KETHLEDGE, J., delivered the opinion of the court, in which KENNEDY, J., joined. CLAY, J. (pp. 380-93), delivered a separate dissenting opinion.
*374OPINION
KETHLEDGE, Circuit Judge.Mark Storey’s principal argument in his federal habeas petition is that he should get a new trial because his lawyer in his first trial was ineffective. It is common ground in this case that Storey’s trial lawyer did a poor job. But the Supreme Court has gone out of its way to make clear that, in order to obtain a new trial on ineffective-assistance grounds, the petitioner must do more than show that he had a bad lawyer- — even a really bad one. Instead, the petitioner must also show prejudice, which means he must show a reasonable likelihood that his lawyer’s bad performance made a difference in the outcome of his trial. The Court’s precedents make clear that the former showing by no means leads inevitably to the latter.
Whether a petitioner can show prejudice depends in large part on the evidence in the case. The evidence here included the testimony of three witnesses who testified that, on separate occasions, Storey had boasted to them about killing Nathan Wilson. There is little likelihood that even an effective defense lawyer could have overcome that testimony, if indeed the trier of fact found it credible. The trial judge who watched each of these witnesses testify specifically found their testimony to be credible on this point. Another trial judge who watched two of these witnesses change their testimony, in a hearing over a decade later, specifically found their recantations not to be credible. And so, in the end, Storey’s petition asks us to set aside the credibility determinations of the two judges who watched these witnesses testify first-hand, in favor of contrary credibility determinations of our own. The record provides us with no basis to take that extraordinary step in this case. We therefore affirm the district court’s denial of the writ.
I.
A.
Wilson worked at the Gold Mine, a Detroit gold and jewelry shop that also reputedly did its share of fencing. A bulletproof plexiglass partition and a sliding-steel door separated the customer counter from the employee-only area of the store. The store’s inventory was displayed in a see-through case on the employee side of the partition. A loaded .357 Magnum revolver typically rested in plain view on top of the case.
Sometime between 2 and 3 p.m. on November 7, 1984, Wilson was found dead in his chair behind the plexiglass at the shop. He had been shot three times in the back of the head. One shot was made with the gun pressed directly against his head. The sliding steel door was open. About $10,000 worth of jewelry, $1,000 cash, and the .357 Magnum were missing from the store.
Mark Storey later admitted that he was in the employee-only area of the store between approximately 1 and 2 p.m. on the day of the shooting. He said that Wilson let him in, along with his friend Shawn Coats, so that Storey could sell some stolen jewelry to the store. Storey and Coats smoked marijuana while they were back there.
About three weeks later, Storey tried to sell the store a gold necklace. The store’s owner, James Floyd, recognized the necklace as one that had been stolen when Wilson was murdered. The police investigated. Eventually, in connection with Wilson’s killing, the State of Michigan charged Storey with first-degree murder and firearm possession during a felony.
B.
Storey was just shy of 16 years old on the date of the murder, so the Wayne *375County Probate Court’s Juvenile Division initially held jurisdiction over his case. But that court waived jurisdiction at the State’s request, thereby sending the case to the Michigan Recorder’s Court for Storey to be tried as an adult. Neither Storey’s counsel at the time, nor his counsel at trial, Charles Campbell, appealed the waiver.
After a four-day bench trial in the Recorder’s Court, Judge James Chylinski found Storey guilty beyond a reasonable doubt on both counts. He emphasized that “the crux of the case” was whether three witnesses — Darin Henderson, David Kidd, and William Walls — were credible when they testified that Storey had told them that he killed Wilson. And Judge Chylinski found that each of these witnesses was credible when he so testified. He sentenced Storey to life in prison, plus two years for the gun charge. Storey appealed, claiming only that the evidence was insufficient to convict him. The Michigan Court of Appeals affirmed the trial court’s judgment, and the Michigan Supreme Court denied leave to appeal.
Storey then pursued state collateral remedies. First he filed a petition for habeas corpus in the Recorder’s Court, which Judge Chylinski denied in 1989. Five years later, Storey filed a motion for relief from judgment in the same court, claiming for the first time that his trial counsel had been ineffective, among other claims. He also moved for an evidentiary hearing with respect to his ineffective-assistance claim. See generally People v. Ginther, 390 Mich. 436, 212 N.W.2d 922 (1973). By then Judge Harvey Tennen had replaced Judge Chylinski. Judge Tennen granted the motion for a Ginther hearing, at which a criminal-defense lawyer offered expert testimony that Campbell had done a poor job before and during trial.
Judge Tennen thereafter ordered a new trial based on Campbell’s failure to prepare adequately for trial and to obtain discovery regarding the State’s witnesses. But the State appealed, and the Michigan Court of Appeals promptly remanded the case for a determination as to whether Storey had shown cause for failing to present his ineffective-assistance claim on direct appeal. On remand, Judge Tennen found that Storey had not shown cause for that failure, so he vacated his earlier order and denied Storey’s motion for a new trial.
Storey then filed two applications for leave to appeal to the Michigan Court of Appeals. The first application included a claim that the Juvenile Division’s waiver of jurisdiction had been proeedurally defective. The Court of Appeals denied that application in a stock order entered under Michigan Court Rule 6.508(D). The second application included a claim that Campbell had been ineffective as a result of his failure to challenge the juvenile-waiver order. In addition, that application challenged Judge Tennen’s determination that Storey had not shown cause for his failure to present a claim of ineffective assistance in his direct appeal. The Court of Appeals denied that application for lack of merit. The Michigan Supreme Court denied leave to appeal from either denial.
In 1999, Storey filed a second motion for relief from judgment before the state trial court. This motion included a new claim that two of the State’s witnesses at his trial — Henderson and Kidd — had perjured themselves when they testified that Storey had told them he shot Wilson. Judge Tennen held another Ginther hearing with respect to the motion. Although both Henderson and Kidd recanted their trial testimony during the hearing, Judge Tennen found that neither recantation was credible. Judge Tennen also held that Storey’s other claims were proeedurally *376barred. He thus denied Storey’s motion for a new trial. The Michigan Court of Appeals and Michigan Supreme Court each denied leave to appeal. .
C.
Storey turned finally to the federal courts. In 2001, he filed a federal habeas petition comprising eight claims. In September 2003, the district court granted relief on one of them: that Storey’s appellate counsel had been ineffective by failing to argue that Campbell had been ineffective at trial. (The 2001 petition did not itself include, however, a claim that Campbell had been ineffective.) As relief, the district court ordered that Storey be granted a new direct appeal. The district court thereafter closed the case without staying, dismissing, or otherwise taking any action with respect to Storey’s other claims.
Per the district court’s order, the Michigan Court of Appeals granted Storey a new appeal. Before proceeding with that appeal, however, Storey filed yet another motion for a new trial before the trial court, again presenting his ineffective-assistance and perjured-testimony claims. Judge Michael Hathaway — who was by then presiding — denied the motion, finding among other things that Judge Chylinski had been better positioned than anyone to judge each witness’s credibility.
So Storey proceeded with his second direct appeal before the Michigan Court of Appeals, raising ten claims. The court rejected all of them and affirmed the trial court’s original judgment. The Michigan Supreme Court denied leave to appeal.
Storey then returned to federal district court, filing a new habeas petition in 2006. That petition included several claims that were in his 2001 petition, but four that were not. One of the latter claims was that Campbell had been ineffective at trial. Based on Storey’s inclusion of the new claims, however, the district court held in a June 2009 order that Storey’s new petition was “second or successive” within the meaning of 28 U.S.C. § 2244(b). The district court gave Storey the choice of dropping the new claims or seeking permission from us to file a second or successive petition. See generally id. § 2244(b)(3). Storey did neither, and instead filed a motion for reconsideration. The district court granted the motion, reached the merits of Storey’s claims, and — in an extensively reasoned opinion — rejected all of them. The court therefore denied his petition, though it did grant a certificate of appealability as to each of his claims.
This appeal followed.
II.
A.
A threshold question is whether Storey’s 2006 petition is “second or successive” under § 2244(b) because it added claims not included in Storey’s 2001 petition. If it is, we must order the petition dismissed because Storey did not obtain permission to file it. See id.
Whether a petition (a term we use interchangeably with “application”) is “second or successive” within the meaning of § 2244(b) does not depend merely on whether the petitioner filed a prior application for habeas relief. The phrase is instead “a ‘term of art’ that is ‘given substance’ by the Supreme Court’s habeas cases.” In re Salem, 631 F.3d 809, 812 (6th Cir.2011) (quoting Slack v. McDaniel, 529 U.S. 473, 486, 120 S.Ct. 1595, 146 L.Ed.2d 542 (2000)). Accordingly, in a number of cases, the Court has held that an application was not second or successive even though the petitioner had filed an earlier one. In Stewart v. Martinez-Villareal, 523 U.S. 637, 118 S.Ct. 1618, 140 L.Ed.2d 849 (1998), the petitioner filed a *377second petition that presented a claim identical to one that had been included in an earlier petition. The claim had been unripe when presented in the earlier petition. The Court treated the two petitions as “only one application for habeas relief.]” Id. at 643, 118 S.Ct. 1618. In Panetti v. Quarterman, 551 U.S. 930, 127 S.Ct. 2842, 168 L.Ed.2d 662 (2007), the Court held that an application that presented a claim that had not been presented in an earlier application, but that would have been unripe if it had been presented then, was not second or successive. Id. at 945, 127 S.Ct. 2842. In Magwood v. Patterson, — U.S.-, 130 S.Ct. 2788, 177 L.Ed.2d 592 (2010), the Court made clear that an application challenging an earlier criminal judgment did not count for purposes of determining whether a later application challenging a new judgment in the same case was second or successive. Id. at 2797-98.
The posture of this ease is somewhat different. In response to Storey’s 2001 petition, the district court ordered the Michigan courts to grant him a new direct appeal. The Michigan courts did so, but ultimately affirmed the original judgment in his case. Storey filed a second petition in 2006, which included some new claims that were undisputedly ripe for inclusion in his earlier petition. The issue, then, is whether the fact of Storey’s new appeal, standing alone, serves to “reset” the “counter” of his applications to zero. In re Williams, 444 F.3d 233, 235 (4th Cir.2006) (internal quotation marks omitted).
There is a circuit split on this issue, though perhaps now a stale one. These cases involve applications under both §§ 2254 and 2255, though courts do not distinguish between those sections for purposes of the second-or-successive determination. See, e.g., id. (“This court and others have repeatedly cited § 2254 and § 2255 interchangeably in such circumstances”). Seven circuits — the Second, Third, Fourth, Seventh, Ninth, Tenth, and Eleventh — hold that the tally of applications is reset to zero after the petitioner concludes a direct appeal that was ordered in response to an earlier petition. See Urinyi v. United States, 607 F.3d 318, 321 (2d Cir.2010) (per curiam) (collecting cases). In those cases, as in this one, the reason why the petitioner was granted a new appeal is that his counsel was ineffective with respect to his first direct appeal; and these courts reason that such a petitioner should be given the same clean slate with respect to habeas review as a defendant whose counsel did not “bunglef ]” his first direct appeal. See, e.g., In re Goddard, 170 F.3d 435, 437 (4th Cir.1999). The upshot of this rule is that the petitioner can eventually seek habeas relief on all of the claims presented in his remedial appeal, in which presumably he will have competent counsel. See, e.g., id. at 437 & n. 1. (One important limitation on this rule, however, is that the petitioner cannot “resurrect” claims that the district court “denied on the merits” in his first petition. See In re Williams, 444 F.3d at 236.)
Two circuits — the First and the Fifth-hold that a post-remedial-appeal petition is second or successive if it includes claims that could have been included, but were not, in the first petition. See United States v. Orozco-Ramirez, 211 F.3d 862, 870-71 (5th Cir.2000); Pratt v. United States, 129 F.3d 54, 61 (1st Cir.1997). These courts invoke what they call “the spirit of AEDPA’s restrictions^]” and they reason that their rule “serves the singularly salutary purpose of forcing federal habeas petitioners to think through all potential post-conviction claims and to consolidate them for unitary presentation to the district court” in the first petition filed. Pratt, 129 F.3d at 61; see also Orozco-Ramirez, 211 F.3d at 870-71. To which one might respond that, in many *378cases, the person doing the thinking is the petitioner himself.
We think there is a reason why no circuit has adopted the minority rule since 2000: namely, the Supreme Court’s decision the same year in Slack Slack’s first federal habeas petition included claims that had not been exhausted in state court. The district court dismissed the petition without prejudice and Slack returned to state court to exhaust those claims. He then filed in federal court a second petition that included claims not included in his first petition. That petition, the district court and Ninth Circuit held, was “second or successive” as a result of those claims. Slack, 529 U.S. at 479-80, 120 S.Ct. 1595.
The Supreme Court reversed. As an initial matter, the Court noted that “preAEDPA law govern[ed]” that case, though it hastened to add that “we do not suggest the definition of second or successive would be different under AEDPA.” Id. at 486, 120 S.Ct. 1595. In any event, what matters for our purposes was the reasoning of the Court’s decision:
The State contends that the prisoner, upon his return to federal court, should be restricted to the claims made in his initial petition. Neither Rose v. Lundy [, 455 U.S. 509, 102 S.Ct. 1198, 71 L.Ed.2d 379 (1982) ] nor Martinez-Villareal requires this result, which would limit a prisoner to claims made in a pleading that is often uncounseled, handwritten, and pending in federal court only until the state identifies one unexhausted claim. The proposed rule would bar the prisoner from raising nonfrivolous claims developed in the subsequent state exhaustion proceedings contemplated by the Rose dismissal, even though a federal court had yet to review a single constitutional claim.
Slack, 529 U.S. at 487,120 S.Ct. 1595.
This reasoning is virtually identical to that followed by the majority of circuits with respect to the question presented here. And it effectively rejects the contrary reasoning of the two circuits that follow the minority rule. It is true that, in these cases, unlike Slack, the district court by definition has ruled on at least “a single constitutional claim”: namely, whether the petitioner is entitled to a new direct appeal. Id. But we think the rest of the quoted passage makes clear that, in these circumstances, the Court would not want the petitioner to be “restricted to the claims made in his initial petition[.]” Id. Instead, we think it clear enough that the petitioner should be able to “rais[e] non-frivolous claims developed in the subsequent” remedial appeal. Id.
So it ultimately does not matter what we think, as an original matter, of the respective rationales offered by the majority and minority circuits for their rules. The Supreme Court has told us what it thinks of those rationales. Thus we join the majority of circuits, and hold that a petition filed after a remedial appeal, ordered in response to an earlier petition, is not second or successive within the meaning of § 2244(b) — even if it includes claims that could have been included, but were not, in the first petition.
Storey’s 2006 petition is not second and successive. We proceed to consider its merits.
B.
1.
We first consider Storey’s claim that Campbell’s lack of preparation and generally weak presentation at trial rendered him ineffective in the constitutional sense of the term. The Michigan Court of Appeals rejected this claim on the merits. In doing so, the court correctly recited the test for showing prejudice; but the court later concluded that Storey had not shown *379prejudice because he had “not demonstrated that the outcome of the trial would have been different” if Campbell had performed effectively. People v. Storey, No. 251271, 2005 WL 711756 (Mich.Ct.App. Mar. 29, 2005). We need not decide whether the latter statement- — -which is an arguably incorrect statement of the Strickland standard for prejudice — requires us to review the state court’s decision on this issue de novo rather than under the deferential standard set forth in § 2254(d). For Storey’s claim fails even under a de novo standard of review.
Strickland’s test for prejudice is a demanding one. “The likelihood of a different result must be substantial, not just conceivable.” Richter, 131 S.Ct. at 792. So we turn to the question whether the test is met here. Broadly stated, Storey’s argument is that, absent Campbell’s failure to prepare for trial, obtain discovery, and uncover an alleged deal between the prosecution and David Kidd, there was a reasonable likelihood that he would have been acquitted. (Storey does not argue before us — and has never claimed in his federal habeas petitions — that Campbell was ineffective as a result of his failure to contest the juvenile-waiver order.)
We first take up the allegation regarding Kidd. “[T]he mere fact that a witness desires or expects favorable treatment in return for his testimony is insufficient; there must be some assurance or promise from the prosecution that gives rise to a mutual understanding or tacit agreement.” Akrawi v. Booker, 572 F.3d 252, 262 (6th Cir.2009) (emphasis in original). Here, as the district court explained at considered length in its opinion denying the writ, there simply is no evidence of any such agreement between the prosecution and Kidd. All there is, rather, is Kidd’s own assertions years after the trial that he “hoped” that he might get a shorter sentence for his own crimes if he testified favorably for the prosecution in Storey’s trial. That is not proof of an “agreement[,]” id., and thus there is no basis to fault Campbell for failing to discover one.
That leaves Campbell’s failures to prepare for trial and to argue effectively once he was there. Those failures are serious, but they are simply not enough to show prejudice in this case, given the record before us. The evidence of Storey’s guilt included the following: Storey’s own admission that he had entered the store’s employee-only area shortly before Wilson was murdered; Floyd’s testimony — specifically found credible by Judge Chylinski— that Storey tried to sell back to Floyd a gold necklace stolen in the murder-robbery; and Henderson’s and Kidd’s testimony that, before the murder, Storey had told each of them that it would be easy to rob the Gold Mine by getting behind the partition and then shooting Wilson.
And finally there was the testimony of three witnesses — Henderson, Kidd, and Walls — each of whom testified that Storey had told him, on separate occasions, that he shot Wilson. The judge who observed that testimony found it credible, notwithstanding Campbell’s extensive cross-examinations of these witnesses; and a second judge who observed Henderson and Kidd change their stories years later found their recantations incredible. For good reason, we give “great deference” to those credibility determinations. See Felkner v. Jackson, — U.S. -, 131 S.Ct. 1305, 1307, 179 L.Ed.2d 374 (2011) (internal quotation marks omitted); see also 28 U.S.C. § 2254(e)(1). Storey makes a number of assertions about the motives of these witnesses and the extent to which the judges who assessed their credibility were aware of them; but none of those assertions meaningfully distinguish this case from the raft of habeas cases in which witnesses change their stories years after the peti*380tioner is sent to prison. We have no lawful basis — or inclination — to displace the credibility findings of the trial judges who observed the testimony of these witnesses first-hand. And that means that Storey cannot show prejudice as the Supreme Court has defined the term. The Michigan Court of Appeals and the district court were both correct to reject this claim.
2.
We make shorter work of Storey’s remaining claims. “The extent to which these claims are procedurally defaulted is a nettlesome question; the extent to which they are meritless, much less so.” Benton v. Booker, 403 Fed.Appx. 984, 985 (6th Cir.2010). So we cut to the merits here.
Storey claims that his rights under various clauses of the Constitution were violated by the prosecution’s nondisclosure of its alleged agreement with Kidd and by the trial court’s failure to order disclosure of the agreement to Storey. Those claims fail because there is no proof that any agreement existed in the first place. Storey next claims that the prosecution violated his Brady rights by failing to disclose testimony from other cases in which, he says, Kidd falsely accused other defendants of murder. That claim fails because “Brady [v. State of Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963) ] does not apply when the information is available from another source,” e.g., by “looking at public records.” Owens v. Guida, 549 F.3d 399, 418 (6th Cir.2008).
Storey also claims that the judges who presided over his state collateral proceedings violated his Sixth Amendment rights by making credibility determinations. That claim fails because the writ’s scope does not extend to state collateral proceedings. Cornwell v. Bradshaw, 559 F.3d 398, 411 (6th Cir.2009). And we likewise cannot grant relief on Storey’s claim that “the various errors that we have individually rejected [are], in the aggregate, a constitutional violation.” Hoffner v. Bradshaw, 622 F.3d 487, 512-13 (6th Cir.2010).
* * *
The district court’s judgment is affirmed.