady duty extended to evidence "known only to police investigators" that had gathered evidence and interviewed witnesses in the investigation against the defendant. Id. at 428-30, 437-38, 115 S.Ct. 1555.

Here, it was not unreasonable for the state court to determine that Supreme Court precedent is uncertain as to whether a mental health professional is encompassed within the Brady fold. Hoffman was not involved in investigating or preparing the case against Pitonyak. Although the affidavit of J.W. Thompson states that the "Travis County Sheriff's Office assists in homicide investigations when it finds possibly incriminating evidence at the jail," there is no evidence in the record that jail mental health professionals communicated to police any information learned within the scope of mental health services, in this case or any other. Indeed, Hoffman's additional notation to call Hall's lawyer — not the prosecutor — regarding Hall's self-incriminating statements suggests that she was not working on the prosecutor's behalf in this case. Of course, the Brady duty falls on the prosecutor to learn incriminating information in the hands of covered third parties, not on third parties to communicate it. Kyles, 514 U.S. at 437-38, 115 S.Ct. 1555. But these facts are relevant to determining whether Hoffman in fact was acting on the prosecutor's behalf in Pitonyak's case, as required by the clearly established precedent in Kyles. This evidence provides a reasonable basis for the state court's conclusion that Pitonyak failed to demonstrate Hoffman was acting on the government's behalf in her provision of mental health services within the jail.

The Pitonyak and Hall investigations were so intertwined as to leave us uncertain whether, had Hoffman's notation in Hall's jail record tended to inculpate rather than exculpate Pitonyak for first-degree murder, procedures and regulations would have ensured it was communicated to Pitonyak's prosecutor. But on the facts in the state habeas record, it was not unreasonable for the state court to conclude that Hoffman's notation falls outside the ambit of the prosecutor's Brady duty in his case against Pitonyak. Because the state court reasonably could find that the evidence thus was not withheld within the meaning of Brady, we deny Pitonyak's claim.

Even if our threshold determination on suppression is an insufficient basis to dispose of Pitonyak's claim, it fails on Brady's prejudice prong. As Brady makes clear, suppression by the government of evidence favorable to the accused violates due process only "where the evidence is material either to guilt or to punishment, irrespective of the good faith or bad faith of the prosecution." 373 U.S. at 87, 83 S.Ct. 1194. And, "evidence is material only if there is 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). Importantly, the materiality standard "is not a sufficiency of evidence test," and thus, "[a] defendant need not demonstrate that after discounting the inculpatory evidence in light of the undisclosed evidence, there would not have been enough left to convict." Kyles, 514 U.S. at 434-35, 115 S.Ct. 1555. Instead, the defendant must show "that the favorable evidence could reasonably be taken to put the whole case in such a different light as to undermine confidence in the verdict." Id. at 435, 115 S.Ct. 1555.

While the materiality of Hoffman's notation presents a difficult question, we cannot say that the state court was unreasonable in finding Hoffman's notation to be immaterial to Pitonyak's guilt or punishment.

There is substantial evidence that an alternate perpetrator defense would not have succeeded. In addition to his amnesic admission of guilt, Pitonyak testified that he did recall Hall arriving at his apartment the morning of August 17 after he found Cave's body in his bathtub; that Hall asked why Cave's purse was in Pitonyak's living room; and that he led her to the bathroom to show her Cave's body. Moreover, undisputed telephone records show that Pitonyak called Hall at 5:59 a.m. and that they spoke again at 6:57 a.m. and 7:24 a.m., corroborating Pitonyak's testimony that Hall was not present at the time of Cave's death. Both the phone records and Pitonyak's testimony in turn corroborate the alibi evidence that Martindill presented, albeit with uncertainty about the date, for Hall's whereabouts the night of the murder and her arrival at Pitonyak's apartment around 8:00 a.m. the following morning. Additionally, Pitonyak's neighbor, Sullivan, recounted that Pitonyak, alone, knocked on her apartment door at approximately 3:00 a.m. the night of the murder. She testified that he told a rambling story about a gunfight with "some Mexicans" in his apartment and that he had a smudge of blood on his arm. The state court reasonably could interpret this evidence as showing that Pitonyak was alone immediately following Cave's death and made a transparent attempt to explain away the sound of the gunshot that killed her, gunshots he claimed later he could not recall.

In light of this evidence, it was not unreasonable for the state court to conclude that Pitonyak's counsel would not have offered an alternate-perpetrator defense at trial rather than seeking conviction on a lesser offense. Given the jailhouse context of Hall's confession and its lack of corroboration and detail, the state court reasonably could have concluded that Hall's statement could not overcome the overwhelming problems with an alternate-perpetrator theory.

On these facts, fairminded jurists of reason could conclude that Hall's "confession" is immaterial within the meaning of Brady because it would not have cast the "whole case in such a different light as to undermine confidence in the verdict," Kyles, 514 U.S. at 435, 115 S.Ct. 1555.

IV.

We affirm the district court's denial of Pitonyak's federal habeas petition.