SOUTER, Associate Justice.

Federal law mandates a minimum ten-year prison sentence for a convicted member of a drug conspiracy responsible for more than 280 grams of crack. 21 U.S.C. §§ 841(b)(1)(A)(iii); 846. So far as it matters to this appeal, the district court made a finding of drug quantity, by a preponderance of the evidence: that the admitted conspirator Ryan Morris was personally responsible for 765.5 grams of crack. The court consequently imposed the mandatory ten-year sentence. While judicial fact-finding of drug quantities sufficient by statute to trigger mandatory minimum sentences was permissible at the time of the sentencing hearing, during the pendency of Morris's appeal the Supreme Court held that the Sixth Amendment guarantees that such qualifying fact issues are subject to jury findings beyond a reasonable doubt. Alleyne v. United States, ___ U.S. ___, 133 S.Ct. 2151, 2160, 186 L.Ed.2d 314 (2013).

The question here is whether the minimum sentence imposed under the district court's judgment may nevertheless be affirmed as resting on harmless constitutional error falling short of affecting the defendant's substantial rights. We conclude that the error is ultimately harmless, in light of concessions made by Morris's counsel and overwhelming evidence that Morris was responsible for at least 280 grams of crack, and thus affirm.

I.

In December 2010, after investigating the activities of a drug ring operating in Dorchester, Massachusetts, the government charged nineteen individuals, including Ryan Morris, with conspiracy to distribute more than 500 grams of cocaine and more than 280 grams of crack, as well as offenses stated in fifteen additional counts. Shortly before the indictment was returned, investigators legally searched Morris's apartment, which yielded up 123.5 grams of crack. In October 2012, Morris pleaded guilty to the conspiracy count,1 but he did not admit that the conspiracy collectively or he individually was responsible for a particular quantity of either form of drug, the questions of quantity being expressly left for later determination by the sentencing judge.

In advance of Morris's sentencing hearing, the probation office prepared a presentence report concluding, based on the government's investigation, that Morris himself was responsible for 10 kilograms of cocaine, and 123.5 grams of crack. Because responsibility for 5 kilograms of cocaine triggers a mandatory minimum ten-year sentence, see 21 U.S.C. §§ 841(b)(1)(A)(ii), 846, the report recommended that Morris be sentenced accordingly. He objected to the conclusion about the cocaine quantity and the ensuing recommendation.

At the hearing, Morris took the stand and disputed that he had ever dealt in kilograms of cocaine. He said that he bought cocaine in quantities never greater than 62 grams, which he would cook into crack and then sell. Between direct and cross-examination, Morris admitted to four specific transactions between May and July 2010 involving 62 grams of cocaine each, for a total of 248 grams. When pressed on cross-examination to state the total number of transactions, he acknowledged more, albeit less exactly:

The district court found that Morris was not responsible for any kilogram transactions of cocaine, but because he had disputed being a cocaine dealer by admitting to being a crack dealer, the judge proceeded to consider what crack quantity he should be found responsible for.

Morris argued that he should be responsible only for the amounts converted from four specifically identified cocaine purchases, that is, a total of 248 grams of crack. He argued that the details of the remaining transactions were speculative guesses, and he suggested that the 123.5 grams of crack found in the search might be a leftover portion of the 248 grams.

The district court rejected Morris's position, and found by a preponderance of the evidence that he was responsible for crack cooked from the quantities of cocaine procured in at least twelve transactions, nine of 62 grams and three of 28. To this, the court added the stash of 123.5 grams of crack, which the district court found was not derived from the admitted transactions, given the "time frame between" between the purchases (May-July 2010) and the seizure (December 2010). The court thus calculated that Morris was responsible for 765.5 grams of crack, calling that conclusion "conservative." Because this exceeded the 280 gram threshold, the judge imposed a ten-year mandatory minimum sentence, although he said that he would impose a lower one if that were open to him.

While Morris's appeal was pending, the Supreme Court handed down Alleyne, which held that the Sixth Amendment requires any fact mandating the imposition (or an increase) of a particular minimum sentence to be treated as an element of the crime. 133 S.Ct. at 2160-63. Accordingly, under the principle of Apprendi v. New Jersey, 530 U.S. 466, 484, 120 S.Ct. 2348, 147 L.Ed.2d 435 (2000), "the Sixth Amendment provides defendants with the right to have a jury find those facts beyond a reasonable doubt," in the absence of a defendant's admission. Alleyne, 133 S.Ct. at 2160.

II.

Because Morris preserved no Sixth Amendment claim in the trial court, we review for plain error, the burden being on Morris to show (1) an error (2) that is clear and obvious, (3) affecting his substantial rights, and (4) seriously impairing the integrity of judicial proceedings. United States v. Santiago, 775 F.3d 104, 106 (1st Cir.2014). The government concedes the first two prongs of plain error in imposing the mandatory minimum based on the judge's finding of crack quantity by a preponderance, rather than a jury's finding beyond a reasonable doubt or Morris's specific admission.2

As for the third prong of plain error review, in substance it is harmless error analysis, except that the defendant bears the burden of persuasion. Ramirez-Burgos v. United States, 313 F.3d 23, 29 (1st Cir.2002). An Alleyne error is harmless when "it can fairly be said beyond any reasonable doubt that the assigned error did not contribute to the result of which the appellant complains." United States v. Harakaly, 734 F.3d 88, 95 (1st Cir.2013) (quoting United States v. Pérez-Ruiz, 353 F.3d 1, 17 (1st Cir.2003)). In drug cases, "overwhelming evidence of the requisite drug types and quantities" generally serves as a proxy for determining whether the Alleyne error contributed to the result. Harakaly, 734 F.3d at 95 (quoting Pérez-Ruiz, 353 F.3d at 18) (preserved Alleyne error); see also United States v. Razo, 782 F.3d 31, 40-41 (1st Cir.2015) (same); United States v. Paladin, 748 F.3d 438, 453 (1st Cir.2014) (third prong of plain error, substantial rights-harmlessness); United States v. Delgado-Marrero, 744 F.3d 167, 189 (1st Cir.2014) (fourth prong of plain error, integrity of proceedings). By "overwhelming evidence," we mean here a corpus of evidence such that no reasonable jury could find, based on the record, that the crack quantity was less than that required for the mandatory minimum to apply.

Much of Morris's brief is devoted to disputing the pertinence of the "overwhelming evidence" standard, but he cites no persuasive authority to support his position.3 To begin with, the cases applying the overwhelming evidence standard to address the harmlessness of Alleyne and Apprendi errors are legion. E.g., Razo, 782 F.3d at 40-41; Paladin, 748 F.3d at 453; Delgado-Marrero, 744 F.3d at 189; United States v. Correy, 570 F.3d 373, 377 (1st Cir.2009); United States v. Casas, 425 F.3d 23, 65-66 (1st Cir.2005); United States v. Morgan, 384 F.3d 1, 8 (1st Cir. 2004); United States v. Soto-Beníquez, 356 F.3d 1, 46 (1st Cir.2003); United States v. Nelson-Rodriguez, 319 F.3d 12, 45-46 (1st Cir.2003).

Morris nonetheless contends that this court should apply a "causal-connection" test, which "asks whether the district court might have imposed a lower sentence if it had complied with the Sixth Amendment's restrictions on judicial factfinding." Appellant's Reply Br. 2. But Morris presents a false choice. On the assumption that this issue had been entrusted to a properly instructed, rational jury, the district court could not have imposed a lower sentence, given overwhelming evidence that Morris was responsible for at least 280 grams of crack. Thus it comes as no surprise that the principal cases Morris cites in support of his causal-connection test are fully consistent with the overwhelming evidence test. See United States v. Barnes, 769 F.3d 94, 99 n. 5 (1st Cir.2014) (challenge to a sentence above the mandatory minimum; citing Harakaly); United States v. Pena, 742 F.3d 508, 514 (1st Cir.2014) (government concedes Alleyne error not harmless; same); United States v. Delgado-Marrero, 744 F.3d 167, 189-90 (1st Cir.2014) ("scant evidence" of fact mandating minimum sentence; same).

At oral argument, Morris sought to invoke a different standard for harmlessness that this court has applied in the context of error under United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005). See, e.g., United States v. Vázquez-Rivera, 407 F.3d 476, 490 (1st Cir.2005). But the applicability of such precedents to Alleyne errors is foreclosed by Harakaly and its progeny, as cited earlier.

In a supplemental filing, Morris seeks to benefit from United States v. Pizarro, 772 F.3d 284, 294-95 (1st Cir.2014), which distinguished between two forms of Alleyne error, at "trial" versus at "sentencing." The former is subject to harmless or plain error review, whereas the latter requires automatic reversal. Id. at 296. Morris contends that this case involves Alleyne "sentencing" error.

But he misreads Pizarro, which calls for reversal as "sentencing" error when the quantity issue had been submitted to a jury that rejected a finding in the government's favor. Id. "Trial" error in Pizarro, on the other hand, was simply a failure to instruct the jury on the quantity issue. Id. at 294-296. The Alleyne error in this case, determining a mandatory sentence on the basis of a fact not admitted in connection with a guilty plea (which here was expressly reserved, without objection, for the judge at sentencing), is akin to that of failing to instruct the jury on an element of the crime; in each circumstance, a crucial but unadmitted fact has escaped the required opportunity for a jury's determination. Thus, plain error review is in order, and we apply the overwhelming evidence test under the third prong.

III.

Under this test, we have no hesitation in concluding that the evidence is overwhelming that Morris is responsible for at least 280 grams of crack. Morris made a critical concession in the district court. His lawyer said to the judge, "[I]f you are going to find any grams of cocaine base or attribute to Mr. Morris, I would ask that you find that the four transactions between May 30th and July 5th of 2010...." Soon thereafter, the attorney added, "I would just ask the Court to just attribute four transactions of 62 grams each." We think these statements are most reasonably read as an admission on Morris's part that not only should he be held responsible for four transactions of 62 grams of cocaine but also that the court could attribute "cocaine base" (i.e., crack) quantities to him based on a 1:1 ratio with cocaine.4 Morris thus conceded responsibility for 248 grams of crack. See United States v. Etienne, 772 F.3d 907, 923 (1st Cir.2014) (where the defendant agrees to having conducted certain drug transactions, this "clearly establishe[s]" drug quantity sufficient to trigger a mandatory minimum sentence). The only question remaining, then, is whether the evidence is overwhelming that Morris is responsible for at least another 32 grams.

We believe that it is. The district court found that Morris was responsible for another 517.5 grams of crack,5 well above the additional 32 grams necessary to trigger the mandatory minimum sentence. While that finding was based on a preponderance standard, the evidence underlying the district court's calculations was Morris's own testimony, as quoted earlier, and obviously a defendant's admissions can support an inference of drug quantity beyond a reasonable doubt. See Harakaly, 734 F.3d at 96-97. Although not all testimony offered by a defendant should be treated as a reliable admission, there is good reason to treat Morris's testimony that way. He took the stand to dispute the government's cocaine quantity recommendations, and so had every incentive to minimize (and very likely did minimize) the degree to which he was involved in purchasing drugs. He testified that he was involved in about eight other drug transactions of at least 28 grams each (the twelve to which he testified, less the four conceded). Accordingly, his own testimony, despite its imprecise aspects, establishes beyond any doubt that he is responsible for far more than another 32 grams of crack.

Morris raises various questions about the district court's calculations of the drug quantity, noting that there was no direct evidence, only circumstantial evidence; suggesting that his own testimony was potentially unreliable; contending that the district court made possibly suspect inferences about quantity loss in the cocaine/crack conversion and the temporal gap between the cocaine transactions and the residence search; and observing that the district court acknowledged its own lack of certainty and cited the preponderance standard in reaching its conclusions. These criticisms would have some force and could be persuasive if we were asked whether the evidence overwhelmingly establishes that Morris was responsible for another 517.5 grams of crack above the 248 grams admitted. But, as mentioned earlier, the question is only whether Morris is responsible for another 32 grams. As to that, Morris's criticisms do not raise doubt in our mind.6

In sum, we conclude there is overwhelming evidence that Morris is responsible for at least 280 grams of crack, and has thus failed to meet his burden of persuasion under the third prong of plain error review.

IV.

The judgment of the district court is affirmed.