GORSUCH, Circuit Judge.

Iqbal Makkar and Gaurav Sehgal ran the "Gitter Done," a small town convenience store in northeastern Oklahoma. When questions surfaced about the incense they carried on their shelves, the men spoke with state law enforcement officers, offered to have the officers test the incense to determine its legality, and offered as well to stop selling the product until the results came in. But this cooperation with state authorities apparently won the men little admiration from federal investigators: soon enough they found themselves under indictment and convicted for violating the Controlled Substance Analogue Enforcement Act (Analogue Act), conspiracy, and money laundering. In this appeal Mr. Makkar and Mr. Sehgal contend that the government overreached at trial — in the jury instructions it sought and won, and again in its successful efforts to exclude evidence of their cooperation with law enforcement. After our own review of the record, we have to agree.

*

Take first Mr. Makkar's argument about the jury instructions. The Analogue Act is a curious animal. It's familiar learning that the Controlled Substances Act (CSA) proscribes the knowing possession and distribution of certain listed substances (marijuana, cocaine, heroin, and the like). What's less well known is that the Analogue Act picks up where the CSA leaves off, forbidding the possession and distribution of substances analogous to those listed in the CSA. In this way, the relationship between the two statutes is not unlike the relationship between the different sections of the Armed Career Criminal Act (ACCA). Much as here, one part of that statute lists certain specific violent felonies and imposes special punishments for their commission. Meanwhile, another part of that statute — what's called its residual clause — extends the statute's punishments to other, unspecified offenses that can claim similarity to listed ones.

The resemblance between the Analogue Act and the residual clause of the ACCA might raise some questions in your mind. After all, the Supreme Court in Johnson v. United States, ___ U.S. ___, 135 S.Ct. 2551, 192 L.Ed.2d 569 (2015), recently declared the ACCA's residual clause too vague to permit its constitutional application. But so far at least the Court hasn't reached a similar judgment about the Analogue Act. In fact, the Court only recently gave the Analogue Act a narrow construction that may go some way to alleviating potential concerns about the vagueness of its terms. In McFadden v. United States, ___ U.S. ___, 135 S.Ct. 2298, 192 L.Ed.2d 260 (2015), the Court accepted the government's concession that to establish a violation of the Analogue Act it must prove the drug in question bears two features: (1) it must be substantially similar in chemical structure to a schedule I or II CSA controlled substance, and (2) it must have, or be represented or intended to have, an effect on the central nervous system that is substantially similar to that of a schedule I or II CSA controlled substance. Id. at 2305 n. 2. When it comes to mens rea, the Court further explained, the government must show that the defendant knew the drug he possessed either (1) had both of these features, or (2) was controlled by the CSA or Analogue Act. Id. at 2305. The Court seemed to suggest that this narrow construction would help alleviate potential vagueness concerns, at least in the face of a facial challenge. Id. at 2307. But whether this construction will suffice to save the Analogue Act from the same fate as the ACCA's residual clause may still remain to be seen. It's an open question, after all, what exactly it means for chemicals to have a "substantially similar" chemical structure — or effect. And whether terms like those will admit of fair application and afford citizens fair notice, or whether we will find ourselves wading incrementally, in one as-applied challenge after another, deeper into an analytical swamp much as we did with the ACCA's residual clause litigation.

Still, in this case we face a much more prosaic problem. The government sought and secured an ambitious mens rea instruction that just will not square with the text of the statute or McFadden. At trial, the government didn't attempt to show that Mr. Makkar or Mr. Sehgal knew the incense they sold was unlawful under the CSA or Analogue Act — the second method of proving mens rea McFadden prescribed. Instead, the government attempted to show mens rea only in the first manner McFadden, discussed — by showing that the defendants knew that the incense they sold had (1) a substantially similar chemical structure to JWH-18 (a synthetic cannabinoid with marijuana-like effects listed as a CSA controlled substance) and (2) a substantially similar effect to that of marijuana (another CSA listed substance). So far, at least for current purposes we may assume, so good.1 The difficulty is that, after choosing to proceed this way, the government sought to shrug off the first of the mens rea requirements it had just agreed to shoulder. In fact, as far as we can tell, at trial the government introduced no evidence suggesting that the defendants knew anything about the chemical structure of the incense they sold. And as a way around this shortcoming the government sought — and the district court agreed to issue — an instruction permitting the jury to infer that the defendants knew the incense they sold had a substantially similar chemical structure to JWH-18 from the fact they knew the incense had a substantially similar effect to marijuana. Coming at the point another way, the government asked for and won the right to collapse its two separate elemental mens rea burdens into one. Under the inferential instruction it secured, the government was able to argue to the jury that it should find the first mens rea element satisfied beyond a reasonable doubt merely (and without more) because it found the second satisfied beyond a reasonable doubt.

This surely made trial easier for the government, but just as surely it means we must undo the judgment now. Before a district court may issue an instruction permitting the jury to infer the presence of even a single essential element from a set of facts the inference must — at the least — be shown capable of leading a rational trier of fact to the conclusion that the element in question is proven to the level demanded by the applicable standard of proof. United States v. Berry, 717 F.3d 823, 829 (10th Cir.2013). Neither may a district court ever issue instructions that effectively relieve the government of proving each essential element specified by Congress. See United States v. Gaudin, 515 U.S. 506, 511, 115 S.Ct. 2310, 132 L.Ed.2d 444 (1995). Both of these principles were violated here. The government's instructional inference invited the jury to infer the presence of one essential element from another, effectively collapsing two independent statutory inquiries into one. And it did so only by resort to a logical fallacy, a hasty generalization or associational error — an unwarranted assumption that because certain things share one characteristic they must share others. As a matter of common experience and logic, the fact that one drug produces a similar effect to a second drug just doesn't give rise to a rational inference — let alone rationally suggest beyond a reasonable doubt — that the first drug shares a similar chemical structure with the second drug. And it most certainly does not give rise to a qualifying inference that the first drug shares a similar chemical structure with a third drug — as the government was, in fact, allowed to argue to the jury in this case.

The government now admits all this, expressly acknowledging on appeal that different drugs can bear similar effects while lacking similar chemical structures. As Mr. Makkar notes by way of example, Tylenol and Aspirin may do much the same job on your aches and pains but most scientists do not place them in the same chemical group. See Harold E. Doweiko, Concepts of Chemical Dependency 208-09 (9th ed.2015). Cocaine and methamphetamine are also sometimes said to have similar effects, but their chemical structures are rarely said to be substantially similar even by the government. See Drug Enforcement Administration, Drugs of Abuse 51 (2015); 2 Encyclopedia of Applied Developmental Science 1062 (Celia B. Fisher & Richard M. Lerner eds., 2005). Even closer to home, the DEA has issued published guidance acknowledging that, while JWH-18 and marijuana have similar effects, the former "is not related in chemical structure to [THC] or other cannabinoids contained in the cannabis plant." Drug Enforcement Administration, JWH-018 1-Penty1-3-(1-naphthoyl)indole (2013). And closer to home still, the government's own chemistry experts advised the government and court in another case decided before Mr. Makkar's trial that the very inferential instruction it sought here "is not scientifically sound." United States v. Lane, No. CR-12-01419-PHX-DGC, 2013 WL 3716601, at *1 (D.Ariz. July 13, 2013).

But while conceding now that the jury instruction it sought and won in this case was wrong, the government also insists that the error isn't consequential enough to worry over. The government points to the fact that Mr. Makkar didn't object to the instruction at trial. And the government reminds us that, in these circumstances, we apply the plain error standard of review. Under that standard, before reversing we must be persuaded that (1) the district court erred; (2) the error was plain; (3) the error affected a defendant's substantial rights; and (4) the error seriously affected the fairness, integrity, or public reputation of a judicial proceeding. United States v. Olano, 507 U.S. 725, 732, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993). The government accepts that the first element — error — is present in this case, but pitches an avid battle on the second, insisting that the instructional error here can't fairly be described as plain. By way of support it points us to United States v. Turcotte, 405 F.3d 515 (7th Cir.2005) — a decision that arguably once endorsed the government's inferential instruction. From the fact that another court once committed the same error it did in this case, the government asks us to reason, the error it invited doesn't qualify as a plain one.

This argument depends on a hasty generalization of its own. The fact other circuits have committed an error can supply "strong evidence" that it doesn't qualify as a "plain" one. United States v. Story, 635 F.3d 1241, 1248 (10th Cir.2011). But another circuit's commission of an error doesn't necessarily and always "control" the plain error inquiry. United States v. Ahidley, 486 F.3d 1184, 1193 n. 7 (10th Cir.2007); see also United States v. Overholt, 307 F.3d 1231, 1256 (10th Cir. 2002). The fact that able judges elsewhere once committed an error may go some way toward showing it isn't plain, but it doesn't always and necessarily prove the point as the government seems to suppose. Neither could it sensibly do so. After all, to err is human — and to plainly err is too. Despite our aspirations (and maybe sometimes our pretenses) we judges can hardly claim to escape that fact of life. Certainly far greater minds than ours haven't managed to evade its grasp. See, e.g., Atom Energy Hope Is Spiked by Einstein, Pittsburgh Post-Gazette, Dec. 29, 1934, at 13 ("The idea that man might some day utilize the atom's energy brought the only emphatic denial from [Einstein]."); Lord May, Royal Society Anniversary Address, Dec. 2003, quoting Lord Kelvin (1895), ("Heavier than air flying machines are not possible.").

When it comes to the particular error before us, too, we harbor no doubt that it was plain by the time of the trial in this case. Yes, Turcotte appears to have fallen into the same trap as the government did here. But even in the government's telling, no other circuit court repeated Turcotte's mistake in all the years between that decision in 2005 and the trial in this case in 2014. And surely those years of precedential silence chirping around Turcotte — the fact that over the span of nearly a decade no other circuit apparently endorsed its reasoning — must be heard and accounted for. Even more dramatically, we know that in those intervening years the government in Lane expressly disavowed Turcotte's inference as scientifically unsound. And we know the DEA effectively rejected the inference as irrational too when it acknowledged that JWH-18 and marijuana bear similar effects but do not share a similar chemical structure. These developments, moreover, only confirmed what's been true all along — that the government's inference depends on what can only be described as an assumption inconsistent with logic and experience — an assumption that just because one drug shares a particular trait in common with a second drug the first also shares a different trait in common with the second drug — or even a third drug. And while we cannot ignore the holding of another circuit when assessing the plainness of an error, we equally cannot ignore so many countervailing considerations like these. In this case, those countervailing considerations seem to us to speak clearly and loudly to the plainness of the government's error and in a way even Turcotte cannot fairly obscure. Indeed, we find it hard to see how we might fairly describe an error as anything less than plain when the government itself concedes the error on appeal and did so (effectively twice) even before the trial in this case.

Of course, that still leaves the remaining two elements of the plain error test. But our discussion on these points can be brief. The government doesn't dispute either element and it's easy to see why. At trial, the government sought to prove mens rea by way of the first route McFadden approved. Yet, it made no effort to carry the burden of producing evidence suggesting that the defendants knew the chemical composition of the incense they sold, relying instead entirely and admittedly on an unsound inference. And surely a defendant's substantial rights and the integrity of judicial proceedings are both implicated when he is relegated to federal prison even though the government concedes it hasn't proven what the law demands it must prove to send him there. United States v. Sabillon-Umana, 772 F.3d 1328, 1333-34 (10th Cir.2014).2

*

With the plain error test satisfied, it might seem we've reached the end of the road on Mr. Makkar's challenge to the jury instructions. But at oral argument on appeal the government sought to introduce a new argument in defense of the instructions it sought and won. To bear its burden of showing mens rea under McFadden, the government contended, it must prove only that the defendant knew the drug in question has a substantially similar effect to a class I or II CSA controlled substance. And, the government contended, it proved at least that much at trial.

This new argument concerns us on various fronts. Not only does it appear too late in these proceedings to be taken seriously — presented for the first time at argument. And not only is it inconsistent with the jury instructions the government itself sought and the trial court issued in this case. It also disregards the Supreme Court's recent teachings and the statutory text. After all and as we've seen, McFadden imposes a far more challenging mens rea requirement than the government is willing now to admit. McFadden requires the government to show that the defendant: (1) knew the drug in question had both a similar chemical structure and similar effects to a controlled substance, or (2) knew the drug in question was unlawful under the Analogue Act or CSA. 135 S.Ct. at 2305. Proof that the defendant merely knew the drug he sold had a similar effect to a controlled substance is never enough. And the plain language of the statute underscores and confirms what McFadden clearly explained. As written, the Analogue Act makes it a crime to possess or distribute a drug that both (1) is substantially similar in chemical structure to a schedule I or II CSA controlled substance, and (2) has, or is represented or intended to have, an effect on the central nervous system that is substantially similar to that of a schedule I or II CSA controlled substance. Both elements are essential to a conviction and the government offers no sensible way or reason why a mens rea requirement applied to the statute might take an olympian leap over the first essential element and touch down only on the second. Neither can we discern for ourselves any plausible argument along these lines. Cf. Flores-Figueroa v. United States, 556 U.S. 646, 650, 129 S.Ct. 1886, 173 L.Ed.2d 853 (2009) (noting that when a statute introduces elements of a crime with the word "knowingly," that means it applies to "all the subsequently listed elements").

*

Where Mr. Makkar identifies an instructional error, Mr. Sehgal focuses our attention on an evidentiary error. As part of their defense at trial Mr. Makkar and Mr. Sehgal sought to introduce evidence showing that they asked state law enforcement agents to test the incense to assure its legality under state law — and that they offered to stop selling the incense until the results came in. In the defendants' view, this evidence was relevant to the questions whether they knew the chemical structure of the incense and whether they knew its structure was substantially similar to the structure of controlled substances. After all, they sought to argue, why run a test if you already know the answer? Let alone agree to have law enforcement administer the test if you know the answer will be unfavorable to you under either state or federal law? Especially when state and federal agents often cooperate with one another (as, in fact, they did in this case)? In reply to the defendants' trial plans, the government filed a written motion in limine seeking to exclude the evidence as irrelevant under Rule 401 of the Federal Rules of Evidence. And here again the district court adopted the government's position. In doing so it rejected the defendants' suggestion that the evidence was relevant because "[i]f they knowingly sold [the incense], and it is an analogue, it's illegal. [They] don't have to know the chemical composition." Aplt.App. Vol. II, at 321.

While we normally afford a district court a good deal of discretion on evidentiary rulings, we do not defer to rulings that rest on legal error. United States v. Toombs, 713 F.3d 1273, 1278 (10th Cir.2013). And the government's motion in limine, adopted by the district court, clearly depends on a mistaken view of the Analogue Act. As we've seen, McFadden's first path for proving mens rea — the path the government chose to pursue at trial in this case — required it to prove a good deal more than that the defendants knowingly sold the incense. The government had to prove that the defendants knew the incense had both similar effects and a similar chemical structure to a controlled substance. Neither does anyone before us even attempt to suggest that the defendants' proffered evidence was irrelevant to these properly controlling questions. The government did not attempt to suggest so much at trial, the district court did not contend so much in its ruling, and the government does not make any effort in this direction on appeal.

Instead and again the government replies that, even if it invited error at trial, the error it invited was (again) immaterial. In doing so, the government this time points to harmless error doctrine and reminds us that, even when a defendant rightly identifies and objects to an error at trial, federal appellate courts aren't in the business of supplying remedies where they aren't needed — where the error in question did not affect the defendant's substantial rights. See Fed.R.Crim.P. 52; 28 U.S.C. § 2111.

All that's true enough but we don't see how it might fairly describe the error here. It's the government's burden of showing any error it invited is only harmless. And toward that end in this case the government stresses that Mr. Makkar and Mr. Sehgal were allowed to introduce other evidence about their mens rea. From this fact, the government would have us conclude that the erroneously excluded evidence was merely cumulative. But this represents another leap of logic longer than we can safely take. As a matter of common sense and our collective experience, we have a hard time imagining more powerful proof that a defendant didn't know the chemical composition of a drug, and didn't know it was substantially similar to an unlawful substance, than evidence that he turned to law enforcement for information about the drug's composition and offered to suspend sales until tests could be performed. Indeed, it's difficult for us to imagine why the government would have opposed the introduction of this evidence and taken the trouble to present a written motion on that score but for its extraordinary power to persuade the jury on questions like these. See United States v. Yarbrough, 527 F.3d 1092, 1103 (10th Cir.2008) (When an error deprives a defendant of "important evidence relevant to a sharply controverted question going to the heart of [the] defense, ... substantial rights [are] affected.").

*

Mr. Makkar and Mr. Sehgal present several other arguments for reversal — at least some of which raise some potentially troubling questions in their own right. For example, they complain that the government endorsed and the district court admitted unreliable and inflammatory testimony about the health effects of the incense. They point by way of illustration to one government witness who testified before the jury that the incense killed his friend — even though the friend was alive and apparently well — and scheduled to testify later the same day. They point as well to the calculation of their advisory guidelines sentences. In most cases, they note, inert materials associated with illicit drugs are added into guidelines calculations at the "drug quantity table" stage, with the aim of capturing the actual "street weight" of the drugs as sold. See U.S.S.G. § 2D1.1. But in this case, they tell us, the district court included inert materials at an earlier stage, when it applied the "drug equivalency table." And this choice meant the inert materials exerted something like a geometric effect on their advisory guidelines sentencing calculations. Thus calculated, they say, the guidelines attributed to them not the actual "street weight" of the inert materials they sold but 167 times that weight. See id. The men argue that in following this path the district court misread the guidelines, which most expressly reference inert materials in a note to the drug quantity table, not the drug equivalency table. And they argue their resulting sentences were irrationally inconsistent with the sentencing commission's goal of seeking sentences that account for the actual street weight of illicit drugs as sold. We see no need, however, to resolve the defendants' additional objections along these and other lines. Whether or not their arguments bear merit, we think the essential point by now amply evident: their convictions rest on legal errors that cannot be easily dismissed as harmless. Neither do we assume the additional putative errors they complain of will necessarily recur: to the contrary and as we've seen, it's unclear at this point whether the men can be lawfully retried consistent with the law's demands.

Mr. Makkar's and Mr. Sehgal's convictions for violating the Analogue Act are vacated. Because the putative violation of the Analogue Act was the only predicate crime to support their convictions for conspiracy and money laundering, Mr. Makkar's and Mr. Sehgal's convictions on these counts are vacated as well. The case is remanded and any further proceedings shall conform with this opinion.