Appellant Patricio Paladin is serving a life sentence following his conviction on a series of drug charges. Subsequent to Paladin’s conviction, but before his sentencing, Paladin learned of the existence of certain evidence that the government had failed to disclose to him, and that he suggests may have been used to impeach the credibility of the key government witness against him at trial. Relying on Brady v. Maryland, 378 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), Paladin moved to set aside the verdict and for a new trial. The district court denied the motion in a thorough bench decision, and this timely appeal followed.
Because we agree with the district court’s determination that the evidence in question was immaterial, we AFFIRM the district court’s denial of Paladin’s motion. And, as Paladin concedes that we are precedent-bound to do, we reject a series of separate constitutional challenges to Paladin’s life sentence, the imposition of which was made mandatory by the quantities of cocaine at issue and by virtue of this being Paladin’s third felony drug conviction.
I. Facts
A second superseding indictment charged Paladin with one count of conspiracy to distribute cocaine, three counts of distribution of cocaine and one count of possession of cocaine with intent to distribute. Following a jury trial in December 2010, Paladin was convicted on all counts.
The testimony of FBI Agent Mark Alford and a confidential informant by the name of Kevin Vega was of central importance to the government’s case. We briefly overview that testimony here, but will return to it in greater detail later. Alford was the first government witness and testified regarding his oversight of the investigation into Paladin’s narcotics activities and about his supervision of Vega in the initiation and completion of a series of meetings and controlled purchases with Paladin.
Vega testified later in the trial and told jurors that he had met Paladin in 2004 and thereafter routinely purchased cocaine from him until late 2008, in a total aggregate amount that Vega estimated was in excess of 100 kilograms. Of central importance to this appeal, Vega testified that during the summer of 2009, he made the decision to cease his participation in the drug trade and to inform law enforcement of his prior activities. Vega told jurors that, in July 2009, he walked into FBI headquarters in New Hampshire, confessed, and agreed to serve as a confidential informant.1 This somewhat unusual decision, Vega testified, resulted from Vega’s guilty conscience, fear of arrest and incarceration, and the fact that he had a newborn son. Vega stated unequivocally that, aside from controlled buys that he performed at the direction of the government, the last time that he dealt drugs was during the summer of 2009.
Because this was Paladin’s third felony drug conviction and because the quantities of cocaine at issue were sufficient to trigger a mandatory life sentence, see 21 U.S.C. § 841(b)(1)(A)(ii), a significant delay preceded sentencing as Paladin sought to overturn at least one of his prior convictions.2 It was during this period of time, *443in March 2012, that the attorney who had represented Paladin at trial received a letter from one Jordan Manning, an inmate at a state correctional facility in New Hampshire. Manning’s letter suggested that Vega had “got[ten] busted for a drug charge” and had “received something in exchange for testimony.”
Defense counsel contacted the Assistant United States Attorney who had led the prosecution and asked that the matter be investigated. In the course of the ensuing investigation, it was discovered that, on November 12, 2010, prior to the start of Paladin’s trial, representatives of various law enforcement agencies and the New Hampshire United States Attorney’s Office had interviewed an individual named Angel Andino in relation to a separate narcotics investigation.
Notes from this proffer session, disclosed in redacted form to defense counsel in April 2012, contain the following passage:
“Andino said that he supplied Vega with 200 [0]xycontin tablets every 2 weeks for an unspecified amount of time. An-dino said that he stopped supplying Vega with cocaine and [OJxycontin tablets approximately 1-2 months before his (Andino’s) arrest.”
The United States Attorney’s Office also informed defense counsel that Andino had been arrested in February 2010. If the contents of the Andino proffer were true, then it would suggest that Vega had lied to the jury when he testified that he had ended his involvement in the drug trade on his own accord during the summer of 2009, because a transaction with Andino one to two months prior to Andino’s arrest would have necessarily taken place during the winter of 2009 to 2010.
With this information in hand, Paladin filed a motion to set aside the verdict and for a new trial, contending that the government had failed to disclose exculpatory evidence that would have allowed Paladin to undermine Vega’s credibility. The government did not dispute that the Andino proffer should have been disclosed, but maintained that Paladin was not entitled to a new trial because the contents of the Andino proffer were immaterial. After a lengthy hearing in August 2012, the district court denied Paladin’s motion in an oral decision.
At a separate hearing shortly after the denial of Paladin’s motion, the district court imposed a life sentence on the conspiracy count, noting that such a sentence was “excessive,” but “legally required” under the circumstances.3 This appeal followed.
II. Evidentiary Suppression
A. Standard of Review
The district court’s denial of a motion for a new trial is properly reviewed for abuse of discretion. United, States v. Hall, 557 F.3d 15, 19 (1st Cir.2009). We conduct our review mindful that “[t]he trial judge, having seen and heard the witnesses at first hand, has a special sense ‘of the ebb and flow of the recently concluded trial.’ Thus,- his views about the likely impact of newly disclosed evidence deserve considerable deference.” United States v. Mathur, 624 F.3d 498, 504 (1st Cir.2010) (quoting United States v. Natanel, 938 F.2d 302, 313 (1st Cir.1991)).
B. An Introduction
prosecution of evidence favorable to an accused ... violates due process where the evi-*444denee is material either to guilt or to punishment, irrespective of the good faith or bad faith of the prosecution.” Brady, 373 U.S. at 87, 83 S.Ct. 1194. A “Brady ” claim, then, has three elements: (1) the evidence at issue must be favorable to the accused, either because it is exculpatory, or because it is impeaching; (2) that evidence must have been suppressed by the government either willfully or inadvertently; and (3) prejudice must have resulted. Strickler v. Greene, 527 U.S. 263, 281-82, 119 S.Ct. 1936, 144 L.Ed.2d 286 (1999); see also United States v. Aviles-Colón, 536 F.3d 1, 19 (1st Cir.2008). The government concedes that the Andino proffer was potentially impeaching and that it was suppressed inadvertently. Thus, the determinative issue is whether the suppression resulted in prejudice.
“Impeachment evidence must be material before its suppression justifies a new trial.” Conley v. United States, 415 F.3d 183, 188 (1st Cir.2005). Evidence is material when a “reasonable probability [exists] that the result of the trial would have been different” if the suppressed evidence had been disclosed. Strickler, 527 U.S. at 289, 119 S.Ct. 1936 (internal quotation marks omitted). Reasonable probability does not require that “the defendant ‘would more likely than not have received a different verdict with the evidence,’ only that the likelihood of a different result is great enough to ‘undermine[] confidence in the outcome of the trial.’ ” Smith v. Cain, — U.S. —, 132 S.Ct. 627, 630, 181 L.Ed.2d 571 (2012) (alteration in original) (quoting Kyles v. Whitley, 514 U.S. 419, 434, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995)). “This somewhat delphic ‘undermine confidence’ formula suggests that reversal might be warranted in some cases even if there is less than an even chance that the evidence would produce an acquittal.” Conley, 415 F.3d at 188 (quoting United States v. Sepulveda, 15 F.3d 1216, 1220 (1st Cir.1993)).
The strength of impeachment evidence and the effect of suppression are evaluated in the context of the entire record to determine materiality. Id. at 189. Evidence is immaterial where it is cumulative or merely impeaches a witness on a collateral issue. United States v. Dumas, 207 F.3d 11, 16 (1st Cir.2000). Likewise, “suppressed impeachment evidence has little probative value if additional evidence strongly corroborates the witness’s testimony the suppressed evidence might have impeached.” Conley, 415 F.3d at 189.
C. The Materiality of the Andino Proffer
The materiality of undisclosed evidence ultimately turns on the factors outlined above. As such, we assess the Andi-no proffer in terms of: (1) its evidentiary strength; (2) whether it was cumulative of other evidence offered at trial; (3) whether the matters on which it would have allowed impeachment were collateral; and (4) whether the matters on which impeachment would have been made possible were otherwise corroborated.
i. Evidentiary Strength
While the impact of withholding evidence is severe when that evidence is highly impeaching, the failure to disclose evidence whose impeachment value is merely marginal is manifestly insufficient to place the trial record in “such a different light as to undermine confidence in the verdict.” Mathur, 624 F.3d at 505 (quoting Kyles, 514 U.S. at 435, 115 S.Ct. 1555).
The district court’s finding that the Andino proffer was immaterial was based, in part, on the conclusion that its contents were “ambiguous.” This conclusion was well-founded, because while the *445Andino proffer suggests that Vega was involved in the drug trade months after he had professed otherwise, the full story is considerably more complicated. At the hearing on the motion for a new trial, the government represented to the district court that Vega’s service as a confidential informant extended beyond the Paladin investigation. More specifically, the government indicated that Vega had executed a controlled buy with Andino in December 2009 at the direction of agents involved in a separate investigation of Andino. This buy, of course, would exactly correspond with the information contained in the Andi-no proffer.
Based on the government’s representations, it seems likely that Vega did engage in a drug transaction with Andino during the period of time that Andino described. And, because Andino was merely under investigation at the time, one might reasonably infer that Andino would have had no reason to believe that he was dealing with a government informant.4 But, because Paladin chose to leave the record before the district court undeveloped, this panel is left to speculate as to what Vega might have said if confronted with the Andino proffer on cross-examination at trial.
We can envision three scenarios. Vega could have confirmed that he purchased Oxycontin from Andino in or around December 2009, but indicated that he did so at the direction of government agents investigating Andino; second, Vega could have admitted to lying on direct examination about his motivation for turning himself in to the FBI and about the cessation of his drug dealing; and third, Vega could have flatly denied engaging in narcotics transactions of any kind with Andino. We simply cannot know which of these scenarios would have played out because Paladin declined the district court’s invitation to hold an evidentiary hearing to learn more.
As the movant seeking a new trial under Brady, the burden to demonstrate the materiality of undisclosed evidence rests squarely with Paladin. See Strickler, 527 U.S. at 289, 119 S.Ct. 1936. Yet Paladin made the decision not to call either Vega or Andino as witnesses before the district court.5 Of the potential scenarios that we outline above, only two of three are potentially helpful to Paladin. In scenario one, were Vega to confirm that he purchased Oxycontin from Andino at the direction of the government, the impeachment value of the Andino proffer would be nonexistent because it would not suggest a discrepancy in Vega’s testimony.
If, on the other hand, in scenario two, the Andino proffer prompted Vega to admit that he had lied to the FBI and to the jury, the impeachment value of such testimony would be most significant. Likewise, in scenario three, were Vega to flatly deny dealing with Andino, it is possible that the Andino proffer could have been used to highlight the factual discrepancy and undermine Vega’s credibility, assuming that the district court admitted the *446proffer under an exception to the rule against hearsay.6
In effect, Paladin asks this panel to infer (or, more accurately, guess) that either scenario two or scenario three would have unfolded and, furthermore, were it scenario three, that the district court would have resolved the various hearsay complications in Paladin’s favor. This is a bridge too far and we decline to cross it because we conclude that Paladin has not carried his burden to demonstrate that the Andino proffer would have been at all helpful to him. We reach the same conclusion as the district court: the contents of the Andino proffer are at best highly ambiguous. Based on the record before us, we can only conclude that the impeachment value of the undisclosed evidence was likely minor, and thus it is insufficient to undermine our confidence in the jury’s verdict.
ii. Was the Andino Proffer Cumulative?
Suppressed evidence that is cumulative of evidence presented at trial is immaterial. Avilès-Colón, 536 F.3d at 19; see also Moreno-Morales v. United States, 334 F.3d 140, 148 (1st Cir.2003). Where, as here, suppressed evidence would have been used for impeachment purposes, the key issue is whether the defense had an adequate opportunity to impeach the witness by other means. Zeigler v. Callahan, 659 F.2d 254, 266 (1st Cir.1981). “Impeachment evidence, even that which tends to further undermine the credibility of the key Government witness whose credibility has already been shaken due to extensive cross-examination, does not create a reasonable doubt that did not otherwise exist where that evidence is cumulative.... ” Dumas, 207 F.3d at 16 (quoting United States v. Shelton, 588 F.2d 1242, 1248 (9th Cir.1978)); see also United States v. Connolly, 504 F.3d 206, 217 (1st Cir.2007). Nevertheless, “suppressed impeachment evidence can be immaterial because of its cumulative nature only if the witness was already ... impeached at trial by the same kind of evidence.” Conley, 415 F.3d at 192 (quoting United States v. Cuffie, 80 F.3d 514, 518 (D.C.Cir.1996)) (internal quotation marks omitted).
The district court’s immateriality finding was based, in part, on its conclusion that the Andino proffer would have provided avenues of impeachment that were cumulative of others already available to the defense. We agree that, at best, the Andino proffer would have provided Paladin with more of the same kind of evidence that was already available to him to undermine Vega’s credibility.
Paladin relies principally on the notion from Conley that evidence is cumulative only insofar as the witness was already impeached by the “same kind of evidence.” See id. In Conley, the defendant police officer was convicted of perjury and obstruction charges stemming from his involvement in (and subsequent cover-up of) the accidental beating of an undercover officer. Id. at 187. A key government witness — a fellow officer involved in the pursuit that led to the beating — testified regarding his perception of the chain of *447events. However, the prosecution had failed to disclose an FBI interview with the witness during which he expressed uncertainty regarding the events, and even asked that he be hypnotized in order to better recall what had happened. Id. at 185-86. This Court found that a Brady violation had occurred because the defendant had been unaware of any evidence suggesting that the witness was uncertain as to his recollection of events. Id. at 191 (“Prior to trial ... Petitioner did not know the Government’s key witness previously suggested he be hypnotized to ‘truly recall’ the events.... Without any other similar material, Petitioner did not impeach [the witness’s] ability to recall at trial.”).
A prejudicial Brady violation has not been effected, however, where the defendant already had available to him evidence that would have allowed for impeachment on the same or similar topics. In Moreno-Morales, for example, a key government witness implicated the defendant police officer in the murder of an unarmed suspect. 334 F.3d at 143-44. On appeal, the defendant argued that the government had turned over only a subset of polygraph test results demonstrating that the witness changed his story numerous times prior to trial. This Court disagreed, concluding that the additional test results would have been cumulative because the defense had ample opportunity to impeach the witness’s credibility with other evidence of his prior inconsistencies. Id. at 148; see also Connolly, 504 F.3d at 217 (“Given [ the witness’s] extensive criminal history, it would not have been an abuse of discretion for the district court to find that the absence of additional cross-examination on essentially the same well-developed theme would not undermine confidence in the jury’s verdict.”).
Paladin’s rebanee on Conley is misplaced. While the Andino proffer had the potential to lead to a line of questioning regarding Vega’s truthfulness with law enforcement and with the jury, Paladin already had available — and used — the same kind of evidence to undermine Vega’s credibility. For example, on cross-examination, the defense elicited from Vega the concession that, despite earning upwards of a million dollars from the sale of Oxycontin tablets, Vega paid virtually nothing in taxes and laundered the narcotics proceeds through a small business that he owned. Vega was also questioned about a home invasion in which several masked intruders broke into his house, bound Vega and his wife in front of their children, and stole various items. Vega admitted that he lied to police officers after the incident when he told them that he was not a drug dealer. In continued questioning, the defense suggested that Vega was similarly lying about his relationship with Paladin in order to avoid prosecution.
In sum, the principal focus of the defense on cross-examination sought to undermine Vega’s credibility by suggesting to the jury that Vega was generally dishonest and was willing to lie to serve his own interests. In the best case scenario for Paladin, the Andino proffer would have permitted one additional avenue to accomplish this same objective.7 This, we believe, necessarily means that the Andino proffer was cumulative because it is ultimately the same kind of evidence already in the record. See Conley, 415 F.3d at 192.
*448iii. Were the Issues for Impeachment Collateral?
“Impeachment evidence ... does not create a reasonable doubt that did not otherwise exist where that evidence is ... collateral.” Dumas, 207 F.3d at 16 (quoting Shelton, 588 F.2d at 1248). A matter is considered collateral if “the matter itself is not relevant in the litigation to establish a fact of consequence, i.e., not relevant for a purpose other than mere contradiction of the in-court testimony of the witness.” United States v. Beauchamp, 986 F.2d 1, 4 (1st Cir.1993) (quoting 1 McCormack on Evidence § 45, at 169 (4th ed.1992)).
The district court touched on this issue when it drew a distinction between “evidence the government has produced to support the elements of the case” and “evidence that bears on the credibility of a witness.” We read the district court’s oral decision as concluding, based on this distinction, that the Andino proffer was collateral because while it would have possibly permitted impeachment of Vega’s general credibility, it would not have permitted impeachment of the factual evidence underlying the government’s case. In light of the weakness of the suppressed evidence and its cumulativeness, any error in the district court’s finding — if indeed there was any at all — was harmless.
iv. Were the Issues for Impeachment Corroborated?
“[Suppressed impeachment evidence has little probative value if additional evidence strongly corroborates the witness’s testimony the suppressed evidence might have impeached.” Conley, 415 F.3d at 189; see also Hall, 557 F.3d at 19. Nevertheless, “[c]onfidence in the outcome is particularly doubtful when the withheld evidence impeaches a witness whose ‘testimony is uncorroborated and essential to the conviction.’ ” Norton v. Spencer, 351 F.3d 1, 9 (1st Cir.2003) (quoting United States v. Martinez-Medina, 279 F.3d 105, 126 (1st Cir.2002)).
This Court has previously found undisclosed evidence to be immaterial by virtue of having been independently corroborated. See, e.g., United States v. González-González, 258 F.3d 16, 18-19 (1st Cir. 2001) (despite government’s nondisclosure of evidence tending to undermine the credibility of two witnesses, numerous other witnesses testified as to defendant’s involvement in a drug conspiracy and the government introduced documentary evidence including recordings of conversations, travel records and surveillance photographs); Connolly, 504 F.3d at 214 (aside from a government witness who later purportedly recanted his testimony, multiple other witnesses testified as to defendant’s corrupt dealings with organized crime figures); Mathur, 624 F.3d at 505 (despite delayed disclosure of evidence defendant argued could be used to shift blame to one of the witnesses against him, numerous other witnesses testified as to their victimization by defendant’s financial scheme and the government introduced voluminous corroborative records). Nevertheless, at the other end of the spectrum, this Court has found that a trial court erred when it declined to grant a new trial where evidence emerged post-conviction that the government’s sole witness had fabricated allegations and where no additional corroborative evidence was introduced at trial. See Norton, 351 F.3d at 9.
Our focus with respect to corroboration is on the conspiracy charge. The district court concluded, and we agree, that the evidence of Paladin’s guilt on the distribution and possession with intent to distribute charges was overwhelming and did not depend on Vega’s credibility. Jurors heard testimony from federal, state and *449local law enforcement personnel who variously investigated Paladin, conducted surveillance on a series of controlled buys with Vega, tested and confirmed as cocaine the substances that Vega purchased from Paladin, executed a search warrant at Paladin’s residence where a significant cocaine stash was uncovered, and arrested Paladin while he was traveling in Georgia and carrying some ten thousand dollars in cash. This evidence was not meaningfully contested.
As the district court recognized, however, despite the strength of the government’s case with respect to the other counts, the conspiracy charge depended in large part on the jury’s willingness to accept Vega’s version of events that had taken place prior to his becoming a confidential informant.8
We believe that there was sufficient evidence corroborative of Vega’s testimony on the conspiracy count. We begin with Paladin’s own statements. The jury heard a series of audio recordings of meetings and controlled buys between Vega and Paladin, as well as the testimony of members of law enforcement who surveilled these interactions. Paladin is heard to make a series of comments indicative of a long-standing drug dealing relationship with Vega. For example, in one conversation in which Vega and Paladin are discussing drug quantities, the following exchange took place:
Paladin: That’s what you need, we don’t need, it’s not like [expletive deleted] to go around like we used to do, you know what I mean.
Vega: [Expletive deleted], what are we doing [expletive deleted] that one summer like 100 keys.9
Paladin: [Inaudible] ... I’m going beyond that.
During a separate conversation on the topic of conducting their meetings in parking lots, Paladin is heard to say: “Yeah, [expletive deleted], you gotta move in close so we can meet up like we did before, you feel me?” We view these (and other) statements by Paladin as corroborating Vega’s testimony that Vega and Paladin had previously conspired to distribute cocaine prior to Vega’s service as an informant.
There was also corroboration of Vega’s testimony by other witnesses, principally Agent Alford, the FBI agent who oversaw the investigation of Paladin and who testified first for the government. In one instance of corroboration, among others, Vega testified that in approximately 2007, he paid Paladin for a shipment of cocaine by giving him a customized Suzuki motorcycle. Agent Alford testified regarding the FBI’s seizure of the same motorcycle in 2010 and the government offered in evidence the motorcycle’s title history which showed Vega’s past ownership and present ownership by Paladin’s girlfriend, Cristy Baez.10
*450Next, Agent Alford corroborated Vega’s testimony with respect to the nature of the “fronting” and payment practices. Vega testified that his independent dealings with Paladin spanned from 2004 to 2008, and that there was a gap between Vega’s last purchase in 2008 and his reinitiation of contact with Paladin at the direction of the FBI in November 2009. Vega testified regarding the details of the first controlled buy, and jurors heard an audio recording of what transpired. During this meeting, Paladin provided Vega with six ounces of cocaine, and Vega and Paladin agreed that Vega would pay for it approximately a week later. This practice, in which the buyer effectively purchases drugs on credit, then pays the seller later with proceeds from the resale, is known as fronting. Vega testified that in two additional controlled buys in November and December 2009, Vega and Paladin used similar fronting arrangements.
Agent Alford corroborated this testimony by detailing for jurors his oversight of the controlled purchases. Specifically, Agent Alford testified that in the case of each controlled purchase, Paladin fronted Vega the drugs, then Vega later paid for them with funds provided by the government. As the district court recognized, Paladin’s willingness to front significant quantities of cocaine to Vega beginning with the first controlled purchase in November 2009 is indicative of a prior drug dealing relationship. We believe it unlikely that Paladin would have done so were he dealing with Vega for the first time.
Witnesses other than Agent Alford corroborated Vega’s testimony as well. For example, Vega testified that Paladin used the Lowell, Massachusetts home of an associate by the name of Juan Burgos (nicknamed “Indio”) as a stash house for guns, drugs and bulletproof vests. Vega testified about a conversation that he had with Paladin in 2007 in which Paladin described arriving at Indio’s home to collect a package of marijuana, knocking on the front door, and it being opened by police officers and FBI agents who happened to have just recently raided the home in connection with an investigation into Indio. Vega testified that Paladin told him that the officers questioned but ultimately released him. This testimony was corroborated by members of the Lowell Police Department who testified as to the execution of the raid at Indio’s home and Paladin’s ill-timed arrival, questioning and release.
D. Conclusion
Because of the questionable evidentiary strength of the Andino proffer, the fact that we believe it to provide avenues of impeachment that are merely cumulative of others that were already available to the defense, and because Vega’s testimony was sufficiently corroborated, we agree with the district court that the Andino proffer was immaterial. As such, we AFFIRM the denial of Paladin’s motion to set aside verdict and for a new trial.
III. Constitutional Challenges
Paladin raises a series of constitutional challenges to the district court’s imposition of a life sentence. As we noted previously, the life sentence was made mandatory by the government’s filing of an information pursuant to 21 U.S.C. § 851(a)(1) regarding two prior felony drug convictions that Paladin had sustained.11 These convic*451tions, combined with the fact that the instant charges involved more than the necessary five kilograms of cocaine, triggered the requirement set forth at 21 U.S.C. § 841 (b)(1)(A)(viii) that “[i]f any person commits a violation of this subparagraph ... after two or more prior convictions for a felony drug offense have become final, such person shall be sentenced to a mandatory term of life imprisonment without release.... ” Paladin raises these constitutional challenges for further appellate review because, as he concedes, most of them are foreclosed by binding precedent. We consider each of Paladin’s arguments in turn, but they need not detain us for long.
A. Does Alleyne v. United States Require Submission of Prior Convictions to the Jury?
Paladin maintains that because the indictment did not reference his two prior felony convictions, and because the jury was not required to find beyond a reasonable doubt that he had been convicted of these crimes, Paladin’s Fifth and Sixth Amendment rights to have all of the elements of an offense included in an indictment and found by a jury beyond a reasonable doubt were violated. Because this argument was preserved, our review is for harmless error, see United States v. Harakaly, 734 F.3d 88, 94 (1st Cir.2013), cert. denied, — U.S.—, 134 S.Ct. 1530, 188 L.Ed.2d 462 (2014), but we conclude based on binding Supreme Court and Circuit precedent that there was no error at all.
In Almendarez-Torres v. United States, the Supreme Court upheld the constitutionality of a statute that permitted district courts to enhance certain sentences based on the defendant’s status as a recidivist even where allegations of the defendant’s prior offenses were not set forth in the indictment. 523 U.S. 224, 228, 118 S.Ct. 1219, 140 L.Ed.2d 350 (1998) (“An indictment must set forth each element of the crime that it charges. But it need not set forth factors relevant only to the sentencing of an offender found guilty of the charged crime.”) (citations omitted). Four years later, in Harris v. United States, the Supreme Court affirmed the conviction of a defendant where the district court had found by a preponderance of the evidence that the defendant had “brandished” a firearm in relation to a drug crime, but the question of brandishing had not been submitted to the jury. 536 U.S. 545, 550-52, 122 S.Ct. 2406, 153 L.Ed.2d 524 (2002). Under the applicable statute, the district court’s finding that the defendant had brandished the firearm resulted in an enhanced mandatory minimum sentence. Id. at 550-51, 122 S.Ct. 2406.
Alleyne v. United States overruled Harris, reasoning that “[f]acts that increase the mandatory minimum sentence are ... elements and must be submitted to the jury and found beyond a reasonable doubt.” — U.S. —, 133 S.Ct. 2151, 2158, 186 L.Ed.2d 314 (2013). Nevertheless, Alleyne recognized an exception to this rule that a defendant’s prior convictions need not be submitted to the jury even where those convictions form the basis for an increased sentence. Id. at 2160 n. 1. In Alleyne, the Supreme Court expressly declined to revisit Almendarez-Torres. See id.; see also United States v. Carrigan, 724 F.3d 39, 51 n. 4 (1st Cir. 2013), cert. denied, — U.S. —, 134 S.Ct. 668, 187 L.Ed.2d 441 (2013) (“[Appellant] ... ask[ed] this court to find that his sentence is unconstitutional because the question of his status as an [armed career criminal] should have been submitted to *452the jury pursuant to [Alleyne ]. We disagree. In Alleyne, the Supreme Court stated that [Almendarez-Torres ] remains good law”) (citations omitted). This being the case, we must reject Paladin’s argument that his Fifth and Sixth Amendment rights were implicated when the indictment did not set forth his prior convictions and the jury was not required to pass upon them.
B. Does Alleyne Require Submission of Drug Quantity to the Jury?
Paladin contends that the district court erred by not submitting to the jury the question of whether Paladin was individually responsible for the charged quantity of cocaine (five kilograms or more). Because Paladin did not preserve this argument, our review is for plain error. See Harakaly, 734 F.3d at 94. To prevail, Paladin must show that the error was “prejudicial and affected his substantial rights, and ... caused a miscarriage of justice or seriously undermined the integrity or public reputation of judicial proceedings.” United States v. Carpenter, 736 F.3d 619, 632 (1st Cir.2013), cert, denied, — U.S. —, 134 S.Ct. 901, 187 L.Ed.2d 776 (2014) (quoting United States v. Henderson, 320 F.3d 92, 105 (1st Cir. 2003)) (internal quotation marks omitted).
As we noted earlier, Alleyne requires that any fact that serves to increase the mandatory minimum sentence be submitted to the jury and found beyond a reasonable doubt. 133 S.Ct. at 2158. Prior to Alleyne, this Court had held that “when a district court determines drug quantity for the purpose of sentencing a defendant convicted of participating in a drug-trafficking conspiracy, the court is required to make an individualized finding as to drug amounts attributable to, or foreseeable by, that defendant.” United States v. Colón-Solís, 354 F.3d 101, 103 (1st Cir.2004). Paladin urges a collective reading of Colón-Solís and Alleyne to require that the jury make an individualized finding as to the quantity of drugs attributable to a particular defendant. Against that backdrop, Paladin argues that the district court’s jury instructions were insufficient to glean an individualized finding as to the quantity of cocaine attributable to him.
This argument misconstrues the district court’s jury instructions and overlooks the nature of the charged conspiracy. The district court instructed the jury that “[i]n order for the defendant to be found guilty of the charged conspiracy, the government must prove that the defendant conspired to distribute and possess with intent to distribute five or more kilograms of cocaine.” (emphasis added). These instructions required the jury to find that Paladin directly conspired with respect to the applicable minimum quantity in order to sustain a conviction.
What is more, we distinguish the instant conspiracy from the facts underlying this Court’s holding in Colón-Solís. There, this Court was faced with one defendant involved in a large-scale Puerto Rican drug cartel responsible for massive quantities of cocaine and heroin. Id. at 102. Of course, in these contexts, the automatic attribution of the full scope of the conspiracy’s dealings to a particular defendant without an individualized quantity finding is problematic. See id. at 103-04. Here, however, the charged five kilogram quantity was based solely on the conspiratorial dealings of two men: Paladin and Vega. Paladin cannot reasonably maintain that his substantial rights were affected when there are no third parties to whom a material portion of the cocaine in question could be attributed.12
*453Even if we were to look beyond the sufficiency of the jury instructions and the nature of the charged conspiracy and conclude that the district court had erred, Paladin would be unable to demonstrate the requisite prejudice necessary to prevail on plain error review. This Court has previously “treated the presence of overwhelming evidence of the requisite drug types and quantities as a proxy for harmlessness.” Harakaly, 734 F.3d at 95 (quoting United States v. Pérez-Ruiz, 353 F.3d 1,18 (1st Cir.2003)).
Vega testified that, in his estimate, he and Paladin dealt in more than 100 kilograms of cocaine between 2004 and 2008. We acknowledge that this figure is an estimate and nothing more. But the volume of cocaine attributable to Paladin in the mere five weeks between the first controlled purchase with Vega and the raid on Paladin’s home confirms that he dealt in significant quantities, lending credibility to Vega’s estimate.
Agent Alford testified that Vega purchased six ounces of cocaine from Paladin in each of two controlled buys on November 12 and December 8, 2009, and a kilogram in a third buy on December 15. Then, Agent Alford testified that in the raid on Paladin’s home on December 17, an additional three and a half kilograms of cocaine were found and seized. That Paladin was responsible for nearly five kilograms of cocaine during this short period of time suggests that Vega’s estimate of their prior dealings between 2004 and 2008 was not wildly inaccurate.13
C. Does the Five-Year Limitation on Challenging Prior Convictions under 21 U.S.C. § 851(e) Violate Due Process and Equal Protection?
“No person who stands convicted of an offense ... may challenge the validity of any prior conviction alleged under this section which occurred more than five years before the date of the information alleging such prior conviction.” 21 U.S.C. § 851(e). Paladin’s argument with respect to this issue may be summarized as follows: the 1998 felony drug conviction that he sustained and that was one of two prior felonies referenced in the government’s information filed with the district court was tainted due to ineffective assistance of counsel, but Paladin was barred from raising this issue before the district court because the conviction was more than five years old. Thus, his right to due process and equal protection was violated. Our review is de novo. See United States v. Robinson, 137 F.3d 652, 653 (1st Cir.1998).
As Paladin acknowledges, this argument has been squarely refuted by Circuit precedent, and we must reject it. See Henderson, 320 F.3d at 104 (“The ban against challenging convictions over five years need only be supported by a rational legislative purpose because no fundamental right or suspect class is at issue in this case. The five year limitation ... has a rational basis in light of both the administrative difficulties inherent in challenges to prior convictions ... and the interest in finality. We therefore have no difficulty concluding that section 851(e) does not vio*454late [defendant’s] right to due process and equal protection of the law.”) (citations omitted).
D. Is the Life Sentence Arbitrary and Capricious?
Paladin contends that broad prosecutorial discretion results in the arbitrary and capricious application of sentencing enhancements under § 851(e). This argument too is foreclosed by binding precedent as the Supreme Court has reviewed and deemed constitutional the applicable practices under § 851. See United, States v. LaBonte, 520 U.S. 751, 761-62, 117 S.Ct. 1673, 137 L.Ed.2d 1001 (1997).
D. Does the Life Sentence Violate the Eighth Amendment?
Paladin’s final challenge suggests that his life sentence violates the Eighth Amendment because it is cruel and unusual and is contrary to an emerging national consensus on sentencing for nonviolent drug crimes.14 Our review is de novo. See United States v. Raymond, 697 F.3d 32, 40 (1st Cir.2012). This Court recently considered, and rejected, virtually identical arguments in United States v. Jones, 674 F.3d 88, 96-97 (1st Cir.2012), cert. denied, — U.S.—, 133 S.Ct. 363, 184 L.Ed.2d 215 (2012) (citing Supreme Court cases upholding life and other lengthy prison sentences for non-violent repeat drug offenders), and we must do the same here.
While we may well agree with the sentiment of the district court that the sentence here is excessive, like the district court, we cannot but hold that it is lawful and constitutional. Relief in cases such as this — if there is any — must come, in the first instance, in the exercise of restraint and wisdom in the charging decision of the prosecutor, or in the exercise of the clemency power; both are executive not judicial functions and leave us powerless to intercede to grant relief.
Affirmed.