Opinion for the Court filed by Circuit Judge MILLETT.
Cable operators’ retransmission of religious and devotional programming from 2000 to 2003 produced a pool of royalties that Congress charged the Copyright Royalty Judges with distributing to the copyright owners. The Appellant Settling Devotional Claimants (“Devotional Claimants”) and Intervenor Independent Producers Group (“IPG”) vigorously contested their respective shares of that pool before the Royalty Judges. The Devotional Claimants now appeal, arguing that the Royalty Judges wrongly calculated their share of the pie by allowing IPG to press claims without proper authority and refusing to accept the Devotional Claimants’ evidence regarding how the relative value of claims should be calculated. They also argue that, after the Royalty Judges rejected both their and IPG’s proposed methodologies, the Royalty Judges’ final allocation simply split the difference between the two parties, and that decision was arbitrary and capricious and unsupported by substantial evidence.
We agree with the Devotional Claimants’ latter claim. King Solomon was not subject to the Administrative Procedure Act; the Royalty Judges are. Congress thus required that the Royalty Judges’ determinations rest on a focused analysis of the record, not an arbitrary splitting of the baby. We affirm the Royalty Judges’ *1110procedural rulings' resolving which IPG claims could go forward and whether the Devotional Claimants’ methodological evidence could be properly considered.
I
Statutory Background
The statutory framework governing the distribution of royalties for the retransmission of copyrighted material by cable system operators is discussed in detail in Independent Producers Group v. Librarian of Congress (“IPG II”), 792 F.3d 132, 134-36 (D.C.Cir.2015); and Independent Producers Group v. Library of Congress (“IPG I”), 759 F.3d 100, 101-103 (D.C.Cir.2014). As relevant here, the Copyright Act, 17 U.S.C. §§ 101 et seq., is designed to further two potentially competing statutory policies — protecting intellectual property while also ensuring that information flows freely. One way the Copyright Act balances those interests is by providing in appropriate circumstances for compulsory licensing accompanied by royalty payments to the copyright holder. See IPG I, 759 F.3d at 101; see also 17 U.S.C. §§ 107-122.
Cable retransmission is an area for which Congress designed such a compulsory licensing scheme. Under 17 U.S.C. § 111(c), a cable system operator may retransmit to its viewers copyrighted material initially aired on a broadcast station without obtaining the permission of the relevant copyright owners for that second transmission. The cable system operators, however, must deposit a statutorily prescribed royalty fee with the Register of Copyrights. IPG II, 792 F.3d at 134-35. Congress charged the Copyright Royalty Judges with annually distributing the pool of funds that accrues to the copyright holders. Id. (citing 17 U.S.C. § 801(b)(3)).
Copyright owners and their agents who assert entitlement to those royalties must file a claim with the Royalty Judges in July of the year following the retransmission of their programming. See 17 U.S.C. § 111(d)(4)(A); 37 C.F.R. § 360.2. Once the claims have been filed, the Royalty Judges “determine whether there exists a controversy concerning the distribution of royalty fees.” 17 U.S.C. § 111(d)(4)(B). If all claimants have agreed on the proper distribution, the Royalty Judges may conclude that no controversy exists and distribute the fees consistent with the claimants’ agreement. See IPG II, 792 F.3d at 134-35 (citing 17 U.S.C. §§ 111(d)(4)(B)— (d)(4)(C), 801(b)(7)). In the absence of such agreement, the Royalty Judges must “conduct a proceeding to determine the distribution of royalty fees.” 17 U.S.C. § 111(d)(4)(B).
That proceeding has two phases. In Phase I, the Royalty Judges apportion the total pool of royalties collected for a year across broad categories of retransmitted programming — such as sports, public television, or devotional (religious) shows — and assign a percentage of the overall fund to each category based on its comparative value. See IPG II, 792 F.3d at 134-35. In Phase II, the Royalty Judges divide up the amount allotted to each category among the individual claimants within that category. Id.
At the outset of each phase, the Royalty Judges announce the commencement of dispute proceedings in the Federal Register, which puts interested claimants on notice to file petitions to participate. See IPG II, 192. F.3d at 135-36; see also 17 U.S.C. § 803(b)(1)(A). The statute then provides for a three-month “voluntary negotiation period,” during which the parties attempt to reach agreement. See IPG II, 792 F.3d at 135-36; see also 17 U.S.C. § 803(b)(3). For claimants who have not *1111resolved their disputes during this period, the Royalty Judges accept written submissions, oversee a period for discovery, and provide for a post-discovery settlement conference period. See IPG II, 792 F.3d at 135-36; IPG I, 759 F.3d at 102.
If the claimants remain at loggerheads, the Royalty Judges conduct a hearing and issue a final determination allocating payments. See IPG II, 792 F.3d at 135-36. That decision is subject to a 60-day period of review by the Register of Copyrights and then published in the Federal Register. IPG I, 759 F.3d at 102-103 (citing 17 U.S.C. §§ 802(f)(1)(D), 803(c)(6)). A disappointed claimant may seek judicial review of the final decision in this court by appealing within 30 days of the Federal Register publication. IPG I, 759 F.3d at 103 (citing 17 U.S.C. § 803(d)(1)).
Factual and Procedural Background
As in IPG II, this appeal challenges the Phase II determination of the Copyright Royalty Judges for the years 2000 to 2003. IPG II involved the sports programming and program suppliers categories, see 792 F.3d at 135-36, while this opinion addresses the allocation of royalties within the devotional programming category.
The Devotional Claimants are twenty-three religious ministries that own copyrights for devotional television programming. The group participated in the Royalty Judges’ Phase I proceeding, which resulted in a partial settlement (including as to the devotional programming category) and, for the non-settling claimants, a final determination allocating royalties among the other programming categories. See Distribution of the 2000, 2001, 2002, and 2003 Cable Royalty Funds, 78 Fed. Reg. 64,984, 64,988 (Oct. 30, 2013).
The Phase II proceeding to allocate the devotional programming royalties began in February 2011, with only the Devotional Claimants and IPG filing petitions to participate. Both of them subsequently filed written direct statements summarizing their claims. IPG, employing a methodology of its own devising, asserted entitlement to a share of the royalties ranging from 37.30% to 53.10% for each year from 2000 to 2003. The Devotional Claimants claimed they were due 100% of the royalties because they knew of no other “valid, compensable claims” within the devotional category. J.A. 7638. Seeing no need to share the royalties, the Devotional Claimants’ direct statement set forth no specific methodology or calculations to govern apportionment of the royalties between them and IPG.
The closest the Devotional Claimants came to an allocation methodology was the proposed testimony of Dr. William Brown, who discussed “potential quantifiable criteria” that the Royalty Judges should consider in making any allocation that may be required. J.A. 7650. While Dr. Brown endorsed a survey of cable system operators called the Bortz survey for making allocations among Phase I categories, he acknowledged that the Bortz survey did not include data needed to make the pending Phase II allocation. In the absence of such Phase Il-relevant data, Dr. Brown suggested that viewership ratings could be “[a] valuable tool” in determining the “relative marketplace value of particular programs.” J.A. 765Í. He noted that a different group had previously shown how such data could be used to help project relevant viewership levels. But Dr. Brown attempted no such projections or calculations for this case. Rather, his testimony ended with the general observation that, in a Phase II determination, “[t]he most useful quantifiable data is Nielsen viewing data, projected to distant cable households, supplemented, where applicable with Bortz study data.” J.A. 7652.
*1112Both the .Devotional Claimants and IPG challenged each other’s authority to represent the claims of certain copyright holders. As relevant here, the Devotional Claimants objected in particular to IPG’s asserted representation of seven of its eight claimants.1
After conducting hearings in November and December 2012, the Royalty Judges issued a Memorandum Opinion and Order on March 21, 2013, addressing the preliminary evidentiary and procedural objections of the 'claimants. The decision dismissed IPG’s claims as to three claimants, but rejected the Devotional Claimants’ argument that IPG did not properly represent Benny Hinn Ministries, Creflo A. Dollar Ministries, and Life Outreach International. The Royalty Judges concluded that IPG had come forward with sufficient evidence of representational authority for each claimant “for purposes of this preliminary stage of the proceeding.” Memorandum Opinion and Order Following Preliminary Hearing on Validity of Claims 6 (March 21, 2013) (“March 21 Op.”), J.A. 3177.
With respect to IPG’s seventh claimant, Billy Graham Evangelistic Association (“Billy Graham”), the Royalty Judges found that IPG had been authorized to file claims on Billy Graham’s behalf for the years 2001 to 2003, but that the agreements for 2002 and 2003 had been terminated by Billy Graham in June 2005. Other correspondence, however, supported the conclusion that, following a remonstrative letter from IPG, Billy Graham later revoked this termination of IPG’s authority. Taking a “dim view” of what they concluded were IPG’s “mischaracterization of [Billy Graham’s] rights under the Copyright Act” and “strong-arm tactics [IPG] used to seek to prevent [Billy Graham] from severing the principal/agency relationship that [Billy Graham] had clearly revoked,” March 21 Op. at 8, J.A. 3179, the Royalty Judges nonetheless declined to dismiss the claims IPG had filed on behalf of Billy Graham, since doing so would unfairly punish Billy Graham. The Royalty Judges ordered IPG to obtain written confirmation of Billy Graham’s continuing interest in the royalty dispute and agreement to IPG’s representation.
At the close of their opinion, the Royalty Judges stated that they would not hear any further testimony or review any further exhibits on the denied claims and “d[id] not request and w[ould] not accept revised Written Direct Statements from the parties, relying instead on their respective abilities to filter admissible from inadmissible material.” March 21 Op. at 18, J.A. 3189.
Royalty Judge Roberts filed a dissenting opinion. In his view, the March 21 Opinion had improperly left open the status of certain claims and contemplated additional challenges, while at the same time “expressly forbid[ding] the parties from amending their Written Direct Statements prior to hearings.” J.A. 6220. He further disagreed with the majority opinion’s analysis of which claims could go forward and, in particular, with the suggestion that Billy Graham could pursue its own claims.
Following those two opinions, Billy Graham submitted a letter on April 19 acknowledging IPG’s authority to represent it for the 2002 and 2003 royalty years. With the last outstanding representation issues thus resolved, the parties exchanged rebuttal statements in the lead-up to a hearing on the Phase II royalty distribu*1113tion. The Devotional Claimants’ response included proposed testimony from a new witness, Alan Whitt, who had previously worked with the Motion Picture Association of America (“MPAA”) in developing a viewership-based model for assigning relative values to individual programs. His proposed testimony offered data derived from the database he had helped develop reflecting the projected viewership of devotional programming in distant markets for the years 2000 to 2003.
The Devotional Claimants also presented additional testimony from Dr. Brown, who attested that IPG’s methodology was unreliable, “premised on faulty and unsupported contentions for valuation of devotional programming,” and “riddled with calculation errors.” J.A. 6445. Using the viewership projections compiled in Whitt’s testimony, Dr. Brown also'calculated the proportion of the total viewership of devotional programming attributable to the Devotional Claimants. He proposed that the Devotional Claimants should receive between 65 and 75 percent of the royalties in the devotional category for each year from 2000 to 2003, with IPG receiving the remainder.
At the allocation hearing, the Royalty Judges excluded Whitt’s testimony, concluding that it constituted “testimony regarding the development of information or data” that should have been disclosed in the Devotional Claimants’ direct case. J.A. 3839. Dr. Brown was allowed to testify without objection, and his proposed rebuttal testimony was introduced as an exhibit.
On July 10, 2013, the Royalty Judges issued their initial allocation decision. 78 Fed.Reg. at 64,985. In assigning royalty amounts for the separate program suppliers category, the Royalty Judges “ultimately relied heavily” on the viewership-based methodology put forward by the MPAA. IPG II, 792 F.3d at 141-42; see also' id. at 142-43. In doing so, the Royalty Judges extensively criticized IPG’s proposed methodology. They noted that IPG’s methodology was advanced not by an econometric or statistical expert, but by the “imperfect messenger” Raul Galaz, an IPG employee who had no econometric or statistical expertise, but did have a fraud conviction arising out of a previous copyright royalty proceeding. 78 Fed.Reg. at 65,000. They also noted that the methodology produced numerous results that could not be justified. Id. at 65,000-65,-001.
For similar reasons, the Royalty Judges dismissed the validity of IPG’s model for the devotional category, finding that “IPG’s formula produced absurd results in the Devotional category, as it did in the Program Suppliers category.” 78 Fed. Reg. at 65,003. The Royalty Judges also dismissed the Devotional Claimants’ argument that they should use the same viewership-based model employed in the program suppliers category as untimely raised and thus procedurally barred. Id. In particular, the Royalty Judges noted that the Devotional Claimants waited until their rebuttal case to present Whitt’s testimony and the data on which Dr. Brown relied to apply the methodology in question. Id. at 65,004. The Royalty Judges stressed that, by providing Whitt’s testimony just three weeks before the hearing, the Devotional Claimants had “prejudiced IPG and, in essence, engaged in trial by ambush, in violation of the letter and spirit of the Judges’ procedural rules * * * [and] deprived IPG of the opportunity to review the work undertaken by Mr. Whitt.” Id.
The Royalty Judges’ rejection of both parties’ proposed methodologies left them empty-handed in allocating the devotional-programming royalties. See 78 Fed.Reg. at 65,004. Explaining that they were *1114“nevertheless[ ] obligated to reach a determination based on the existing record,’' the Royalty Judges observed that, for the year 2000, the allocations proposed by IPG happened to fall within a range proposed by the Devotional Claimants. Id. They ruled that this “some degree of agreement” provided a basis for awarding to IPG the 37.14% of royalties it proposed. Id. For the year 2002, the Royalty Judges similarly found that IPG’s proposed allocations were “almost equal to the lower bound” of the range proposed by the Devotional Claimants. Id. They thus concluded that it would be within the “zone of reasonableness” to give effect to IPG’s number, holding that IPG would receive 41.02% of the royalties. Id.
For the years 2001 and 2003, the Royalty Judges recognized that the parties’ proposals were not even in the same ballpark, 78 Fed.Reg. at 65,004, but stated that “there is no record evidence explaining why the percentage allocations for 2001 and 2003 should be so markedly different in those years compared to 2000 and 2002.” Id. at 65,004-65,005. The Royalty Judges accordingly allocated royalties for 2001 and 2003 by simply averaging the allocations for the two other years. Under that approach, the Devotional Claimants received 60.92% of the royalties for each year, and IPG received 39.08%. Id. at 65,-005.
The Devotional Claimants subsequently filed a petition for rehearing, which the Royalty Judges denied. With respect to the Devotional Claimants’ objection to the exclusion of Whitt’s and Dr. Brown’s rebuttal testimony, the Royalty Judges reiterated that 37 C.F.R. § 351.4(b) required that Whitt’s testimony be included in the written direct statement. They concluded that the Devotional Claimants had been afforded “ample time” to submit an amended written direct statement with this additional information and simply did not do so. J.A. 6877. Finally, the Judges rejected a challenge to the final allocation of royalties, emphasizing the overlap or near-overlap in the parties’ proposals for two years and insisting that, for the other two years, the Royalty Judges had not relied on IPG’s proposed allocations.
The Royalty Judges issued their final determination on August 13, 2013. It was approved by the Librarian of Congress and published in the Federal Register on October 30, 2013. This appeal followed.
II
Analysis
We have jurisdiction over the Devotional Claimants’ timely filed appeal pursuant to 17 U.S.C. § 803(d)(1). That jurisdiction includes the Devotional Claimants’ objections to the Royalty Judges’ March 21 procedural and evidentiary order, because that earlier interlocutory order merges into and is renewable as part of the Royalty Judges’ final determination. See IPG II, 792 F.3d at 138-39.
We review decisions of the Copyright Royalty Judges under the familiar standards of the Administrative Procedure Act, reversing only if their decision is arbitrary, capricious, contrary to law, or not based on substantial evidence. See 17 U.S.C. § 803(d)(3) (incorporating by reference 5 U.S.C. § 706). Our review is “highly deferential,” Intercollegiate Broadcast Sys., Inc. v. Copyright Royalty Board, 571 F.3d 69, 79 (D.C.Cir.2009), and, in reviewing royalty distribution decisions specifically, we ask only whether the Royalty Judges’ assigned allocation percentages are “within a zone of reasonableness,” Christian Broadcasting Network, Inc. v. Copyright Royalty Tribunal, 720 F.2d 1295, 1304 (D.C.Cir.1983) (internal quotation marks omitted).
*1115 IPG’s Authority to Represent Four Claimants
The Devotional Claimants repeat on appeal their objections to IPG’s representation of Benny Hinn Ministries, Creflo A. Dollar Ministries, Life Outreach International, and the Billy Graham Evangelistic Association. Those largely factual inquiries implicating the Royalty Judges’ own proceedings, however, fall squarely within the Royalty Judges’ area of expertise. The only question before us is whether the Royalty Judges’ rulings were supported by substantial evidence and were not arbitrary or capricious. Their decision crosses that deferential threshold.
The Devotional Claimants do not dispute that representational authority turns on a factual inquiry into “whether the claimant intended for its claim to be filed on its behalf by another.” March 21 Op. at 4, J.A. 3175. Such intent must be expressed prior to the filing of the relevant claim. See Distribution of 1993, 1994, 1995, 1996 and 1997 Cable Royalty Funds, 66 Fed. Reg. 66,433-66,435 (Dec. 26, 2001) (noting Library of Congress’s determination that “what the law requires! ] is a factual determination as to which of the owners and distributors identified [in a claim] were in fact represented by [the filer] at the close of the filing period for 1997 cable claims”) (quoting Order in Docket No. 2002-2 CARP CD 93-97 at 7 (June 22, 2000)); see also 37 C.F.R. § 360.3(b)(l)(vi), (2)(ii), (2)(vii) (requiring declaration or statement of authorization to be filed with claims for compulsory license royalties).
For the Benny Hinn and Creflo Dollar Ministries, the Royalty Judges rested their judgment on IPG’s evidence of agreements in which each ministry authorized IPG “to apply for and collect any and all monies distributed by audiovisual copyright collection societies throughout the world (e.g., monies derived from rights set forth on Exhibit ‘A’ hereto)” for works that the ministry owned or distributed. J.A. 6195, 6198. Exhibit A, in turn, included among the list of recoverable monies “Cable and Satellite Retransmission Royalties,” defined as “[r]oyalties and charges imposed by law with respect to the retransmission by cable or satellite of terrestrial broadcast signals.” J.A. 6197, 6201. IPG also produced email correspondence in which the ministry identified for IPG programs for which claims could be made.
The Devotional Claimants argue that there was insufficient evidence that the agreements were signed before the claims were filed and that IPG needed to present confirmatory testimony from the representatives of the ministries. Looking through the highly deferential lens of substantial evidence review, however, we hold that the documentary agreements themselves provided sufficient evidence that IPG was authorized to file claims on behalf of the two ministries. The Devotional Claimants suggest that the Royalty Judges’ decision contravened earlier precedent requiring a heftier evidentiary showing in the distribution of the 1997 cable royalty funds. But that distribution involved a case in which there was actual evidence of backdating the dates on which the contracts were signed. See 66 Fed.Reg. at 66,438-66,439. There was no analogous evidence here, and thus it was not unreasonable for the Royalty Judges to rely on the written representation agreements themselves, as corroborated by subsequent email correspondence.
Finally, invoking our decision in National Broadcasting Co. v. Copyright Royalty Tribunal (“NBC”), 848 F.2d 1289 (D.C.Cir.1988), the Devotional Claimants argue that the Royalty Judges’ reading of the phrase “audiovisual copyright collection societfy]” in the two agreements was the product of impermissible contract in*1116terpretation that went beyond the “face of these [contracts]” to consider their exhibits. (Devotional Claimants’ Br. 9). That argument misunderstands NBC. Disputes regarding who may claim royalties generally raise two issues: “(1) the proper dis-tributee of the royalties allocated by the [agency] to [a particular program]; and (2) the oumership of the copyright to which the royalties attach.” NBC, 848 F.2d at 1293. NBC made clear that the Royalty Judges have no authority to decide the second question — the proper ownership of the copyright. Id. Accordingly, when the Royalty Judges assign a claim to a particular claimant as part of their allocation process (the first question), the ruling “should not be seen at all as adjudicating the contractual entitlement rights of [the parties], but rather as setting forth a general rule for the distribution of cable royalties in these cases,” and the disposition “leavés the parties free to litigate their contractual claims in an appropriate forum.” Id. at 1296.
That holding does not preclude the Royalty Judges from looking at a contract in resolving the first question or other antecedent questions concerning whether to allocate royalties to a particular program in the first place. Instead, the statute’s silence on how to resolve a dispute over an agency relationship for purposes of royalty-allocation proceedings leaves the Royalty Judges discretion to which we must defer. See NBC, 848 F.2d at 1296 (deferring to the Copyright Royalty Tribunal’s “presumption * * * in the face of congressional silence” regarding allocation as “permissible interpretation of the statute”).
In short, then, the Copyright Act does not confine the Royalty Judges to a face-of-the-contract-only analysis of representational authority in proceedings before them, and nothing in the Royalty Judges’ consideration of the exhibits to the Benny Hinn and Creflo Dollar agreements contravened NBC. While the Royalty Judges’ decision would not bind the parties in any future contractual dispute, see NBC, 848 F.2d at 1293, 1296, it was an entirely appropriate step for the Royalty Judges to take here.
For largely the same reasons, the Royalty Judges’ decision to allow IPG to continue' to represent Life Outreach also passes muster. Indeed, the agreement with Life Outreach specifically identified monies associated with cable and retransmission royalties under 17 U.S.C. §§ 111 and 119 as within IPG’s authority to collect. While the Devotional Claimants object to the absence of a date associated with IPG’s signature on the agreement, we cannot say that the Royalty Judges’ reliance on the date that Life Outreach’s representative signed was unreasonable or based on insubstantial evidence.
The Devotional Claimant’s objection to IPG’s representation of Billy Graham focuses less on the evidence of initial authorization than on Billy Graham’s subsequent effort to terminate IPG’s representation. While the Royalty Judges’ decision could have been clearer, it was clear enough to be sustained on this point, too.2
Billy Graham terminated IPG as its agent in June 2005, which was after the filing date for claims for each of the royal- ty years at issue in this proceeding. Accordingly, for purposes of this argument, *1117IPG’s authority to have filed the claims in the first instance is not at issue. The question, instead, is whether IPG had authority to petition to participate in the Phase II proceeding on behalf of Billy Graham. To do so, IPG would again have had to be authorized to represent Billy Graham at least by the latest point at which Billy Graham could have filed its own petition to participate. See 37 C.F.R. § 351.1(b)(2)(ii)(E) (joint petition to participate filed by copyright owners’ representative must include certification that, “as of the date of submission of the joint petition, such * * * representative has the authority and consent of the participants to represent them in the royalty distribution proceeding.”).3
In finding such authority, the Royalty Judges relied in part on correspondence between Denise Vernon, an IPG executive, and Billy Graham reflecting that, at some unspecified date, Vernon employed what the Royalty Judges termed “strong-arm tactics” to induce Billy Graham to allow continued representation by IPG. March 21 Op. at 8, J.A. 3179. Subsequent emails, which at least extended to after the date by which Billy Graham would have had to file a petition to participate, indicate that Billy Graham acceded to IPG’s efforts to collect royalties for the years at issue. A witness for IPG also testified that Billy Graham had “recanted” its initial termination of IPG. J.A. 2907.
The record before the Royalty Judges certainly does not demand the conclusion that Billy Graham’s “recantation” was timely. But all that matters is that we cannot say that the Royalty Judges lacked substantial evidence in finding that Billy Graham’s reauthorization of IPG was timely ipade. See, e.g., Christian Broadcasting Network, 720 F.2d at 1313 (concluding substantial evidence existed for the Copyright Royalty Tribunal’s conclusion that it had made awards only to bona fide copyright owners).
The Royalty Judges also obtained contemporaneous corroboration from Billy Graham of IPG’s authority, and we see nothing in the Copyright Act or the Royalty Judges’ rules that precluded that measure. Indeed, given the questionable representations that IPG made in obtaining that reauthorization, the reconfirmation made sense. Accordingly, given the very narrow scope of our review, we uphold the Royalty Judges’ determinations concerning the scope of IPG’s representation authority.
Exclusion of the Devotional Claimants’ Allocation Methodology
The Devotional Claimants next object to the Royalty Judges’ exclusion of testimony and evidence they presented in their rebuttal case suggesting a possible viewership-based valuation methodology for devotional programming royalties. We conclude that the Royalty Judges’ application of their own procedural regulations was reasonable.
The regulations governing proceedings before the Royalty Judges provide that all parties who have filed a petition to participate in a proceeding “must file a written direct statement.” 37 C.F.R. § 351.4(a). A written direct statement is defined by statute to mean “witness statements, testimony, and exhibits to be presented in the proceedings, and such other information that is necessary to establish * * * the distribution of royalty payments * * * as *1118set forth in regulations issued by the Copyright Royalty Judges.” 17 U.S.C. § 803(b)(6)(C)(ii)(II). The Judges’ regulations, in turn, provide that “[t]he written direct statement shall include all testimony, including each witness’s background and qualifications, along with all the exhibits.” 37 C.F.R. § 351.4(b)(1). In a royalty distribution proceeding, the parties must further include in their direct statement their “percentage or dollar claim to the fund,” but “[n]o party will be precluded from revising its claim * * * at any time during the proceeding up to, and including, the filing of the proposed findings of fact and conclusions of law.” Id. § 351.4(b)(3).
Focusing on the requirement that the written direct statement include “all testimony” a party intends to present, the Royalty Judges concluded that the Devotional Claimants’ direct statement should have included analysis of how the viewership-based methodology that they endorsed in general terms would apply in a relative valuation of the claims brought by the Devotional Claimants and IPG. The Devotional Claimants challenge both that interpretation of Section 351.4 and the reasonableness of its application.
We generally accord deference to an agency’s interpretation of its own regulation “unless that interpretation is plainly erroneous or inconsistent with the regulation.” Decker v. Northwest Envtl. Defense Ctr., — U.S. -, 133 S.Ct. 1326, 1337, 185 L.Ed.2d 447 (2013) (internal quotation marks omitted). Such deference is perhaps particularly appropriate here because the rule concerns the Copyright Royalty Judges’ own procedures. Cf. National Ass’n of Broadcasters v. Copyright Royalty Tribunal, 675 F.2d 367, 375 n. 8 (D.C.Cir.1982) (noting that the Copyright Act “gives the [Copyright Royalty Judges’ predecessor] considerable freedom to determine its own procedures”). Such deference may not even be needed, though, because the Royalty Judges’ interpretation wholly comports with the plain text of the regulation. It should be no surprise that a requirement that a party present in its initial direct statement “all testimony” necessary to establish its claimed entitlement to royalties would obligate the party to include in that statement the testimony applying its proposed methodology to the case at hand. Indeed, linking the proposed methodology to the facts of the case is the testimonial heart of the matter. Abstract discussion of potential methodologies, without any application, would do nothing to support the party’s desired outcome.
Nor do the rules provide any reason to think that this weighty evidence may be saved for the rebuttal case. Indeed, in civil litigation generally, courts have recognized that a trial court “has the discretion to ‘limit the scope of rebuttal testimony to that which is directed to rebut new evidence or new theories proffered in the defendant’s case-in-chief.” Toth v. Grand Trunk R.R., 306 F.3d 335, 345 (6th Cir.2002) (quoting Martin v. Weaver, 666 F.2d 1013, 1020 (6th Cir.1981)). When, as here, the proposed rebuttal testimony “could have been included in the same witness’ direct examination,” it may be excluded. Waterview Mgmt. Co. v. FDIC, 203 F.3d 54, 1999 WL 503921, at *2 (D.C.Cir. May 20, 1999) (unpublished) (citing Geders v. United States, 425 U.S. 80, 86, 96 S.Ct. 1330, 47 L.Ed.2d 592 (1976)). Nothing in the text of the regulations suggests a different operation here. The Royalty Judges thus acted within the bounds of reasonable discretion in affording their rule an interpretation grounded in text and practice.
The Devotional Claimants point out that a predecessor to the Royalty Judges once allocated royalties relying in part on a *1119study presented only in the rebuttal phase of the proceeding. See Distribution of 1998 and 1999 Cable Royalty Funds, 69 Fed.Reg. 3,606, 3,619 (Jan. 26, 2004). In that case, however, the Librarian turned aside an objection to the consideration of the evidence in part by noting the very limited use that was made of it. See id. Indeed, the Librarian specifically noted that, had the Copyright Arbitration Royalty Panel “fully credited [the study in question] and used it as the basis for determining [the objecting party’s] award,” the outcome may have been different. Id. That one-time dispensation hardly evidences a binding precedential blessing of parties lobbing in critical evidence, like the determinative application of a deci-sional methodology, only at the rebuttal phase.
The Devotional Claimants’ reliance on National Association of Broadcasters as another case in which evidence introduced during rebuttal was used is no help either. No one even raised the admissibility issue in the case. See 675 F.2d at 376.
We also hold that the Royalty Judges’ application of the rule in this case was neither arbitrary nor capricious. The Devotional Claimants assert that they did not know until after the March 21 order that IPG in fact had valid claims to assert, but by that point, the Royalty Judges had closed the door on amended direct statements. See March 21 Op. at 18, J.A. 3189; see also 37 C.F.R. § 351.4(c) (“A participant in a proceeding may amend a written direct statement based on new information received during the discovery process, within 15 days after the end of the discovery period.”). The Devotional Claimants’ bad strategy call does not make enforcement of written rules of which they had fair notice arbitrary or capricious.
First, the Devotional Claimants had full notice prior to the time an amended written direct statement would have been due that IPG claimed a share of the royalties as well. Any uncertainty about which claims IPG might be able to represent — or hope that they would all be dismissed— was no excuse for failing to prepare for the chance of a disputed allocation. Indeed, the Devotional Claimants did not even specifically object to one of IPG’s representations, so it was more than likely that a contested allocation would go forward. In any event, there was certainly nothing to prevent them from timely identifying testimony by the August 2012 deadline, rather than making a tactical choice to wait until May 2013 — just 20 days before the hearing — to disclose their anticipated testimony.
Tellingly, the MPAA was faced with similar uncertainties in the program suppliers category. But when the MPAA received IPG’s direct statement claiming a share of the royalties, the MPAA timely filed an amended written direct statement specifying how a relative valuation should be calculated under its proposed methodology. The Devotional Claimants' could have done the same.
Second, the Devotional Claimants point out that, on the eve of the hearing, IPG was permitted to provide a great deal of additional data, reflecting recalculations performed in response to an error identified in its initial methodology. See 78 Fed.Reg. at 65,001-65,002. They argue that the disparate treatment of their late evidence and IPG’s was arbitrary and capricious. We disagree. IPG’s submission was a correction to data that had been presented along with IPG’s applied methodology in IPG’s direct statement, affording the Devotional Claimants fair notice. The Royalty Judges could reasonably conclude that such amendments were far less prejudicial than interjecting the application of an allocation methodology for the *1120very first time in rebuttal just weeks before the hearing was to begin.
Third, the Devotional Claimants are wrong to argue that, because Dr. Brown’s written rebuttal testimony was accepted into evidence without objection, the Royalty Judges were also obligated to consider the portions of this testimony applying- a viewership-based methodology to the devotional programming category. “In for a penny, in for a pound” is not an evidentia-ry rule binding the Royalty Judges. That is especially true when, as here, the foundation for Dr. Brown’s rebuttal testimony was the excluded testimony of Whitt. With- Dr. Brown’s testimony stripped of its proffered factual basis, the Royalty Judges reasonably concluded that they could not give it any weight.
To be sure, as the Devotional Claimants note, experts may base their opinion on facts that may otherwise be inadmissible. See Fed.R.Evid. 703 (“If experts in the particular field would reasonably rely on those kinds of facts or data in forming an opinion on the subject, they need not be admissible for the opinion to be admitted.”). But the Devotional Claimants do not claim to have made any showing before the Royalty Judges that Dr. Brown’s testimony had been based on the kinds of facts on which an expert would reasonably rely.
The Ultimate Allocation Decision
The Devotional Claimants’ final objection is to the Royalty Judges’ decision to go forward with the allocation of royalties despite having rejected every proposed methodology making that decision for the devotional programming royalties. As to this final challenge, we agree that, despite its Solomonic pedigree, the Royalty Judges’ approach was quintessentially arbitrary and capricious.
With respect to the royalties from the year 2000, the Royalty Judges ruled that there was a functional agreement — or at least “some degree” of agreement — between the parties to the extent that IPG’s proposed allocation percentages happened to fall within the range proposed by the Devotional Claimants. 78 Fed.Reg. at 65,-004. We are told on appeal that the parties “effectively” agreed on this point. Gov’t Br. 50.
We cannot agree. Settling royalty distributions by agreement reflects a separate avenue for resolving royalty distributions under the Copyright Act, subject to its own requirements. See 17 U.S.C. § 801 (b) (7) (A) (i) (providing that Royalty Judges may use agreement as the basis for a royalty distribution provided that the parties that would be bound by the determination have an opportunity to comment). In this case, any intersection of the two parties’ numbers was the product of accident, not agreement, as evidenced by the fact that the Devotional Claimants were awarded almost 5% less of the total fund than they had requested. Indeed, even to find overlap in the parties’ numbers at all, the Royalty Judges adopted a “two wrongs make a right” methodology, (i) crediting IPG percentages that were derived from a discredited methodology and partially based on claims that the Royalty Judges had ruled IPG could not bring, and ©giving effect to allocations derived from a methodology that the Royalty Judges had refused to consider. In the face of that actual non-agreement, simply picking a number out of IPG’s flawed and otherwise-rejected proposal just because it happened to roughly coincide with the lowest bound proposed by the Devotional Claimants falls beyond the bounds of reasoned decision-making.
It gets still worse for subsequent royalty years, in which even the fig leaf of “some degree” of agreement fell away. For 2002, the Royalty Judges chose IPG’s (otherwise *1121infirm) proposed allocation solely because it was deemed to be close enough to the lower bound proposed by the Devotional Claimants. The decision gives no hint as to how close is close enough, nor any explanation for why they picked IPG’s proposal in lieu of the supposedly close-enough lowest bound of the range proposed by the Devotional Claimants. And for 2001 and 2003, the Judges, with a blank slate of an evidentiary record, simply split the difference of the allocations from the two other years.
That cannot be sustained. The Royalty Judges’ obligation is to make reasoned decisions supported by the written record before them. See 17 U.S.C. § 803(c)(3). They do not satisfy that burden by bridging over a lacuna in the record with a purported agreement that does not actually exist.
That is not to say that royalty distributions must be perfect. “[A]n agency’s choice of a particular number or percentage is not reviewable for exact precision, but simply for broad reasonableness.” National Ass’n of Broadcasters, 675 F.2d at 374. Indeed, “mathematical exactitude in these matters appears well-nigh impossible; rough justice in dividing up the royalty pie seems to be the inevitable result of the process that Congress ordained.” National Ass’n of Broadcasters v. Copyright Royalty Tribunal, 772 F.2d 922, 926 (D.C.Cir.1985) (citation omitted). And we have on rare occasion sustained a superficially similar rough-justice approach. But in those cases, the administrative body relied on some relevant and creditable methodological evidence, even if it was “far from perfect,” National Cable Television Ass’n, Inc. v. Copyright Royalty Tribunal, 724 F.2d 176, 184 (D.C.Cir.1983), or “fairly but not wholly satisfactory,” Association of American Publishers, Inc. v. Governors of U.S. Postal Service, 485 F.2d 768, 773 (D.C.Cir.1973). In this case, even that minimal foundation is lacking.
We thus face here on a larger scale the type of situation we confronted in Intercollegiate Broadcast System, Inc. v. Copyright Royalty Board, 571 F.3d 69 (D.C.Cir.2009). In that case, the Royalty Judges’ rate-setting decision adopted a certain minimum fee to be charged to each noncommercial webcasting channel or station to cover administrative costs. See id. at 86-87. There the Royalty Judges relied on a single party’s proposed fee amount, notwithstanding the lack of any demonstrated link between the proposal and the expenses it was designed to compensate. Id. at 87. We rejected that approach, holding that the Royalty Judges’ decision was “inconsistent with rational decision-making.” Id.
A reasoned justification “requires more than an absence of contrary evidence; it requires substantial evidence to support a decision.” Intercollegiate Broadcast, 571 F.3d at 87. That is missing here. And making matters still worse, the Royalty Judges’ approach to allocation was “first presented in the Judges’ determination and not advanced by any participant.” Id.
Perhaps the Royalty Judges’ decision was driven by the statutory mandate to decide the case within eleven months of the end of the settlement conference period. See 17 U.S.C. § 803(c)(1). In an appropriate case, the conditions imposed on the Royalty Judges and the resource constraints they face may be relevant to our consideration of the reasoned nature of their decisionmaking. See National Cable Television Ass’n, 724 F.2d at 187. But the bottom-line obligation to produce a reasoned decision remains. Moreover, the Royalty Judges were not limited to the deficient methodological evidence the parties put before them. The Copyright Act *1122permits the Royalty Judges to request information even from a nonparticipant if relevant , to a material issue of fact. 17 U.S.C. § 803(b)(6)(C)(ix). Indeed, the Royalty Judges have made use of similar initiatives before to obtain additional methodological evidence. See 66 Fed.Reg. at 66,441.
For those reasons, we vacate the Royalty Judges’ allocation of royalties in the devotional programming category for 2000 to 2003. We leave to the Royalty Judges on remand how to balance their legitimate interest in preventing parties before them from engaging in trial by ambush with the need to have a sufficient factual basis to make a reasoned decision.
Ill
Conclusion
We hold that the Royalty Judges reasonably determined that IPG had the authority to represent the four claimants challenged by the Devotional Claimants on appeal. The Royalty Judges also reasonably declined to consider the Devotional Claimants’ methodological evidence given its untimely presentation. We conclude, however, that the Royalty Judges’ ultimate royalty allocation, in the wake of the evi-dentiary gap left by their rejection of all proffered methodologies, was arbitrary and capricious. We consequently vacate that portion of the determination and remand for further proceedings consistent with this opinion.
So ordered.