
530 F.3d 300 (2008)
UNITED STATES of America, Plaintiff-Appellee,
v.
Sabri BENKAHLA, Defendant-Appellant.
Council on American-Islamic Relations; Muslim American Society Freedom Foundation, Amici Supporting Appellant.
No. 07-4778.
United States Court of Appeals, Fourth Circuit.
Argued April 11, 2008.
Decided June 23, 2008.
*302 ARGUED: William Benjamin Moffitt, Moffitt & Broadnax, Reston, Virginia, for Appellant. Gordon Dean Kromberg, Office of the United States Attorney, Alexandria, Virginia, for Appellee. ON BRIEF: Andrew L. Hurst, Reed Smith, L.L.P., Washington, D.C., for Appellant. Chuck Rosenberg, United States Attorney, Alexandria, Virginia, for Appellee. Mara Verheyden-Hilliard, Partnership for Civil Justice, Washington, D.C., for Muslim American Society Freedom Foundation; John Kenneth Zwerling, Zwerling, Leibig & Moseley, P.C., Alexandria, Virginia, Nadhira F. Al-Khalili, Washington, *303 D.C., for Council on American-Islamic Relations, Amici Supporting Appellant.
Before WILKINSON, MOTZ, and DUNCAN, Circuit Judges.
Affirmed by published opinion. Judge WILKINSON wrote the opinion, in which Judge MOTZ and Judge DUNCAN joined.

OPINION
WILKINSON, Circuit Judge:
Sabri Benkahla was part of a network of people the government was investigating for crimes connected to radical Islamic terrorism and violent jihad. The FBI questioned him and prosecutors twice called him before grand juries. Then he was prosecuted himself for false declarations, false statements, and obstructing justice. He raises three main issues on appeal. First, he claims his prosecution violated the collateral estoppel component of the Double Jeopardy Clause, as he had already been prosecuted and acquitted for some of the activities he was questioned about. Second, he claims the trial court admitted irrelevant and unduly prejudicial evidence about terrorism and violent jihad. Third, he claims the trial court erred in determining his sentence by applying the Sentencing Guidelines' terrorism enhancement. For the reasons below, we reject all three claims and affirm the judgment of the district court.

I.
An organization in Falls Church, Virginia, known as the Dar al-Arqam Islamic Center, has figured in no fewer than fourteen terrorism prosecutions so far. See, e.g., United States v. Chandia, 514 F.3d 365 (4th Cir.2008); United States v. Khan, 461 F.3d 477 (4th Cir.2006). Some of those prosecutions centered on a group of young men who assembled at Dar al-Arqam and, in early 2000, started training together for violent jihad. The group escalated stepwise from an ideological attraction to religious violence to actually taking up arms against nations they saw as enemies of Islam: Russia in Chechnya, India in Kashmir, and the United States. They began by talking with some of the more militant leaders at Dar al-Arqam. Then they started conducting quasi-military exercises with paintball guns in the Virginia woods and practicing marksmanship with AK-47 style rifles on Virginia shooting ranges. A few members traveled to Pakistan or Afghanistan to train at jihadist camps run by Lashkar-e-Taiba (a designated terrorist organization since December 2001).
Then came the attacks of September 11th and a schism at Dar al-Arqam between those who condemned and those who condoned the attacks. Within a few days, the leader of the violent wing, a Dar al-Arqam founder named Ali Al-Timimi (later convicted of solicitation to levy war against the United States), held a secret meeting at which the core of the paintball group formally dedicated itself to violence. More members went abroad to the jihadist camps. Some who went, upon returning to the United States, purchased sophisticated aerial surveillance technology to send to Lashkar-e-Taiba overseas. Then, in 2003, the group was arrested and eleven men indicted together. Six of the men pled guilty. Three were convicted. One was acquitted. The instant case concerns the eleventh man: Sabri Benkahla.
Benkahla's case was severed from the other ten defendants. The indictment charged the other members of the group chiefly with a conspiracy, beginning in 2000, to engage in armed hostilities against the United States, take part in military expeditions against nations with which the United States was at peace, and provide *304 material support to terrorists. See 18 U.S.C. §§ 960, 2339A, 2390 (2000). But Benkahla was not charged with that conspiracy. He had taken a trip to England in the summer of 1999, and, from there, had bought a ticket to Pakistan, where he traveled with a man called "Abdullah." According to the government, in August 1999 he crossed from Pakistan into Afghanistan and there attended a Lashkar-e-Taiba jihadist training camp, where he fired an AK-47 and a rocket-propelled grenade launcherconduct charged (since attending a Lashkar-e-Taiba jihadist training camp was not necessarily illegal at the time) as supplying services to the Taliban and using a firearm in furtherance of a crime of violence. See 50 U.S.C. § 1705 (2000); Exec. Order No. 13,129 (July 4, 1999), 31 C.F.R. § 545.204 (prohibiting transactions with the Taliban); 18 U.S.C. §§ 924(c)(1)(B)(ii), 3238 (2000). Benkahla was arrested in Saudi Arabia in 2003, where he had been studying Islamic law and traveling with Ahmed Omar Abu Ali, a friend from Dar al-Arqam and a member of al Qaeda (eventually convicted of conspiracy to assassinate the President of the United States, among other crimes). Ultimately, having waived his right to a jury trial, Benkahla appeared before the U.S. District Court for the Eastern District of Virginia for a bench trial in March 2004.
It was clear in the trial that Benkahla was drawn to violent jihad, had traveled to Pakistan in August 1999, and had cultivated relationships with various individuals connected to terrorist organizations and jihadist training. In its decision, the trial court indicated that it thought he had attended a jihadist camp somewhere, either in Pakistan or Afghanistan, and fired an AK-47 and rocket-propelled grenade launcher while there. The court stated that "[i]f the standard of proof for the government were by a preponderance of the evidence, I would be able to find this defendant guilty." But the nature of the charges required that the camp be located in Afghanistan and that Benkahla have provided some meaningful form of support to the Taliban while there. In the court's judgment, there simply was not enough evidence on those points to convict beyond a reasonable doubt.
Within a few weeks of his March 2004 acquittal, Benkahla was subpoenaed. The government had been unable to prove that he had attended a jihadist training camp in Afghanistan, but it was by no means convinced that he hadn't attended a jihadist training camp at all. Indeed, it was still investigating such camps, the individuals who facilitated training at them, and several militants associated with Dar al-Arqam. Specifically, the government had convened two grand juries to investigate violations of 18 U.S.C. §§ 2339A and 2339B, which concern the provision of material support to terrorists and terrorist organizations. Thus over the next few months, the government compelled Benkahla to testify before each of the grand juries and to meet with the FBI several times in ancillary proceedings, with immunity from criminal prosecution for truthful testimony.
The questions throughout the proceedings focused anew on whether Benkahla had attended a jihadist training camp during that August 1999 trip. But they no longer centered on the camp's location, and the government took the approach of asking about the camp in the disjunctive (as in "Did you participate in any training... during your trip to Pakistan or Afghanistan in the summer of 1999?"). The questions also concerned the individuals with whom Benkahla had communicated in the course of exploring violent jihad and planning the 1999 trip abroad. For his part, Benkahla consistently denied attending any such camp anywhere, or knowing anything substantial about the individuals.
*305 The proceedings ended in November 2004. A little over a year later, in 2006, the government indicted Benkahla for making false material declarations to the two grand juries, 18 U.S.C. § 1623 (2000), obstructing justice on account of the false declarations, 18 U.S.C. § 1503 (2000), and making false material statements to the FBI, 18 U.S.C. § 1001(a) (2000). Specifically, Benkahla stood accused of a set of false denials: that he had participated in a jihadist training camp somewhere in August 1999; that he had handled weapons while there and observed others doing the same; and that he knew about the various people he had communicated with about training for jihad (such as "Abdullah," Ali Al-Timimi, and other persons of interest to the FBI in terrorism-related investigations). Benkahla moved to dismiss on collateral estoppel grounds and lost, and the case went to a jury in January and February 2007.
The jury trial lasted four days and included a good deal of background testimony on terrorism and violent jihad worldwide from the government's expert witness, Evan Kohlmann, as well as further testimony on the subject from an FBI agent working on Benkahla's case, Sarah Linden. After a day-and-a-half of deliberation, the jury convicted Benkahla on all counts, though it acquitted him of certain particular allegations in its special verdict form (each count alleged multiple falsehoods). Post-trial motions also led the court to strike Count II (concerning whether Benkahla handled weapons at the camp) for being based on uncorroborated admissions. Benkahla's convictions under Counts I, III, and IVfalse declarations to the grand jury, obstruction of justice by virtue of the false declarations, and false statements to the FBIstood.
At Benkahla's sentencing, the pivotal issue was whether to apply the terrorism enhancement of U.S. Sentencing Guidelines Manual § 3A1 .4 (2007): "If the offense is a felony that involved, or was intended to promote, a federal crime of terrorism," increase the offense level to at least a 32 (and by no less than 12 levels) and the criminal history category to a Category VI (the maximum). According to Application Note 2, "[f]or purposes of this guideline, an offense that involved ... obstructing an investigation of a federal crime of terrorism, shall be considered to have involved, or to have been intended to promote, that federal crime of terrorism." In Benkahla's case, applying the enhancement meant a Guidelines sentence over six times longer than he otherwise would getjumping from a range of 33 to 41 months to a range of 210 to 262 months.
The court held that Benkahla qualified for the enhancement. See United States v. Benkahla, 501 F.Supp.2d 748, 751-57 (E.D.Va.2007). First, the investigations Benkahla obstructed concerned "federal crimes of terrorism" under the statutory definition of the term. See 18 U.S.C. § 2332b(g)(5) (2006). The investigations concerned offenses enumerated in § 2332b(g)(5), namely the provision of material support to terrorists and terrorist groups. And the investigations were sufficiently particular to satisfy § 2332b(g)(5), being oriented to a set of people (particularly Abu Ali and Al-Timimi) who would ultimately be prosecuted. Furthermore, the court thought Benkahla's false statements had genuinely impeded the government's investigation, preventing it from finding out about Lashkar-e-Taiba training camps or uncovering the true identities of Benkahla's correspondents.
Benkahla's Guidelines range was thus 210 to 262 months. But the court thought the case called for a downward departure *306 under § 4A1.3 or (in the alternative) a variance under 18 U.S.C. § 3553(a). "Sabri Benkahla is not a terrorist," the court stated. Benkahla, 501 F.Supp.2d at 759. He "has not committed any other criminal acts" and his likelihood of doing so upon release is "infinitesimal." Id. Also, Benkahla's former co-defendants, the other ten members of the Dar al-Arqam paintball group, had received lesser sentences for what were more dangerous and more violent offenses, a disparity the court found "staggering." Id. at 762. The court thus treated Benkahla as having a Category I criminal history and sentenced him to 121 months.
Benkahla has appealed chiefly three issues: whether his second prosecution violated principles of collateral estoppel; whether the trial court admitted irrelevant or unduly prejudicial evidence concerning radical Islamic terrorism; and whether the court improperly applied the terrorism enhancement of Sentencing Guideline § 3A1.4. A fourth issueBenkahla's claim that the government's evidence was insufficient as a matter of lawreceived only passing attention in the briefs and argument, and we find it unpersuasive.[1] For its part, the government takes issue with the district court's characterization of Benkahla and calls its calculation of his criminal history category "baseless" and "mystifying," but, on the grounds that the sentence imposed "does not constitute an abuse of discretion" when treated as a variance, filed no cross-appeal. Appellee's Br. 62.

II.
We begin with Benkahla's collateral estoppel claim. Collateral estoppel, in the criminal context, is one part of the Fifth Amendment's guarantee against double jeopardy. The guarantee as a whole prohibits twice prosecuting or punishing a person for the same offense. See United States v. Dixon, 509 U.S. 688, 695-96, 113 S.Ct. 2849, 125 L.Ed.2d 556 (1993). Its collateral estoppel strand "means simply that when an issue of ultimate fact has once been determined by a valid and final judgment, that issue cannot again be litigated between the same parties in any future lawsuit." Ashe v. Swenson, 397 U.S. 436, 443, 90 S.Ct. 1189, 25 L.Ed.2d 469 (1970). Benkahla argues that, since he was acquitted of attending a jihadist training camp in Afghanistan, he could not afterwards be prosecuted for falsely denying that he attended such a camp in (as the government framed it) "Pakistan or Afghanistan."
Benkahla's claim arises in an area of legal tension. On the one hand, there is some potential for abuse in the government's procedure of acquittal, questioning on matters related to the acquittal, and second prosecution for some form of perjury. Accusing someone of a false denial does indeed assume the truth of the matter denied, and prosecutors frustrated at an acquittal should not lightly be able to take a second bite at the apple by bringing perjury charges afterwards. Nor may the device of the "or" be used to create a spurious appearance of dissimilarity between prosecutions with a common core.
On the other hand, a defendant does not win with acquittal a license to commit perjury. Cf. United States v. Dunnigan, 507 U.S. 87, 96, 113 S.Ct. 1111, *307 122 L.Ed.2d 445 (1993) ("[A] defendant's right to testify does not include a right to commit perjury."); Bryson v. United States, 396 U.S. 64, 72, 90 S.Ct. 355, 24 L.Ed.2d 264 (1969) ("Our legal system provides methods for challenging the Government's right to ask questionslying is not one of them."). Law enforcement is entitled to keep investigating a criminal enterprise even after one defendant is acquitted, and if that defendant is presented with a subpoena and cloaked with the dual protections of court-ordered immunity and the guarantee against double jeopardy, he may well be required to admit the very conduct he successfully denied at trial.
Given this tension, a mechanical approach to the collateral estoppel rule will not doand the law has never required one. "[T]he rule of collateral estoppel in criminal cases is not to be applied with the hypertechnical and archaic approach of a 19th century pleading book, but with realism and rationality.... The inquiry must be set in a practical frame and viewed with an eye to all the circumstances of the proceedings." Ashe, 397 U.S. at 444, 90 S.Ct. 1189 (quotation omitted). A practical approach means closely examining the record of both trials to discern whether an "issue of ultimate fact" resolved in the first was indeed reopened in the second. Id. at 443, 90 S.Ct. 1189. Thus our circuit has asked whether the two trials involved an "identical" issue "necessarily adjudicated" in the first, United States v. Nash, 447 F.2d 1382, 1386 (4th Cir.1971); or whether an "identical" issue was "actually" and "necessarily decided" in the first trial, after "a full and fair opportunity to litigate" and a "final and valid" judgment, United States v. Fiel, 35 F.3d 997, 1006 (1994); or again whether "certain facts were necessarily determined in the first trial" that "constituted ultimate issues" in the second, United States v. Yearwood, 518 F.3d 220, 229 (4th Cir.2008) (quotation omitted). Consistent throughout these cases over nearly forty years is the detailed, trial-specific look at the factual issues asked and answered at each prosecution.
Benkahla's first prosecution was a bench trial that concluded with a full judicial explanation of the verdicta welcome thing, for a good deal of the law in this area stems from the mysteries of the general jury verdict. See, e.g., United States v. Ruhbayan, 325 F.3d 197, 203 (4th Cir. 2003) ("[D]oubt as to what was decided by a prior judgment should be resolved against using it as an estoppel." (quotation omitted)). There is no mystery as to why the court acquitted in the first proceedings. What disturbed it was a lack of evidence showing that the jihadist training camp was in Afghanistan rather than Pakistan, and, if it was in Afghanistan, that Benkahla provided any serious form of support to the Taliban while there. "I would find ... that at some point, Mr. Benkahla has fired an automatic AK-47 and RPG" while abroad, the court said. But much of the evidence "is equally consistent in my view with going to a training camp in Pakistan as it would be to go to Afghanistan." The question was nonetheless close; the court stated that it "would be able to find this defendant guilty" on a preponderance of the evidence standard. But the charges brought required a high degree of certainty that the camp was located in Afghanistan and that by attending Benkahla was actually fighting or preparing to fight for the Taliban. At the factual heart of Benkahla's acquittal was a measure of uncertainty about those matters. In no way did the court's decision turn on doubt about whether Benkahla attended a jihadist training camp somewhere.
The second prosecution, Benkahla argues, once again put whether he attended *308 a jihadist training camp in Afghanistan at issue. Asking the second jury whether he falsely denied attending a camp in "Pakistan or Afghanistan," he claims, invited conviction on a thinly veiled forbidden ground. But this argument mistakes the factual issues at stake in both trials. An analogy helps: One may be quite uncertain whether a man who always wears a blue or grey suit to work wore the blue one on Mondaybut quite certain that he wore one or the other. In the same sense, the charges in the first trial required that the court determine with some certainty whether Benkahla attended a camp in Afghanistan and fought for the Taliban while there (the equivalent of the "blue suit on Monday" question). It could not. But the court was confident that Benkahla had attended a camp somewhere in the "Pakistan or Afghanistan" collectiveand what mattered factually in the second trial, given the new charges, was that collective. Indeed, the form the government's questioning took, asking about "Pakistan or Afghanistan," was not in this context a device with which to evade a matter already decided. It was exactly the question suggested by the first acquittal, a question to which both the first court and the second jury were prepared to answer "Yes."
On that note, it is worth observing that the investigations in which Benkahla was interviewed and the questions he was asked show no sign of having been manufactured for the sake of a second prosecution. Given the character of the first court's acquittal, the government had every right to think Benkahla had attended a jihadist training camp somewhere (it would have been anomalous for it to have thought otherwise), and, for the best of reasons, the government was still investigating those camps and the people involved in them. Indeed, the two grand juries to which Benkahla testified were convened to investigate violations of 18 U.S.C. §§ 2339A and 2339B, which concern the provision of support to terrorists and terrorist organizations. And those investigations or connected ones eventuated in indictmentsincluding ones for serious terrorists like Al-Timimi (whom Benkahla had been questioned about, leading to one of the grounds for Benkahla's false statements conviction), Abu Ali, and Chandia. It was legitimate to ask Benkahla, even post-acquittal, about his jihadist training in Pakistan or Afghanistan, and it was legitimate to prosecute him when he spoke falsely about it.
Thus the issue of ultimate fact in Benkahla's two prosecutions was distinct, and collateral estoppel presents no bar to the second.

III.
Benkahla next claims that the trial court admitted irrelevant and unduly prejudicial testimony and exhibits. He takes special exception to the government's terrorism expert, Evan Kohlmann, who runs a consulting company on Islamic terrorism and regularly testifies in terrorism-related prosecutions. Kohlmann's testimony was proffered mainly under Federal Rule of Evidence 702 to "assist the trier of fact to understand the evidence or to determine a fact in issue," and he gave background information on radical Islam and jihad generally rather than discussing Benkahla individually. As to that background, he spoke at length, beginning with the nature and history of the Taliban government in Afghanistan but ultimately touching on many of the individuals, ideologies, and organizations underlying the current conflict. Benkahla takes special note of a passage in which Kohlmann remarked that, for Osama bin Laden and al Qaeda, "Americans, no matter where they are on earth, whether they're civilian or military, *309 are considered to be a target. There are no innocent civilians." Benkahla argues that this sort of testimony inflamed the jury and condemned him by association. It was too far afield from the issues in his case to be relevant under Rules 401, 402, and 702, he claims, and to the extent it was relevant, its probative value was substantially outweighed "by the danger of unfair prejudice, confusion of the issues, ... or needless presentation of cumulative evidence" under Rule 403.[2]
Benkahla makes a similar argument against the testimony and exhibits admitted to prove that his false statements were material to the government's investigationmateriality being an element of both the false declarations and false statements charges. See 18 U.S.C. §§ 1001(a), 1623 (2000); United States v. Sarihifard, 155 F.3d 301, 306 (4th Cir.1998) ("An essential element in both grand jury perjury and the crime of making false statements is materiality."). This testimony came mainly from Sarah Linden, an FBI agent working on Benkahla's case, who spoke at length about the set of investigations connected to Dar al-Arqam and the network of terrorists and terrorist organizations facilitating jihadist training abroad. The exhibits included several dozen videos, photographs, and documents (how many exactly is in dispute) in support of Linden's testimony. Benkahla argues that this evidence went well beyond what was necessary to establish materiality and became "a vehicle for placing irrelevant and prejudicial statements and events before the jury." Appellant's Br. 41. He particularly objects to a taped confession from Abu Ali that was played for the jury.
Judgments of evidentiary relevance and prejudice are fundamentally a matter of trial management, for "[t]rial judges are much closer to the pulse of a trial than we can ever be and broad discretion is necessarily accorded them." United States v. Simpson, 910 F.2d 154, 157 (4th Cir.1990) (quotation omitted). The standard of review therefore counsels deference to the discretion of trial courts: "In a criminal appeal, we will not vacate a conviction unless we find that the district court judge acted arbitrarily or irrationally" in admitting evidence. United States v. Ham, 998 F.2d 1247, 1252 (4th Cir. 1993); see also United States v. Udeozor, 515 F.3d 260, 265 (4th Cir.2008) (requiring "extraordinary circumstances" where discretion was "plainly abused") (quotation omitted).
For Kohlmann, the relevance inquiry turns on Rule 702: His testimony had to "assist the trier of fact to understand the evidence or to determine a fact in issue." See Daubert v. Merrell Dow Pharm., Inc., 509 U.S. 579, 591, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993). The trial court could fairly conclude that it did. The evidence in this case was complicated, touching by necessity on a wide variety of ideas, terms, people, and organizations connected to radical Islam. The indictment alone refers to "jihad"; "jihad training camp[s]" and the "organizations" participating in them; "Lashkar-e-Taiba"; "other jihad organizations" espousing violence toward "the United States and countries with whom the United States was at peace"; "the Taliban"; "the territory of Afghanistan controlled by the Taliban"; "the territory ... across the border in Pakistan"; "the area north of Peshawar in Pakistan"; "lectures at Dar Al-Arqam"; the Arabic word "salaam"; and men *310 named (or identified as) "Haroon," "Myunus," "Abdullah," "Ibrahim Buisir," "Muhammad Siddique," "Abdel Atti Al Bakai," "Emara Al Bashir Binkaid," and "Ali Al-Timimi." Of course, the evidence required to evaluate the indictment involved a broader frame of reference still. In these circumstances, the trial judge could well conclude that lengthy testimony about various aspects of radical Islam was appropriate, and indeed necessary, for the jury "to understand the evidence" and "determine [the] fact[s]."
For Linden and the exhibits that supported her testimony, the relevance inquiry turns on materiality under 18 U.S.C. §§ 1001(a) and 1623. A statement is "material," the Supreme Court has held, "if it has a natural tendency to influence, or was capable of influencing, the decision of the decisionmaking body to which it was addressed." Kungys v. United States, 485 U.S. 759, 770, 108 S.Ct. 1537, 99 L.Ed.2d 839 (1988). Thus it was Linden's task, with the help of the exhibits, to "show[ ] a nexus between the false statements and the scope of the grand jury's [and FBI's] investigation," and she had some leeway in doing so: "Given the wide-ranging investigative function of the grand jury, the materiality of any line of inquiry pursued by a grand jury must be broadly construed." United States v. Farnham, 791 F.2d 331, 333-34 (4th Cir. 1986) (citation omitted). Here, Linden essentially testified to the FBI's and grand juries' investigations of jihadist camps abroad, the people who facilitate training at them, and the militants associated with Dar al-Arqam. These are exactly the subjects of Benkahla's alleged false denials and false statements. The relevance is clear.
Thus we come to the Rule 403 heart of Benkahla's argument against the witnesses and exhibits. At base, his contention is that the subject of terrorism arouses so much passion these days that its prominent presence at his trial put him at risk of a jury "excited to irrational behavior." Udeozor, 515 F.3d at 264 (quotation omitted). He cites as analogy United States v. Ham, where we reversed a trial judge under Rule 403 for admitting evidence of child molestation in a fraud case. 998 F.2d at 1251-54.
Undoubtedly, some of what Kohlmann and Linden had to say, and some of what the exhibits showed, was alarming. Undoubtedly the evidence's scope was wide. But the jury in this case acquitted Benkahla of several alleged falsehoods not exactly the mark of irrational fervor. Cf. United States v. Chandia, 514 F.3d 365, 375 (4th Cir.2008) (finding the claim that evidence of terrorism spurred an emotional conviction "discredited to a significant extent by the jury's rejection of one of the four counts charged"). And more importantly, Rule 403 is not an injunction to exclude prejudicial evidence but a mandate, entrusted mainly to the trial court, to weigh prejudice against probative value. See United States v. Hammoud, 381 F.3d 316, 341 (4th Cir.2004) ("The mere fact that the evidence will damage the defendant's case is not enoughthe evidence must be unfairly prejudicial, and the unfair prejudice must substantially outweigh the probative value of the evidence." (quotation omitted, emphasis in original)), vacated on unrelated grounds, 543 U.S. 1097, 125 S.Ct. 1051, 160 L.Ed.2d 997 (2005). In this case, the relevance of the challenged evidence cannot be doubted, and the same qualities that made it relevant gave it a probative value that the trial judge could fairly think outweighed its prejudicial risk. To reverse that judge for an abuse of discretion would betray too much distrust of the ability of the adversary process to *311 reach just results when the evidence on both sides is in.
The trial judge managed the proceedings in this case with care and skill, and we see no abuse of discretion in the testimony and exhibits admitted.

IV.
Finally, Benkahla claims that the district court should not in his case have applied the terrorism enhancement of U.S. Sentencing Guidelines Manual § 3A1.4 (2007). Under Gall v. United States, "a district court should begin all sentencing proceedings by correctly calculating the applicable Guidelines range" and then decide whether "an outside-Guidelines sentence is warranted." 552 U.S. ___, 128 S.Ct. 586, 596-97, 169 L.Ed.2d 445 (2007). The appellate court must then "review the sentence under an abuse-of-discretion standard" with an eye toward both "procedural" and "substantive reasonableness." Id. at 597. Here, Benkahla argues that the terrorism enhancement led the district court to the wrong starting point.
The Guideline, adopted November 1, 1995, states:
§ 3A1.4. Terrorism
(a) If the offense is a felony that involved, or was intended to promote, a federal crime of terrorism, increase by 12 levels; but if the resulting offense level is less than level 32, increase to level 32.
(b) In each such case, the defendant's criminal history category from Chapter Four (Criminal History and Criminal Livelihood) shall be Category VI.
Two relevant application notes follow:
1. "Federal Crime of Terrorism" Defined. For purposes of this guideline, "federal crime of terrorism" has the meaning given that term in 18 U.S.C. § 2332b(g)(5).
2. Harboring, Concealing, and Obstruction Offenses.For purposes of this guideline, an offense that involved... obstructing an investigation of a federal crime of terrorism, shall be considered to have involved, or to have been intended to promote, that federal crime of terrorism.
Application Note 1 took shape in response to the Antiterrorism and Effective Death Penalty Act of 1996. See U.S. Sentencing Guidelines Manual app. C, amend. 539 (2007). The statute it refers to, 18 U.S.C. § 2332b(g)(5) (2006), states:
[T]he term "Federal crime of terrorism" means an offense that
(A) is calculated to influence or affect the conduct of government by intimidation or coercion, or to retaliate against government conduct; and
(B) is a violation of ... 2339A (relating to providing material support to terrorists), 2339B (relating to providing material support to terrorist organizations)....
Application Note 2 was passed in response to the USA Patriot Act of 2001 "to clarify that § 3A1.4 may apply in the case of offenses that occurred after the commission of the federal crime of terrorism." U.S. Sentencing Guidelines Manual app. C, amdt. 637 (2007).
In light of these provisions, applying § 3A1.4 in Benkahla's case seems straightforward. He was convicted of obstruction offenses. Application Note 2 says that obstruction offenses qualify for the enhancement so long as the thing obstructed qualifies as an investigation into a "federal crime of terrorism." Application Note 1 tells us that the term "federal crime of terrorism" has a statutory definition. The statute, 18 U.S.C. § 2332b(g)(5), defines it in two parts. Benkahla obstructed *312 a grand jury investigation into violations of §§ 2339A and 2339B, which satisfies the second part. And the violations involved jihadist camps training people to fight the governments of India, Russia, and the United States, which satisfies the first.
Nonetheless, Benkahla presents several arguments against applying the enhancement to him. First, with reference to the United States v. Booker line of cases, Benkahla contends that applying the enhancement depended on finding as a matter of fact that he "actually obstructed an investigation of a federal crime of terrorism." Appellant's Br. 22. The trial court, not the jury, found that fact. His sentence plainly would not survive reasonableness review, he argues, but for that fact. And therein lies the problem, for if a sentence depends on judge-found facts to survive reasonableness review, he contends, it violates the Sixth Amendment.
This argument is too creative for the law as it stands. Sentencing judges may find facts relevant to determining a Guidelines range by a preponderance of the evidence, so long as that Guidelines sentence is treated as advisory and falls within the statutory maximum authorized by the jury's verdict. Indeed, "many individual Guidelines apply higher sentences in the presence of special facts" and "[i]n many cases, the sentencing judge, not the jury, will determine the existence of those facts." Rita v. United States, 551 U.S. ___, 127 S.Ct. 2456, 2465, 168 L.Ed.2d 203 (2007). That "does not violate the Sixth Amendment," however, because "[a]s far as the law is concerned, the judge could disregard the Guidelines and apply the same sentence ... in the absence of the special facts." Id. at 2465-66; see also United States v. Battle, 499 F.3d 315, 322-23 (4th Cir.2007) ("When applying the Guidelines in an advisory manner, the district court can make factual findings using the preponderance of the evidence standard."). The point is thus that the Guidelines must be advisory, not that judges may find no facts. Here, in a case in which the district court slashed the defendant's Guidelines sentence in half, no one could doubt that the Guidelines were advisory. There is no Sixth Amendment violation.
Benkahla next argues that Application Note 2 contradicts Guideline § 3A1.4, and we should therefore hold Application Note 2 invalid. See Stinson v. United States, 508 U.S. 36, 43, 113 S.Ct. 1913, 123 L.Ed.2d 598 (1993) ("[If] commentary and the guideline it interprets are inconsistent in that following one will result in violating the dictates of the other, the Sentencing Reform Act itself commands compliance with the guideline."). The two must contradict, he claims, because a person can obstruct an investigation into a federal crime of terrorism without necessarily committing, in § 3A1.4's words, "a felony that involved, or was intended to promote, a federal crime of terrorism."
In general, we have a duty to harmonize Guidelines and commentary. See Stinson, 508 U.S. at 44-46, 113 S.Ct. 1913; United States v. Pedragh, 225 F.3d 240, 244-45 (2d Cir.2000). One might wonder, in the abstract, whether obstructing an investigation into a federal crime of terrorism necessarily "involve[s]" a federal crime of terrorism, but plainly one could think that it does, and Application Note 2 decides the matter. Indeed, the language of Application Note 2 is identical in all material respects to the language of § 3A1.4 itself. There is no inconsistency of any kind.
Finally, Benkahla claims that Application Note 2, even if constitutional and consistent with § 3A1.4, does not apply to him. The argument rests less on Application *313 Note 2 itself than on two conclusions of law the district court came to in the course of its sentencing decision. The court reasoned that to qualify for the enhancement, Benkahla had to actually obstruct a terrorism investigation (not just attempt to obstruct one), and that the investigation itself had to be specific and targeted, oriented to particular terrorism offenses rather than to general intelligence-gathering. See United States v. Benkahla, 501 F.Supp.2d 748, 751-54, 756 (E.D.Va.2007). Benkahla echoes these conclusions of law, but contrary to the district court, insists that the investigation in his case was too general, and his obstruction too ineffectual, for the enhancement to apply.
There is no need to review the district court's legal conclusions. Whether those conclusions are correct or incorrect, the court's factual findings clearly support applying the enhancement. See Gall, 128 S.Ct. at 597 (holding that appellate courts review district courts' sentencing facts for clear error). All the evidence indicates that Benkahla attended a jihadist training camp abroad, was acquainted with a net-work of people involved in violent jihad and terrorism, and lied about both. Moreover, the sentencing judge, after a meticulous review of the evidence, concluded that "[i]n the same investigation in which Defendant was questioned, eight individuals to whom he was connected went to foreign jihad training camps and one was convicted of soliciting treason to fight against the United States." Benkahla, 501 F.Supp.2d at 755. Testimony secured from some of those individuals led to "convictions for specific terrorist acts in Australia, France, and England." Id. And "[m]ost notable... at the time Defendant was questioned, Ali Al-Timimi and Ahmed Abu-Ali were yet to be indicted." Id. Benkahla "had relationships with both." Id. The district court also found that Benkahla's falsehoods not only delayed some parts of the investigation, but wholly frustrated others. When Benkahla was questioned, "the Government did not know the details about Lashkar-e-Taiba training camps, training techniques, curriculum, and locations," the court wrote. Id. at 757. In addition, "because of Defendant's false or intentionally misleading answers, the Government still does not know the identity or whereabouts of the persons about whom Defendant was questioned, their involvement with Lashkar-e-Taiba, and their role in aiding persons to obtain jihad training." Id.
This factual amplitude stands in contrast to the situation in United States v. Chandia, 514 F.3d 365 (4th Cir.2008), on which Benkahla relies. There we vacated and remanded an application of the terrorism enhancement because the district court "appeared to assume (erroneously) that the enhancement automatically applies to a material support conviction" and so "did not make any factual findings related to the intent element." Id. at 376. Here, the district court made extensive factual findings and the terrorism enhancement is doing just what it ought to do: Punishing more harshly than other criminals those whose wrongs served an end more terrible than other crimes.
We therefore approve the application of the terrorism enhancement to Benkahla. As the government has not appealed the district's court's downward departure or variance, we do not address it.

V.
For the foregoing reasons, the judgment is in all respects
AFFIRMED.
NOTES
[1]  Only one part of Benkahla's insufficiency of evidence argument is substantive: the claim that the uncorroborated admissions that led the trial court to strike Count 2 were repeated in Counts 1 and 4. In our judgment, however, the admissions were corroborated, and, in any case, Counts 1 and 4 rest mainly on other, unrelated false denials and false statements.
[2]  Benkahla also attacks Kohlmann's qualifications as an expert, but those qualifications were obviously substantial and the district court acted well within its discretion in determining that they were sufficient.
