
555 F.3d 303 (2009)
UNITED STATES of America, Appellee,
v.
IONIA MANAGEMENT S.A., Defendant-Appellant.
Docket Nos. 07-5801-cr, 08-1387-cr.
United States Court of Appeals, Second Circuit.
Argued: November 21, 2008.
Decided: January 20, 2009.
*305 Sandra S. Glover, Assitant United States Attorney (William M. Brown, Jr., Assistant United States Attorney, on the brief), for Nora R. Dannehy, Acting United States Attorney, District of Connecticut, New Haven, CT, and John L. Smeltzer, Attorney (Lana N. Pettus, Attorney, on the brief), for Ronald J. Tenpas, Assistant Attorney General, Environment and Natural Resources Division, Washington, DC, for Appellee.
Irwin H. Schwartz, Sheryl Gordon McCloud, Seattle, WA, for Defendant-Appellant.
Andrew Weissmann, Jenner & Block LLP, New York, N.Y., for Amici Curiae The Chamber of Commerce of the United States, The Washington Legal Foundation, The Association of Corporate Counsel, The National Association of Criminal Defense Lawyers, The National Association of Manufacturers, and The New York State Association of Criminal Defense Lawyers.
Before: McLAUGHLIN, CALABRESI, LIVINGSTON, Circuit Judges.
PER CURIAM:
Defendant-Appellant Ionia Management S.A. ("Ionia") appeals from a jury verdict convicting it of, inter alia, violating the Act to Prevent Pollution from Ships ("APPS") by failing to "maintain" an oil record book ("ORB") while in U.S. waters. The charges in this case derive from events aboard the M/T Kriton ("Kriton"), a 600-foot oil tanker managed (but not owned) by Ionia that flies under the flag of the Bahamas.[1] Ionia is incorporated in Liberia and headquartered in Greece. During the period named in the indictment (January 2006 to April 2007), the Kriton delivered oil and petroleum products to ports along the east coast of the United States. While making these deliveries, Ionia's engine room crew, under the direction and participation of the Chief Engineers and Second Engineer, routinely discharged oily waste water into the high seas through a "magic hose" designed to bypass the vessel's Oily Water Separator, which would have cleaned the waste to prepare it for disposal as required by law. Furthermore, the Kriton's crew made false entries in the ORB to conceal such discharges, and obstructed a federal investigation (a) by hiding the "magic hose" from Coast Guard inspectors during a March 20, 2007, inspection and (b) by lying to Coast Guard officials. The Government brought four separate indictments against Ionia in June 2007  in the District of Connecticut, the Eastern District of New York, the Southern District of Florida, and the District of the United States Virgin *306 Islands. The indictments, which were consolidated for trial in Connecticut, collectively charged Ionia with one count of conspiracy (18 U.S.C. § 371), thirteen counts under the APPS (33 U.S.C. § 1908(a)), three counts of falsifying records in a federal investigation (18 U.S.C. § 1519), and one count of obstruction of justice (18 U.S.C. § 1505). After trial, the jury found Ionia guilty on all counts. Ionia now appeals that conviction, arguing: (1) that the District Court erred in instructing the jury that Ionia could be found liable for violating the APPS, 33 U.S.C. § 1908(a), for failing to "maintain" an ORB when the Kriton's crew only possessed the falsified ORB and did not make any false entries when it was in U.S. waters; (2) that the jury instruction on vicarious corporate liability was erroneous and constructively amended the indictment, and that the evidence was insufficient to establish respondeat superior criminal liability; (3) that the District Court constructively amended the indictment when it failed to instruct the jury that it had to find "material" falsification to convict under 18 U.S.C. § 1519; and (4) that the District Court erred in sentencing by not properly grouping the crimes, relying on disputed facts, and considering extraterritorial information.
Ionia's argument about the APPS concerns an issue that this Court has not yet addressed. We therefore interpret for the first time the regulation that requires subject ships to "maintain" an ORB. 33 C.F.R. § 151.25(a). In doing so, we join the Fifth Circuit in holding that this provision imposes a duty on ships, upon entering the ports or navigable waters of the United States, to ensure that its ORB is accurate (or at least not knowingly inaccurate). We find that this requirement complies with international law as required by 33 U.S.C. § 1912, which provides that "[a]ny action taken under [the APPS] shall be taken in accordance with international law." In addition, it is supported by the regulation's plain text and is necessary to advance the aims of the international treaties governing pollution on the high seas. Accordingly, we conclude that the District Court did not err in its jury instruction.
With respect to the remaining issues on appeal, we address them summarily as Ionia has failed to demonstrate that there were any errors based on the established precedents of our Circuit.

I. The Act to Prevent Pollution on Ships

A. Statutory Framework
Congress enacted the APPS, 33 U.S.C. §§ 1901-1912, to implement two related marine environmental treaties to which the United States is a party: (1) the 1973 International Convention for the Prevention of Pollution from Ships, Nov. 2, 1973, 1340 U.N.T.S. 184, and (2) the Protocol of 1978 Relating to the International Convention for the Prevention of Pollution from Ships, Feb. 17, 1978, 1340 U.N.T.S. 61. United States v. Jho, 534 F.3d 398, 401 (5th Cir.2008). These two international conventions, jointly referred to as MARPOL, aim "to achieve the complete elimination of international pollution of the marine environment by oil and other harmful substances," MARPOL, 1340 U.N.T.S. at 128, and consequently address how oceangoing vessels are to dispose of wastes generated onboard.
Specifically, MARPOL's Annex I contains regulations for the prevention of oil pollution. United States v. Abrogar, 459 F.3d 430, 432 (3d Cir.2006). During normal operation, vessels like the Kriton accumulate large volumes of oily waste water in their bilges, engine rooms, and mechanical spaces. The provisions of Annex I prohibit ships from discharging these *307 wastes at sea, except under certain prescribed conditions. For instance, a vessel may only discharge en route if the discharged material is processed through specified oil filtration equipment, such as an Oily Water Separator, that traps most of the oil. MARPOL, reg. 4(c), 1340 U.N.T.S. at 67; Id. Reg. 9, 1340 U.N.T.S. at 202. Moreover, Annex I mandates that vessels record all oil transfer operations, including overboard discharge of bilge water, in an ORB that is retained on board and available for inspection by the "competent authority" of any government party to MARPOL. Id. reg. 20, 1340 U.N.T.S. at 211-12; see also Abrogar, 459 F.3d at 432 (explaining that Annex I requires that the ORB include records of all transfers of oil, disposals of oily waste generated on the vessel, the disposal of sludge, emergency discharges, among others).
MARPOL is not a self-executing treaty; instead, each party agrees to "give effect" to it. MARPOL, art. 1(1), 1340 U.N.T.S. at 63, 184. In particular, each party to the treaty agrees to set up rules for ships that fly that party's flag, and each is responsible for certifying that such ships comply with the treaty rules. Parties to MARPOL are required to set sanctions for violations by ships flying their flag "wherever the violation occurs." Id. Art. 4(1), 1340 U.N.T.S. at 185-86. When another country observes or suspects that a ship is violating the treaty, Article 6(2) provides that a port state should send the flag state a report of those possible violations so that proceedings may be initiated in the flag state. Id. at 187.
The United States law executing MARPOL  the APPS  authorizes the Coast Guard to "prescribe any necessary or desired regulations to carry out the provisions of ... MARPOL." 33 U.S.C. § 1903(c)(1). Pursuant to this authority, the Coast Guard has issued regulations that generally track the requirements set forth in Annex I. The particular APPS regulation at issue here is 33 C.F.R. § 151.25(a), which states that "[e]ach oil tanker of 150 gross tons and above ... shall maintain an [ORB]" (emphasis added), which it must keep "readily available for inspection," id. at 151.25(i); see also Abrogar, 459 F.3d at 432 (explaining that member states conduct inspections to ensure compliance in their ports and waters, and that a ship's failure to comply with MARPOL requirements can form a basis for refusing to let a ship enter port, referring the matter to the flag state, or, where appropriate, prosecuting the violation in the United States). There are two pertinent limitations to the application of the APPS. First, the record book requirements of 33 C.F.R. § 151.25 apply to U.S.-registered ships wherever they are (including beyond U.S. waters), but to any foreign-flagged ships only "while in the navigable waters of the United States, or while at a port or terminal under the jurisdiction of the United States." 33 C.F.R. § 151.09(a)(1)-(5); see also 33 U.S.C. § 1902(a)(2). Second, the APPS states that "[a]ny action taken under [the APPS] shall be taken in accordance with international law." 33 U.S.C. § 1912; see also Jho, 534 F.3d at 402.

B. Failure to "Maintain" an Oil Record Book
The crux of the dispute before us focuses on the words "shall maintain" in the APPS regulations regarding ORBs. Ionia argues that the word "maintain" obligates it only to keep possession of an ORB, while the Government submits that the APPS requires Ionia to keep the ORB accurately. Shortly after Ionia filed its appeal in this case, the Fifth Circuit issued an opinion in United States v. Jho, 534 F.3d 398 (5th Cir.2008), which presented a similar fact scenario and essentially identical *308 arguments. In Jho, a crew member tipped off the Coast Guard that the Chief Engineer had manipulated some of the ship's pollution-detection equipment so that it would not recognize discharges with higher oil content than allowed under U.S. law. Id. at 400. The indictments charged the operators of the foreign-flagged ship with failing to "maintain" an ORB under 33 C.F.R. § 151.25. Id. at 401. As in the case sub judice, the defendants argued that international law prohibited the government from prosecuting these offenses that, they alleged, occurred on the high seas. Id. at 403. The Fifth Circuit, reversing the district court, disagreed. It first held that the district court was incorrect in concluding that the alleged conduct  failing to "maintain" an ORB  occurred outside of U.S. waters. Id. at 402-03. The court reasoned that if the requirement to "maintain" an ORB included only an obligation to record entries when discharges were made (an obligation violated on the high seas), and not to keep the book accurate (an obligation existent at U.S. ports), then the regulation would be "at odds with MARPOL and Congress' clear intent under the APPS to prevent pollution at sea according to MARPOL." Id. at 403. Because "[a]ccurate oil record books are necessary to carry out the goals of MARPOL and the APPS," the court found if record books "did not have to be `maintained' while in the ports or navigable waters of the United States, then a foreign-flagged vessel could avoid application of the record book requirements simply by falsifying all of its record book information just before entry into a port or navigable waters," and thus avoid detection. Id. Consequently, the Fifth Circuit held that the requirement to "maintain" an ORB "impos[es] a duty upon a foreign-flagged vessel to ensure that its oil record book is accurate (or at least not knowingly inaccurate) upon entering the ports of navigable waters of the United States." Id.
Furthermore, the Jho court rejected the defendants' argument that international law  i.e., the law of the flag doctrine embodied in the United Nations Convention on the Law of the Sea ("UNCLOS") and MARPOL  limited the government's jurisdiction to prosecute violations of domestic law committed in port. Id. at 409. Because the failure to maintain the ORB occurred in U.S. waters, there was no obligation to let the flag state prosecute the violation. The Fifth Circuit noted that the "Supreme Court has recognized that the law of the flag doctrine does not completely trump a sovereign's territorial jurisdiction to prosecute violations of its laws: The law of the flag doctrine is chiefly applicable to ships on the high seas, where there is no territorial sovereign; and as respects ships in foreign territorial waters it has little application beyond what is affirmatively or tacitly permitted by the local sovereign." Id. at 406 (quoting Cunard S.S. Co. v. Mellon, 262 U.S. 100, 123, 43 S.Ct. 504, 67 L.Ed. 894 (1923)) (alterations omitted) (internal quotation marks omitted). Accordingly, the court found that the ORB offenses were charged "in accordance with" the law of the flag. Id.
We agree for substantially the reasons stated by the Fifth Circuit in Jho. Any other reading would defeat the purpose of MARPOL and the APPS, and would be inconsistent with international law. The law of the flag doctrine depends on member states being able to report violations to flag states. If ships such as the Kriton did not have to maintain an accurate ORB, member states would be severely hampered in their ability to report violations to the flag state for enforcement, and the international system of reporting and accountability under MARPOL would collapse.
*309 The reading the Fifth Circuit adopted in Jho also is strongly supported by the plain text of the regulation. Webster's Third New International Dictionary, Unabridged (2002) defines "maintain" as, inter alia, "to keep in a state of repair, efficiency, or validity...." In the context of a regulation imposing record-keeping requirements, the duty to "maintain" plainly means a duty to maintain a reasonably complete and accurate record. No reasonable reader of this regulation could conclude, given the context, that the regulation merely imposes an obligation to preserve the ORB in its existing state.
We therefore hold that the APPS's requirement that subject ships "maintain" an ORB, 33 C.F.R. § 151.25, mandates that these ships ensure that their ORBs are accurate (or at least not knowingly inaccurate) upon entering the ports or navigable waters of the United States. This requirement is in compliance with international law, supported by the plain text of the regulation, and necessary to advance the aims of the international treaties governing international pollution in marine environments. Accordingly, the District Court did not err in instructing the jury on the APPS charges.

II. Remaining Claims
Ionia's remaining claims do not require the Court to interpret new areas of law and, hence, can be addressed summarily. We assume the parties' familiarity with the additional facts, procedural history, and scope of the issues pertinent to these remaining claims.


A. Corporate Criminal Liability (Respondeat Superior)
We find that Ionia's claim that there was not sufficient evidence to convict it on a respondeat superior theory to be meritless. The Court reviews challenges to the sufficiency of the evidence de novo, United States v. Yannotti, 541 F.3d 112, 120 (2d Cir.2008), affirming if "after viewing the evidence in the light most favorable to the prosecution, any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt," Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979). The record reflects that there was ample evidence for a jury to have reasonably found that the Kriton's crew acted within the scope of their employment. The crew members acted within their authority to maintain the engine room, discharge waste, and record relevant information, when they used the "magic hose." They testified that they acted at the direction of their supervisors not just (a) in the discharging of the oil, and (b) in the making and maintaining of the false ORB entries, but also (c) in lying to the Coast Guard. The jury could, moreover, infer from the expert testimony about the maintenance and expense involved in using the Oil Water Separator that the crew used the bypass hose to benefit Ionia and subsequently lied to protect the company.
Ionia's appeal that the jury charge on respondeat superior was erroneous and constructively amended the indictment also fails. Ionia did not challenge the instruction below, so we review for plain error. Johnson v. United States, 520 U.S. 461, 466-67, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997) (explaining that showing "plain error" involves proving an (1) error, (2) that is plain, and (3) affected the appellant's substantial rights). Ionia contends that the District Court erred because, inter alia, it failed to instruct the jury that corporate criminal liability can only stem from the actions of so-called "managerial" employees. That contention seems at odds with our precedents in United States v. Twentieth Century Fox Film Corp., 882 *310 F.2d 656, 660 (2d Cir.1989), and United States v. George F. Fish, Inc., 154 F.2d 798, 801 (2d Cir.1946) (per curiam). But regardless of these, there was overwhelming evidence that the Kriton's Chief Engineers specifically directed crew members to use the "magic hose," and so Ionia's argument is without merit in any event.
Furthermore, we refuse to adopt the suggestion that the prosecution, in order to establish vicarious liability, should have to prove as a separate element in its case-in-chief that the corporation lacked effective policies and procedures to deter and detect criminal actions by its employees. We note that this argument is made only by amici curiae and not by Ionia, and so we are not obligated to consider it. But the argument, whoever made it, is unavailing. Adding such an element is contrary to the precedent of our Circuit on this issue. See Twentieth Century Fox Film Corp., 882 F.2d at 660 (holding that a compliance program, "however extensive, does not immunize the corporation from liability when its employees, acting within the scope of their authority, fail to comply with the law"). And this remains so regardless of asserted new Supreme Court cases in other areas of the law. As the District Court instructed the jury here, a corporate compliance program may be relevant to whether an employee was acting in the scope of his employment, but it is not a separate element.

B. Falsification Under 18 U.S.C. § 1519
Ionia next asserts that the District Court's jury charge as to falsification under 18 U.S.C. § 1519 constructively amended the indictment by failing to instruct the jury that the falsification had to be "material." Because Ionia requested the instruction given by the District Court on the § 1519 counts, it has likely waived any argument concerning a constructive amendment. See United States v. Giovanelli, 464 F.3d 346, 351 (2d Cir.2006) (per curiam) ("[I]f a party invited the charge... she has waived any right to appellate review of the charge." (internal quotation marks omitted)). In any event, however, a constructive amendment occurs only when the trial evidence and jury instructions "so modify" the terms of an indictment that "there is a substantial likelihood that the defendant may have been convicted of an offense other than that charged in the indictment." United States v. Mollica, 849 F.2d 723, 729 (2d Cir.1988) (internal quotation marks omitted). In considering this question, courts "have constantly permitted significant flexibility in proof, provided that the defendant was given notice of the core of criminality to be proven at trial." United States v. Rigas, 490 F.3d 208, 228 (2d Cir.2007) (footnote omitted) (internal quotation marks omitted). Here, Ionia was on notice of the "core criminality" to be proven at trial. Moreover, the evidence shows that the falsification in this case was indisputably material, and thus Ionia's rights were not substantially affected.

C. Sentencing
Lastly, Ionia appeals its sentence, arguing that the District Court made several procedural errors that warrant a remand. Ionia primarily argues that the District Court erred in its grouping analysis by not following Sentencing Guidelines Section 3D1.2. The District Court, however, was not sentencing pursuant to this section, and was not required to do so because there were no guidelines applicable to the organizational offense at issue. Accordingly, the District Court "determine[d] an appropriate fine by applying the provisions of 18 U.S.C. §§ 3553 and 3572." See U.S.S.G. §§ 8C2.1, 8C2.10. In announcing the sentence, the District Court stated: "And I want you to understand *311 that I in no way felt constrained and confined or mandated [by the Guidelines]." The District Court also explained how the sentence was appropriate to serve the "deterrent[,] punishment and public message function" for a company that had "previously been convicted of a like crime" and "apparently failed to see the seriousness" of its obligations. Thus, the District Court was not distracted by any alleged erroneous Guidelines calculation or grouping. The other sentencing challenges are similarly without merit. Given that a fine of $4.9 million is also substantively reasonable in this case, there is no reason to disturb the District Court's sentence.

III. Conclusion
We have carefully considered all of Ionia's claims, and we find them to be without merit. Accordingly, the judgment of the District Court and the jury verdict are AFFIRMED.
NOTES
[1]  We recite only an abbreviated version of the facts here, as the full background is not necessary for reaching the legal issue presented in this case. Because we review a jury verdict, we view the facts and evidence in the light most favorable to the verdict.
