HARTZ, Circuit Judge.

Defendants William L. Pickard and Clyde Apperson were convicted of drug-related crimes in the United States District Court for the District of Kansas. An important witness for the prosecution was informant Gordon Todd Skinner, a criminal associate of Defendants. After the convictions were affirmed on appeal, see United States v. Apperson, 441 F.3d 1162 (10th Cir.2006), Defendants filed motions for relief under 28 U.S.C. § 2255 claiming, among other things, that the prosecution had violated their rights under Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), and Giglio v. United States, 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), by suppressing evidence of Skinner's criminal and informant background. The district court rejected the claims. See United States v. Pickard, No. 00-40104-01/02-RDR, 2009 WL 939050 (D.Kan. Apr. 6, 2009). Defendants applied for certificates of appealability (COAs) to appeal the district court's decision, but we denied their applications. See United States v. Pickard, 396 Fed.Appx. 568 (10th Cir.2010); 28 U.S.C. § 2253(c)(1)(B) (requiring COA to appeal denial of § 2255 relief). They also filed district-court motions under Fed.R.Civ.P. 60(b) to set aside the court's judgment in the § 2255 proceedings, raising multiple claims. The claims relevant to this appeal are that evidence newly discovered by them through Freedom of Information Act (FOIA) requests shows (1) that the prosecution violated its Brady/Giglio duties at trial and (2) that the prosecution made a false statement in the § 2255 proceedings that forestalled the discovery from which they could have established that there had been a Brady/Giglio violation at trial.

The district court ruled that the claims of prosecutorial misconduct amounted to second-or-successive claims under § 2255, which it could not consider without authorization from this court, see 28 U.S.C. §§ 2244(b)(3)(A); 2255(b). It therefore transferred the claims here. See In re Cline, 531 F.3d 1249, 1252 (10th Cir.2008) (district court may transfer rather than dismiss the matter if "it is in the interest of justice to do so"). We agree with the district court that Defendants' claims of Brady/Giglio violations at trial are second-or-successive claims; and because Defendants have not established the requisites for authorizing a second-or-successive claim, we deny authorization. On the other hand, Defendants' claims that prosecutorial misconduct in the § 2255 proceedings affected the integrity of those proceedings are proper Rule 60(b) claims; and we remand those claims to the district court for resolution.

I. BACKGROUND

In 2003 Defendants were convicted of conspiracy to manufacture lysergic acid diethylamide (LSD), see 21 U.S.C. §§ 841(a)(1), (b)(1)(A), and 846, and possession with intent to distribute LSD, see 21 U.S.C. § 841(a)(1) and (b)(1)(A). We affirmed the convictions on direct appeal. See Apperson, 441 F.3d 1162. In 2008 Defendants filed motions under 28 U.S.C. § 2255 to set aside their convictions and sentences. The "centerpiece" of these motions was the claim that "the government violated its Brady/Giglio obligations by suppressing the criminal and informant backgrounds of certain witnesses, including Gordon Todd Skinner." Pickard, 2009 WL 939050, at *5. "[T]he majority of this claim [was] directed at [impeachment and exculpatory] evidence that was purportedly suppressed by the government concerning Skinner." Id. at *6. In particular, Defendants argued that the government had failed to disclose relevant files from agencies other than the federal Drug Enforcement Administration (DEA). In response, the prosecution asserted (1) that no agency other than the DEA was involved in the LSD investigation and (2) that the prosecution was not aware of Skinner's involvement with any agency besides the DEA. Defendants nevertheless sought an order requesting the government to identify the federal agencies other than the DEA that had participated in their case. The district court denied these requests, stating that Defendants "have failed to point to any evidence showing any involvement by [agencies other than the DEA] in the investigation of this case." Aplee. Supp.App. at 159 (Mem. & Order at 41, United States v. Pickard, No. 00-40101-01/02-RDR (D.Kan. Apr. 6, 2009)). It ultimately denied the § 2255 motions, ruling that the new evidence presented to the court by Defendants was cumulative and would not have caused a different result at trial. We declined to issue COAs for Defendants to appeal the denial. See Pickard, 396 Fed.Appx. 568.

While awaiting our ruling on the district-court denial of their § 2255 motions, Defendants filed in district court two Rule 60(b) motions to set aside the denial of their § 2255 motions. The first motion (Doc. 637) listed five matters that the district court had allegedly failed to consider. It also contended that the district court should reconsider Defendants' Brady/Giglio and prosecutorial-misconduct claims in light of newly discovered evidence referenced in their second Rule 60(b) motion (Doc. 639).

The second 60(b) motion alleged that evidence of "substantive undisclosed FBI and IRS records" newly obtained through Defendants' FOIA requests demonstrated that the prosecution had committed fraud because the FOIA documents showed that federal agencies other than the DEA were involved in investigating Defendants' drug-related activities. Aplee. Supp.App. at 207 (Rule 60(b)(2) & 60(b)(3) Mots. Based on Newly Discovered Evid. of Non-Disclosure of Records at 1, Pickard, No. 00-40104-01/02-RDR (D.Kan. Apr. 5, 2010)). They argued that these agencies might have additional undisclosed evidence that could have been used at trial to impeach Skinner.

The district court ruled that the first five claims in Doc. 637 were proper under Rule 60(b) because they challenged the integrity of the habeas proceedings; but it rejected them as moot or without merit. Defendants sought to appeal this ruling, but we denied a COA. See United States v. Pickard, 445 Fed.Appx. 61 (10th Cir. 2011).

As for the remaining claims — the sixth claim in Doc. 637 and those in Doc. 639 — the district court ruled that they "invite further review of the substantive habeas claims already decided on the merits and, thus, must be treated as a second or successive habeas petition requiring authorization by the Tenth Circuit Court of Appeals." Aplee. Supp.App. at 283 (Mem. & Order at 9-10, Pickard, No. 00-40104-01/02-RDR (D.Kan. Jan. 24, 2011)). It transferred these claims to this court to give Defendants an opportunity to seek authorization under 28 U.S.C. § 2255(h). Defendants filed with us a motion to remand the claims to the district court on the ground that they are not second-or-successive § 2255 motions but are proper motions before the district court under Fed.R.Civ.P. 60(b).

II. ANALYSIS

The primary issue on appeal is whether the claims transferred by the district court were, as characterized by the district court, second-or-successive § 2255 claims, see 28 U.S.C. § 2244(b)(3), or rather claims properly brought by Defendants under Fed.R.Civ.P. 60(b). The factual predicate for all the claims is the same: Defendants allege that through FOIA requests they have obtained evidence that agencies other than the DEA were involved in the LSD investigation and have had records related to Skinner. Their claims, as we understand them, are that this evidence establishes that the prosecution withheld exculpatory evidence at trial and that it then committed fraud in the § 2255 proceedings by falsely denying the involvement of other agencies. As we now explain, the claim of prosecutorial misconduct at trial was properly transferred to this court as a second-or-successive claim (although we decline to authorize Defendants to pursue the claim), but the claim of prosecutorial misconduct in the § 2255 proceedings is a proper Rule 60(b) claim that should be addressed in the first instance by the district court.

It is not unusual for defendants who have failed to obtain relief in federal habeas proceedings to attempt to bring new habeas claims in the guise of Rule 60(b) motions. Guided by the Supreme Court's decision in Gonzalez v. Crosby, 545 U.S. 524, 125 S.Ct. 2641, 162 L.Ed.2d 480 (2005), we have explained how to distinguish a true Rule 60(b) motion from a new (that is, a second-or-successive) § 2255 claim:

United States v. Nelson, 465 F.3d 1145, 1147 (10th Cir.2006) (applying analysis in Gonzalez, which arose in the § 2254 context, to the § 2255 context).

Applying this law, we first consider Defendants' claims that the prosecution violated its Brady/Giglio duties at trial. These claims are certainly second-or-successive claims because they assert a basis for relief from the underlying convictions. See id. Defendants contend that the alleged Brady/Giglio violation entitles them to a new trial on the LSD charges.

The district court properly characterized the claims as second or successive and refused to consider them because it could do so only after certification by this court under 28 U.S.C. § 2255(h). See 28 U.S.C. § 2244(a). It transferred the claims to this court for consideration, and we decline to certify them. We can certify the claims only if they contain either:

28 U.S.C. § 2255(h). Yet Defendants do not even assert either ground for certification. They invoke no new constitutional rule. And they do not contend that the new evidence would compel a reasonable factfinder to find them not guilty. They merely suggest that the jury and the court "would have found the government's motivation for this withholding, and the withheld evidence itself, of concern." Aplee. Supp.App. at 236 (Rule 60(b)(2) & 60(b)(3) Mots. Based on Newly Discovered Evid. of Non-Disclosure of Records at 30, Pickard, No. 00-40104-01/02-RDR (D.Kan. Apr. 5, 2010)).

We draw a different conclusion, however, regarding Defendants' claim that the prosecution improperly withheld information during the § 2255 proceedings when the prosecutor said that no agency other than the DEA participated in the investigation leading to Defendants' trial. Defendants moved for an order requiring the government to disclose all agencies that had participated in the investigation, but the district court denied the motion on the ground that they had no evidence of participation by any agency other than the DEA. As we understand Defendants' claim, they are asserting that the prosecutor's statement prevented their discovery of the involvement of other agencies and, most pertinent to their § 2255 claim, thereby prevented them from showing that those agencies had additional information about Skinner that could have been used to impeach him at trial. Defendants say they are now entitled to relief because the evidence obtained through their FOIA requests shows that the prosecutor's statement (that only the DEA was involved) was false.

Although the government denies that it made any false statement in the § 2255 proceedings, our task is not to ascertain the truth of Defendants' allegations but to decide which tribunal should resolve the matter. In our view, the matter should be heard by the district court because Defendants' claim challenges the integrity of the § 2255 proceedings and is therefore properly presented under Rule 60(b). To be sure, this claim is closely related to the Brady/Giglio claim raised in the original § 2255 proceeding. But the claim is a distinct one. To recapitulate: The claim raised in the § 2255 proceedings is that the prosecution violated its Brady/Giglio duties at trial. In contrast, the claim in the Rule 60(b) motion is that the prosecutor committed fraud in the § 2255 proceedings that prevented Defendants from obtaining discovery to establish their § 2255 claims. If we assume the truth of Defendants' allegations, as we must at this juncture, then Defendants have stated a proper Rule 60(b) motion.

Some confusion may arise from dictum in Spitznas v. Boone, 464 F.3d 1213 (10th Cir.2006). Spitznas, which considered a § 2254 proceeding, was the first opinion of this court interpreting Gonzalez and endeavored to provide comprehensive guidance on how to distinguish between a proper Rule 60(b) motion and a second-or-successive habeas petition. In doing so it stated that a Rule 60(b) motion in a habeas proceeding is a "`true'" 60(b) motion if it "challenges a defect in the integrity of the federal habeas proceeding, provided that such a challenge does not itself lead inextricably to a merits-based attack on the disposition of a prior habeas petition. [Gonzalez, 545 U.S. at 532, 125 S.Ct. 2641]." Id. at 1215-16 (emphasis added). The words lead inextricably should not be read too expansively. They certainly should not be read to say that a motion is an improper Rule 60(b) motion if success on the motion would ultimately lead to a claim for relief under § 2255. What else could be the purpose of a 60(b) motion? The movant is always seeking in the end to obtain § 2255 relief. The movant in a true Rule 60(b) motion is simply asserting that he did not get a fair shot in the original § 2255 proceeding because its integrity was marred by a flaw that must be repaired in further proceedings. The proviso in Spitznas can be best understood by looking at its citation to Gonzalez. The citation is referring to the Supreme Court's statement that a true Rule 60(b) motion does not "attack[] the federal court's previous resolution of a claim on the merits, since alleging that the court erred in denying habeas relief on the merits is effectively indistinguishable from alleging that the movant is, under the substantive provisions of the statutes, entitled to habeas relief." Gonzalez, 545 U.S. at 532, 125 S.Ct. 2641 (footnote and emphasis omitted). Thus, the proviso means only that a Rule 60(b) motion is actually a second-or-successive petition if the success of the motion depends on a determination that the court had incorrectly ruled on the merits in the habeas proceeding. For example, in In re Lindsey, 582 F.3d 1173 (10th Cir.2009), we held that the defendant's challenge to the district court's failure to conduct an evidentiary hearing in the § 2255 proceeding amounted to a challenge to the district court's decision on the merits and was therefore not a proper Rule 60(b) motion. See id. at 1175-76 ("The decision not to hold an evidentiary hearing essentially is the equivalent of a dismissal for failure to state a claim or a summary judgment.").

More problematic, however, is later dictum in Spitznas. Although the case before the panel involved no claim of fraud or deceit of any kind, the opinion stated that if the fraud alleged in a Rule 60(b) motion "includes (or necessarily implies) related fraud on the state court (or the federal district court that convicted and/or sentenced the movant in the case of a § 2255 motion), then the motion will ordinarily be considered a second or successive petition because any ruling would inextricably challenge the underlying conviction proceeding." 464 F.3d at 1216. One could read this statement as requiring us to treat Defendants' claim of fraud in the § 2255 proceedings as a second-or-successive claim because the alleged fraud that obstructed discovery in the § 2255 proceedings paralleled the misconduct that allegedly occurred at trial.

We will not, however, follow that dictum as so interpreted. We cannot accept the proposition that the government has a free pass to deceive a habeas court into denying discovery just because it similarly deceived the trial court. If Defendants' claim of prosecutorial deceit during the § 2255 proceedings must be treated as a second-or-successive § 2255 motion, then the government's alleged misconduct during that proceeding could compound a substantial injustice to Defendants. In a second-or-successive proceeding Defendants must show that their FOIA evidence (which allegedly contradicts the prosecutor's statement in the § 2255 proceedings) would suffice "to establish by clear and convincing evidence that no reasonable factfinder would have found the movant[s] guilty of the offense." 28 U.S.C. § 2255(h)(1). This is a far heavier burden than the burden under Rule 60(b) of showing only that the false statement had improperly forestalled discovery in the § 2255 proceedings. We doubt that the governing procedural rules permit the government to gain such an advantage by its own fraudulent conduct. In Douglas v. Workman, 560 F.3d 1156, 1192 (10th Cir. 2009), we addressed "the prosecutor's conduct in this case in taking affirmative action, after [the defendant's] trial, to conceal the tacit agreement the prosecutor had made in exchange for Smith's testimony that prevented [the defendant] from discovering the Brady claim in time to assert that claim originally in his first habeas petition." Expressing a proposition similar to that animating our present decision, we wrote: "[T]o treat [the defendant's] Brady claim as a second or successive request for habeas relief, subject to the almost insurmountable obstacles erected by 28 U.S.C. § 2244(b)(2)(B), would be to allow the government to profit from its own egregious conduct." Id. at 1192-93.1

III. CONCLUSION

Defendants' claim in their Rule 60(b) motion that the prosecution violated its Brady/Giglio duties at trial was properly characterized by the district court as a second-or-successive claim, and we DENY the implied request to file second-or-successive § 2255 motions. We REMAND, however, for the district court to consider in the first instance Defendants' claim that the prosecutor's false statement improperly prevented them from obtaining relevant discovery in the § 2255 proceedings.