MOORE, J!, delivered the opinion of the court, in which GIBBONS, J., joined, and SUTTON, J. joined in part. SUTTON, J. (pp. 505-07), delivered a separate opinion concurring in all but Section III. A of the majority opinion.
OPINION
KAREN NELSON MOORE, Circuit Judge.
An Ohio state-court jury convicted Petitioner-Appellant, James Frazier, of aggravated murder (with two death-penalty specifications), aggravated burglary, and aggravated robbery. The jury recommended the death penalty, and the state-trial-court judge sentenced him to die by lethal injection. After exhausting his state appeals, Frazier filed a petition for a writ of habeas corpus in federal district court, arguing inter alia that he is ineligible for the death penalty under Atkins v. Virgi[490]*490nia, 536 U.S. 304, 122 S.Ct. 2242, 153 L.Ed.2d 335 (2002), due to his intellectual disability, that his trial counsel provided ineffective assistance, and that Ohio’s lethal-injection regime is unconstitutional. The district court denied Frazier’s petition. Because the state courts’ decisions were not contrary to, nor an objectively unreasonable application of, clearly established federal law as defined by the United States Supreme Court, we AFFIRM the district court’s denial of Frazier’s petition.
I. BACKGROUND
A. Facts
Early in the morning of March 2, 2004, Frazier brutally murdered Mary Stevenson. The police found Stevenson’s body at the foot of her bed later that evening. On March 3, the police searched the dumpster used by Stevenson and other first-floor residents of the apartment complex, found nothing, and proceeded to search the sealed dumpster used by Frazier and other residents on the second through tenth floors. In this second dumpster, the police discovered some of Stevenson’s belongings, as well as a bloody knife, which matched one missing from Stevenson’s knife block. They also found a bloody t-shirt (size XXL) and pieces of mail addressed to Frazier near Stevenson’s belongings. Based on this information, investigators sought, obtained, and executed a search warrant for Frazier’s apartment where they confiscated two t-shirts that matched the bloody one’s brand, style, and size.
On March 4, detectives William Seymour and Denise Knight interviewed Frazier in connection with Stevenson’s murder. They read Frazier his Miranda rights, and he waived them. See Joint Appendix (“J.A.”) Vol. 7 at 3252-54 (Seymour Test.); App’x to Writ Vol. 8 at 195 (Miranda waiver). Det. Seymour testified that Frazier appeared “clearheaded,” that he understood his rights, and that he was not “under the influence of ... crack cocaine or alcohol.” J.A. Vol. 7 at 3252:21-3253:6 (Seymour Test.). During questioning, Frazier gave answers that the detectives found to be inconsistent with other evidence in the case. Additionally, the police conducted forensic tests on the knife, t-shirt, and various hairs and bodily fluid samples, which implicated Frazier.
On March 9, a grand jury indicted Frazier for aggravated murder with two death-penalty specifications, in violation of Ohio Revised Code § 2903.01(B); aggravated robbery, in violation of § 2911.01(A)(3); and aggravated burglary, in violation of § 2911.11(A)(1). J.A. Vol. 1 at 294-95 (Indictment). The state trial court appointed counsel, and counsel, in turn, requested that the court authorize funds for the employment of a psychologist and a mitigation expert. The court granted this motion on December 30, 2004.
Frazier’s counsel hired Dr. Jeffrey L. Smalldon, a board-certified forensic psychologist with death-penalty experience, to determine whether Frazier was mentally retarded.1 Dr. Smalldon met with Frazier [491]*491twice: on January 12 and May 2, 2005. J.A. Vol. 3 at 1075 (Smalldon Report). He administered a battery of tests, including the Wide Range Achievement Test-Revision 3 (“WRAT-3”), the Wechsler Adult Intelligence Scale-Third Edition (‘WAISIII”), the Bender Visual Motor Gestalt Test (“Bender”), the Trail Making Test (Parts A and B), the Aphasia Screening Test, and the Rotter Incomplete Sentences Blank Test. Dr. Smalldon considered administering several other tests, but he “concluded that [Frazier’s] limited reading and comprehension abilities wouldn’t enable him to produce valid profiles on instruments such as those.” Id. at 1075.
On the WRAT-3, which measures “an individual’s ability levels,” J.A. Vol. 8 at 3525:7-8 (Smalldon Test.), Frazier scored “within the range ... typically associated with either mild mental retardation or borderline intellectual functioning,” J.A. Vol. 3 at 1083 (Smalldon Report). He could not spell “circle,” “enter,” or “believe”; he could not do “fairly simple, two- and three-column subtraction problems”; he could not “correctly read words like ‘lame’ and ‘split.’ ” Id.
On the WAIS-III, the standard IQ test, Frazier received a Verbal IQ score of 77, a Performance IQ score of 72, and a Full Scale IQ score of 72.2 Id. On particular WAIS-III subtests, Frazier struggled. He defined “yesterday” to mean “a day after”; “sentence” meant “more than one word”; “confide” meant “to ask questions from someone else”; and “ballad” meant “it’s approved !” Id. at 1083-84 (internal quotation marks omitted). As Dr. Small-don noted, Frazier “had no idea how to define words like ‘ponder,’ ‘tranqil’ [sic], ‘reluctant,’ [or] ‘plagiarize.’ ” Id. at 1084. In his report, Dr. Smalldon noted that Frazier’s scores suggested “intellectual functioning at around the cusp of the ‘mildly retarded’ and ‘borderline intellectual functioning’ ranges.” Id.
The Bender and Aphasia tests did not go well either. On Part B of the Trail Making Test, Frazier made two errors and took seven minutes to complete a task that “the vast majority of unimpaired subjects can easily complete — without sequencing errors — in 60-90 seconds.” Id. at 1085.
Dr. Smalldon also considered Frazier’s school records that had been gathered by the court-appointed mitigation specialist. Frazier’s elementary school records note that his scholastic achievement was “not too bad, but [he] is a B.A. pupil.” Id. at 1120 (Elementary School Record). Dr. Smalldon guessed that “B.A.” stood for “below average.” J.A. Vol. 8 at 3579:16 (Smalldon Test.). According to Frazier’s high school records, he received exclusively “D” grades, except for physical education class in which he received two “Cs” and one “F.” J.A. Vol. 3 at 1103 (High School Record). The records also reflect that the school classified Frazier as a “slow learner,” that he attended special education classes, and that he withdrew from the tenth grade at nineteen years old. Id. at 1103-04.
Despite these tests and records, Dr. Smalldon did not find Frazier to be mentally retarded. Dr. Smalldon reported that Frazier could cash his disability check (which he received based on a finding of mental retardation), use a phone card, cook for himself, and travel on public transportation. Id. at 1085. These ac[492]*492tions led Dr. Smalldon to believe that Frazier functioned “not far above the upper threshold that’s typically used for demarcating the domain of ‘mild mental retardation.’ ” Id. at 1086.
In anticipation of a hearing to determine Frazier’s mental capacity, Dr. Gregory Forgac — a court-appointed clinical psychologist for the state — also met with Frazier for two hours and fifteen minutes. Id. at 1092 (Forgac Report). Dr. Forgac found Frazier to be “attentive,” capable of “express[ing] himself in a clear and effective manner,” and able to “maintain[ ] adequate eye contact.” Id. at 1093. In his report, Dr. Forgac noted that Frazier had stated that he “couldn’t learn when [he] went to school. Couldn’t get it right.” Id. at 1094 (internal quotation marks omitted). Like Dr. Smalldon, Dr. Forgac administered the WAIS — III test. Frazier received a Verbal IQ score of 81, a Performance IQ score of 73, and a Full Seale IQ score of 75. Id. at 1095-96. Dr. Forgac admitted a “95% confidence level” in the scores. Id. at 1096. Based on these scores and Frazier’s ability to use the bus system and appear “neat and clean,” Dr. Forgac found Frazier not to be mentally retarded. Id.
On the basis of Dr. Forgac’s report and discussions with Dr. Forgac and Dr. Small-don,3 Frazier’s counsel withdrew the motion for a determination of mental capacity. See J.A. Vol. 3 at 1447-51 (May 3, 2005 Tr.). The court granted defense counsel’s request and admitted Dr. Forgac’s report into evidence. Id. at 1450:8-13. The following week, the state-trial-court judge had enough worries to review Ohio’s test for mental retardation, as stated in State v. Lott, 97 Ohio St.3d 303, 779 N.E.2d 1011 (2002), with defense counsel and to question counsel and Frazier on the record. Counsel stated that his decision to withdraw the motion was based upon both psychologists finding (1) that Frazier had an IQ above 70; (2) that the Lott presumption against a finding of mental retardation could not be rebutted based on Frazier’s “functional adaptation”; and (3) that the psychologists’ findings would be useful at the mitigation phase. J.A. Vol. 4 at 1512-13 (May 9, 2005 Tr.). The trial court then engaged in a colloquy with Frazier during which the defendant agreed to the withdrawal of the motion. Id. at 1513-14. At this time, Frazier also rejected a plea agreement with the state in which he would plead guilty to a three-judge panel and the state would tell the panel that it was not seeking the death penalty. Id. at 1505-1510.
Frazier went to trial. The state called numerous witnesses and introduced more than one hundred exhibits into evidence. At the close of the state’s case, Frazier’s counsel moved for a judgment of acquittal pursuant to Ohio Rule of Criminal Procedure 29. J.A. Vol. 8 at 3303 (May 17, 2005 Tr.). The court rejected this motion. Id. at 3305:22-23. The defense offered no other evidence or witnesses during the guilt phase. On May 18, the jury convicted Frazier of all counts and specifications. J.A. Vol. 2 at 558-68 (Verdicts).
During the mitigation phase, defense counsel called Dr. Smalldon to testify. Dr. [493]*493Smalldon testified that “consultation to domestic courts” in child-custody cases represents “80 to 85 percent” of his practice. J.A. Vol. 8 at 3513:2-4 (Mitigation Tr.). Dr. Smalldon also stated that he had served as a consultant in “close to 200” death-penalty cases over thirteen years. Id. at 3513:16-19. He reviewed many of his findings from his report discussed above. Id. at 3517-37; 3552-59. He further described Frazier’s difficult upbringing, including his troubling family situation, and his substance-abuse history. Id. at 3537-45; 3559. Dr. Smalldon described the effects of crack-cocaine addiction, though he noted that commenting too much on the physiological effects of crack cocaine “would take [him] beyond [his] expertise.” Id. at 3568:2. Despite this testimony, the jury recommended the death penalty. J.A. Vol. 2 at 571 (State Ct. J.).
B. Procedural History
1. State-Court Proceedings
Frazier appealed to the Ohio Supreme Court, raising twenty-four propositions of law, including (1) a claim that Frazier is not eligible for the death penalty under Atkins; (2) several claims of ineffective assistance of counsel for (a) failure to present evidence of mental retardation, including testimony from a mental-retardation expert, (b) failure to suppress pre-trial statements and evidence collected pursuant to a search warrant, and (c) failure to retain a substance-abuse expert; and (3) a claim that the death penalty, generally, and by lethal injection, in particular, violates the Eighth Amendment. See J.A. Vol. 2 at 617-45 (Br. in Ohio S.Ct.); J.A. Vol. 1 at 109-11 (D.Ct.Op.) (listing the twenty-four propositions). The court addressed all twenty-four propositions in turn. In relevant part: for the first time, the Ohio Supreme Court held that a defendant could waive an Atkins claim, which Frazier did by withdrawing the motion to determine mental capacity, and that he subsequently failed to carry his burden of proving plain error. State v. Frazier, 115 Ohio St.3d 139, 873 N.E.2d 1263, 1291 (2007). Similarly, it denied Frazier’s ineffective-assistance-of-counsel claim related to mental retardation because of Dr. Forgac’s and Smalldon’s IQ test results and a lack of countervailing evidence. Id. at 1291-92.
The court also held that there was otherwise no ineffective assistance of counsel. According to the Ohio Supreme Court, Frazier’s counsel did not err by failing to suppress Frazier’s pre-trial statements to police because there was no evidence of police coercion or that Frazier could not make a voluntary and knowing waiver. Id. at 1285. Likewise, it dismissed his claim regarding failure to challenge evidence obtained according to a search warrant. Id. at 1286. The court rejected Frazier’s final ineffective-assistance claim based on a failure to. retain a substance-abuse expert because it found that Dr. Smalldon had done a sufficient job. Id. at 1301.
The court summarily rejected Frazier’s challenges to the death penalty and lethal injection. Id. at 1302. Finally, after rejecting Frazier’s other, unrelated claims, the court reweighed the evidence and affirmed the trial court’s imposition of the death penalty. Id. at 1305.
While Frazier’s direct appeal was pending in the Ohio Supreme Court, he filed an application for post-conviction relief in the state trial court, asserting nine claims for relief. That court rejected Frazier’s motion, finding that it was barred by principles of res judicata. He appealed to the state court of appeals, arguing that (1) he was ineligible for the death penalty due to being mentally retarded; (2) he was denied effective assistance of counsel because of failure to (a) assert mental retardation, [494]*494(b) present evidence related to his background at mitigation, and (c) present testimony of a substance-abuse expert; and (3) the death penalty by lethal injection is unconstitutional. J.A. Vol. 3 at 985-1002 (Br. in Ohio Ct. of Appeals); J.A. Vol. 1 at 111-12 (D.Ct.Op.) (listing claims). As part of his submissions, Frazier included an affidavit from Dr. Timothy Rheinscheld, asserting that Frazier could be mentally retarded based on the IQ test’s margin of error, J.A. Vol. 2 at 773; affidavits from family members, id. at 793-98; a letter from the Social Security Administration, stating that Frazier received benefits due to his mental retardation, id. at 791-92; and an affidavit and report from a substance-abuse expert, id. at 799-812. State v. Frazier, No. L-07-1388, 2008 WL 4408645, at *6-7 (Ohio Ct.App. Sept. 30, 2008). The Ohio Court of Appeals also rejected his claims. Id. at *15.
First, the court cited Ohio’s doctrine of res judicata, which prohibits a defendant from raising arguments that were made or could have been made at trial or on direct appeal. Id. at *9 (citing State v. Perry, 10 Ohio St.2d 175, 226 N.E.2d 104, 108 (1967)). On this standard, the state court of appeals held that the Ohio Supreme Court had decided Frazier’s claims or that he could have raised them on appeal; therefore, they were barred by res judicata. Id. at *10-11. The state court of appeals, however, also recognized that an exception to this procedural bar exists if the defendant can put forward material evidence outside the record. Id. (citing State v. Cowan, 151 Ohio App.3d 228, 783 N.E.2d 955, 958 (2002)). However, it held that Frazier’s additional submissions were cumulative and did not trigger the recognized exception to res judicata. Id. at *10-12.
2. Federal Habeas Proceedings
On September 11, 2009, Frazier filed a petition for a writ of habeas corpus in federal district court. He asserted fifteen grounds for relief, including (1) an Atkins claim; (2) a claim of ineffective assistance of counsel for (a) failing to raise an Atkins claim, (b) failing to request a mental-retardation expert, (c) failing to move to suppress evidence, (d) failing to investigate Frazier’s background during mitigation, and (e) failing to obtain a substance-abuse expert; and (3) a challenge to lethal injection. See J.A. Vol. 1 at 34-95 (Pet. for Writ); id. at 113-15 (D.CtOp.). The federal district court found that Frazier waived his Atkins claim, id. at 139 (D.Ct. Op.); it also found that Frazier’s Atkins and Atkins-related ineffeetive-assistanceof-counsel claims were barred by res judicata, id. at 141. Despite these procedural defaults, the court evaluated Frazier’s mental-retardation claim de novo because the Ohio Supreme Court reviewed Frazier’s Atkins claim only for plain error. Id. at 142. Nonetheless, the district court found that Frazier could not satisfy his burden even under this standard. Id. at 150. It reviewed Frazier’s other claims on the merits, but it rejected them as well.
This appeal followed. The district court granted a certificate of appealability on Frazier’s Atkins claim, Atkins-based ineffective-assistance-of-counsel claim, and his failure-to-suppress-evidence claim. Id. at 206. This court granted a certificate of appealability on three additional claims: (1) ineffective assistance of counsel for failing to investigate Frazier’s background; (2) ineffective assistance for failing to retain a substance-abuse expert; and (3) the unconstitutionality of Ohio’s lethal-injection system. See Frazier v. Bobby, No. 11^4262 (6th Cir. June 13, 2012) (order).
II. STANDARD OF REVIEW
Prisoners may not be held “in violation of the Constitution or laws or trea[495]*495ties of the United States.” 28 U.S.C. § 2254(a). Despite the simplicity of this command and the proud history of the Great Writ, our review of state-court convictions is carefully circumscribed under the Antiterrorism and Effective Death Penalty Act of 1996 (“AEDPA”), Pub.L. No. 104-132, 110 Stat. 1214. First, for us to consider a state prisoner’s habeas petition in federal court, the prisoner must clear two procedural hurdles. One, a petitioner must exhaust his state-court remedies. 28 U.S.C. § 2254(b)(1)(A). Two, if a state court denies relief on adequate and independent procedural grounds, a federal court may entertain those claims only if the petitioner shows “cause and prejudice” for failing to comply with state procedures, Murray v. Carrier, 477 U.S. 478, 486, 106 S.Ct. 2639, 91 L.Ed.2d 397 (1986), or if refusing to hear the claim would result in a “fundamental miscarriage of justice,” id. at 495-96, 106 S.Ct. 2639 (internal quotation marks omitted).
Second, provided that the petitioner exhausted his claims in state court and the state court rendered a decision on the merits, we must defer to the state court’s decision of a claim unless its decision “was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States.... ” 28 U.S.C. § 2254(d)(1). “Clearly established law,” as the Supreme Court has reminded us, “includes only ‘the holdings, as opposed to the dicta, of [the Supreme] Court’s decisions.’ ” White v. Woodall, — U.S. -, 134 S.Ct.' 1697, 1702, 188 L.Ed.2d 698 (2014) (quoting Williams v. Taylor, 529 U.S. 362, 412, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000)). Moreover, “an unreasonable application of’ clearly established Federal law requires a state-court decision to be “objectively unreasonable,” meaning that “a state prisoner must show that the state court’s ruling on the claim being presented in federal court was so lacking in justification that there was an error well understood and comprehended in existing law beyond any possibility for fairminded disagreement.” Id. (internal quotation marks omitted).
These standards are “difficult to meet, ... [and they were] meant to be.” Harrington v. Richter, 562 U.S. 86, 131 S.Ct. 770, 786, 178 L.Ed.2d 624 (2011). The petitioner’s burden is made even heavier by the fact that we, as a federal court, are “limited to the record that was before the state court that adjudicated the prisoner’s claim on the merits.” Greene v. Fisher, — U.S.-, 132 S.Ct. 38, 44,181 L.Ed.2d 336 (2011) (citing Cullen v. Pin-holster, — U.S. -, 131 S.Ct. 1388, 1398, 179 L.Ed.2d 557 (2011)). We are also limited to the law as it existed at the time of the state court’s decision if we are reviewing a state court’s merits determination under § 2254(d). Id. at 44-45; see also Cunningham v. Hudson, 756 F.3d 477, 485-87 n. 4 (6th Cir.2014) (noting that these restrictions apply only when federal courts are reviewing state-court merits determinations).
In this case, the district court determined that none of Frazier’s claims cleared all of these hurdles and, therefore, denied Frazier’s petition. We review this decision de novo. Murphy v. Ohio, 551 F.3d 485, 493 (6th Cir.2009).
III. ANALYSIS
A. Atkins Claim
Frazier’s first contention is that he is not eligible for the death penalty under Atkins v. Virginia, 536 U.S. 304, 122 S.Ct. 2242, 153 L.Ed.2d 335 (2002), and State v. Lott, 97 Ohio St.3d 303, 779 N.E.2d 1011 (2002), because he is mentally retarded. In Atkins, the Supreme Court held that [496]*496the Eighth Amendment bars execution of the mentally retarded, 536 U.S. at 321, 122 S.Ct. 2242, but it left “ ‘to the [s]tate[s] the task of developing appropriate ways to enforce the constitutional restriction upon [their] execution of sentences,’ ” id. at 317, 122 S.Ct. 2242 (quoting Ford v. Wainwright, 477 U.S. 399, 405, 416-17, 106 S.Ct. 2595, 91 L.Ed.2d 335 (1986)) (second and third alterations in original). In Lott, the Ohio Supreme Court held that a defendant must prove — by a preponderance of. the evidence — three things to be ineligible for the death penalty: “(1) significantly subaverage intellectual functioning, (2) significant limitations in two or more adaptive skills, such as communication, self-care, and self-direction, and (3) onset before the age of 18.” 779 N.E.2d at 1014. Additionally, the Lott court erected a “rebuttable presumption that a defendant is not mentally retarded if his or her IQ is above 70.”4 Id.
Frazier’s claim of mental retardation is complicated by the fact that there has never been a determination on the merits of whether Frazier is mentally retarded because Frazier’s counsel withdrew the motion before the state trial court to determine mental capacity. On direct appeal, the Ohio Supreme Court held that a defendant could forfeit his rights under Atkins. In this case, it concluded that Frazier had done so by withdrawing his request for a hearing on the issue and by not complying with the contemporaneous objection rule. In any event, the Ohio Supreme Court determined that “there [was] no error, plain or otherwise” in imposing the death penalty.5 Frazier, 873 N.E.2d at 1291-92. Moreover, on post-conviction review, the Ohio Court of Appeals applied the doctrine of res judicata and refused on procedural grounds to consider Frazier’s argument.6 See Frazier, 2008 WL 4408645, at *10. The state now argues (and Frazier agrees) that this claim [497]*497is procedurally defaulted. See Pet’r Br. at 35; Resp’t Br. at 31. Nonetheless, Frazier contends that we should review his substantive Atkins claim because (1) Frazier lacked the capacity to waive his rights; (2) his trial counsel’s constitutionally defective performance excuses the failure to comply with the contemporaneous-objection rule; and (3) he is actually innocent of the death penalty — meaning that he is actually mentally retarded. See Pet’r Br. at 35-39.
Frazier’s first argument fails because he did not assert in the district court that his waiver was invalid due to his mental incapacity. See R. 24 (Traverse at 15-32) (Page ID #294-311). Generally, we will not address arguments raised for the first time on appeal. See Bruederle v. Louisville Metro Gov’t, 687 F.3d 771, 777-78 (6th Cir.2012); Spengler v. ADT Sec. Servs., Inc., 505 F.3d 456, 458 (6th Cir. 2007); J.C. Wyckoff & Assocs., Inc. v. Standard Fire Ins. Co., 936 F.2d 1474, 1488 (6th Cir.1991). We decline to do so here.
Frazier’s second argument — that the procedural bar should be excused because his counsel’s ineffective assistance caused the Ohio Supreme Court to enforce the contemporaneous-objection rule and, thus, prejudiced his case — is beside the point because it does not show cause and prejudice with regard to the Ohio Court of Appeals’ res judicata decision.
Frazier’s third argument — that he is actually innocent of the death penalty — cuts through all of the potential procedural bars and is properly before us. McQuiggin v. Perkins, — U.S.-, 133 S.Ct. 1924, 1928, 185 L.Ed.2d 1019 (2013). In run-of-the-mill procedural-default cases, a defendant would need to show that failing to address his claims, despite procedural bars, would result in a “miscarriage of justice,” meaning that the defendant is actually innocent of the crime. Carrier, 477 U.S. at 496, 106 S.Ct. 2639. The United States Supreme Court, in Sawyer v. Whitley, 505 U.S. 333, 112 S.Ct. 2514, 120 L.Ed.2d 269 (1992), extended this exception to the death-penalty context, and as a result, a death-row prisoner can escape procedural default if he can “show by clear and convincing evidence that, but for a constitutional error, no reasonable juror would have found the petitioner eligible for the death penalty under the applicable state law.” Id. at 336, 112 S.Ct. 2514. Here, Frazier needs to show that he satisfies Ohio’s definition of mentally retarded under this heightened standard, which “is not a light burden and should not be confused with the less stringent, proof by a preponderance of the evidence.” Elec. Workers Pension Trust Fund of Local Union # 58 v. Gary’s Elec. Sew. Co., 340 F.3d 373, 379 (6th Cir.2003). The district court found that Frazier could not make the necessary showing, and on the record before us, we must agree with that conclusion.
1. Significantly Subaverage Intellectual Functioning
Frazier must first show that he has “significantly subaverage intellectual functioning,” which Ohio and many states have defined as having an IQ of 70 or lower. Lott, 779 N.E.2d at 1014 (citing other state statutes). Dr. Smalldon and Dr. Forgac found- that Frazier had Full Scale IQ scores of. 72 and 75, respectively. While these scores are slightly above the magic number of 70, Frazier contends that they are within the WAIS — III test’s standard error of measurement of five points and that, as a result, he should be considered to have significantly subaverage intellectual functioning. In support of this argument, Frazier submitted Dr. Rheinscheld’s affidavit, see J.A. Vol. 2 at 774 (Rheinscheld Aff.), and importantly, Dr. Smalldon [498]*498admitted this fact during his testimony during the penalty phase of trial, see J.A. Vol. 8 at 3535 (“There’s wiggle room that goes about five points either way.”).
In response, the state argues that Lott forecloses consideration of the standard error of measurement given that the Ohio Supreme Court erected a presumption against finding a defendant mentally retarded if his IQ scores are above 70. After Hall v. Florida, it is unclear whether this argument holds much water. In Hall, the Supreme Court emphatically stated that “[ijntellectual disability is a condition, not a number” and that “[cjourts must recognize, as does the medical community, that the IQ test is imprecise.” 134 S.Ct. at 2001 (citations omitted). We need not determine the precise reach of Hall in this case, however, because Frazier has not carried his burden of showing that he meets the second Lott prong.
2. Significant Limitations in Two or More Adaptive Skills
Frazier must also demonstrate that he has “significant limitations in two or more adaptive skills....” Lott, 779 N.E.2d at 1014. The Ohio Supreme Court took this prong from the American Association on Mental Retardation (“AAMR”) and the APA’s definitions of mental retardation. Id. Adaptive skills include: “ ‘communication, self-care, home living, social/interpersonal skills, use of community resources, self-direction, functional academic skills, work, leisure, health, and safety.’ ” Atkins, 536 U.S. at 308 n. 3, 122 S.Ct. 2242 (quoting APA, Diagnostic and Statistical Manual of Mental Disorders 41 (4th ed. 2000) (“DSM-4”)). In 2002, the AAMR revised its definition, though there was little substantive change. See Pet’r Br. at 24 (citing AAMR, Mental Retardation: Definition, Classification, and Systems of Supports (10th ed.2002)); Resp’t Br. at 46 n. 7 (same).
In the district court, Frazier argued that his subaverage intellectual functioning resulted in limitations to his functional academic and work skills.7 J.A. Vol. 1 at 39 (Pet. for Writ). The district court found Frazier’s arguments lacking, and we agree that Frazier has not shown by clear and convincing evidence that his subaverage intellectual functioning caused significant limitations in these skills. First, with regard to Frazier’s functional academic skills, the record is mixed. In Frazier’s favor are the facts that he attended special-education classes, that he never progressed beyond the tenth grade, that he earned two Cs, twenty-one Ds, and one F in high school, and that he was labeled a “slow learner.” See J.A. Vol. 1 at 146-47 (D.Ct.Op.). Based on this information and the tests conducted by Dr. Smalldon, Dr. Rheinscheld concluded that Frazier “exhibited] a significant deficit in the area of functional academics.” J.A. Vol. 2 at 775 (Rheinscheld Aff.). However, being placed in special-education classes does not necessarily render someone mentally retarded. See, e.g., Sheeks v. Comm’r of Soc. Sec. Admin., 544 Fed.Appx. 639, 642 (6th Cir. 2013); Eddy v. Comm’r of Soc. Sec., 506 Fed-Appx. 508, 510 (6th Cir.2012). And failing to complete high school does not necessarily result from subaverage intellectual functioning. The school records are also more muddled than Dr. Rheinscheld suggests. For instance, Frazier’s elementary-school principal described his [499]*499performance as “not too bad” and noted that he was “working to his capacity” in response to a probation officer’s request for information. J.A. Vol. 3 at 1120 (School Records). If Frazier needed to prove his limitations only by a preponderance of the evidence, we might be inclined to agree with him; however, at this stage of the litigation, Frazier’s proof must be clear and convincing. The limited and muddled academic records make this impossible.
Second, the record before us does not clearly and convincingly demonstrate that Frazier’s subaverage intellectual functioning resulted in a significant limitation with regard to his ability to work. On this point, Frazier offers the Social Security Administration’s diagnosis of Frazier as mentally retarded and his checkered work history. Courts will consider Social Security Administration records and rulings on mental retardation, see, e.g., Newman v. Harrington, 726 F.3d 921, 932-33 (7th Cir. 2013), but those administrative determinations are not dispositive for purposes of an Atkins finding. In this case, the Social Security Administration’s records actually create doubt as to whether Frazier has a significant limitation due to his subaverage intellectual functioning. None of the Social Security Administration’s records relate to its mental-retardation finding (because they were purportedly destroyed), and the records that have been submitted detail the Social Security Administration’s prior findings contrary to its subsequent finding of mental retardation. See J.A. Vol. 3 at 1162-65 (Soc. Sec. Records) (noting only “slight” limitations in “maintaining social functions” and “restriction of activities of daily living”); id. at 1171 (Soc. Sec. Records) (“No evidence of a sign or symptom [of mental retardation or autism].”). Furthermore, when Dr. Small-don “asked [Frazier] whether he’s ever been ‘fired from’ any of the jobs that he’s held as an adult, [Frazier] replied, ‘I’ve lost jobs ‘cause I’d get in trouble. I’d drink.’ ” Id. at 1082 (Smalldon Report). These statements and evaluations potentially indicate that Frazier’s difficulties maintaining employment are the result of his behavior — not his subaverage intellectual functioning. Again, were we to evaluate this evidence under the preponderance-of-the-evidence standard, we might be inclined to hold that Frazier has demonstrated a significant limitation in this adaptive skill. Under the clear-and-convincing-evidence standard, however, the conflicting information in the record requires us to side against Frazier here.8 As a result, we cannot conclude that Frazier is actually innocent of the death penalty, and his substantive Atkins claim remains procedurally defaulted. Thus, we AFFIRM the district court’s denial of his petition on this claim.
B. Ineffective-Assistance-of-Counsel Claims
Frazier next contends that his Sixth Amendment right to effective assistance of counsel has been violated by counsel’s failure (1) to present evidence of mental retardation at an Atkins hearing; (2) to move to suppress various pieces of evidence; (3) to investigate Frazier’s family life and upbringing; and (4) to employ a substance-abuse expert. Our review of each of these claims is governed by Strickland v. Washington, 466 U.S. 668, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984), and its [500]*500progeny. Under Strickland, a convicted prisoner must make two showings. One, he “must identify the acts or omissions of counsel that are alleged not to have been the result of reasonable professional judgment. [We] must then determine whether, in light of all the circumstances, the identified acts or omissions were outside the wide range of professionally competent assistance.” Id. at 690, 104 S.Ct. 2052. Two, he “must show that there is a reasonable probability that, but for counsel’s unprofessional errors, the result of the proceeding would have been different.” Id. at 694, 104 S.Ct. 2052. If the state court adjudicates a prisoner’s ineffective-assistance claim on the merits, these requirements become even more stringent under AEDPA, because a defendant must show that a state court’s application of Strickland was unreasonable, meaning that fair-minded judges would all agree that the state court erred. Richter, 131 S.Ct. at 786. We evaluate each claim in turn, though we ultimately find none of them successful under AEDPA.
1. Atkins-Based Claim
Frazier first contends that his trial counsel provided constitutionally defective assistance by withdrawing the motion for an Atkins hearing despite record evidence indicating that Frazier might be mentally retarded. In particular, Frazier argues that his IQ scores are within the standard error of measurement, that the Social Security Administration’s finding that he was mentally retarded is particularly persuasive, and that counsel should have employed a mental-retardation expert. When Frazier raised this claim and these arguments on direct appeal, the Ohio Supreme Court rejected them on the merits. Frazier, 873 N.E.2d at 1292. Specifically, it held that counsel’s representation was not constitutionally defective because: (1) Dr. Smalldon testified that Frazier’s Full Scale IQ score of 72 was “pretty accurate”; (2) Frazier failed to present evidence that linked his Social Security benefits to his mental retardation claim; (3) there was no evidence that Frazier’s attorney failed to consult with him prior to withdrawing the motion, particularly given Frazier’s in-court statements before the motion was withdrawn; and (4) counsel was permitted to rely on the judgments of Dr. Smalldon and Dr. Forgac. Id. The Ohio Supreme Court adjudicated this claim on the merits, and thus, we afford that decision substantial deference under AED-PA. Under this deferential standard, we cannot conclude that the Ohio Supreme Court’s ultimate decision was objectively unreasonable.
Frazier’s main assignment of error is that his attorney failed to present evidence of mental retardation at an Atkins hearing. His counsel claimed that they skipped the Atkins hearing because Frazier’s Full Scale IQ scores were above 70, and on the advice of Dr. Smalldon and Dr. Forgac, they did not believe that the Lott presumption could be overcome. J.A. Vol. 4 at 1512-13. Given the record in this case, we find counsel’s performance troubling for two reasons.
First, the Ohio courts have found defendants mentally retarded even though their IQ scores exceeded 70. See, e.g., State v. Gumm, 169 Ohio App.3d 650, 864 N.E.2d 133 (2006). Here, there is plenty of evidence that suggests Frazier had a chance to overcome the Lott presumption that he was not mentally retarded. Frazier had a well-documented history of academic struggles; for instance, the top of his high school records read, “Frazier, James (Slow Learner).” His IQ scores were within the standard error of measurement, and Dr. Smalldon could not complete various tests because Frazier’s limited abilities “wouldn’t enable [Dr. Smalldon] to pro[501]*501duce valid profiles [from those tests].” J.A. Vol. 3 at 1075 (Smalldon Report). Moreover, the Social Security Administration had been paying Frazier disability benefits for years based on a finding of mental retardation. As we explained above, it is not clear from this evidence that Frazier could have overcome the Lott presumption, but there is a non-frivolous chance that the state trial court would have concluded that Frazier met the standard for mental retardation.
Second and more fundamentally, we fail to see the downside in having a non-frivolous Atkins hearing, and it is difficult to ascertain a strategic reason for withdrawing the motion in this case. These hearings are before the judge, not the jury, see Lott, 779 N.E.2d at 1015, and thus, no potentially prejudicial material would be kept from the jury by foregoing the hearing. Moreover, to the extent that the evidence is helpful to Frazier, nothing bars counsel from presenting the same information to the jury during the mitigation stage of trial (as happened during Frazier’s trial). By choosing to withdraw the motion for an Atkins hearing, counsel deprived Frazier of the best opportunity to create a full record on the issue and to allow the state-trial-court judge — the judicial officer with the best sense of Frazier’s actual abilities — to decide whether he met the Lott definition of mental retardation.
While we find counsel’s performance questionable, the Ohio Supreme Court’s application of Strickland in this case was not objectively unreasonable. Lawyers are permitted to rely upon qualified experts, see Murphy, 551 F.3d at 500-01 (citing Hill v. Mitchell, 400 F.3d 308, 319 (6th Cir.2005)), and in this case, Frazier’s own expert found him not to be mentally retarded. Fairminded jurists could find that counsel’s reliance upon Dr. Smalldon’s opinion was consistent with professional norms.
Frazier also argues that counsel should have retained a mental-retardation specialist, but as the district court noted, Dr. Smalldon had consulted in about 200 capital cases over thirteen years, many of them involving issues of mental retardation. J.A. Vol. 1 at 145 (D.Ct.Op.). Certainly, Dr. Rheinscheld may have testified differently, but Frazier has not shown that Dr. Smalldon is unqualified. More importantly, Dr. Smalldon actually met Frazier, while Dr. Rheinscheld merely relied upon Dr. Smalldon’s reports and other materials available at trial. See J.A. Vol. 2 at 773 (Rheinscheld Aff.). Therefore, Dr. Rheinscheld’s analysis is merely another interpretation of Dr. Smalldon’s data. Given that Frazier has not shown that Dr. Small-don’s interpretation was unreasonable or that he was unqualified, counsel could rely upon Dr. Smalldon’s recommendation — or so a reasonable jurist could find. As a result, we do not need to determine whether counsel’s performance prejudiced Frazier, and we AFFIRM the denial of Frazier’s petition on this claim.
2. Failure-to-Suppress-Evidence Claim
Frazier next claims that his trial counsel provided constitutionally defective representation by not moving to suppress oral statements that Frazier made to the police.9 Specifically, Frazier [502]*502argues that his counsel should have objected because Frazier lacked the mental capacity to waive his Miranda rights knowingly and intelligently. On direct appeal, the Ohio Supreme Court rejected this claim, holding, in part, that “there [was] no evidence of police coercion or overreaching rendering Frazier’s statement involuntary” and that “there [was] no evidence that Frazier was incapable of making a voluntary statement.” 10 Frazier, 873 N.E.2d at 1285. This conclusion was not objectively unreasonable, and we must therefore affirm the denial of Frazier’s petition on this claim.
In Miranda v. Arizona, 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966), the Supreme Court recognized that the Fifth Amendment grants suspects in custody “[the] privilege against self-incrimination and [the] right to retained or appointed counsel.” Id. at 475, 86 S.Ct. 1602 (citing Escobedo v. Illinois, 378 U.S. 478, 490 n. 14, 84 S.Ct. 1758, 12 L.Ed.2d 977 (1964)). If the government interrogates a suspect “without the presence of an attorney and a statement is taken, a heavy burden rests on the government to demonstrate that the defendant [voluntarily,] knowingly[,] and intelligently waived his [rights].” Id. Determining whether a waiver is valid requires a two-part inquiry.
First, the relinquishment of the right must have been voluntary in the sense that it was the product of a free and deliberate choice rather than intimidation, coercion, or deception. Second, the waiver must have been made with a full awareness of both the nature of the right being abandoned and the consequences of the decision to abandon it. Only if the “totality of the circumstances surrounding the interrogation” reveals both an uncoerced choice and the requisite level of comprehension may a court properly conclude that the Miranda rights have been waived.
Moran v. Burbine, 475 U.S. 412, 421, 106 S.Ct. 1135, 89 L.Ed.2d 410 (1986) (quoting Fare v. Michael G, 442 U.S. 707, 725, 99 S.Ct. 2560, 61 L.Ed.2d 197 (1979)).
Frazier does not dispute that his Miranda waiver was made voluntarily; rather, he argues that his mental retardation (or diminished mental capacity) precluded him from knowingly or intelligently waiving his rights. The problem for Frazier is that he cites no cases holding that mental retardation or a diminished mental capacity necessarily renders a person’s waiver unknowing or unintelligent. To the contrary, this court and other courts have concluded that “diminished mental capacity alone does not prevent a defendant from validly waiving his or her Miranda rights.” Gamer v. Mitchell, 557 F.3d 257, 264 (6th Cir.2009) (en banc). Nor does Frazier give specific reasons for why his [503]*503particular intellectual disabilities made his waiver invalid. As a result, we cannot conclude that there is a reasonable probability that a motion to suppress would have succeeded, and therefore Frazier has failed to demonstrate that he was prejudiced by his attorney’s failure to file such a motion. Consequently, we AFFIRM the denial of Frazier’s petition on this point.
3. Failure-to-Investigate Claim
Frazier next claims that his trial counsel provided ineffective assistance by failing to make “a full and adequate investigation into [his] background” prior to the penalty phase of his trial. Pet’r Br. at 58. In particular, Frazier objects to his counsel’s failure to interview his siblings and have them testify. Frazier asserts that his siblings would have “humanized him to the jury,” id. at 60, and his siblings submitted affidavits regarding Frazier’s upbringing, trouble in school, and general demeanor, see J.A. Vol. 2 at 793-98 (Affs.). The Ohio Court of Appeals denied this claim on the merits, holding that “the mitigation evidence proposed by [Frazier] is not sufficient to overcome the presumption that the decision not to ask [Frazier’s] siblings to testify at the mitigation hearing was anything other than a tactical trial strategy.” Frazier, 2008 WL 4408645, at *12. Because the state court adjudicated this claim on the merits, we again review Frazier’s assignment of error under AEDPA’s deferential standard.
“The Eighth Amendment requires that the jury be able to consider and give effect to all relevant mitigating evidence offered by petitioner.” Boyde v. California, 494 U.S. 370, 377-78, 110 S.Ct. 1190, 108 L.Ed.2d 316 (1990). Under the Sixth Amendment, “counsel has a duty to make reasonable investigations” in an attempt to find such evidence. Strickland, 466 U.S. at 691, 104 S.Ct. 2052; see also Wiggins v. Smith, 539 U.S. 510, 521-22, 123 S.Ct. 2527, 156 L.Ed.2d 471 (2003). “[T]horough investigation^] of law and facts relevant to plausible options are virtually unchallengeable,” but “a particular decision not to investigate must be directly assessed for reasonableness in all the circumstances, applying a heavy measure of deference to counsel’s judgments.” Strickland, 466 U.S. at 690-91, 104 S.Ct. 2052. After all, “reasonably diligent counsel may draw a line when they have good reason to think further investigation would be a waste.” Rompilla v. Beard, 545 U.S. 374, 383, 125 S.Ct. 2456, 162 L.Ed.2d 360 (2005). Even if counsel’s failure to investigate was proven to be unreasonable, a prisoner must also “establish a reasonable probability that a competent attorney, aware of the available mitigating evidence, would have introduced it at sentencing, and that had the jury been confronted with this mitigating evidence, there is a reasonable probability that it would have returned with a different sentence.” Wong v. Belmontes, 558 U.S. 15, 20, 130 S.Ct. 383, 175 L.Ed.2d 328 (2009) (internal quotation marks, alterations, and ellipses omitted). On this record, we cannot conclude that Frazier has offered proof demonstrating that the Ohio court’s application of this test was objectively unreasonable.
First, Frazier’s counsel received the assistance of a court-appointed mitigation specialist that uncovered Frazier’s school records, juvenile records, health records, and records from the Social Security Administration. See J.A. Vol. 3 at 1103-1331 (various records). These records provide the basis for most of Frazier’s claims and cover a wide range of years and topics. Furthermore, Dr. Smalldon interviewed Frazier, in part, regarding his upbringing. See id. at 1075-87 (Smalldon Report). Despite this information, which was introduced at trial, Frazier argues that counsel [504]*504should have also interviewed his siblings and called them to testify. The problem for Frazier is that two of his three siblings, who Frazier argues should have been interviewed, admitted to speaking with Frazier’s counsel. See J.A. Vol. 2 at 796 (Jean Frazier Aff.); id. at 797 (Nathaniel Frazier Aff.). Moreover, counsel has discretion over deciding which witnesses to call and how to examine them. Gonzalez v. United States, 553 U.S. 242, 249, 128 S.Ct. 1765, 170 L.Ed.2d 616 (2008). As a result, we cannot hold that the Ohio Court of Appeals was objectively unreasonable in concluding that counsel performed an adequate investigation.
Second, even if counsel’s investigation was lacking, Frazier has failed to demonstrate that the Ohio Court of Appeals made an objectively unreasonable decision by holding that Frazier was not prejudiced by such deficient performance. The affidavits submitted by Frazier’s siblings state that they would have testified that Frazier grew up in poverty and that Frazier attended special-education classes. J.A. Vol. 2 at 793 (John Frazier Aff.); see also id. at 795-96 (Jean Frazier Aff.). Dr. Smalldon, however, testified to these facts. See, e.g., J.A. Vol. 8 at 3532:19-22, 3539:1-3540:8, 3569:16-3570:19 (Smalldon Test.). Perhaps the jurors would have been more sympathetic to Frazier if his siblings testified to these facts directly, but as noted above, from whom a jury learns certain information is a decision for trial counsel. Gonzalez, 553 U.S. at 249, 128 S.Ct. 1765. We agree with the Ohio Court of Appeals that the information to which Frazier’s siblings allude in the affidavits is cumulative, and therefore, Frazier was not prejudiced by counsel’s decisions. As a result, we AFFIRM the district court’s denial of Frazier’s petition on this issue.
4. Failure-to-Retain-Substance-Abuse-Expert Claim
Finally, Frazier claims that his counsel provided ineffective assistance by failing to retain a substance-abuse expert for the penalty phase of trial. Frazier argues that a substance-abuse expert would have explained to the jury how “Frazier’s limited cognitive functioning and drug abuse ha[d] a synergistic effect ... [that each one] exacerbated the effects of the other, further impairing Frazier’s-cognitive functioning.” Pet’r Br. at 62. For support, Frazier submits an affidavit from Dr. Robert Smith, testifying to the same. See J.A. Vol. 2 at 799-807 (Smith Aff.). The Ohio Supreme Court rejected this claim on the merits, holding that Dr. Smalldon adequately testified to Frazier’s prior drug and alcohol abuse. Once again, we apply AEDPA deference, and we conclude that the Ohio Supreme Court’s decision was not objectively unreasonable.
The record shows that Dr. Smalldon testified regarding Frazier’s drug and alcohol abuse, particularly his crack-cocaine use. See J.A. Vol. 8 at 3566:10-3569:3 (Smalldon Test.) He explained the behavioral effects of “acute crack cocaine intoxication.” Id. at 3567:3 (Smalldon Test.). Additionally, Dr. Smalldon hypothesized that Frazier’s drug use may have led to “hyperarousal, ... extreme aggressiveness, irritability, ... restlessness[,] and agitation.”11 Id. at 3567:18-20 (Smalldon Test.). As before, the jury might have been more receptive to a different messenger, but who testifies [505]*505is generally a decision for counsel. Because Dr. Smalldon testified — at length and in detail — to the effects of crack-cocaine abuse, the Ohio Supreme Court was not unreasonable to conclude that counsel did not provide ineffective assistance by failing to employ a substance-abuse expert and that Frazier was not prejudiced by counsel’s failure to introduce cumulative evidence. Therefore, we AFFIRM the denial of Frazier’s petition on this claim.
C. Challenge to Ohio’s Lethal-Injection Regime
Finally, Frazier contends that Ohio’s administration of the death penalty is unconstitutional. The Ohio Supreme Court summarily rejected this claim on the merits, and the district court denied Frazier’s petition. Frazier admits that the United States Supreme Court, in Baze v. Rees, 558 U.S. 35, 128 S.Ct. 1520, 170 L.Ed.2d 420 (2008), held that execution by lethal injection is not per se unconstitutional, but he contends that “the implementation of the method of execution could implicate the Eighth Amendment prohibition against cruel and unusual punishment.” Pet’r Br. at 67. In his briefing on appeal, Frazier does not offer an argument on the merits of his Eighth Amendment claim; instead, he asks us to “remand this case back to the district court to permit Frazier to join in the lethal injection litigation at the district court level or await the result of Judge Katz’ findings in Case No. l:03-cv1192.” Pet’r Br. at 68.
At oral argument, Frazier’s counsel notified our panel that Frazier is a party to an ongoing action, filed under 42 U.S.C. § 1983, challenging Ohio’s lethal-injection regime. See Docket, In re Ohio Execution Protocol Litigation, Case No. 2:11-CV-01016-GLF-MRA (S.D.Ohio) (Frost, J.). We think that that litigation is the proper avenue for Frazier to bring this constitutional challenge. As we stated in Scott v. Houk, 760 F.3d 497 (6th Cir.2014), “in order to obtain relief from his sentence, [Frazier] would first have to gather facts showing that Ohio is unable to administer lethal injection in a constitutionally permissible manner. And this is precisely the type of discovery that [Frazier] can pursue in his § 1983 litigation.” Id. at 512. Therefore, we -AFFIRM the district court’s denial of Frazier’s petition on this claim.
IY. CONCLUSION
Frazier’s substantive Atkins claim remains procedurally defaulted because he has not shown by clear and convincing evidence that he is mentally retarded. Under applicable law, the Ohio courts were not objectively, unreasonable in rejecting his ineffective-assistance claims, and Frazier’s constitutional challenge to Ohio’s lethal-injection protocol requires the accumulation of evidence in another court. Therefore, we AFFIRM the district court’s denial of Frazier’s petition for a writ of habeas-corpus.