BROWN, Circuit Judge:

In the late 1990s, Jamal Kifafi, an erstwhile Hilton employee and participant in Hilton's retirement plan ("Plan"), noticed a problem with the benefits calculation on his statement of benefits. Concluding the Plan violated the Employee Retirement Income Security Act ("ERISA") in a number of ways, he sued. Now, almost fifteen years and twelve district court opinions later, we join the fray and force the parties a little closer to final resolution of their dispute. The district court found that the Plan was impermissibly backloaded and that Hilton failed to calculate participants' vesting credit properly, and it imposed relief accordingly. We affirm because the district court's handling of the case was well within its discretion.

I

ERISA guarantees neither a particular benefit nor a particular method for calculating the benefit. Alessi v. Raybestos-Manhattan, Inc., 451 U.S. 504, 511-12, 101 S.Ct. 1895, 68 L.Ed.2d 402 (1981). Yet employers do not have carte blanche.

ERISA defines the universe of permissible benefit-accrual rates by requiring defined benefit plans to satisfy one — and only one — of three rules designed to prevent backloading, which occurs when a plan awards benefits to employees in later years of service at a rate disproportionately higher than the rate for employees in earlier years of service. See 29 U.S.C. § 1054; H.R. REP. NO. 93-807, at 21 (1974), reprinted in 1974 U.S.C.C.A.N. 4670, 4688 (defining "backloading"). The three rules contained in this antibackloading provision are known as the 3% rule, the fractional rule, and the 133 1/3% rule. The 3% rule "prescribes a minimum percentage of the total retirement benefit that must be accrued in any given year." Alessi, 451 U.S. at 512 n. 9, 101 S.Ct. 1895. The fractional rule is "essentially a pro rata rule under which in any given year, the employee's accrued benefit is proportionate to the number of years of service as compared with the number of total years of service appropriate to the normal retirement age." Id. The 133 1/3% rule, meanwhile, "permits the use of any accrual formula as long as the accrual rate for a given year of service does not vary beyond a specified percentage from the accrual rate of any other year under the plan." Id. Specifically, the "rate of benefit accrual in any future year may not be more than one-third greater than the rate in the current year." Lonecke v. Citigroup Pension Plan, 584 F.3d 457, 464 (2d Cir.2009).

ERISA circumscribes pension plans in other ways as well, such as by setting minimum vesting standards. Where accrual relates to "the amount of the benefit to which the employee is entitled," vesting relates to when an employee has a right to the accrued benefit. Holt v. Winpisinger, 811 F.2d 1532, 1536 (D.C.Cir. 1987) (quoting Stewart v. Natl Shopmen Pension Fund, 730 F.2d 1552, 1562 (D.C.Cir.1984)) (internal quotation marks omitted). Benefit accrual and vesting are not coextensive concepts, so an employee might "earn credit toward vesting without accumulating any pension benefits." Id. at 1537. Because vesting is tied to length of employment, this would happen if an employee works without participating in the plan ("nonparticipating service"), although the benefits do not actually vest until the employee begins participating in the plan. ERISA generally requires employers to count all of an employee's years of service when calculating vesting credit, including years completed before the employee began participating in the plan. See 29 U.S.C. § 1053; Holt, 811 F.2d at 1536-37.

This appeal implicates both rules.1 In his complaint, Kifafi alleged that the Plan's benefit accrual formula was impermissibly backloaded and that Hilton, the Plan sponsor and administrator, violated both ERISA and the Plan by failing to credit certain years of service when calculating employees' vesting credit.

Unfortunately, the parties were not content to let the district court decide (relatively) straightforward issues of law. Instead, as the district court later put it, they "shifted their burden to the Court to determine which facts are in dispute between the parties" and they "repeatedly shifted their arguments such that the Court has consistently been presented with moving targets." Kifafi v. Hilton Hotels Ret. Plan, 616 F.Supp.2d 7, 22 (D.D.C.2009) ("Kifafi I"). This included changing the facts of the case.

Before Kifafi filed suit,2 the Plan calculated normal retirement benefits as a function of participants' compensation and years of service, less the participant's "integrated benefits," that is, benefits payable to that participant under another pension plan or government-sponsored system to which Hilton contributed (including half of the participant's social security benefits). This calculation guaranteed, at a minimum, 2% of the participant's average monthly compensation multiplied by the participant's years of service up to twenty-five years, plus 0.5% of the average monthly compensation for each year after twenty-five years, minus the integrated benefits offset. The Plan also included the following language in a separate article titled "Limitation on Benefits and Payments": "The method of computing a Participant's accrued benefit under the provisions of Article IV is intended to satisfy the requirements of the 133-1/3 rule provided in Section 411(b)(1)(B) of the [Internal Revenue] Code."

After Kifafi moved for class certification (but before the district court ruled on the motion), Hilton amended the Plan ("1999 Plan") to eliminate "any controversy regarding the proriety [sic] of the rate of benefit accruals under the Plan." The 1999 Plan modified the benefit accrual formula using a "greater of" approach: Plan participants would receive the greater of the benefit determined under the old Plan or the benefit determined under the 1999 Plan's modified accrual formula, which, as the IRS subsequently determined, satisfied the fractional rule. The 1999 Plan also made two other retroactive changes: first, it decreased the relevant percentage of employees' average monthly compensation during the first twenty-five years of service from 2% to 1.33%; and second, it increased the social security offset.

The district court ultimately certified Kifafi's proposed "benefit-accrual class" for the backloading claim — all former and current Hilton employees whose pension benefits "have been, or will be, reduced" because of the backloading — but not Kifafi's proposed class for the vesting claim. Four years later, after the parties completed discovery, Kifafi renewed his motion to certify the vesting claim and sought to include as class representatives three class members he hoped would cure any deficiencies in the original class certification motion. This time, the court granted the certification motion (though it denied the motion to intervene), certifying for class treatment Kifafi's claim that Hilton "failed to credit employees with years of union service for vesting purposes." The court ultimately granted summary judgment to Kifafi on both claims and, eventually, ordered Hilton (1) to amend the Plan's benefit accrual formula by capping the social security offset, thereby bringing the Plan into retroactive compliance with the 133 1/3% rule, and (2) to administer a claim procedure for crediting participants' years of union service for vesting purposes. Both parties appealed.

II

As an initial matter, Hilton claims the 1999 Plan mooted Kifafi's backloading claim, an argument this Court assesses de novo. Del Monte Fresh Produce Co. v. United States, 570 F.3d 316, 321 (D.C.Cir.2009).3 As most of the issues in this case presuppose a live controversy over the backloading claims, Hilton's victory on this point would resolve those other issues quite tidily. Unfortunately for Hilton, its arguments do not persuade us. See Honeywell Int'l, Inc. v. NRC, 628 F.3d 568, 576 (D.C.Cir.2010) (explaining that the party asserting mootness carries a "heavy burden").

Mootness doctrine "limits federal courts to deciding actual, ongoing controversies." Am. Bar Ass'n v. FTC, 636 F.3d 641, 645 (D.C.Cir.2011) (internal quotation marks omitted). A case is moot when the court's decision "will neither presently affect the parties' rights nor have a more-than-speculative chance of affecting them in the future." Id. (internal quotation marks omitted). Accordingly, a court must dismiss a properly-brought case if it is subsequently rendered moot. Honeywell Int'l, 628 F.3d at 576. Hilton marshals three arguments to show that happened here: the district court relied on bad law, the 1999 Plan did not violate ERISA's "anti-cutback" provision, and the backloading was eliminated and would not likely recur.

There are problems with each argument, but at bottom, the issue comes down to whether a party can deprive a court of jurisdiction with a wave of its hand.4 It cannot, no matter how contritely it apologizes for the conduct giving rise to the litigation. "It is well settled that the voluntary cessation of allegedly unlawful conduct does not moot a case in which the legality of that conduct is challenged." Christian Legal Soc'y Chapter of the Univ. of California, Hastings Coll. of the Law v. Martinez, ___ U.S. ___, 130 S.Ct. 2971, 3009 n. 3, 177 L.Ed.2d 838 (2010) (Alito, J., dissenting). A defendant's voluntary cessation of allegedly unlawful conduct moots a case only if (1) "there is no reasonable expectation ... that the alleged violation will recur," and (2) "interim relief or events have completely and irrevocably eradicated the effects of the alleged violation." Am. Bar Ass'n, 636 F.3d at 648 (quoting HARRY T. EDWARDS & LINDA A. ELLIOTT, FEDERAL STANDARDS OF REVIEW — REVIEW OF DISTRICT COURT DECISIONS AND AGENCY ACTIONS 114-15 (2007)) (internal quotation marks omitted).

Hilton's argument that the backloading will not recur boils down to its promise not to violate the anti-backloading provision. See Hilton Br. at 50-51 (arguing it would be "illogical," "irrational," and "absurd" to further violate the anti-backloading provision because doing so would subject Hilton to further litigation and might entail adverse tax consequences). This is insufficient. See United States v. Concentrated Phosphate Export Ass'n, 393 U.S. 199, 203, 89 S.Ct. 361, 21 L.Ed.2d 344 (1968) (finding insufficient "appellees' own statement that it would be uneconomical for them to engage in [the challenged activity]"). And even were it sufficient, Hilton has given us little reason to think its intentions are a good predictor of reality. When amending the Plan in 1999, for instance, Hilton flatly asserted its belief "that the Plan satisfied ERISA's benefit accrual requirements even without the amendment." See also Kifafi I, 616 F.Supp.2d at 28 (noting that Hilton "has insisted upon the legality of the challenged practices"). As the district court concluded — and Hilton now concedes — this assessment was wrong. Hilton may have made its erroneous statements in good faith, but that does little to reassure us that it is "absolutely clear" the backloading could not reasonably be expected to recur. Friends of the Earth, Inc. v. Laidlaw Envtl. Servs. (TOC), Inc., 528 U.S. 167, 189, 120 S.Ct. 693, 145 L.Ed.2d 610 (2000). This is a complicated area of law in which even the best-intentioned actors may yet do wrong.

Because Hilton cannot meet its burden of showing there is no reasonable likelihood of future backloading, we need not determine whether the 1999 Plan eradicated the effects of the backloading. Hilton's inability to show the first scuttles all need for the second. See Cnty. of L.A. v. Davis, 440 U.S. 625, 631, 99 S.Ct. 1379, 59 L.Ed.2d 642 (1979).

III

The parties challenge the district court's remedy and class certification from a number of angles. We review the district court's class certification decisions and its remedial decisions for abuse of discretion. See, e.g., Garcia v. Johanns, 444 F.3d 625, 631 (D.C.Cir.2006); SEC v. Banner Fund Int'l, 211 F.3d 602, 616 (D.C.Cir.2000). To determine if the district court applied the wrong legal standard or otherwise "misapprehended the underlying substantive law," Brayton v. Office of the U.S. Trade Representative, 641 F.3d 521, 524 (D.C.Cir.2011) (quoting Kickapoo Tribe of Indians of the Kickapoo Reservation in Kan. v. Babbitt, 43 F.3d 1491, 1497 (D.C.Cir.1995)) (internal quotation marks omitted), we consider whether the court "failed to consider a relevant factor" or "relied on an improper factor," as well as whether "the reasons given reasonably support the conclusion." Peyton v. DiMario, 287 F.3d 1121, 1126 (D.C.Cir. 2002).

A

Hilton first challenges the district court's general remedial approach to the backloading claim. In granting Kifafi summary judgment on the issue, the district court relied on the pre-1999 Plan's statement of intent and Hilton's representation of compliance with the 133 1/3% rule to the court and the IRS. Stating that Hilton was "required to comply with the accrual method it expressly selected," the court concluded that "the Plan's participants are entitled to receive the benefits they would have accrued had the Plan complied with the 133 1/3% rule." Kifafi I, 616 F.Supp.2d at 24. It therefore directed the parties to discuss, "at a minimum, the methodolog[y] that should be used to ... provide members of the `benefit-accrual class' with the benefits they would have accrued under the Plan's initial benefit accrual formula, amended only to bring it into compliance with the 133 1/3% rule." Hilton now argues that requiring the Plan to comply with the 133 1/3% rule in particular, rather than the anti-backloading provision generally, was an abuse of discretion. We disagree.

Whether a plan satisfies the anti-backloading provision — and which anti-backloading rule it satisfies — turns on the nature of the plan. In other words, the terms of the plan will or will not be amenable to analysis under any given rule as a matter of fact. See, e.g., Tomlinson v. El Paso Corp., 653 F.3d 1281, 1290 (10th Cir.2011) (noting party concession the plan "must qualify under the `133 1/3 percent' test if it qualifies at all"); Hurlic v. S. Cal. Gas Co., 539 F.3d 1024, 1033 n. 4 (9th Cir.2008) (disclaiming need to consider plan's ability to satisfy fractional or 3% rules because plan satisfies the 133 1/3% rule); Carollo v. Cement & Concrete Workers Dist. Council Pension Plan, 964 F.Supp. 677, 681 (E.D.N.Y.1997) (noting party agreement that "the 133 1/3% Rule is the only standard the Plan is capable of satisfying"). A plan must be measured not against the anti-backloading rule it says it is following, but against ERISA's general provision for plans to satisfy any one of the three anti-backloading rules. As Hilton points out, it would be absurd to find a plan that in fact satisfies one of the antibackloading rules to be backloaded because it "intends" to comply with a different rule.

Such is not the case here because the Plan did not satisfy any of the antibackloading rules, but that does not change the fact that the Plan's statement of intent is irrelevant to the backloading analysis. The 133 1/3% rule delimits only a range of possibility; to comply, plans must still set out the benefit accrual rate. See 29 U.S.C. § 1102(b)(4); Kennedy v. Plan Adm'r for DuPont Sav. & Inv. Plan, 555 U.S. 285, 300, 129 S.Ct. 865, 172 L.Ed.2d 662 (2009); Curtiss-Wright Corp. v. Schoonejongen, 514 U.S. 73, 83, 115 S.Ct. 1223, 131 L.Ed.2d 94 (1995). And once a plan sets out an accrual rate, the rate is directly testable against the various anti-backloading rules. Kifafi's claim that statements of intent are "clearly" enforceable terms therefore overstates the analogy between ERISA plans and contracts where intent is relevant either under the substantive law or to clarify ambiguity. See, e.g., In re Palmdale Hills Prop., LLC, 457 B.R. 29, 44-45 (9th Cir. BAP 2011) (invoking language of intent in order to determine parties' intent). Certainly, no one has claimed the accrual formula is ambiguous.

But this does not mean the district court abused its discretion. Once the court determined the Plan violated ERISA, it entered the world of equity. See 29 U.S.C. § 1132(a)(3); CIGNA Corp. v. Amara, ___ U.S. ___, 131 S.Ct. 1866, 1875, 1879-80, 179 L.Ed.2d 843 (2011) (concluding that 29 U.S.C. § 1132(a)(3) authorized district court's reformation of plan and injunction requiring the plan to pay corresponding benefits). And Hilton's premise that a plan's prospective compliance with ERISA's anti-backloading mandate circumscribes a court's remedial options in the face of past violations cannot be sustained. We see no reason why the court's remedy need be a perfect reflection of the legal violations supporting the remedy; the district court exercised a discretion informed by much more than just the ERISA violation. See Cobell v. Salazar, 573 F.3d 808, 813 (D.C.Cir.2009); Sheaffer v. Warehouse Emps. Union, Local No. 730, 408 F.2d 204, 206-07 (D.C.Cir.1969). Hilton misses this distinction.5 No doubt a plan may alter its accrual formula to comply with a different anti-backloading rule than the one it had hitherto complied with (assuming it does not violate ERISA in other ways); the IRS accordingly concluded the 1999 Plan satisfied the fractional rule, even though the pre-1999 Plan had failed to satisfy any of the anti-backloading rules. But the standard for prospective compliance does not ipso facto establish the adequacy of a retroactive amendment or court — imposed relief. To reduce the district court's remedy to nothing more than a demand that Hilton comply with any of the anti-backloading rules would — like evaluating a movie by analyzing a single frame — ignore too much.

First, the parties did not make things easy for the district court. After Kifafi filed suit, for example, Hilton amended the Plan into compliance with the anti-backloading provision, simultaneously making two changes to the benefit accrual formula unfavorable to Plan participants, and over the next few years, Hilton amended the Plan additional times in response to Kifafi's various legal claims. This, in addition to freezing the Plan's benefit accruals in 1996, which Kifafi's actuarial expert explained had a material impact on the Plan's ability to comply with the various anti-backloading rules. The court expressed its frustration when, in its summary judgment opinion, it referred to the parties' arguments as "moving targets." Kifafi I, 616 F.Supp.2d at 22. In short, more than a decade passed after Kifafi filed suit before the district court granted summary judgment, what the court referred to as "an extraordinary amount of judicial time."

Second, Hilton represented its compliance with the 133 1/3% rule to Kifafi, the IRS, and the district court. We do not, as Hilton charges, think this constitutes the predicate for estoppel. Rather, this suggests the Plan was more amenable to analysis under the 133 1/3% rule than under the fractional or 3% rules. See, e.g., Carrabba v. Randalls Food Mkts., Inc., 145 F.Supp.2d 763, 773 (N.D.Tex.2000) (requiring plan without accrual rate to satisfy 133 1/3% rule, which, the court concluded, "most appropriately recognizes the objectives of the [plan] in an ERISA context," based on the actual progression of benefits), aff'd, 252 F.3d 721 (5th Cir.2001) (per curiam) (calling the district court opinion "conscientious" and "well-reasoned"). Indeed, as Kifafi's actuarial expert attested in an affidavit, "The progression of the Plan's existing accrual rates lends itself to compliance with [the 133 1/3%] rule."

Read in this context, we understand the district court's remedial order to be an attempt to pin Hilton down, denying it the opportunity to avoid the consequences of its ERISA violations. The district court certainly used language suggesting it thought that as a matter of law Hilton could not comply with any anti-backloading rule other than the 133 1/3% rule. But like the court's reliance on the statement of intent, most such statements occurred in the context of the court's conclusion that the original Plan was backloaded under any of the anti-backloading rules. The court also stated, in the context of rejecting Hilton's mootness argument, that Plan participants are entitled to what they would have received had the Plan complied with the 133 1/3% rule. But it continued: "If this were not so, Hilton and all other employers that have unlawfully backloaded benefit accruals could simply `amend away' their ERISA violations." Kifafi I, 616 F.Supp.2d at 25. And as it emphasized in a later proceeding, the 1999 Plan's compliance with the fractional rule came at the expense of "substantial modifications to the benefits that would be paid to participants." Kifafi v. Hilton Hotels Ret. Plan, 736 F.Supp.2d 64, 73 (D.D.C.2010) ("Kifafi II"). This was enough to support its remedy. See, e.g., Shahriar v. Smith & Wollensky Rest. Grp., Inc., 659 F.3d 234, 251-52 (2d Cir.2011); In re Grand Jury Investigation, 545 F.3d 21, 26 (1st Cir.2008). Just as an employer would not remedy its failure to pay overtime by "retroactively revising the base rate of pay down from $10 per hour to $6.50 per hour and offering to multiply the reduced rate by the required `time and one-half,'" Kifafi Br. at 36, the district court did not abuse its discretion by requiring Hilton to provide a remedy it considered meaningful. See, e.g., Hecht Co. v. Bowles, 321 U.S. 321, 329, 64 S.Ct. 587, 88 L.Ed. 754 (1944) (tying the injunctive process to deterrence); H.R. REP. 101-386, at 433 (1989) (Conf. Rep.), reprinted in 1989 U.S.C.C.A.N. 3018, 3036 (stating congressional intent that courts fashion remedies for ERISA violations "that not only protect participants and beneficiaries but deter violations of the law as well").

B

Hilton next invokes the statute of limitations to argue the district court should have dismissed the claims of Plan participants who received benefits more than three years before Kifafi filed this suit. Although ERISA is not entirely silent with respect to statutes of limitations, see, e.g., 29 U.S.C. § 1113, there is no applicable limitations period for the type of claims Kifafi brought. See Kifafi I, 616 F.Supp.2d at 35 (noting Kifafi brought his class claims under 29 U.S.C. § 1132). We accordingly apply "the most closely analogous statute of limitations from the state in which the court sits." Connors v. Hallmark & Son Coal Co., 935 F.2d 336, 341 (D.C.Cir.1991). Fortunately, the parties agree on the appropriate limitations period (three years), as well as the appropriate standard for determining when the limitations period begins (the discovery rule). They disagree, however, about whether the district court properly applied these principles. We think it did.6

Statutes of limitations ordinarily begin running when "the factual and legal prerequisites for filing suit are in place." Norwest Bank Minn. Nat'l Ass'n v. FDIC, 312 F.3d 447, 451 (D.C.Cir.2002). It will not always be obvious when this happens. In such cases, and absent a contrary congressional directive, this Court applies the discovery rule, which provides that the statute of limitations begins when the plaintiff "discovers, or with due diligence should have discovered," the injury supporting the legal claim. Connors, 935 F.2d at 341, 343.

Hilton's sole argument is that its payment of backloaded benefits to Plan participants amounted to a repudiation of the participants' right to additional benefits, the implication being that the participants should have discovered from these payments that their benefits were backloaded. Courts have indeed held that repudiation can trigger ERISA's statute of limitations if it is clear and made known to the plan beneficiary. See, e.g., Thompson v. Ret. Plan for Emps. of S.C. Johnson & Son, Inc., 651 F.3d 600, 604 (7th Cir.2011); Miller v. Fortis Benefits Ins. Co., 475 F.3d 516, 520-21 (3d Cir.2007); Davenport v. Harry N. Abrams, Inc., 249 F.3d 130, 134-35 (2d Cir.2001); Union Pac. R. Co. v. Beckham, 138 F.3d 325, 330 (8th Cir.1998); see also Connors, 935 F.2d at 342 (noting consistency of discovery rule and "time of injury" rule). But the requirement that the repudiation be clear and made known to the plan beneficiaries is not an idle one. Whether repudiation may trigger the limitations period depends on what the prospective plaintiff should have understood from the miscalculated benefit payments. Where the miscalculated benefits comprise a single lump-sum payment, it might make sense to hold plan participants responsible for their failure to notice the miscalculation, although we do not decide the issue. See Thompson, 651 F.3d at 606 (holding that receipt of lump-sum distribution served as an "unequivocal repudiation of any entitlement to benefits beyond the account balance" because it was clear that "no additional benefits would be forthcoming"). The same might be true for miscalculated periodic payments. See Miller, 475 F.3d at 521-22 (explaining that beneficiaries ordinarily should know if a benefit award is improperly low). But if so, the miscalculation must still be such that the beneficiary should recognize it. See id. at 523 ("[T]he need for Miller to be vigilant was triggered only when his receipt of benefits alerted him that his award had been miscalculated."). Miller involved "a simple calculation of sixty percent of [the beneficiary's] salary," id. at 522, and if one thing in this case is clear, it is that Hilton's miscalculation was anything but simple. To catch the backloading, Plan participants would have needed to apply complex law to complex facts. If Hilton itself admits the Plan "did not appear to be backloaded," it makes no sense to ask the participants to navigate the complexity of ERISA's antibackloading provision immediately upon receipt of their first benefits payment. They are the parties least likely "to have a clear understanding of the terms of the pension plan and their application to [the] case." Novella v. Westchester Cnty., 661 F.3d 128, 146 (2d Cir.2011).

C

Having failed to circumscribe the scope of the district court's remedy by excluding participants who received benefits more than three years before Kifafi filed suit, Hilton tries to contain the remedial fallout by challenging the district court's determination that the Plan's retroactive compliance with the 133 1/3% rule should apply to participants who separated from service before the controversial statement of intent was added to the Plan in 1994 (with retroactive effect). This argument also fails. The district court determined that the Plan had never complied with the anti-backloading provision and that the benefits accrual formula did not substantially change in 1994, findings Hilton does not challenge. If the Plan violated the anti-backloading provision after 1994, and it violated the anti-backloading provision essentially the same way before 1994, then we see no reason to distinguish between pre- and post-1994 separation dates, particularly given our conclusion that the statement of intent is irrelevant to the backloading violation.

D

Kifafi, in turn, argues the district court improperly accepted Hilton's theory that participants can "outgrow" the backloading merely by participating in the Plan for a sufficiently long period of time. As Kifafi puts it,

Kifafi Br. at 56-57. According to Kifafi, the district court should have increased the early-year accrual rates without touching the later-year accrual rates. Translated into his hypothetical plan, this would mean offering a $7.50 rate of accrual for the first seven years while leaving everything else about the plan unchanged. We disagree.

Backloading is nothing more than the improper allocation of benefit accrual rates; the concept does not necessarily say anything about the amount of benefits participants ultimately accrue. Fixing a backloaded plan might entail increased benefits, but it need not. A participant in Kifafi's hypothetical plan, for instance, accrues $2,580 after twenty-five years of service. Amending the plan to comply with the 133 1/3% rule by offering $7.50 each month for the first seven years, as Kifafi suggests, would yield participants an additional $210 after twenty-five years of service. But the plan could satisfy the 133 1/3% rule while still yielding only $2,580; this would happen if, say, the plan offered $8 per month rate of accrual in the first seven years and (approximately) $8.83 in the next eighteen, or if it offered $8.60 every month for all twenty-five years. The district court did not abuse its discretion when it chose which of these counterfactual accrual formulas the backloading remedy should track.7 As Revenue Ruling 2008-7, 2008-7 I.R.B. 419, explains, if a plan becomes backloaded because of a decreased interest crediting rate, the plan could be amended into compliance with ERISA by increasing "the hypothetical pay credits at the earlier ages," reducing the credits at the higher ages, or a combination of both.

According to Kifafi, IRS regulations prohibit the district court's "average rate of accrual" methodology. See Treas. Reg. § 1.411(b)-1(b)(2)(iii) (Example 3) (explaining that a plan under which a participant accrues benefits at the rate of 2% for each of the first five years, 1% for each of the next five years, and 1.5% for every year thereafter violates the 133 1/3% rule because 1.5% is more than 133 1/3% of 1%). This is beside the point. The regulations purport to address base compliance with the anti-backloading provision; we now deal only with the remedy for an undisputedly backloaded formula. Though there may be situations where the proper remedy for backloading involves additional benefits, this is not one of them. See Kifafi II, 736 F.Supp.2d at 72 (finding that over time, some participants in fact recovered from any benefits deficiency they may have initially suffered). Given that the benefit-accrual class is limited to "employees whose benefits `have been, or will be, reduced'" because of the backloading, id., and that the Plan could have yielded the same amount of total benefits to participants while complying with the anti-backloading provision, the district court's refusal to apply its remedy to employees whose benefits were not reduced by the backloading — to penalize Hilton for the fact of the backloading — is far from an abuse of discretion. See Mertens v. Hewitt Assocs., 508 U.S. 248, 257 n. 7, 113 S.Ct. 2063, 124 L.Ed.2d 161 (1993) (noting that punitive damages were not a "major issue" when ERISA was enacted). Equitable relief — the only kind of relief at issue here — may very well mean "something less than all relief." Id. at 258 n. 8, 113 S.Ct. 2063.

E

Finally, Kifafi challenges the district court's approach to his union service vesting claim. The court found that Hilton violated the Plan's vesting provisions by failing to credit employees' years of union service, and it molded its remedial relief around the contours of that finding. Kifafi complains this contravenes both ERISA and the Plan itself. His argument goes like this: because both ERISA and the Plan require crediting all nonparticipating service and, under the Plan, union membership is not the only type of nonparticipating service,8 the district court erred by addressing Hilton's failure to count years of union service but not its failure to count years of nonunion nonparticipating service, either in its class certification decision or its remedial order. This is an impractical approach, he continues, because Hilton failed to maintain any records about employees' union service, so participants end up bearing the burden of proving uncredited union service years after the fact when the district court could have avoided this by broadening its perspective to encompass Hilton's general failure to count nonparticipating service.

We reject Kifafi's arguments. The district court's approach to Kifafi's vesting claim is not just a matter of law; it reflects the parties' respective actions throughout the litigation and effected the court's determination about how best to manage the shape-shifter shackled to the parties' dispute. In this light, we see no abuse of discretion.

To start, the court could reasonably have concluded that Kifafi was best able to represent a class limited to union participation. As Kifafi's original complaint made evident, his claim to representative status on the nonparticipating service issue derived from his "service as a union employee prior to 1985." The complaint thus listed as one of the two legal questions "common to the members of the class and subclass" whether Hilton may "fail to credit years in unionized employment with Hilton," and it mentioned only union service in the relevant complaint count. Kifafi subsequently amended the complaint, expanding the scope of the proposed class to include Hilton employees who "have not been credited with all years of service with Hilton, including years in unionized employment," but he continued to suggest that his claim remained tied to union service. For example, when describing the nature of the case at the beginning of the complaint, he alleged that Hilton "violated ERISA by not crediting his years of union service," and in his claim for relief, he alleged only that Hilton "violated the Retirement Plan by not counting years of service in union employment." In his renewed motion to certify the class,9 Kifafi asserted that "the common legal thread that binds the class is Hilton's failure to count all years of service," but he did little else to advance that broader argument. Though referring generally to nonparticipating service and citing Hilton's admission that it failed also to credit certain nonparticipating service other than union service, the motion otherwise focused on union service, tying general nonparticipating-service references to union service. Indeed, the motion's list of "individuals whose union or other non-participating service was not counted for vesting" included only three individuals, each listed for his or her uncredited years of union service; and the three class members Kifafi sought to include as class representatives likewise asserted that their uncredited years consisted of union service. Even if it would have been reasonable to certify a broader nonparticipating service class, the district court's actual certification decision was no less reasonable. The same is true of its later refusals to expand the certified subclass. See, e.g., Kifafi I, 616 F.Supp.2d at 30 n. 18; Kifafi II, 736 F.Supp.2d at 74.

Kifafi cites two cases where a court of appeals found an abuse of discretion in the district court's class certification. Abrams v. Communications Workers of America, 59 F.3d 1373 (D.C.Cir.1995), involved a challenge to the adequacy of a union's notice of the non-union members' right to object to paying dues. We reversed the district court's refusal to certify a class of all nonmember employees — including both employees who had objected and employees who merely might object — explaining that every employee had an interest in adequate notice because "the union must provide notice in advance of an employee's decision to object." Id. at 1378. And in Green v. Ferrell, 664 F.2d 1292 (5th Cir. 1982), the Fifth Circuit required the district court to broaden the class of convicted prisoners to include pretrial detainees because "those rights and the conditions of confinement that impact upon those two groups at the same county jail facility are sufficiently common to warrant contemporaneous consideration in a single judicial proceeding under the circumstances present here." Id. at 1295. Yet in both cases, the excluded group of proposed class members necessarily belonged to the included group: a challenge to notice procedures affects anyone who ought to receive that notice; a challenge to jail conditions affects anyone who is or will be held in that jail. Here, by contrast, Hilton's alleged failure to count nonunion nonparticipating service potentially affects individuals other than those affected by its failure to count union service. That Hilton apparently treated union service as a mere subset of nonparticipating service, ignoring both entirely, does not mean it could not have counted one but not the other. The extent to which Hilton treated the two groups of employees the same is a question of fact, as is whether Hilton's alleged blanket policy affected any employees for reasons unrelated to union service. And where changed jail conditions or notice procedures necessarily provide relief for all potential class members regardless of their identities, this case involves detailed actuarial determinations and individualized remedies.

Rather than agreeing with Kifafi's argument, then, we commend the district court for its exemplary work on this case. The court managed and reasonably disposed of this litigation — juggling a voluminous record and ably balancing competing considerations — despite shouldering much of the burden that should have been carried by counsel, and despite facing arguments it characterized as "moving targets." Along the way, the district court fashioned a remedy that hewed closely to its class certification decision — and that makes sense. Indeed, the court had promised as much when it stated at the summary judgment phase that "[i]n resolving the parties' claims, the Court shall not allow Kifafi to expand his Amended Complaint." Kifafi I, 616 F.Supp.2d at 23. Once it limited the scope of the dispute, the district court could quite reasonably restrict relief to those parameters. See Aviation Consumer Action Project v. Washburn, 535 F.2d 101, 108 (D.C.Cir.1976).

For similar reasons, we reject Kifafi's assertion that the claim procedure ordered by the district court was "completely unnecessary" and improperly shifts the burden onto the plaintiff class. The district court considered using nonparticipating service as a proxy for union service because Hilton's records were incomplete, but it rejected the idea in order to prevent Kifafi from using the procedure to evade the court's class certification decision. The court instead required Hilton to fund and administer a claim procedure open to all participants whose vesting status turns on nonparticipating service. As part of this, the district court not only required Hilton to provide Kifafi information submitted by claimants so Kifafi can challenge Hilton's claim-process decisions, but it expressly permitted Hilton to credit all nonparticipating service if it wants to avoid the administrative costs incident to its mandated record searches and the claim procedure. This seems both fair and reasonable. While it is unfortunate for the burden to fall on innocent parties rather than the employer who failed to perform its statutory duties, that is not enough to turn the district court's otherwise-competent performance into an abuse of discretion.

IV

For the reasons stated, the district court's orders are

Affirmed.