Plaintiff Anthony Waller appeals the district court's Rule 12(b)(6) dismissal of his municipal liability claim against the City and County of Denver for a Denver deputy sheriff's use of excessive force against him.

I.

On September 11, 2012, while in pretrial detention, Mr. Waller was escorted in handcuffs and other restraints to a courtroom located within the Denver City Jail for a first advisement hearing. Mr. Waller remained "very respectful and calm throughout the advisement." (Appellant's App. at 28.) After the judge finished the advisement, Mr. Waller "politely address[ed] the Court in a normal and subdued voice," stating that he thought the investigation should have come before his arrest. ( Id. ) The judge began to respond, but while she was speaking, Deputy Sheriff Brady Lovingier, who had been standing directly behind Mr. Waller, suddenly and "without warning, justification[,] or provocation" grabbed Mr. Waller, spun him around, and threw him face first into a nearby glass wall and metal post, causing *1281 him to sustain "serious and permanent injuries." ( Id. at 25, 29.) Deputy Lovingier's assault on Mr. Waller was captured on video recorded by the courtroom cameras.

Approximately one year later, on September 24, 2013, Deputy Lovingier received a thirty-day suspension for his assault on Mr. Waller. The official who issued the suspension order concluded that Deputy Lovingier had violated "DSD RR-300.22-Inappropriate Force; DSD RR-300.19.1-Disobedience of Rule: Department Order 5011.1J-Use of Force; Career Service Rule 16-60 for Neglect of Duty, Carelessness in Performance of Duties and Responsibilities, and Conduct Prejudicial to the Good Order and Effectiveness of the Department or Conduct That Brings Disrepute on or Compromises the Integrity of the City." ( Id. at 32.) The official explained that Deputy Lovingier's attack on Mr. Waller was "unprovoked," "egregious and unprofessional," and "breached several of the [Sheriff Department's] Guiding principles," as well as compromising the efficiency and reputation of the Sheriff Department and the City and County of Denver. ( Id. at 32-33.) A hearing officer affirmed Deputy Lovingier's thirty-day suspension.

Mr. Waller filed this federal lawsuit under 42 U.S.C. § 1983 in July 2014. His complaint included two claims that are pertinent to this appeal: a claim of excessive force against Deputy Lovingier, and a claim of municipal liability against Denver premised on Deputy Lovingier's use of force. The municipal liability claim alleged that Denver could be found liable for Deputy Lovingier's unconstitutional use of force based on the municipality's "failure to train and to appropriately supervise and discipline officers of the [Denver Sheriff Department]." ( Id. at 67.) Defendants filed separate motions to dismiss the complaint, and Mr. Waller then filed a motion for leave to amend. As pertinent here, Mr. Waller's proposed amended complaint raised two new theories of municipal liability relating to Deputy Lovingier's use of force, asserting that Denver could also be found liable based the Sheriff Department's poor hiring decisions and inadequate investigation of excessive force allegations.

A magistrate judge reviewed these motions and recommended that the district court grant Defendants' motions to dismiss but permit Mr. Waller to amend his complaint in part. Specifically, the magistrate judge recommended that Mr. Waller be permitted to amend his original complaint to clarify the legal basis of his excessive force claim against Deputy Lovingier. However, the magistrate judge concluded that neither the original nor the proposed first amended complaint contained sufficient factual allegations to set forth a plausible claim of municipal liability against Denver. The magistrate judge therefore recommended that the district court deny as futile Mr. Waller's motion for leave to amend his claim against Denver and grant Denver's motion to dismiss. Mr. Waller objected to this recommendation, but the district court ultimately adopted it.

Mr. Waller then filed a second amended complaint and proceeded with his excessive force claim against Deputy Lovingier. This claim was tried before a jury, which found that Deputy Lovingier had violated Mr. Waller's right to be free from excessive force. The jury awarded Mr. Waller $50,000 in actual damages but rejected his request for punitive damages. The district court entered final judgment consistent with this verdict and the earlier Rule 12(b)(6) dismissal, thus entering judgment in favor of Denver and against Deputy Lovingier. In addition to the $50,000 in damages awarded by the jury, the district *1282 court also awarded Mr. Waller $176,226 in attorney's fees under 42 U.S.C. § 1988 (b).

In this appeal, Mr. Waller challenges the Rule 12(b)(6) dismissal of his municipal liability claim against the City and County of Denver. Although he does not directly challenge the district court's denial of his motion for leave to amend his complaint, his arguments regarding the court's Rule 12(b)(6) dismissal are based primarily on the allegations in his proposed first amended complaint.

II.

"We review de novo a district court's decision on a Rule 12(b)(6) motion for dismissal for failure to state a claim." Alvarado v. KOB-TV, L.L.C. , 493 F.3d 1210 , 1215 (10th Cir. 2007). Under this standard, we "must accept all the well-pleaded allegations of the complaint as true and must construe them in the light most favorable to the plaintiff." Id. (internal quotation marks omitted). However, "mere 'labels and conclusions' and 'a formulaic recitation of the elements of a cause of action' will not suffice." Khalik v. United Air Lines , 671 F.3d 1188 , 1191 (10th Cir. 2012) (quoting Bell Atl. Corp. v. Twombly , 550 U.S. 544 , 555, 127 S.Ct. 1955 , 167 L.Ed.2d 929 (2007) ). "Accordingly, in examining a complaint under Rule 12(b)(6), we will disregard conclusory statements and look only to whether the remaining, factual allegations plausibly suggest the defendant is liable." Id. Stated differently, "[a] claim has facial plausibility when the plaintiff pleads factual content that allows the court to draw the reasonable inference that the defendant is liable for the misconduct alleged." Ashcroft v. Iqbal , 556 U.S. 662 , 678, 129 S.Ct. 1937 , 173 L.Ed.2d 868 (2009).

"In reviewing a Rule 12(b)(6) motion to dismiss, our first step is to review the factual allegations that should have been considered by the district court." Alvarado , 493 F.3d at 1215 . As an initial matter, we note a potential issue regarding which complaint is the operative complaint for us to consider in this appeal. Mr. Waller assumes that our review should be based on the allegations in his proposed first amended complaint; however, his motion for leave to amend the complaint was denied, at least as it related to his municipal liability claim, and Mr. Waller does not challenge that decision on appeal. Nevertheless, we need not resolve this potential issue because we agree with the magistrate judge and district court that neither complaint states a plausible claim of municipal liability, for the reasons discussed below.

Mr. Waller also contends that we may consider evidentiary materials outside his complaints. Specifically, his appellate briefs refer extensively to two reports regarding the Denver Sheriff Department that were released in May 2015, after he had filed both his original complaint and his proposed first amended complaint. Mr. Waller attached these reports to his objection to the magistrate judge's report and recommendation, and he argues both that the district court should have considered these reports below and that we should consider them on appeal because they "substantiate" his municipal liability claims against Denver. (Appellant's Br. at 19.)

The "usual rule" is "that a court should consider no evidence beyond the pleadings on a Rule 12(b)(6) motion to dismiss." Alvarado , 493 F.3d at 1215 . We have recognized a limited exception to this rule: "[T]he district court may consider documents referred to in the complaint if the documents are central to the plaintiff's claim and the parties do not dispute the documents' authenticity." Jacobsen v. Deseret Book Co. , 287 F.3d 936 , 941 (10th Cir. 2002). Mr. Waller argues that the reports *1283 may be considered because they are central to his claim of municipal liability. However, he ignores the clear language of our precedents, under which this exception applies only to "documents referred to in the complaint." Id. The reports are not referred to in either the original or the first amended complaint, and thus we are not persuaded that this exception would have permitted the district court to consider them in evaluating Mr. Waller's Rule 12(b)(6) motion to dismiss.

Mr. Waller also contends that the district court was permitted-and even mandated-to consider the 2015 reports because Rule 72(b)(3) of the Federal Rules of Civil Procedure and 28 U.S.C. § 636 (b)(1) both provide that district courts "may ... receive further evidence" in conducting a de novo review of objected-to portions of a magistrate judge's report and recommendation. We are not persuaded that the district court's general discretionary authority to receive further evidence in reviewing a magistrate judge's report and recommendation-which could entail, for instance, proposed findings made following an evidentiary hearing, see 28 U.S.C. § 636 (b)(1)(B) -trumps our specific holding in the Rule 12(b)(6) context that, with the limited exception of documents both referred to in the complaint and central to the plaintiff's claims, "a court should consider no evidence beyond the pleadings," Alvarado , 493 F.3d at 1215 . Moreover, even assuming for purposes of argument that Rule 72 or § 636(b)(1) would permit a district court to consider additional evidence in this context, this permission would still be discretionary, not mandatory, and it would extend, at best, only to the district court. Thus, the district court did not err in excluding these reports from consideration below, and we are neither required nor permitted to consider them in our review on appeal.

We accordingly turn to the primary question before us in this appeal-whether the factual allegations in either or both of Mr. Waller's complaints were sufficient to state a plausible claim of municipal liability against the City and County of Denver. The Supreme Court has made clear that "a local government may not be sued under § 1983 for an injury inflicted solely by its employees or agents." Monell v. Dep't of Soc. Servs. , 436 U.S. 658 , 694, 98 S.Ct. 2018 , 56 L.Ed.2d 611 (1978). "[I]n other words, a municipality cannot be held liable under § 1983 on a respondeat superior theory." Id. at 691 , 98 S.Ct. 2018 . Instead, "the government as an entity" may only be held liable "when execution of a government's policy or custom, whether made by its lawmakers or by those whose edicts or acts may fairly be said to represent official policy, inflicts the injury." Id. at 694 , 98 S.Ct. 2018 .

Thus, to establish municipal liability, a plaintiff must first demonstrate a "municipal policy or custom," which may take one of the following forms:

(1) a formal regulation or policy statement; (2) an informal custom amounting to a widespread practice that, although not authorized by written law or express municipal policy, is so permanent and well settled as to constitute a custom or usage with the force of law; (3) the decisions of employees with final policymaking authority; (4) the ratification by such final policymakers of the decisions-and the basis for them-of subordinates to whom authority was delegated subject to these policymakers' review and approval; or (5) the failure to adequately train or supervise employees, so long as that failure results from deliberate indifference to the injuries that may be caused.

Bryson v. City of Okla. City , 627 F.3d 784 , 788 (10th Cir. 2010) (internal quotation *1284 marks and brackets omitted). In addition to the failure to train and to supervise, other alleged supervisory shortcomings-for instance, failing to adequately screen job applicants-can also constitute an official policy if the plaintiff meets the stringent "deliberate indifference" standard of fault. Bd. of Cty. Comm'rs v. Brown , 520 U.S. 397 , 410-12, 117 S.Ct. 1382 , 137 L.Ed.2d 626 (1997) ; see also Dodds v. Richardson , 614 F. 3d 1185 , 1212 (10th Cir. 2010) (Tymkovich, J., concurring).

After establishing a municipal policy or custom, a plaintiff must demonstrate "a direct causal link between the policy or custom and the injury alleged." Bryson , 627 F.3d at 788 (internal quotation marks omitted). "Where a plaintiff claims that the municipality has not directly inflicted an injury, but nonetheless has caused an employee to do so, rigorous standards of culpability and causation must be applied to ensure that the municipality is not held liable solely for the actions of its employee." Brown , 520 U.S. at 405 , 117 S.Ct. 1382 . "The causation element is applied with especial rigor when the municipal policy or practice is itself not unconstitutional, for example, when the municipal liability claim is based upon inadequate training, supervision, and deficiencies in hiring." Schneider v. City of Grand Junction Police Dep't , 717 F.3d 760 , 770 (10th Cir. 2013) (internal quotation marks omitted).

Finally, at least for claims of inadequate hiring, training, or other supervisory practices, a plaintiff "must demonstrate that the municipal action was taken with 'deliberate indifference' as to its known or obvious consequences." Brown , 520 U.S. at 407 , 117 S.Ct. 1382 . " 'Deliberate indifference is a stringent standard of fault, requiring proof that a municipal actor disregarded a known or obvious consequence of his action,' " Connick v. Thompson , 563 U.S. 51 , 61, 131 S.Ct. 1350 , 179 L.Ed.2d 417 (2011) (quoting Brown , 520 U.S. at 410 , 117 S.Ct. 1382 ) (internal quotation marks and brackets omitted), as "[a] less stringent standard of fault for a failure-to-train claim 'would result in de facto respondeat superior liability on municipalities,' " id. at 62, 131 S.Ct. 1350 (quoting City of Canton v. Harris , 489 U.S. 378 , 392, 109 S.Ct. 1197 , 103 L.Ed.2d 412 (1989) ). "The deliberate indifference standard may be satisfied when the municipality has actual or constructive notice that its action or failure to act is substantially certain to result in a constitutional violation, and it consciously or deliberately chooses to disregard the risk of harm." Barney v. Pulsipher , 143 F.3d 1299 , 1307 (10th Cir. 1998). "In most instances, notice can be established by proving the existence of a pattern of tortious conduct." Id. Deliberate indifference "may be found absent a pattern of unconstitutional behavior" only in "a 'narrow range of circumstances' " where "a violation of federal rights is a 'highly predictable' or 'plainly obvious' consequence of a municipality's action or inaction." Id. at 1307-08 (quoting Brown , 520 U.S. at 409 , 117 S.Ct. 1382 ).

With this legal background in mind, we will separately consider each of Mr. Waller's alleged theories of liability: (1) inadequacies in Denver's hiring practices; (2) failure to train; (3) failure to supervise; (4) failure to investigate; and (5) failure to discipline.

We begin with Mr. Waller's argument that Denver is responsible for the injuries inflicted by Deputy Lovingier because Denver's hiring practices were inadequate and improper. The original complaint contains no allegations relating to Denver's hiring practices, but the first amended complaint contains the following allegations regarding Denver's hiring practices: "The City of Denver fails to perform sufficient *1285 background checks for hiring of deputies and hires deputies with criminal records including persons with misdemeanor and felony convictions"; "The hiring of deputies by the City of Denver is not based on a merit system but is influenced by the office of the Mayor including executive staff, administration[,] and members of Denver City Council"; and, "The hiring of deputies by the City of Denver is achieved through nepotism. The Defendant Brady Lovingier is the son [of] Bill Lovingier who headed the Sheriff's Department from 2006 to 2010." (Appellant's App. at 114-15.)

These allegations fail to state a plausible claim for relief. As an initial matter, we note that the first amended complaint does not plausibly allege "a direct causal link," Brown , 520 U.S. at 404 , 117 S.Ct. 1382 , between the violation of Mr. Waller's constitutional rights and the hiring of any deputies other than Deputy Lovingier. As for the hiring of Deputy Lovingier, the complaint does not plausibly allege that Denver acted with deliberate indifference to a known or obvious risk. The Supreme Court has made clear that the failure to conduct a sufficient background check on a job applicant is insufficient in itself to satisfy the deliberate indifference element. Id. at 410-11 , 117 S.Ct. 1382 . "The fact that inadequate scrutiny of an applicant's background would make a violation of rights more likely cannot alone give rise to an inference that a policymaker's failure to scrutinize the record of a particular applicant produced a specific constitutional violation." Id. "Only where adequate scrutiny of an applicant's background would lead a reasonable policymaker to conclude that the plainly obvious consequence of the decision to hire the applicant would be the deprivation of a third party's federally protected right can the official's failure to adequately scrutinize the applicant's background constitute 'deliberate indifference.' " Id. at 411 , 117 S.Ct. 1382 . Although the complaint alleges that Denver has hired some deputies with criminal records, it does not allege that Deputy Lovingier is one of these deputies, nor does it otherwise include any allegations that would plausibly suggest the individuals who hired him should have concluded that his "use of excessive force would be a plainly obvious consequence of the hiring decision." Id. at 412-13 , 117 S.Ct. 1382 . We therefore affirm the district court's holding that Mr. Waller did not set forth a plausible municipal liability claim based on Denver's hiring practices.

We next consider Mr. Waller's "failure to train" theory of liability. In considering this theory, we are mindful of the Supreme Court's warning that "[a] municipality's culpability for a deprivation of rights is at its most tenuous where a claim turns on a failure to train." Connick , 563 U.S. at 61 , 131 S.Ct. 1350 . To satisfy the stringent deliberate indifference standard, "[a] pattern of similar constitutional violations by untrained employees is 'ordinarily necessary' ": "Without notice that a course of training is deficient in a particular respect, decisionmakers can hardly be said to have deliberately chosen a training program that will cause violations of constitutional rights." Id. at 62 , 131 S.Ct. 1350 (quoting Brown , 520 U.S. at 409 , 117 S.Ct. 1382 ). Evidence of "a pre-existing pattern of violations" is only unnecessary " 'in a narrow range of circumstances,' " "however rare," in which "the unconstitutional consequences of a failure to train" are " 'highly predictable' " and "patently obvious." Id. at 63-64, 131 S.Ct. 1350 (quoting Brown , 520 U.S. at 409 , 117 S.Ct. 1382 ).

Mr. Waller argues that he has shown the necessary pattern of similar constitutional violations to establish deliberate indifference here. However, we are *1286 not persuaded that the factual allegations in either complaint are sufficient to make this showing. First, we disregard the original complaint's allegations regarding Denver police officers, who are employed by the Denver Police Department rather than the separate Denver Sheriff Department. These allegations were removed from the original complaint at Mr. Waller's request, and, even if they remained in the complaint, they would not be relevant to the question before us in this appeal-whether Denver was deliberately indifferent in September 2012 to a known or obvious risk that deputy sheriffs employed by the Denver Sheriff Department would use excessive force against detainees such as Mr. Waller. Other allegations in the original and first amended complaint are similarly irrelevant to this question because they allege only that certain inmates died or were injured in custody, with no indication that these incidents were related in any way to a deputy sheriff's use of force. 1 Without a connection between these incidents and a deputy sheriff's improper use of force, these allegations fail to suggest any deficiency in Denver's excessive-force training for deputy sheriffs.

As for Mr. Waller's allegations of a handful of excessive-force incidents that allegedly occurred in the months and years after he was assaulted by Deputy Lovingier, even if these allegations could arguably constitute circumstantial evidence that Denver's training program is deficient in some way, they fail to show that Denver acted with deliberate indifference in September 2012 to a risk that was known or obvious to the city at that time. Incidents that occurred subsequent to the incident at issue in this case cannot have provided Denver with notice of a deficiency in its training program before that incident, and thus they cannot be used as evidence that, prior to Deputy Lovingier's use of force against Mr. Waller, Denver "decisionmakers ... deliberately chose[ ] a training program that w[ould] cause violations of constitutional rights." Connick , 563 U.S. at 62 , 131 S.Ct. 1350 ; see also ibr.US_Case_Law.Schema.Case_Body:v1" id="p1287" href="#p1287" data-label="1287" data-citation-index="1" class="page-label">*1287 id. at 64 , 131 S.Ct. 1350 (noting that deliberate indifference generally requires "proof of a pre-existing pattern of violations" (emphasis added)). 2 Likewise, Mr. Waller's allegation that Deputy Lovingier was a trainer for the Sheriff Department on the use of force when he assaulted Mr. Waller in September 2012 does not support the inference that Denver was on notice before this incident of a risk of known or obvious constitutional harm.

The original and first amended complaint each contain only two allegations of a deputy sheriff's pre-September-2012 use of excessive force. The first of these allegations does not involve similar conduct to the conduct at issue in this case. Specifically, Mr. Waller alleges in both complaints that a civilian protesting the Iraq War in public was thrown to the ground by a Denver police officer, whose beating of the civilian was later joined by a Denver deputy sheriff. We are not persuaded that this civilian incident would have put Denver on notice of a need to train deputy sheriffs regarding their use of force against detainees held in the custody of the Denver Sheriff Department. The complaints allege only one prior incident involving a deputy sheriff's alleged use of excessive force against an individual in the Sheriff Department's custody. In that incident, as here, a deputy sheriff allegedly launched an unprovoked and unjustified attack on a restrained, compliant inmate who was simply standing where he had been directed to stand-there, next to an elevator.

Although Mr. Waller has thus alleged one similar prior incident, we have found no cases suggesting that a single prior incident can constitute a "pattern" of conduct giving rise to an inference of deliberate indifference. To the contrary, we have expressly held that "[o]ne prior incident, even if it was a constitutional violation sufficiently similar to put officials on notice of a problem, does not describe a pattern of violations." Coffey v. McKinley Cty. , 504 F. App'x 715 , 719 (10th Cir. 2012) ; see also Martin v. Malhoyt , 830 F.2d 237 , 255 (D.C. Cir. 1987) ("One instance, however egregious, does not a pattern or practice make.").

In the analogous case of Schneider , which involved an alleged sexual assault by a police officer, the plaintiff presented evidence both that the officer who raped her had been the subject of a prior sexual assault complaint and that the police department had received "one other report of non-consensual sexual misconduct" involving a different officer. 717 F.3d at 774 . We held: "Although this report of police misconduct along with the [prior] complaint are troubling, they were not enough to make it obvious to [the police department] that officers were likely to engage in non-consensual sexual conduct" without training on this issue. Id.

Our reasoning in Schneider weighs against any conclusion that the allegations in this case would fall within the " 'narrow range of circumstances' " in which deliberate indifference may be found "without proof of a pre-existing pattern of violations."

*1288 Connick , 563 U.S. at 63-64 , 131 S.Ct. 1350 (quoting Brown , 520 U.S. at 409 , 117 S.Ct. 1382 ). In Schneider , we rejected the plaintiff's municipal liability claim based in part on our conclusion that " 'specific or extensive training hardly seems necessary for [officers] to know that sexually assaulting [individuals] is inappropriate behavior.' " 717 F.3d at 774 (quoting Barney , 143 F.3d at 1308 ). As we had previously held in Barney , even assuming that a jail's training course on inmates' rights was inadequate, "we [we]re not persuaded that a plainly obvious consequence of a deficient training program would be the sexual assault of inmates." 143 F.3d at 1308 ; see also Andrews v. Fowler , 98 F.3d 1069 , 1077 (8th Cir. 1996) ("In light of the regular law enforcement duties of a police officer, we cannot conclude that there was a patently obvious need for the city to specifically train officers not to rape young women.").

With this reasoning in mind, we note that the conduct at issue here, unlike in many excessive force cases, did not involve an officer making the wrong call regarding the level of force to employ against an individual who posed a threat to the officer or other individuals or was actively resisting arrest. Cf. Connick , 563 U.S. at 63-64 , 131 S.Ct. 1350 (suggesting that police officers might need to be trained on the use of deadly force when they are "arm[ed] ... with firearms and deploy[ed] ... into the public to capture fleeing felons," as untrained novice officers are unlikely to be "familiar with the constitutional constraints on the use of deadly force"). Deputy Lovingier's use of force was improper not because of the amount of force he used, but because no force was warranted in the first place. Even an untrained law enforcement officer should have been well aware that any use of force in this situation-where a restrained detainee was simply addressing a judge at a hearing in a polite, calm voice-was inappropriate. This case does not involve technical knowledge or ambiguous "gray areas" in the law that would make it "highly predictable" that a deputy sheriff in Deputy Lovingier's position would need "additional specified training" to know how to handle the situation correctly. Id. at 71 , 131 S.Ct. 1350 (internal quotation marks omitted); see also R.A. v. City of N.Y. , 206 F. Supp. 3d 799 , 803 (E.D.N.Y. 2016) ("To establish deliberate indifference, Plaintiffs must show that Defendant City was on notice that it was highly predictable that Defendant Avalos would face 'a difficult choice of the sort that training or supervision will make less difficult.' The complaints against Defendant Avalos show a conscious decision by Defendant Avalos to commit sexual assault, which does not present a 'difficult choice' [that] further training would prevent." (quoting Walker v. New York , 974 F.2d 293 , 297 (2d Cir. 1992) ) (citations omitted)). Rather, as in Schneider and Barney , " 'specific or extensive training hardly seems necessary,' " Schneider , 717 F.3d at 774 (quoting Barney , 143 F.3d at 1308 ) (internal brackets omitted), to put deputy sheriffs on notice that they may not violently assault a restrained detainee who is not acting in a threatening manner.

Mr. Waller accordingly has not shown either that there was a pattern of prior similar misconduct or that " 'the need for more or different training [was otherwise] so obvious, and the inadequacy so likely to result in the violation of constitutional rights, that the policymakers of the city can reasonably be said to have been deliberately indifferent to the need.' " Id. at 773 (quoting Canton , 489 U.S. at 390 , 109 S.Ct. 1197 ). We therefore affirm the district court's dismissal of Mr. Waller's failure-to-train theory of municipal liability.

We next consider Mr. Waller's failure-to-supervise theory of liability. Although *1289 both complaints allege in conclusory fashion that Denver knew or should have known that its employees were inadequately supervised, neither complaint alleges any facts regarding any supposed supervisory deficiencies in the Denver Sheriff Department, much less any facts describing how Deputy Lovingier's supervision was inadequate or how his purportedly inadequate supervision caused Mr. Waller's injury. Mr. Waller has accordingly failed to "state a claim to relief that is plausible on its face," Twombly , 550 U.S. at 570 , 127 S.Ct. 1955 , on this theory of liability.

We turn then to Mr. Waller's failure-to-investigate theory of municipal liability. Mr. Waller asserts that Denver has created an atmosphere in which excessive force is likely to occur by consistently failing to investigate misconduct by deputies in the Denver Sheriff Department. The complaints contain the following specific allegations regarding Denver's allegedly inadequate investigation of deputy misconduct: (1) in a 2013 report, the Office of the Independent Monitor "found that inmate grievances that allege serious deputy misconduct are often not referred to or investigated by DSD internal affairs, as required by DSD policy"; (2) "[t]he OIM found that from January 1, 2011[,] to June 30, 2013, only 6% of inmate allegations of serious misconduct were investigated by [internal affairs]"; (3) "[t]he OIM found that DSD's failure to investigate allegations of serious misconduct 'deviates from both DSD policy and national standards on law enforcement accountability' "; (4) "[t]he OIM was concerned about the length of time that [internal affairs] investigations took. [Mr. Waller's] Complaint took over one year to investigate"; and (5) "[t]he OIM found that Denver's policy of failing to investigate serious allegations of inappropriate force 'deviates from both DSD policy and national standards on law enforcement accountability.' " (Appellant's App. at 112-13; see also id. at 33-34.)

These allegations are not sufficient to state a plausible failure-to-investigate claim. The first four allegations relate to deputy misconduct generally, rather than to uses of force specifically, and we are not convinced that they establish the requisite "direct causal link between the municipal action and the deprivation of federal rights," Brown , 520 U.S. at 404 , 117 S.Ct. 1382 . Moreover, all five of these allegations describe findings from an OIM report issued in 2013, the year after the assault at issue here, and, to the extent these allegations establish a time period for the underlying findings, they indicate that the OIM was reporting on the period between "January 1, 2011[,] to June 30, 2013." (Appellant's App. at 34, 113.) "[B]asic princip[le]s of linear time prevent us from seeing how conduct that occurs after the alleged violation could have somehow caused that violation," Cordova v. Aragon , 569 F.3d 1183 , 1194 (10th Cir. 2009), and thus only investigation deficiencies occurring prior to September 11, 2012, could help Mr. Waller demonstrate the requisite "direct causal link between the municipal action and the deprivation of federal rights," Brown , 520 U.S. at 404 , 117 S.Ct. 1382 . Mr. Waller's allegations in this case are too general, encompassing too "wide [a] swath of conduct," to "nudge[ his] claims across the line from conceivable to plausible." Khalik , 671 F.3d at 1191 (internal quotation marks omitted). We accordingly affirm the dismissal of this theory of municipal liability as well.

The last theory of municipal liability raised in the original and first amended complaints is that Denver caused Mr. Waller's injuries by failing to properly discipline prior acts of excessive force, thus giving rise to a custom or policy of excessive force within the Sheriff Department.

*1290 Mr. Waller alleges that Denver has not imposed "appropriate discipline for the egregious use of excessive force by any officer of the DSD." ( Id. at 33, 112.) Besides conclusory allegations such as this one, however, the original and first amended complaints are virtually silent on the question of discipline. To the extent Mr. Waller's complaints describe the discipline of deputy sheriffs besides Deputy Lovingier, the incidents he discusses either occurred after the conduct at issue in this case, or he does not allege that these incidents involved a deputy's use of excessive force. As discussed above, the complaint alleges only one prior incident of similar misconduct within the Denver Sheriff Department, and the complaint does not say whether or not this deputy sheriff was disciplined for his conduct, thus permitting no conclusions as to the appropriateness of any discipline that may have been imposed. Moreover, to the extent Mr. Waller argues that Deputy Lovingier's discipline was itself too lenient, we note that a subsequent failure to discipline cannot be the cause of a prior injury. See Cordova , 569 F.3d at 1194 . We therefore hold that Mr. Waller has not plausibly alleged that he was injured by any municipal failure to properly discipline deputy sheriffs for excessive force.

On appeal, Mr. Waller also argues broadly that he can prevail because the allegations in his complaint in general establish "that Denver has a custom, policy, or practice of tolerating and ratifying the use of excessive force." (Appellant's Br. at 48.) Assuming without deciding that this argument was properly preserved below and supported on appeal, we see no error. As noted above, a municipal policy or custom may take the form of a formal regulation or policy; a widespread, permanent, and well-settled custom; a decision by an employee with final policymaking authority; a final policymaker's ratification of both an employee's unconstitutional actions and the basis for them; or the deliberately indifferent failure to appropriately hire, train, supervise, or discipline employees. See Bryson , 627 F.3d at 788 . First, Mr. Waller does not allege that Denver has a formal regulation or policy permitting the use of excessive force. Second, Mr. Waller's allegations-describing only one similar incident of excessive force prior to his own injuries-fall far short of plausibly alleging a "widespread practice" of excessive force, much less a practice "so permanent and well settled as to constitute a custom or usage with the force of law." Id. (internal quotation marks omitted). We are not persuaded by Mr. Waller's argument that we can infer the existence of such a custom or practice because nothing else would explain why Deputy Lovingier would brazenly launch a wholly unprovoked attack on a detainee in front of a judge-and multiple cameras-in a courtroom. This argument assumes that Deputy Lovingier's conduct involved a level of deliberation and rational thought that the allegations of the complaint do not support. Additionally, even assuming for purposes of argument that Deputy Lovingier only assaulted Mr. Waller because he apparently perceived that such conduct would be accepted by city officials, we would need to "stack inference upon inference," In re Level 3 Commc'ns, Inc. Sec. Litig. , 667 F.3d 1331 , 1345 (10th Cir. 2012), to conclude that this perception was accurate and reflected a custom or practice of the Sheriff Department. Indeed, rather than supporting such an inference, Mr. Waller's allegations indicate that any such perception was wrong, as Deputy Lovingier was in fact suspended for his behavior, which the suspending official described as "egregious" and contrary to the Department's "[g]uiding principles." (Appellant's App. at 32-33.) Deputy Lovingier's actions, *1291 no matter how egregious, cannot in themselves give rise to an inference that the city must have been at fault, "for the officer's shortcomings may have resulted from factors other than a faulty training program" or other municipal deficiency. Canton , 489 U.S. at 390-91 , 109 S.Ct. 1197 . "To adopt lesser standards of fault and causation would open municipalities to unprecedented liability under § 1983." Id. at 391 , 109 S.Ct. 1197 .

As for the other potential forms of municipal policy that Mr. Waller could be attempting to invoke here, he does not allege that Deputy Lovingier was a municipal policymaker or that any municipal policymaker made a final decision that caused Mr. Waller's injuries. Nor is there any indication that a final decisionmaker ratified Deputy Lovingier's use of force; to the contrary, Mr. Waller alleges that Deputy Lovingier was criticized and suspended for using force in violation of Sheriff Department policies. Finally, as discussed above, we are persuaded that Mr. Waller has not plausibly alleged a claim of deliberately indifferent hiring, training, or other supervisory inadequacy.

Lastly, Mr. Waller argues that because he alleged a plausible municipal liability claim, he should have been permitted to conduct discovery. However, for the reasons discussed above, we are not persuaded that he has stated a plausible claim for relief. "Aside from conclusory statements, no allegations in the complaint give rise to an inference that the municipality itself established a deliberate policy or custom that caused [Mr. Waller's] injuries. Consequently, the complaint 'stops short of the line between possibility and plausibility of entitlement to relief.' " Mocek v. City of Albuquerque , 813 F.3d 912 , 934 (10th Cir. 2015) (quoting Twombly , 550 U.S. at 557 , 127 S.Ct. 1955 ). We therefore see no error relating to discovery. See Iqbal , 556 U.S. at 678-79 , 129 S.Ct. 1937 ; see also Roth v. Wilder , 420 F. App'x 804 , 805 (10th Cir. 2011) ("As the Supreme Court made clear in Twombly , ... Rule 12(b)(6) does not allow a plaintiff to file a complaint devoid of supporting facts as a vehicle to commence discovery on the off chance [s]ome facts might exist which could support a plausible claim.").

III.

The district court's decision is accordingly AFFIRMED .