Since 1948, federal prisoners who contend that they were convicted or sentenced in violation of the Constitution or laws of the United States have been required in most cases to present that claim through a motion under 28 U.S.C. § 2255. The motion must be filed in the district of conviction. As a rule, the remedy afforded by section 2255 functions as an effective substitute for the writ of habeas corpus that it largely replaced. See 28 U.S.C. § 2241; United States v. Hayman, 342 U.S. 205, 72 S.Ct. 263, 96 L.Ed. 232 (1952). But Congress recognized that there might be occasional cases in which “the remedy by motion is inadequate or ineffective to test the legality of [the applicant’s] detention.” 28 U.S.C. § 2255(e). The question before us is whether petitioner Bruce Webster has presented such a case. If so, then he may proceed to the merits of his petition; if not, then his case must be dismissed at the threshold.
Webster was convicted in the Northern District of Texas of the federal crimes of kidnapping resulting in death, conspiring to commit kidnapping, and using and carrying a firearm during a crime of violence. United States v. Webster, 162 F.3d 308 (5th Cir.1998) (Webster I).. He was sentenced to death on the first count, after the district court rejected his argument *1125that he was ineligible for the death penalty-on account of mental retardation (now' termed “intellectual disability” by the Supreme Court, see Hall v. Florida, — U.S. —, 134 S.Ct. 1986, 1990, 188 L.Ed.2d 1007 (2014)). The Fifth Circuit later rejected Webster’s motion for relief under section 2255, United States v. Webster, 421 F.3d 308 (5th Cir.2005) (Webster II), and his application for an order authorizing a successive 2255 proceeding. In re Webster, 605 F.3d 256 (5th Cir.2010) (Webster III).
Webster is now seeking the opportunity to present newly discovered evidence that would demonstrate that he is categorically and constitutionally ineligible for the death penalty under the Supreme Court’s decisions in Atkins v. Virginia, 536 U.S. 304, 122 S.Ct. 2242, 153 L.Ed.2d 335 (2002), and Hall. A panel of this court concluded that new evidence can never satisfy the demanding standard of section 2255(e) and thus that Webster cannot be heard. Webster v. Caraway, 761 F.3d 764 (7th Cir.2014) (Webster IV). In light of the'importance of the question, the full court decided to rehear the case en banc. We conclude that there is no such absolute bar to the use of the safety valve found in section 2255(e) for new evidence that would demonstrate categorical ineligibility for the death penalty. We therefore reverse the district court’s judgment and remand for further proceedings.
I. Background Facts and Proceedings
A. Facts
There is no doubt that Webster and his co-defendants committed a horrible crime. We take our account of the underlying facts from the Fifth Circuit’s opinion in Webster I. Those facts are largely undisputed at this stage; the only question is what they show, or do not show, about Webster’s intellectual functioning.
Webster, along with Orlando Hall and Marvin Holloway, ran a marijuana business in Pine Bluff, Arkansas, a city of approximately 50,000 that lies about 45 miles south of Little Rock and 330 miles east of Dallas, Texas. The group used suppliers in the Dallas/Fort Worth area with the help of a local contact, Steven Beckley.
On September 21, 1994, Holloway drove Hall from Pine Bluff to the Little Rock airport, and Hall flew to Dallas; Beckley and Hall’s brother Demetrius picked Hall up at the other end. Later that day, Hall and Beckley met two local dealers, Stan-field Vitalis and Neil Rene, at a car wash and gave them $4,700 as payment in advance for some marijuana. Beckley and Demetrius then returned to the car wash, but Vitalis and Rene never appeared. Hall phoned them to find out what happened, and they told him that both The car they had been driving and the money had been stolen from them. Hall figured out that the telephone number he had used was associated with the Polo Run apartments in Arlington, Texas (a Dallas suburb). Hall, Demetrius, and Beckley began watching the apartment. When they spotted Vitalis and Rene in the supposedly stolen car, they concluded that the story about the stolen money was also false.
Three days later, Hall contacted Holloway and told him to arrange for Webster to fly to Dallas. Webster complied with Holloway’s instructions. That evening, Hall, Demetrius, Beckley, and Webster went to the Polo Run apartments in a Cadillac owned by Hall’s sister, Cassandra Ross. Hall and Webster were armed with handguns; Demetrius had a small souvenir baseball bat; and Beckley had duct tape and a jug of gasoline. So equipped, the group approached the apartment they had *1126seen Vitalis and Rene use, and they knocked on the door. The occupant, Lisa Rene (the 16-year-old sister of Neil Rene), refused to let them in and called her sister and the police emergency number. Webster unsuccessfully trie.d to kick in the door. When that did not work, he and Demetrius looked through a sliding glass door and saw Lisa on the telephone. Demetrius shattered the door with the bat, and Webster entered the apartment, seized Lisa, and dragged her to the car.
In the meantime, Hall and Beckley had returned to the car. Webster, with Lisa in tow, met them there. He forced Lisa onto the floorboard and the group drove to Ross’s apartment nearby. Once there, they left the Cadillac and shoved Lisa into the back seat of Beckley’s car. Hall climbed into the back seat with her, and Webster sat in the front passenger seat. Beckley drove around looking for a secluded spot; while he did so, Hall raped Lisa and forced her to perform oral sex on him.
Eventually Beckley drove them back to Ross’s apartment. From there, Beckley, Demetrius, and Webster drove Lisa, still a prisoner, the 330 miles to Pine Bluff. En route, Webster and Demetrius took turns raping Lisa. Once they reached Pine Bluff, they rented a motel room, where they tied Lisa to a chair and continued to assault her sexually.
The next morning, September 25, Hall and Holloway showed up at the motel room. They took Lisa into the bathroom for about 20 minutes. When they came out again, Hall told Beckley that “she know too much.” Hall, Holloway, and Webster then left the motel. Later that afternoon, Hall and Webster went to a park and dug a grave. That evening, Hall, Beckley, and Webster took Lisa to the park, but they could not find the grave site in the dark and so they returned to the motel room. They shifted Lisa to another room the next morning.
Later that morning, Hall, Beckley, and Webster took Lisa back to the park. They covered her eyes with a mask. Hall and Webster led the way to the grave site, while Beckley guided Lisa along. At the grave site, Hall turned Lisa’s back to the grave, placed a sheet over her head, and hit her in the head with a shovel. She tried to run away, but Beckley grabbed her and they both fell down. Beckley hit her twice with the shovel and handed it to Hall. At that point, Webster and Hall took turns hitting her with the shovel. Webster then gagged her, dragged her to the grave, stripped her, poured gasoline on her, pushed her in, and shoveled dirt over her. The record indicates that, although she was unconscious by then, Lisa was probably still breathing when she was buried.
It did not take long for the authorities to find out who was responsible for Lisa’s hideous death. Lisa’s brothers gave information leading to Demetrius’s arrest to the police, and Hall and Beckley surrendered soon thereafter. Beckley confessed to his role in the kidnapping; his confession also implicated Hall and someone he called “B-Love.” Beckley also said that he had last seen Lisa at the motel with B-Love, and a security guard at the hotel told the officers that Webster went by that name. When Webster pulled into the motel parking lot early on September 30, he was arrested.
B. Trial and Direct Appeal
In November 1994, Webster (along with Hall, Demetrius, Beckley, and Holloway) was indicted by a federal grand jury on charges of kidnapping in which a death occurred (Count 1, 18 U.S.C. § 1201(a)(1)), conspiracy to commit kidnapping (Count 2, 18 U.S.C. § 1201(c)), traveling in interstate *1127commerce with intent to promote extortion (Count 5, 18 U.S.C. § 1952), and using and carrying a firearm during a crime of violence (Count 6, 18 U.S.C. § 924(c)). In February 1995, the government filed a notice of intent to seek the death penalty against Webster, pursuant to the Federal Death Penalty Act of 1994, 18 U.S.C. § 3593(a) (which had taken effect just 12 days before the murder). Webster’s trial later was severed from that of his co-defendants.1
The jury returned guilty verdicts on Counts 1, 2, and 6; Count 5 was dismissed on the government’s motion. The court conducted a separate sentencing hearing before the same jury, which returned special findings that Webster satisfied the statute’s intent requirement, 18 U.S.C. § 3591(a), and that three statutory and two non-statutory aggravating factors were present. 18 U.S.C. § 3592(c). Varying numbers of jurors found nine mitigating factors, some statutory and some non-statutory. See 18 U.S.C. § 3592(a); Webster I, 162 F.3d at 319 & n. 2. The court sentenced Webster to death on Count 1; to life imprisonment on Count 2; and to 60 months’ imprisonment on Count 6.
On direct appeal, Webster raised four grounds for reversal that the Fifth Circuit had already rejected in Hall’s separate appeal, see United States v. Hall, 152 F.3d 381 (5th Cir.1998), and 16 additional grounds, some of which related to his conviction and some to his sentence. Only one of them remains relevant at this stage of the game: point 13, which asserted that the district court “plainly erred and violated Webster’s constitutional rights by entering a factual finding that he is not mentally retarded.” Webster I, 162 F.3d at 321. Before turning to the Fifth Circuit’s resolution of that point, it is essential to review the evidence of intellectual disability that was presented at the sentencing phase of the trial; virtually none came in at the guilt phase. Without this background, it is impossible to decide whether the newly discovered evidence would have made a difference.2
The defense relied primarily on the testimony of three experts: Dr. Raymond Finn, a clinical psychologist; Dr. Denis Keyes, a professor of special education and a certified school psychologist with expertise in mental retardation; and Dr. Robert Fulbright, a clinical neuropsychologist. At the most general level, those three agreed with the experts for the United States that a finding of mental retardation is appropriate if the person’s I.Q. is roughly 70 or below on one of the accepted tests, and if the person has a deficit in at least one of three areas of adaptive functioning (communication, socialization, and daily living skills). (A third criterion — onset before the age of 18 — was not contested.) Dr. Finn testified about Webster’s I.Q.; Dr. Keyes testified about his adaptive functioning; and Dr. Fulbright testified more specifically about his level of mental functioning.
Before Dr. Finn personally administered an I.Q. test to Webster, he received copies of tests that had been performed in 1992, *1128approximately two years before the murder, at the Southeast Arkansas Mental Health Clinic (the Clinic). Webster had gone there after his brother-in-law stabbed one of his brothers to death. The Clinic gave him the Wechsler Adult Intelligence Scale (WAIS) test, which is widely used. Webster scored 56 on the verbal part of the test, 48 on the performance part, and received a full-scale I.Q. score of 48. The 1992 results, both defense and prosecution witnesses said, had to be taken with a grain of salt. Dr. Finn reported that the psychologist who conducted the test noted that Webster was confused, preoccupied, and poorly oriented at the time, and so the scores may have been depressed for that reason. The I.Q. tests were also viewed at the time as secondary to the possibility that Webster was suffering from schizophrenia, depression, or some other mental disorder. With that possibility in mind, the Clinic prescribed the anti-psychotic drug Haldol for him, although in the end it did not conclude that he was schizophrenic. A full-scale score of 48, however, easily meets the threshold of an I.Q. of 70 for intellectual disability recognized by the American Psychiatric Association’s Diagnostic and Statistical Manual of Mental Disorders (DSM), both in the DSM-IV (the 4th edition that was in force at the time of trial) and in the current DSM-V (which took effect in 2013). It also meets the standard recognized by the American Association for Mental Retardation, now called the American Association on Intellectual and Developmental Disabilities.
After collecting background information on Webster, Dr. Finn administered the WAIS test to him in January 1995. That test revealed a verbal I.Q. score of 59, a performance I.Q. score of 60, and a full-scale I.Q. of 59, still well below the 70 mark. Dr. Finn explained that this score put Webster in the 0.3 percentile in the country, meaning that 99.7 percent of test-takers would do better. Taking into account Webster’s agitation at the time.of the 1992 test, Dr. Finn found the results of his test to be consistent with the earlier one, though likely more reliable.
Dr. Finn administered the WAIS test again to Webster in June 1996, just before the trial started. He noted that there is some learning effect from repeated exposure to the same test, and so one would predict a somewhat better performance. And that is what happened. Webster’s verbal I.Q. rose to 72; his performance I.Q. dropped one point to 59; and his full-scale I.Q. was now 65, which (like the earlier findings) put Webster in the “mildly retarded” group.
Other aspects of Webster’s behavior confirmed this conclusion, in Dr. Finn’s view. Among other things, he noted Webster’s marginal school achievement (he dropped out in 9th grade), marginal employment history (almost none, apart from one job he lost after a week), lack of independent living (he lived with his mother), and concrete speech patterns (meaning that he was not able to deal with abstract. concepts). Webster did have good knowledge of words, but Dr. Finn thought he was better at speaking than understanding.
On cross-examination, the government urged Dr. Finn to consider whether a person such as Webster, charged with a capital crime, would have a motive to lie or manipulate while being tested. The Assistant U.S. Attorney (AUSA) also suggested that someone might have a motivation to lie if a finding of mental retardation would establish eligibility for various governmental benefits. In addition, the AUSA noted that Dr. Finn had mentioned in a letter he wrote to defense counsel that Webster told him that Webster had been in special education classes through most of his school *1129career. The AUSA then said “you know that’s not true now, don’t you,” and Dr. Finn agreed that he did “know that now.” As for the questions about motivation, Dr. Finn rejected the AUSA’s suggestion, insisting instead that Webster had put forth a good effort on all of the tests, that an experienced administrator could readily detect an effort to manipulate, and that he saw no such thing.
Dr. Keyes, who was a specialist in mental retardation, also relied on the DSM-IV. He testified about adaptive functioning, which he explained covered “the main areas of adaptive skills ... communication, socialization, and daily living skills....” It is important, he stated, to look at adaptive functioning, because “it is possible for a person to be intellectually retarded but not necessarily adaptively retarded.” Both are necessary to meet the definition of mental retardation under the law.
Several important points came up during Dr. Keyes’s testimony. The first relates to whether adaptive functioning should be assessed in relation to the world outside institutional walls, or in the prison environment. He argued that the former is what counts, because of the strictures of institutional living. The second point relates to assessment methodology: are accepted psychological tests necessary, or is it acceptable to rely exclusively on anecdotal evidence and perceptions? Dr. Keyes, as well as one of Webster’s rebuttal experts (Dr. George Denkowski), took the position that professional testing of this type is critical, because less rigorous measures give one no idea of where a person stands relative to the rest of the population. He accordingly administered the well-known Vineland test, which involved interviews of many people who had known Webster before the age of 18 (the age by which intellectual disability must appear) and in the “real world” as opposed to an institutional setting. He concluded from the results that Webster’s verbal skills somewhat exceeded what one might predict from his I.Q., but that overall his adaptive functioning was at the level of a six- to seven-year-old. Dr. Keyes firmly denied that Webster could have manipulated the results of the several tests that he had administered. Finally, Dr. Keyes recalled that Dr. Finn had told him about Webster’s special-education classes. On cross-examination, the AUSA asked whether Dr. Finn had ever mentioned that Webster had lied when he said he was in special education classes; Dr. Keyes had no memory of such a comment.
Next, Dr. Fulbright, the clinical neuropsychologist, discussed the tests he had performed to evaluate Webster’s attention, memory, problem-solving abilities, and general mental functioning. He too administered standardized tests. Those tests did not include an I.Q. test, because by that time Webster had already undergone multiple I.Q. tests while he was in the detention facility. The tests Dr. Fulbright administered included ones for attention and concentration, memory, information-processing speed, distractibility, visual and verbal memory, abstract reasoning, and logical analysis.
Webster performed poorly on many of these tests. He was unable to learn even the basics of an auditory addition test (thinking speed); he was very distractible; he did very badly on a visual memory test, but better on the verbal test; and his performance was severely impaired on higher-level thinking, problem-solving, and logical analysis tests. Dr. Fulbright found, consistently with a comment Dr. Finn made, that Webster is “extremely concrete” and not able to think abstractly. Yet, as others had also noted, his verbal fluency was surprisingly good. Even there, however, the testing revealed prob*1130leras. In a test where he was read sentences of increasing length and complexity and asked to say them back, Webster could manage only fairly short sentences. He was unable to listen to long complex communications and understand fully what was being said. Finally, Webster’s lawyer asked Dr. Fulbright whether someone could fake results in the tests that had been described. Dr. Fulbright’s response was “not convincingly,” because the tests are somewhat redundant, the subject would not know how to fake the results, and inconsistencies would be evident if someone were trying to manipulate them. He concluded that Webster’s testing revealed him to be a person who is mildly to moderately retarded. On cross-examination, he reiterated that faking is “quite easy” to detect on the tests he gave.
The government followed two strategies on rebuttal: first, it called a large number of lay witnesses (police officers, school administrators, school teachers, an employer, jailers) who all testified that Webster did not seem mentally retarded to them; second, it offered two experts, Dr. George Parker and Dr. Richard Coons, to rebut Webster’s experts. When the topic of special education came up, the government’s witnesses all denied that Webster had been in those classes. We will not review all of the lay testimony, other than to note a number of points that some or all of the witnesses made. Webster had managed to pass his Arkansas driving test with an almost perfect score (though other testimony indicated that he had done so by cheating). Several noted that Webster was “street smart.” Both teachers said that he did not seem to be mentally retarded, though Pat Drewett, one of them, said, “[He] performed at a slower level. He did read slower. On the days that he did perform, which most of the days he did sleep. He didn’t perform a lot. He could read. He could do. He chose not to do on a lot of days. He did sleep a lot.” School counselor E.C. Turner reported that in the 7th grade, Webster scored in the 43rd percentile of a national achievement test known as the M.A.T. 6 Survey. On cross-examination, Turner admitted that Webster’s grades began to fall off between the 6th and 7th grades. Tom McHan, a general contractor for whom Webster worked about a week, testified that he saw nothing to indicate that Webster was mentally impaired, but that he fired Webster from his job on a clean-up crew after one week for sleeping on the job.
The government also presented, as evidence of Webster’s functional capabilities, several facts about his life in prison. An inmate testified that he and Webster would communicate in “pig Latin” and that Webster was capable of quoting lai-ge portions of scripture. He described Webster as having a “photographic” memory for the Bible. Other testimony, consistent with Webster’s penchant for bragging about his sexual prowess, recounted Webster’s successful effort to crawl through “the bean chute” in the jail to get to the women’s area. Finally, witnesses reported that Webster visited the law library, obtained books from it, and on one occasion spotted that the commissary had given him the wrong change for a purchase.
Dr. Parker, a psychologist with a general practice, was the government’s first expert witness. At the government’s request, he evaluated Webster and gave him a truncated version of the WAIS I.Q. test. His results showed a verbal I.Q. estimate of 77, a performance I.Q. of 67, with a full-scale I.Q. of 72. He agreed with the AUSA that motivation plays a significant role in the results, and that this would be a particular problem in a “forensic” setting. Dr. Parker did not attempt formally to assess Webster’s adaptive functioning, either through the Vineland test or other*1131wise. He challenged the utility of the Vineland test, based on the fact that Webster (then age 23) had been incarcerated for five years since the age of 15. He agreed with the AUSA that Webster was able to communicate effectively, to speak in full sentences, to read simple stories aloud, and to keep his cell clean and tidy.
On cross-examination, defense counsel brought out the fact that the WAIS test has 11 parts, all of which must be administered to obtain a valid result, but that Dr. Parker had deliberately omitted some. He gave Webster four of the sub-tests in the verbal area, and three of the sub-tests in the performance area, omitting two in each. Worse than that, counsel suggested, the tests that he omitted (arithmetic and information) are traditionally the ones in which the intellectually disabled do the worst. Dr. Parker claimed ignorance of that fact (although on surrebuttal a defense expert supported Webster’s counsel’s assertion). What Dr. Parker did instead was to assign an average score, based on the tests that were administered, to the omitted sub-tests. This created an upward bias, counsel charged. Thus, for instance, when Dr. Finn administered the information sub-test, Webster scored a 2; Dr. Parker’s average assigned a score of 6. Dr. Parker also acknowledged that the DSM-IV and the American Association on Mental Retardation recommend using a recognized instrument like the Vineland test for adaptive functioning, but that he had chosen not to do so. Finally, Dr. Parker administered a test called the Schretlen Malingering Scale, and he admitted that the results did not support a finding of malingering.
The government’s final expert was Dr. Coons, a forensic psychiatrist whose practice focused on assessments of competency to stand trial, but who occasionally looked at questions of intellectual disability. Upon a personal examination, he found that Webster was well oriented to time, place, and person, and that he could give a detailed chronological account of events. Relying on the accounts of Webster’s day-to-day life, Dr. Coons concluded that his adaptive functioning was not consistent with a finding of mental retardation.
After entering the sentence of death on the verdict, the district court filed a separate document entitled “Factual Finding Regarding Mental Retardation.” It concluded that “Webster is not mentally retarded and ... he possesses the requisite mental capacity to understand the death penalty and why it will be imposed on him. As a result, the defendant Webster is not exempt under 18 U.S.C. § 3596(c) from implementation of the death penalty.”
Webster challenged this finding on direct appeal, but the Fifth Circuit “conclude[d] that the court took proper action, and the finding was supported by the evidence.” Webster I, 162 F.3d at 351. It applied plain error review, because Webster had failed to object to the factual finding in a timely way. Given how recent the statute was at the time of Webster’s trial, the court found no reversible error in the procedures the district court followed in carrying out its responsibilities. It rejected in particular Webster’s argument that the issue of mental retardation should have been decided by the jury, not the court. With respect to the sufficiency of the evidence on intellectual disability, the court had almost nothing to say; we reproduce its discussion in its entirety:
Webster contends that the finding that he is not mentally retarded is against the greater weight and credibility of the evidence. The standard of review for a finding that a defendant is not mentally retarded under § 3596 presents an issue of first impression. Because it is a fac*1132tual finding, we adopt the clearly erroneous standard.
The government presented substantial evidence to support the finding. Furthermore, only four of the twelve jurors found that Webster is or may be mentally retarded and that he suffers from low intellectual functioning.3 We cannot say the court clearly erred in deciding that Webster is not mentally retarded.
Id. at 352-53 (footnote omitted). Based on the information then available to Webster, this marked the end of Webster’s direct appeal on the issue of intellectual disability-
C. First Motion under Section 2255
At that point, Webster filed a motion for post-conviction relief under 28 U.S.C. § 2255. Webster’s counsel had sought additional discovery on the question of mental retardation, but two days before Atkins was decided and well before it decided the motion, the district court denied the request for additional discovery and required Webster’s motion to be filed within 60 days. Webster III, 605 F.3d at 259 n. 1 (Wiener, J., concurring). Webster’s motion raised 16 grounds for relief, but the district court rejected all of them. Seconded by the Fifth Circuit, it granted a certificate of appealability on only two claims: “first, that the evidence presented at trial was insufficient to warrant the district court’s finding that Webster is not mentally retarded; and second, that his alleged retardation renders him ineligible for a death sentence.” Webster II, 421 F.3d at 310.
Although the court of appeals had briefly considered these points on direct appeal, it agreed with Webster that a fresh look was warranted, in light of the Supreme Court’s intervening decision in Atkins, which held that the Eighth Amendment— not just a statute — prohibits the execution of the intellectually disabled. The Fifth Circuit saw little difference between the governing standards under the Constitution and section 3596(c), however, and so it was not persuaded that Atkins required a different result. It was willing to accept that Webster had a low I.Q., but it found that the government’s evidence of his adaptive functioning had effectively countered those numbers. The Vineland test that Dr. Keyes had conducted was all well and good, the court of appeals thought, but it accepted the “direct evidence” of adaptive functioning that the government had proffered. Id. at 313. It therefore affirmed the district court’s judgment denying Webster’s motion under section 2255.
D. Application for Successive Relief under Section 2255
-After losing that round, nothing of legal significance happened for four years. Some 13 years after Webster’s conviction, and four years after his section 2255 motion was denied, new counsel uncovered previously undisclosed evidence revealing that Webster had been diagnosed as mentally retarded a year before the commission of the crime. With those records in hand, counsel filed an application with the Fifth Circuit for permission to file a successive motion under section 2255; the proposed motion was directed exclusively at the death sentence. Before turning to the Fifth Circuit’s disposition of that application, we describe the new evidence and why it had not come to light earlier.
Among many other things that they did, Webster’s trial counsel had submitted a *1133request to the Social Security Administration for any records that it might have. There had been hints in the record that he, or he and his mother, had sought some kind of benefit, and counsel were following up on that clue. (The government challenges Webster’s contention that defense counsel actually tried to locate the records, either before the trial, when the government contends Webster must have known they existed, or for the original 2255 motion. But the facts about these old records are contested, to say the least.) The Social Security Administration produced nothing. Webster’s new lawyers contend that trial counsel, having hit a dead end, reasonably dropped the inquiry at that point. They also stress that when most of the records were produced in response to their own request, it was by mistake. Equally troubling is the fact that the remainder of the records were destroyed.
The newly produced records, which Webster’s current lawyers received on February 9, 2009, showed that Webster applied for Social Security benefits based on a sinus condition when he was 20 years old, approximately a year before the crime. The agency’s attention was evidently quickly redirected to Webster’s mental capacity. Two psychologists and one physician examined him. On December 22, 1993, Dr. Charles Spellman, a psychologist, evaluated him for the purpose of ascertaining his eligibility for Social Security benefits. He noted that “[i]deation was sparse and this appeared to be more of a function of his lower cognitive ability than of any mental illness.” Dr. Spellman also observed that Webster’s intellectual functioning was quite limited: he could not register three objects (meaning that he could not remember three objects a short time after they were shown to him); he could not do simple calculations; and he did not know what common.sayings meant. With respect to adaptive functioning, Dr. Spell-man stated that Webster lived with his mother; that he watched television, listened to the radio, and went walking; that he did no chores around the house; and that he was idle both in the house and on the streets. Taking into account both his estimate that Webster’s I.Q. was 69 or lower and his assessment of adaptive functioning, Dr. Spellman concluded that Webster was mentally retarded and antisocial. He foupd no evidence of exaggeration or malingering.
A few months earlier, in October 1993, Dr. Edward Hackett conducted a full-scale WAIS I.Q. test on Webster. He came up with a verbal I.Q. of 71, a performance I.Q. of 49, and a full-scale I.Q. of 59. He evaluated Webster as “mildly retarded, but ... also antisocial.” Pertinent to the central question of adaptive functioning, Dr. Hackett noted in a later report that “[Webster] was viewed as a somewhat mild[ly] retarded con man, but very street wise.... [H]e could not be functional in a community setting.... He would also not function well in the work place.” Dr. Hackett did not believe that Webster was capable of managing his own benefits; He found Webster’s behavior somewhat bizarre. Finally, he commented that on the I.Q. tests, Webster’s performance was estimated to be lower than his verbal score, and that some organic function might be involved.
The last .professional to examine Webster in conjunction with the 1993 Social Security application was Dr. C.M. Rittelmeyer, a physician. Dr. Rittelmeyer found Webster’s physical health to be fine, 'but he also had this to say: “Mental retardation. Flat feet.- Chronic sinus problems and allergies by history.”
The Social Security records included an intriguing letter that strongly suggested that Webster in fact had been in special *1134education classes. It was dated November 8, 1993, and had been written by Lou Jackson, the Special Education Supervisor for the school system Webster had attended, Watson Chapel Schools. Jackson’s letter explained that Webster’s special education records had been destroyed in 1988, after the family did not respond to a letter “telling them they could have the records if they wanted them.”
The Social Security records also provide some direct evidence about Webster’s abilities. The form Webster completed, for example, is rife with errors in syntax, spelling, punctuation, grammar, and thought. In response to a question asking him to describe his pain or other symptoms, Webster wrote “it causEs mE to gEt up sEt Easily hEadhurtsdiffiErnt of brEdth.” When asked about the side effects of his medication, he wrote “Is lEEp bEttEr.” When asked about his usual daily activities, Webster wrote (consistently with the comments from his teacher and employer) “I slEEps look at. cartoon.” He repoi'ted that he “ain’t got no chang” in his condition since its onset.
These records, new counsel urged, raised serious questions about the linchpins of the government’s case at trial with respect to intellectual disability. Counsel argued that they strongly refuted the consistent theme that Webster was malingering on the I.Q. tests he took after the crime was committed, since they showed a level consistent with those tests from a time (a) before the crime, and (b) when he was not under the emotional stress that tainted the 1992 tests at the Clinic. They also provided direct evidence .of adaptive functioning consistent with the I.Q. test scores — evidence that might have changed the minds of experts had they seen a more complete picture.
The Fifth Circuit, however, concluded that Webster’s proposed new evidence did not meet the stringent standards imposed by section 2255(h), which reads as follows:
(h) A second or successive motion must be certified as provided in section 2244 by a panel of the appropriate court of appeals to contain—
(1) newly discovered evidence that, if proven and viewed in light of the evidence as a whole, would be sufficient to establish by clear and convincing evidence that no reasonable fact-finder would have found the movant guilty of the offense; or
(2) a new rule of constitutional law, made retroactive to eases on collateral review by the Supreme Court, that was previously unavailable.
The Webster III judges concluded that a petitioner seeking only to challenge his eligibility for the death penalty cannot do so under section 2255(h)(1), because that section requires evidence that shows that the movant could not be found guilty of the offense.4 Webster’s application did not attack his guilt of the offense of murder but instead challenged only his sentence. Section 2255(h)(2) requires a new rule of constitutional law that previously was unavailable, but Atkins had already been decided at the time of Webster’s initial section 2255 *1135motion, and nothing else came close to satisfying that criterion.
Judge Wiener concurred in that disposition, but he wrote separately “to emphasize the absurdity of its Kafkaesque result: Because Webster seeks to demonstrate only that he is constitutionally ineligible for the death penalty — and not that he is factually innocent of the crime — we must sanction his execution.” Webster III, 605 F.3d at 259. He went on to say that “[i]f the evidence that Webster attempts to introduce here were ever presented to a judge or jury for consideration on the merits, it is virtually guaranteed that he would be found ,to be mentally retarded.” Id. Nevertheless, under section 2255(h) the court was required to “turn a blind eye to this evidence, as it speaks to Webster’s constitutional eligibility for the death penalty and not his factual guilt or innocence of the crime.” Id. at 260. He concluded by expressing his “deep and unsettling conviction” that, under the strictures of section 2255(h), “we today have no choice but to condone ... an unconstitutional punishment.” Id.
II. The 2241 Petition
Accepting, as they had to,5 the Fifth Circuit’s conclusion that a successive motion under section 2255 was not available, Webster’s counsel filed the current proceeding in the Southern District of Indiana, where Webster resides on the federal death row in Terre Haute. See 28 U.S.C. §§ 2242, 2243 ¶2 (directing the writ to run against the person having custody of the person detained). He argued that he was entitled to do so by virtue of the last sentence of section 2255(e), which permits an application for a writ of habeas corpus under section 2241 by someone who otherwise would be required to use the motion under 2255 and has failed in that effort, if “it also appears that the remedy by motion is inadequate or ineffective to test the legality of his detention.” 28 U.S.C. § 2255(e). This is often called the “savings clause”; we will follow that practice.
The district court found that Webster was not entitled to take advantage of the savings clause, because it understood the clause to apply only to changes in the law, not to new facts. It relied for that proposition primarily on Garza v. Lappin, 253 F.3d 918 (7th Cir.2001), and In re Davenport, 147 F.3d 605 (7th Cir.1998). The court read our cases to hold that section 2255 is “inadequate or ineffective” only when “a structural problem in § 2255 forecloses even one round of effective collateral review — and then only when as in Davenport the claim being foreclosed is one of actual innocence.” Taylor v. Gilkey, 314 F.3d 832, 835 (7th Cir.2002); see also Unthank v. Jett, 549 F.3d 534, 536 (7th Cir.2008). The court also commented in a footnote that it was not clear whether Webster’s trial counsel had followed up on their request to the Social Security Administration for its files, and that it was too late now to consider those materials. As we noted at the outset, in Webster TV a panel of this court affirmed the district court’s judgment, but the full court vacated that decision and reset the case for en banc consideration.
III. The Savings Clause and Section 2241
Our most extensive treatment of the relation between 2255’s savings clause and *1136section 2241 appears in Davenport. There, the petitioner had been convicted of the use of a firearm during the commission of a drug offense, in violation of 18 U.S.C. § 924(c). Davenport, 147 F.3d at 607. After he had filed his first motion under section 2255, the Supreme Court held that mere possession of a firearm, under the statute as it then read, did not amount to a prohibited “use.” See Bailey v. United States, 516 U.S. 137, 116 S.Ct. 501, 133 L.Ed.2d 472 (1995). The Seventh Circuit, as well as its sister circuits, had not construed the term “use” so restrictively. Like Webster, however, Davenport was barred from filing a successive motion under 2255 because he had neither new evidence of innocence of the offense nor a new Supreme .Court constitutional ruling. It would have been futile for him to have raised this point in his first section 2255 motion, as the law was squarely against him. We therefore held that section 2255 was inadequate within the meaning of sub-part (e) and that Davenport could raise his claim under section 2241. See Davenport, 147 F.3d at 610-12. In so doing, we said that whether section 2255 is inadequate or ineffective depends on whether it allows the petitioner “a reasonable opportunity to obtain a reliable judicial determination of the fundamental legality of his conviction and sentence.” Id. at 609. We also recognized that arguments addressing “the fundamental legality of [a] sentence[']” could be entertained, not just those attacking a conviction. Id.
Later cases have followed Davenport, albeit with somewhat different emphases. Compare Garza, 253 F.3d at 918, and Brown v. Caraway, 719 F.3d 583 (7th Cir.2013) (both with a broader understanding), with Unthank, 549 F.3d at 534, and Taylor, 314 F.3d at 832 (inadequacy or ineffectiveness exists only when a petitioner presents a claim of actual innocence and might exist only for claims related to retroactive Supreme Court statutory decisions). All of these decisions hold, nevertheless, that there must be some kind of structural problem with section 2255 before section 2241 becomes available. In other words, something more than a lack of success with a section 2255 motion must exist before the savings clause is satisfied. (Our dissenting colleagues would throw the entire line of cases that began with Davenport out the window, post at 1151-54, but their position essentially reads the savings clause of section 2255(e) out of the statute. We are not willing to take that step.)
In Garza, we found that “something more” in the form of an intervening decision of an international tribunal. Juan Garza had been sentenced to death in federal court under 18 U.S.C. § 848(e). After exhausting both his direct appeals and his opportunity for collateral relief under section 2255, Garza filed a petition with the Inter-American Commission on Human Rights. 'After reviewing the case, the Commission concluded that the introduction of evidence of murders in Mexico— crimes for which Garza had never been charged but which were necessary predicates to his death sentence — violated Garza’s rights under the American Declaration of the Rights and Duties of Man. Shortly after the Commission issued its report, Garza filed a habeas corpus petition under section 2241 in the Southern District of Indiana, the place of his incarceration. He argued in that petition that the United States was bound by treaty to abide by the Commission’s decision. Construing Garza’s petition as an unauthorized effort to file a successive motion under section 2255, the district court held that it lacked jurisdiction to decide the case and dismissed the action.
Garza appealed that unfavorable ruling to this court. We identified two issues *1137that had to be resolved: first, whether Garza qualified for section 2255’s savings clause, thereby enabling him to bring a petition under section 2241; and second, if 2241 was available, whether he was entitled to relief. We began by recognizing that “the mere fact that Garza’s petition would be barred as a successive petition under § 2255 ... is not enough to bring the petition under § 2255’s savings clause; otherwise, the careful structure Congress has created to avoid repetitive filings would mean little or nothing.” Garza, 253 F.3d at 921. We concluded, however, that in rare circumstances “the operation of the successive petition rules [would] absolutely prevent[ ] the petitioner from ever having an opportunity to raise a challenge to the legality of his sentence.” Id. at 922. Garza presented such a case, we conclud-r ed. On the merits, we decided that the Commission’s report was advisory only and thus did not support the relief Garza sought.
Garza thus offers one illustration of a situation in which a petitioner was entitled under the savings clause to use section 2241 to attack a sentence, even though he was not making a claim of actual innocence of the offense. In fact, it would be more difficult for someone making the latter kind of argument, because section 2255(h)(1) expressly addresses guilt or innocence of the offense and sets out the evidentiary standard that a successive petition must meet. That is enough to explain why, in Unthank and Taylor, we held that the' petitioners were not entitled to turn to section 2241. It is true that in describing Davenport we used more general language that could be read as an absolute restriction on the savings clause, but in neither of those cases did the court have before it an argument that a particular sentence was constitutionally forbidden (either as a matter of law or as a matter of fact), rather than just outside the scope of a statute. We therefore think it best to understand those holdings as appropriate applications of the law to the facts before the court.
Pointing to some legislative history, the government in this case argues that Congress did not intend to permit the savings clause to be used for anything other than claims of actual innocence of the underlying offense. It is worth noting that this contention is inconsistent with the position that the Solicitor General took in Persaud v. United States. See Brief for the United States, Persaud v. United States,— U.S. -, 134 S.Ct. 1023, 188 L.Ed.2d 117 (2014) (mem.), 2013 WL 7088877. In Persaud, the question was whether certain prior felonies qualified as predicates for an enhanced sentence pursuant to 21 U.S'.C. § 851(a)(1). As the Solicitor General’s brief put it, “The question presented here is whether petitioner is entitled to challenge the sentencing error by way of a petition for a writ of habeas corpus under 28‘U.S.C. 2241. The lower courts held that Section 2241 was not available because petitioner was challenging his sentence rather than his conviction. That holding is incorrect.” Id. at *13. The Solicitor General noted that the Supreme Court has not spoken on when the motion under section 2255 is “inadequate or ineffective.” In the circumstances presented in Persaud, the brief continued, the savings clause is available even though the challenge is to the sentence, not to the underlying conviction. Id. at *18-19. Responding to that point, the Supreme Court vacated the judgment and remanded the case to the court of appeals “for further consideration in light of the position asserted by the Solicitor General in his brief for the United States filed on December 20, 2013.” Persaud, 134 S.Ct. at 1023 (mem.). The Court’s action indicates that it does not take as narrow a view of the *1138savings clause as our dissenting colleagues do.
Before this court the government points to House of Representatives Report no. 104-23 (part of AEDPA’s legislative history), which discusses a proposal for a provision on stays of execution. The proposal, which was not enacted, stated that a federal court could not issue a stay merely because a state prisoner files a second habeas corpus petition, unless the petition set forth newly discovered facts showing that the petitioner is not guilty of the underlying offense. H.R. REP. NO. 104-23, at 4-5 (1995). It would have barred “claims that go only to the validity of the capital sentence and claims that go only to the petitioner’s eligibility for a capital sentence.” Id. at 16-17.
Putting to one side the fact that the comity concerns that exist with respect to state-court proceedings are not present for federal prosecutions, we do not find this snippet of legislative history to be too helpful. The language of section 2255(h) already makes it clear that Congress was aware of the difference between claims of innocence of the underlying offense and claims relating to a sentence. The problem before us, quite simply, is not one that Congress could have contemplated. At the time AEDPA was under consideration, the Supreme Court had not yet held it unconstitutional to execute either an intellectually disabled person (Atkins) or a minor (Roper v. Simmons, 543 U.S. 551, 125 S.Ct. 1183, 161 L.Ed.2d 1 (2005)). We thus consider this question to be an open one, both from the standpoint of legislative history and that of Supreme Court rulings.
Several considerations persuade us that in the circumstances presented here the savings clause permits Webster to resort to a petition under section 2241. The first is the language of the statute. Section 2255 motions are available to “[a] prisoner in custody under sentence of a court established by Act of Congress claiming the right to be released upon the ground that the sentence was imposed in violation of the Constitution or laws of the United States....” 28 U.S.C. § 2255(a) (emphasis added). The language we have highlighted does not distinguish between a sentence that is unlawful because of a flaw in the underlying conviction and a sentence that is unlawful because of a constitutional or statutory rule pertaining to sentences. Moving down to the savings clause, we see that it applies when “the remedy by motion is inadequate or ineffective to test the legality of [the prisoner’s] detention.” Id. § 2255(e) (emphasis added). Once again, the statute-focuses on the detention as a whole, not on the underlying offense. Only when one reaches section 2255(h)(1) does Congress single out the underlying offense. It was certainly free to do so, but there would have been no need for the specification if the rest of section 2255 (including subpart (e)) were already so limited.
Second, the fact that the Supreme Court had not yet decided Atkins and Roper at the time AEDPA was passed supports the conclusion that the narrow set of cases presenting issues of constitutional ineligibility for execution is another lacuna in the statute. In that respect it is similar to the problem we faced in Davenport, where we found the remedy under section 2255 to be “inadequate or ineffective” for a case in which the Supreme Court had definitively ruled that the conduct for which the person was convicted and imprisoned was not an offense under the statute. The provisions of section 2255 permitting successive motions speak exclusively in terms of constitutional problems, and so left someone who wished to show that a new Supreme Court decision clarifying as a matter of statutory interpretation that he had com*1139xnitted no offense without anywhere to turn.
In Webster’s case, the problem is that the Supreme Court has now established that the Constitution itself forbids the execution of certain people: those who satisfy the criteria for intellectual disability that the Court has established, and those who were below the age of 18 when they committed the crime.6 In virtually all other situations, Congress has almost unlimited discretion to select the penalty, or the range of penalties, that go along with a particular crime. If Congress selects 20 years, but because of some error that went undetected through direct appeals and an initial motion under section 2255 the defendant receives 25 years, there is no doubt a problem, but it is likely not one of constitutional dimension. Congress could have chosen 25 years to begin with, and the defendant would have had nothing to complain about.
If Atkins had never been decided, Webster would have been left with an argument that he now has new evidence that would demonstrate that the statute forbidding the execution of a mentally retarded person would be violated by the implementation of his sentence. See 18 U.S.C. § 3596(c). We can assume that this would not be enough to trigger the savings clause. But Atkins, and later Roper, were decided by the Supreme Court, and they must guide our understanding of the law. Judge Wiener, in speaking of the unavailability of section 2255 for Webster, spoke powerfully of the “Kafkaesque” nature of a procedural rule that, if construed to be beyond the scope of the savings clause, would (or could) lead to an unconstitutional punishment.
If one were to disagree with our view, stated earlier, that ordinary principles of statutory interpretation lead directly to the result that the savings clause applies here, then the next step would be to take into account the fact that a core purpose of habeas corpus is to prevent a custodian from inflicting an unconstitutional sentence. As Judge Wiener implied, there is no reason to assume that our procedural system is powerless to act in such a case. It is fairly possible to read section 2255(e) as encompassing challenges to both convictions and sentences that as a structural matter cannot be entertained by use of the 2255 motion. To hold otherwise would lead in some cases — perhaps Webster’s— to the intolerable result of condoning an execution that violates the Eighth Amendment.7 We decline to endorse such a reading of the statute.
IV. Application to Webster’s Case
Having found that there is no categorical bar against resort to section 2241 in cases where new evidence would reveal that the Constitution categorically prohibits a certain penalty, it remains for us to *1140consider whether Webster has presented a proper case for its use, and if so, how we should proceed from here. For this purpose, it is necessary to go back and compare the evidence he • has proffered and decide whether he has shown enough to proceed to the merits of his 2241 petition, or at a minimum to have a hearing to resolve predicate issues of fact.
We have established thus far that a person who proposes to show that he is categorically ineligible for the death penalty, based on newly discovered evidence, may not be barred from doing so by section 2255. But this rule cannot apply to all newly discovered evidence, or else there would never be any finality to capital cases involving either the intellectually disabled or minors.8 Looking particularly at the intellectually disabled, it would always be possible to conduct more I.Q. and adaptive functioning tests in the prison. Those new scores would have no bearing on the initial conviction and sentence, though they would be highly pertinent to the ultimate ability of the government to carry out the sentence. But our concern is with the former, not the latter.
What distinguishes Webster’s case from the one we just hypothesized are two facts: first, the newly discovered evidence that current counsel have proffered- existed before the time of the trial, and is relevant for precisely that reason; second, although the facts are disputed, there is evidence indicating that they were not available during the initial trial as a result of missteps by the Social Security Administration, not Webster’s counsel. (We would say “not Webster” too, except that the question whether Webster can be held responsible for those records is intimately tied to the general question of his intellectual abilities. The dissent, post at 1150-51, assumes the answer to this question when it states that “Webster has long known” of the missing Social Security evidence. There is no evidence in the record that Webster had any personal awareness of this evidence, much less that he was capable of appreciating its significance.) This limitation answers most of the concerns about creating too large an exception to the exclusivity of section 2255. It will be a rare case where records that predate the trial are found much later, despite diligence on the part of the defense, and where those records bear directly on the constitutionality of the death sentence.9
*1141Rare, but not impossible. Suppose, for instance, that the United States tries a man in central Texas for a capital crime. Suppose further that he is convicted and sentenced to death, based on the jury’s and court’s acceptance of substantial oral testimony confirming that he was 19 years old when he committed the crime. Now imagine that after he has exhausted his first post-conviction motion under section 2255, irrefutable evidence comes to light revealing that he was born in Germany and that his birth certificate confirms that he was actually only 17 years old at the time he committed the crime. (According to the Texas State Historical Association, “[a]fter Anglos, Mexican-Americans and African-Americans, the ethnic group with the largest impact on Texas has been the Germans.” See http://www.texasalmanac. com/topics/culture/german/german-texans.) Under the government’s view of the relation between section 2255 and 2241, our hypothetical defendant would be executed despite the plain violation of the Eighth Amendment as described in Roper. Under the view we adopt, this defendant is entitled to be heard under section 2241. At any such hearing, the government would be entitled to challenge the authenticity of the birth certificate; the defendant would be entitled to support it; and the district court would decide the factual question on which everything hinges: whether the defendant was below the age of 18 at the time of the crime and thus constitutionally ineligible for the death penalty.
Webster’s case is no different. At the threshold stage, he has the burden of proffering convincing evidence of intellectual disability to show that he may not, consistently with Atkins and Hall, be under a sentence of death. In order to relate to the findings at trial, that evidence cannot be newly created; instead, it must be previously existing evidence of his intellectual disability that counsel did not uncover despite diligent efforts. If he succeeds in this prima facie showing, then he has satisfied the savings clause and may proceed with a section 2241 petition. At a hearing on the merits, he would need to introduce the evidence and allow the government to refute it. The government here has indicated that it would challenge the diligence of counsel; that it would argue that the allegedly new evidence is actually just cumulative; that it would review the earlier evidence; and that even with the new evidence Webster’s adaptive functioning is high enough that it negates his low I.Q. scores (an extremely rare and largely untested scenario, as far as we can tell from post-Atkins cases). Webster would respond to those points. In the end, the district court would decide whether, as a matter of fact, Webster is constitutionally ineligible for the death penalty.
Although we do not want to prejudge the district court’s disposition of this matter, a few observations are in order. First, we reiterate that the question whether the evidence we described earlier in Part I.D of our . opinion qualifies as newly discovered is contested. Both Webster and the government have a right to be heard on this point. The record presently before-us is inconclusive about what happened at the Social Security Administra*1142tion. Webster’s trial counsel provided an affidavit stating that he requested records from the agency but that to the best of his recollection he never received them. We do not know at this juncture whether counsel never followed up, whether there was a technical problem with his request, or if the agency deliberately or accidentally informed counsel that the records did not exist. Webster’s current legal team requested and received some of the records in 2009. It later realized that the file was incomplete and sent another request for the missing records. A representative from the agency told them that it would not provide any more records and that “normal procedures” had not been followed when the partial records had been sent earlier that year. Webster’s attorneys interpret that statement as an admission that the records were produced by accident; other evidence indicates that the representative may have been speaking only about a supposed failure to pay (contested again by Webster’s lawyers) or a lack of specificity. In any event, Webster’s current lawyers followed up with a new request, only to be told, mysteriously, that Webster’s remaining records had been destroyed. If the agency was aware that active efforts were underway to obtain them, this is troublesome indeed. But our main point here is that we do not know important details.10
Another issue that the court will need to explore is the significance of the records. As Part I.B of our opinion reviews in detail, there was a good deal of evidence about intellectual disability at the -trial: Drs. Finn, Keyes, and Fulbright for the defense, and Drs. Parker and Coons for the government. That evidence, however, included only one I.Q. test that had been performed on Webster before the crime'— the 1992 test done at the Southeast Arkansas Mental Health Clinic. Webster’s full-scale score was then just 48, but Webster’s own experts agreed that he had been under severe stress at the time that test was administered and so the score was unreliable. Although other tests were performed (two by Dr. Finn, showing full-scale I.Q.’s of 59 and 65, and an incomplete one by Dr. Parker, showing a full-scale I.Q. of 72), the government argued strongly that Webster was motivated to underperform on the later tests because of the risk of the death penalty. The tests and estimates from the Social Security Administration records, however, were immune from that argument about manipulation. Webster scored 59 on Dr. Hackett’s test; Dr. Spellman estimated his I.Q. to be 69 or lower; and Dr. Rittelmeyer commented that Webster was “mentally retarded.” It is significant in this respect that Webster was not trying to obtain Social Security benefits on the basis of his intellectual disability, and so an argument about manipulation for purposes of benefits would have been weak. He asserted instead that he was disabled from a sinus condition. In short, had the I.Q. tests and opinions from the Social Security file been' available to the experts at the trial, to the jury, and to the district court, their assessments of Webster may have been different.
Equally importantly, the Social Security records shed additional light on Webster’s adaptive functioning. New counsel found evidence that contradicted everyone’s assumption at the trial that Webster had been lying when he said that he was in special education classes. If it had been clear that he was telling the truth, or even if objective evidence had supported his *1143assertion, the jury and the district court may have viewed Webster’s poor school performance, sleeping in class, and dropping out in the 9th grade in a different light.
At a more general level, the government treated this case as the reverse of the one the Supreme Court discussed in Hall v. Florida. In Hall, the defendant’s I.Q. score was 71, but the Florida courts refused to allow him to introduce evidence of adaptive functioning that would have shown him to be intellectually disabled. In Webster’s case, virtually all of the I.Q. scores put him in the range of mild to moderate intellectual disability, but the government argued strongly (and both the court and the jury were persuaded) that his adaptive functioning was good enough to demonstrate that he was not, in fact, so disabled. (It is of some interest that the evidence of adaptive functioning that the Supreme Court criticized in Hall looks very much like the evidence that the government used here. Although there was substantial evidence that Hall had been mentally retarded his entire life, the trial court was suspicious because “[njothing of which the experts testified could explain how a psychotic, mentally-retarded, brain-damaged, learning-disabled, speech-impaired person could formulate a plan whereby a car was stolen and a convenience store was robbed.” Hall, 134 S.Ct. at 1991.)
Looking at the trial evidence of adaptive functioning solely for the purpose of assessing how much of a difference the new evidence might make, we note that there was some consensus at the outset. All agreed that Webster’s verbal abilities were relatively strong, at least in terms of his spoken communications. His written work was more questionable, and the Social Security records reveal a person who is barely literate. And there were other problems as well with the government’s evidence of adaptive functioning. Some of the government’s evidence came straight from laypersons, who more or less said “he seemed fine to me.” It also used experts, but neither of its experts administered any kind of accepted adaptive functioning test before reaching his opinion.
We have often pointed out the dangers of relying on “common sense” when social science reveals that common assumptions are wrong. As we noted in United States v. Williams, 522 F.3d 809 (7th Cir.2008), in the context of a discussion of eyewitness identification, “[t]he problem with ‘common sense’ is that experience tells us what leads to confidence about whether we have seen a given person before but does not provide reliable ways to test whether that confidence is justified. People confuse certitude with accuracy and so are led astray. Psychologists have established that certitude often is unwarranted. It takes data rather than intuition to answer questions such as ‘can non-uniform foot-gear in a lineup lead to misidentification?’ ” Id. at 811; see also United States v. Bartlett, 567 F.3d 901, 906 (7th Cir.2009) (“That jurors have beliefs about [the fallibility of memory] does not make expert evidence irrelevant; to the contrary, it may make such evidence vital.”); Murillo v. Frank, 402 F.3d 786, 792 (7th Cir.2005) (no social science verification of the proposition that emotionality enhances the reliability of a statement). That principle applies here as well. It is one thing to describe what Webster did and to believe, as a layperson, that these acts revealed that his I.Q. tests understated his intellectual functioning; it is quite another for a qualified professional to test whether such a discrepancy exists.
There was no testimony at trial correlating Webster’s day-to-day skills with the intellectual age that his I.Q. tests suggested. Counsel for the government, at oral *1144argument, pointed to Webster’s ability to come to the Dallas area, to lie about being an F.B.I. agent at Lisa’s door, to travel back to Pine Bluff, to dig the grave in advance, and to kill and bury her, as evidence of his competence. But as we have just pointed out, that is a lay opinion. Dr. Finn put Webster’s mental age at somewhere between six and seven. Common experience shows that children of that age can do quite a few things: they can lie; they can plan an immediate event; they can carry out instructions. Dr. Keyes testified that the results of the Vineland test he performed showed that Webster’s adaptive functioning level was consistent with his low I.Q. scores. The government’s experts offered conclusions, but little in the way of reasons for their conclusions.
The government also relied on the fact that Webster complained on one occasion that he received the wrong change from the commissary. But studies indicate that adults with mild retardation can learn the essentials of paying bills. See George S. Baroff with J. Gregory Olley, Mental Retardation: Nature, Cause, and 'Management 308-09 (3d ed.1999) (citing John LaCampagne & Ennio Cipani, Training Adults with Mental Retardation to Pay Bills, 25 Mental Retardation 293 (1987)). Webster’s ability to squirm through the bean chute to reach the women’s section of the detention center also may or may not be consistent with the behavior of a seven-year-old child. The government’s experts did not use any recognized methodology to connect those dots. And that is not because there are no measurement tools. Several well-accepted adaptive-functioning tests are available: the Vineland Adaptive Behavior Scales that Dr. Keyes used, which is for primary caregivers and others familiar with the person, rather than for the person; the Diagnostic Adaptive Behavior Scale, which measures adaptive behavior skills in the three main categories of conceptual, social, and practical life skills; and the Supports Intensity Scale, which is used to determine what a person needs to live independently.
This evidence may or may not carry the day for Webster, but we believe that it does qualify as the kind of “clear and convincing” evidence that would be required to earn a hearing if we were evaluating new factual evidence of guilt or innocence under section 2255(h)(1). We therefore do not need to decide whether that is the correct standard, or if a lesser showing would suffice.
V. Concluding Observations
Webster filed his section 2241 petition in the district where he is incarcerated, as 28 U.S.C. §§ 2242 and 2243 require. The Supreme Court held in Rumsfeld v. Padilla, 542 U.S. 426, 124 S.Ct. 2711, 159 L.Ed.2d 513 (2004), that this is the one and only proper venue:
The federal habeas statute straightforwardly provides that the proper respondent to a habeas petition is “the person who has custody over [the petitioner].” 28 U.S.C. § 2242; see also § 2243 (“The writ, or order to show cause shall be directed to the person having custody of the person detained”). The consistent use of the definite article in reference to the custodian indicates that there is generally only one proper respondent to a given prisoner’s habeas petition. This custodian, moreover, is “the person” with the ability to produce the prisoner’s body before the habeas court. Ibid.
Id. at 434-35, 124 S.Ct. 2711. Webster’s custodian is Charles A. Daniels, the warden of the United States Penitentiary in Terre Haute, Indiana. Terre Haute lies within the Southern District of Indiana, *1145and so Webster’s petition under section 2241 was properly filed in that district.
There are obviously costs in requiring a new court to delve into the question of Webster’s intellectual disability (or lack thereof), although after 21 years the comparative advantage of the district court in the Northern District of Texas has inevitably faded. But whatever costs remain are nothing more than the price of the limitations on relief by motion under section 2255. When that statute was passed, Congress not only relieved the district courts where the major federal prisons were located from a heavy load of petitions for collateral relief; it also enhanced the efficiency of the system by assigning these cases to the judges who were familiar with the records. Nonetheless, the Fifth Circuit has decided that Webster’s current action falls outside the scope of section 2255(h), and we have no quarrel with that assessment. That means, for better or for worse, that the remaining work on Webster’s case must be conducted by the district court for the Southern District of Indiana.
We say “must,” because we understand Padilla to hold that the only proper district is the one containing the prisoner’s immediate custodian. That is why, despite the fact that Padilla was initially detained in the Southern District of New York, once the President had him moved to South Carolina, the only district in which his habeas corpus petition could be brought was the District of South Carolina. Bra-den v. 30th Judicial Circuit Court of Kentucky, 410 U.S. 484, 93 S.Ct. 1123, 35 L.Ed.2d 443 (1973), is not to the contrary. It dealt with a situation in which the petitioner (then incarcerated in Alabama) wanted to challenge a future detainer issued by Kentucky. In that situation, the Court held, the effective custodian was Kentucky, because it was the source of the detainer. Webster, however, is not challenging his detention by anyone other than the warden at Terre Haute, and so it follows that the only permissible respondent is Warden Daniels.
This also means that there is no other judicial district, from a venue standpoint, in which the claim “might have been brought,” as that term is used in the general venue statute, 28 U.S.C. § 1404(a). The district court thus has no power under that statute to transfer the case either on its own initiative or upon the motion of either party. (Section 1406(a) is also inapplicable, because there is nothing wrong with the district in which Webster is proceeding.) If both Webster and Warden Daniels were to consent to a transfer to another district, they might have an enforceable agreement that the court could implement, by analogy to choice-of-forum agreements. See Atl. Marine Constr. Co. v. U.S. Dist. Court for W. Dist. of Texas, — U.S.-, 134 S.Ct. 568, 187 L.Ed.2d 487 (2013). The only thing that seems clear is that in the absence of any such consent, the case must stay in Indiana. That result also avoids knotty problems that could arise if there is another appeal. We have no authority over the district court in the Northern District of Texas, which lies within the Fifth Circuit. 28 U.S.C. § 41. That court would be put in an awkward position, to say the least, if it were responsible for carrying out a Seventh Circuit mandate with any appeals going to the Fifth Circuit. In contrast, if the case stays in Indiana (as we believe it must), should there be a new appeal, it will be a straightforward matter for this court to decide whether the district court has properly carried out our mandate.
In summary, therefore, we hold that Webster is' not barred as a matter of law from seeking relief under section 2241. Further proceedings are necessary, how*1146ever, before the district. court can reach the merits of his habeas corpus petition. The district court must hold a hearing for the purpose of deciding whether the Social Security records were indeed unavailable to Webster and his counsel at the time of the trial. In considering that question, the district court must also evaluate trial counsel’s diligence. If the court concludes, as Webster’s lawyers urge, that the records were unavailable and all due diligence was exercised, and that Webster has them now only because of a fluke, then it must turn to the merits of the petition: is Webster so intellectually disabled that he is categorically ineligible for the death penalty under Atkins and Hall?
Both the government and Webster would be entitled at a hearing to offer evidence relevant to that determination; if Webster is not limited to the record before the original trial court, there is no reason to impose such a limit on the government. Webster, as the petitioner, bears the burden of proving intellectual disability. If he gets this far, what this will be as a practical matter is a partial new sentencing hearing. At this later stage, the proper burden of proof is a preponderance of the evidence, not “clear and convincing” evidence. (The-'latter standard serves as a gateway to the hearing; it would be wrong to apply it twice.) Finally, our mandate does not extend to any other issue that Webster has raised over the years. In particular, we have found no reason to revive the question of the abuse he suffered as a child.
If the district court concludes that Webster meets the requirements of Atkins and Hall, then it should issue the writ stating that Webster is entitled to relief from the death penalty. This is the relief that Webster requested in his section 2241 petition.11 As happens with writs issued on behalf of state prisoners under 28 U.S.C. § 2254, the writ should give the prosecutor (here, the Attorney General) a reasonable period within which to take appropriate action in light of the writ. Any such action, including correction of the judgment, would properly occur in the court that entered the judgment and sentence (the Northern District of Texas), which would be responsible for further sentencing proceedings in line with both this opinion and the district court’s order. If, on the other hand, Webster fails to show that he has met the Atkins and Hall requirements in the Indiana proceedings, then the writ should be denied.
The judgment of the district court is Reversed and the case is Remanded for further proceedings consistent with this opinion.
. Hall went to trial, was convicted, and was also sentenced to death. See United States v. Hall, 152 F.3d 381 (5th Cir.1998). Like Webster, he is currently housed at the U.S. Penitentiary in Terre Haute. Demetrius Hall, Steven Beckley, and Marvin Holloway all pleaded guilty and testified at Webster's trial; they received varying sentences for terms of years.
. Although our account may seem detailed, it is only a summary of more than 600 pages of trial transcript. We have attempted to hit the high points, without mentioning every time someone said something of interest.
. As this comment illustrates, the court took the position that the jury did not need to be unanimous on the question of mental retardation. Webster has not separately attacked this ruling in his section 2241 petition.
. Our dissenting colleagues disagree with this aspect of the Fifth Circuit's ruling. See post at 1148 n. 1. They argue that the rule of Sawyer v. Whitley, 505 U.S. 333, 112 S.Ct. 2514, 120 L.Ed.2d 269 (1992), recognizing that a person can be "innocent of the death penalty,” survived the enactment four years later of the Antiterrorism and Effective Death Penalty Act (AEDPA). They acknowledge, however, that this was not the Fifth Circuit's view. Since the Fifth Circuit panel in Webster III unanimously found that only factual innocence of the crime would do, the majority did not express any opinion on whether Webster's new evidence undermined the findings underpinning his death sentence.
. The dissent appears to agree with the position that Webster’s lawyers were required to accept the Fifth Circuit’s ruling that section 2255 addresses only innocence of the underlying crime; they note that section 2244(b)(3)(E) cuts off any right to further review of such a ruling. See post at 1148. We have treated this ruling as part of the law of the case, both in recognition of its unreviewability and out of respect for our sister circuit.
. The dissent takes the position that Atkins and Hall add nothing to the protection against the death penalty in 18 U.S.C. § 3596(c). With respect, we disagree. Collateral relief is primarily used for constitutional violations, not violations of federal law that could and should be raised on direct appeal. The dissent paints a picture of the floodgates opening up as a result of our decision, but, as we explain in more detail below, that cannot happen.
. The dissent suggests that the savings clause is available only when the application of se'ction 2255 would conflict with the Suspension Clause. See post at 1152-54. We have never contended that denying Webster the right to use section 2241 would violate the Suspension Clause. But this is precisely the point: we do not have to reach this constitutional issue because Congress included section 2255(e) in the statutory framework, thus allowing us to resolve the question of the availability of section 2241 on statutory — rather than constitutional — grounds.
. In fact, given the rule in Ford v. Wainwright, 477 U.S. 399, 106 S.Ct. 2595, 91 L.Ed.2d 335 (1986), which holds that "[t]he Eighth Amendment prohibits the State from inflicting the penalty of death upon a prisoner who is insane,” id. at 410, 106 S.Ct. 2595, there will always be some lack of finality for a person whose mental condition "prevents him from comprehending the reasons for the penalty or its implications.” Id. at 417, 106 S.Ct. 2595. Both parties have assumed, however, that the Ford standard is higher than the one imposed in Atkins and Hall. We assume for present purposes that this is correct.
. The dissent is concerned that, contrary to Congress's intention in enacting AEDPA, our interpretation of section 2255 returns us to the pre-AEDPA standard for successive habeas corpus petitions. See post at 1148-49 (the majority’s interpretation "open[s] the door to any proceedings that do not abuse the writ— the pre-1996 standard”). In so contending, however, it disregards the array of limitations, both legal and factual, on which our ruling' depends. We have highlighted at least three principal reasons why our finding that the savings clause is available here will have a limited effect on future habeas corpus proceedings. First, the evidence sought to be presented must have existed at the time of the original proceedings. (A free-standing claim that an execution would violate Ford v. Wainwright might involve later-acquired evidence, but such a claim is quite different from the one now before us.) Second, the evidence must have been unavailable at the time of trial despite diligent efforts to obtain it. Third, and most importantly, the evidence must show that the petitioner is constitution*1141ally ineligible for the penalty he received. Because the Supreme Court has declared only two types of persons (minors and the intellectually disabled) categorically ineligible for a particular type of punishment, our ruling is as a matter of law limited to that set of people— those who assert that they fell into one of these categories at the time of the offense. These three limitations are more than adequate to prevent the dissent’s feared flood of section 2241 petitions, and they in no way represent a return to the abuse of the writ standard.
. The dissent’s discussion of these points, post at 1150-51, rests on factual assumptions that are contested. It merely highlights the need for the evidentiary hearing we contemplate.
. This does not create a conflict with the Fifth Circuit. That court ruled that it could not even consider Webster's new evidence, because of the limitations in section 2255(h). If Webster wins relief in the district court in Indiana, and if any such ruling is upheld on appeal in this court, then the mandate will give the sentencing court clear and uncontradicted instructions for moving forward.