Jaime Castillo, Brad Eugene Branch, Renos Lenny Avraam, Graeme Leonard Craddock, and Kevin A. Whitecliff ("the defendants") appeal their convictions and sentences under 18 U.S.C. § 924(c)(1). We affirm.

* The defendants are Branch Davidians convicted for, among other things, violating 18 U.S.C. § 924(c)(1) because of their involvement in the events that occurred at the Mount Carmel compound near Waco, Texas, in early 1993.1 At the time of the defendants' conviction, § 924(c)(1) stated:

Whoever, during and in relation to any crime of violence or drug trafficking crime (including a crime of violence or drug trafficking crime which provides for an enhanced punishment if committed by the use of a deadly or dangerous weapon or device) for which he may be prosecuted in a court of the United States, uses or carries a firearm, shall, in addition to the punishment provided for such crime or drug trafficking crime, be sentenced to imprisonment for five years, ... and if the firearm is a machinegun, or a destructive device [e.g., a hand grenade], or is equipped with a firearm silencer or muffler, to imprisonment for thirty years.

18 U.S.C. § 924(c)(1) (1994) (superseded).

Pursuant to § 924(c)(1), the district court sentenced Castillo, Branch, Avraam, and Whitecliff to thirty years imprisonment and Craddock to ten years imprisonment.2 Underlying the sentences were findings of fact that the defendants had actually or constructively possessed enhancing weapons (i.e., machine guns; destructive devices; firearms equipped with silencers or mufflers) during and in relation to a crime of violence--that is, a conspiracy to murder federal agents. With the exception of Avraam, Craddock, and another co-defendant, Ruth Riddle, the district court did not base its findings of fact on direct evidence of actual possession. Rather, it attributed the possession of enhancing weapons--specifically, machine guns, hand grenades, and firearms equipped with silencers--to the defendants based on the "fortress theory"3 and the Pinkerton doctrine.4

The defendants appealed. We affirmed on all issues but the sentences for the § 924(c)(1) convictions. See United States v. Branch, 91 F.3d 699, 745 (5th Cir.1996), cert. denied, 520 U.S. 1185, 117 S. Ct. 1466, 1467, 137 L. Ed. 2d 681 (1997). In regard to the sentences, we noted that Bailey v. United States, 516 U.S. 137, 146-50, 116 S. Ct. 501, 507-09, 133 L. Ed. 2d 472 (1995), establishes that, in the context of § 924(c)(1), "use" means "active employment,"5 and held that the district court's finding of use based on evidence of actual or constructive possession did not meet Bailey's definition of "use." See Branch, 91 F.3d at 740. At the end of our discussion, we stated:

As we have explained, there is evidence from which it could be found that machine-guns and other enhancing weapons [e.g., destructive devices, firearms equipped with firearm silencers] were used by one or more members of the conspiracy in the firefight of February 28[, 1993]. The jury was not required to do so and the district court only entered those findings then required. With Bailey, the district court must take another look and enter its findings regarding "active employment." Should the district court find on remand that members of the conspiracy actively employed machine-guns, it is free to reimpose the 30-year sentence. We vacate the defendants' sentences on [the count of the indictment that charges a violation of § 924(c)(1) ] ... and remand for re-sentencing on that count.

We note that, on remand, the district court should consider whether the defendants actively employed a weapon during and in relation to the conspiracy to murder federal agents.

Id. at 740-41. In accordance with these comments, we entered a mandate that stated, in part: "[The sentences for the § 924(c)(1) convictions] are vacated and remanded for findings and re-sentencing."

On remand, the district court found that one or more persons involved in the conspiracy to murder federal agents had actively employed machine guns and other enhancing weapons in the firefight on February 28, 1993, and then applied the Pinkerton doctrine to attribute the active employment of machine guns and other enhancing weapons to the defendants on February 28, 1993. Alternatively, it found that Branch and Avraam each had used (i.e., actively employed) and carried a machine gun on February 28, 1993, and that Castillo and Craddock each had carried a hand grenade on April 19, 1993. The district court re-sentenced Castillo, Branch, Avraam, and Whitecliff to thirty years imprisonment and Craddock to ten years imprisonment.6 The defendants timely appealed.

This appeal involves the following contentions: (1) we erred in holding that, in the context of § 924(c)(1), the type of firearm used or carried during and in relation to a crime of violence or a drug trafficking crime is a sentencing enhancement, and not an element of the offense; (2) we erred in holding that a conviction for violating § 924(c)(1) stands despite the absence of a conviction for the predicate offense; (3) the district court's jury instruction on the "use" of a firearm was improper; (4) the district court's application of the Pinkerton doctrine was improper; (5) the district court clearly erred in finding that Branch and Avraam each had used and carried a machine gun on February 28, 1993; (6) the district court clearly erred in finding that Castillo and Craddock each had carried a hand grenade on April 19,1993; (7) the district court clearly erred in considering conduct other than that which occurred on February 28, 1993, in re-sentencing Castillo and Craddock; and (8) the district court applied the wrong standard of proof in sentencing the defendants.7 The Government asserts that either the law-of-the-case doctrine or the waiver doctrine precludes the first, second, third, fourth, and eighth contentions.

The law-of-the-case doctrine "posits that when a court decides upon a rule of law, that decision should continue to govern the same issues in subsequent stages in the same case." Arizona v. California, 460 U.S. 605, 618, 103 S. Ct. 1382, 1391, 75 L. Ed. 2d 318 (1983). Based on this general principle, we have developed specific rules about how our earlier decisions affect later proceedings and cases. See Crocker v. Piedmont Aviation, Inc., 49 F.3d 735, 739 (D.C.Cir.1995).

The law-of-the-case doctrine follows from the "sound policy that when an issue is once litigated and decided, that should be the end of the matter." United States v. United States Smelting Ref. & Mining Co., 339 U.S. 186, 198, 70 S. Ct. 537, 544, 94 L. Ed. 750 (1950). It is an exercise of judicial discretion, not a limit on judicial power. See Messinger v. Anderson, 225 U.S. 436, 444, 32 S. Ct. 739, 740, 56 L. Ed. 1152 (1912).

The waiver doctrine bars consideration of an issue that a party could have raised in an earlier appeal in the case. See Brooks v. United States, 757 F.2d 734, 739 (5th Cir.1985). It "serves judicial economy by forcing parties to raise issues whose resolution might spare the court and parties later rounds of remands and appeals." Hartman v. Duffey, 88 F.3d 1232, 1236 (D.C.Cir.1996), cert. denied, 520 U.S. 1240, 117 S. Ct. 1844, 137 L. Ed. 2d 1048 (1997). The waiver doctrine differs from the law-of-the-case doctrine in that it arises as a consequence of a party's inaction, not as a consequence of a decision on our part. See Crocker, 49 F.3d at 739.

Only plain error justifies departure from the waiver doctrine. See id. at 740; cf. McCann v. Texas City Ref., Inc., 984 F.2d 667, 673 (5th Cir.1993) ("It is the unwavering rule in this Circuit that issues raised for the first time on appeal are reviewed only for plain error."). We find plain error when (1) there is an error (2) that is clear and obvious and (3) that affects substantial rights. See United States v. Olano, 507 U.S. 725, 732, 734, 113 S. Ct. 1770, 1776, 1777, 123 L. Ed. 2d 508 (1993). In most cases, the party asserting plain error bears the burden of making a specific showing of prejudice (i.e., the error affected the outcome of the district court proceedings) to satisfy the affects substantial rights prong. See id. at 735, 113 S.Ct. at 1778. Even if we find plain error, we reverse only if the error seriously affects the fairness, integrity, or public reputation of judicial proceedings. See id. at 732, 113 S.Ct. at 1776.

We now consider the applicability of the law-of-the-case doctrine and the waiver doctrine to this appeal.

* The defendants argue that, in the context of § 924(c)(1), the type of firearm used or carried during and in relation to a crime of violence or a drug trafficking crime is an element of the offense. In disposing of the defendants' first appeal, we rejected this contention, and found that the type of firearm is a sentencing enhancement.8 See Branch, 91 F.3d at 740.

The defendants' argument implicates the law-of-the-case rule that when we resolve a legal issue and remand to the district court, our decision binds subsequent proceedings in the district court and on later appeal. See White v. Murtha, 377 F.2d 428, 431-32 (5th Cir.1967). The rule covers issues that we have decided explicitly and by necessary implication. See Terrell v. Household Goods Carriers' Bureau, 494 F.2d 16, 19 (5th Cir.1974). It furthers judicial efficiency and discourages panel shopping at the circuit level. See Lehrman v. Gulf Oil Corp., 500 F.2d 659, 662 (5th Cir.1974). We depart from the rule only if (1) the evidence presented at trial following remand is substantially different, (2) a controlling authority has made a contrary decision of law since our earlier decision, or (3) our earlier decision was clearly erroneous and would work a manifest injustice. See White, 377 F.2d at 431-32. The party seeking a reversal of our earlier decision bears the burden of showing the applicability of one of these exceptions. See Morrow v. Dillard, 580 F.2d 1284, 1292 (5th Cir.1978).

The defendants maintain that the law-of-the-case rule is inapplicable here because our prior finding that the type of firearm is a sentencing enhancement, and not an element of the offense, was dictum. We agree with the assumption underlying this assertion that the rule reaches earlier decisions but not earlier dictum.9 See Society of Roman Catholic Church of Diocese of Lafayette, Inc. v. Interstate Fire & Cas. Co., 126 F.3d 727, 735 (5th Cir.1997); see also In re United States, 60 F.3d 729, 731 (11th Cir.1995) (per curiam). However, we disagree that our prior finding was dictum. As the Government says in its brief, our prior finding "was clearly necessary and essential to the decision to remand the case for further findings, for there would have been no need for further findings if the type of weapon was not a sentencing enhancement."

The defendants also attempt to persuade us to discard the law-of-the-case rule. However, they are unsuccessful. We reject the insinuation that our refusal to depart from the rule will result in manifest injustice because Supreme Court review is no longer available to the defendants as a result of the previous denial of their petition for a writ of certiorari on our prior decision. See Forsyth v. Hammond, 166 U.S. 506, 513, 17 S. Ct. 665, 668, 41 L. Ed. 1095 (1897) (characterizing the discretion to grant or deny a petition for a writ of certiorari as "comprehensive and unlimited"). We also find unavailing the various arguments suggesting that our prior decision was clearly erroneous--that is, was "dead wrong." City Pub. Serv. Bd. v. General Elec. Co., 935 F.2d 78, 82 (5th Cir.1991) (following Parts & Electric Motors, Inc. v. Sterling Electric, Inc., 866 F.2d 228, 233 (7th Cir.1988)). While most of the arguments merit no discussion, we wish to respond to two of them. First, our prior decision need not be reversed because it conflicts with the holdings of several other circuits. See In re Korean Air Lines Disaster of Sept. 1, 1983, 829 F.2d 1171, 1183 n. 22 (D.C.Cir.1987) (D.H.Ginsburg, J., concurring) (discussing the law-of-the-case doctrine), aff'd, 490 U.S. 122, 109 S. Ct. 1676, 104 L. Ed. 2d 113 (1989). Second, our prior decision does not contravene Almendarez-Torres v. United States, 523 U.S. 224, 118 S. Ct. 1219, 140 L. Ed. 2d 350 (1998), and Jones v. United States, --- U.S. ----, 119 S. Ct. 1215, --- L.Ed.2d ---- (1999), two intervening Supreme Court decisions. It accords with the directive in Almendarez-Torres, 523 U.S. at ----, 118 S. Ct. at 1223, to look at "language, structure, subject matter, context, and history" in determining whether or not Congress intended for a statute to define a separate crime or to set forth a separate sentencing factor. See Branch, 91 F.3d at 738-40 (examining § 924(c)(1)'s text, structure, and legislative history). Our prior decision also does not conflict with Jones. This case and Jones differ in a critical way. In Jones, the legislative history contained conflicting indications of whether Congress intended for 18 U.S.C. § 2119, the statute at issue, to lay out three distinct offenses or a single crime with three maximum penalties. See Jones, --- U.S. at ----, 119 S. Ct. at 1221. In contrast, the legislative history of § 924(c)(1) discloses that Congress consistently referred to the machine gun clause as a penalty and never indicated that it intended to create a new, separate offense for machine guns. See Branch, 91 F.3d at 739. Accordingly, we decline to reconsider our prior decision that the type of firearm used or carried is a sentencing enhancement, and not an element of the offense.

The defendants further assert that the doctrine of constitutional doubt10 and the rule of lenity11 require that the type of firearm be deemed an element of the offense. This contention implicates the waiver doctrine. The defendants did not raise the doctrine of constitutional doubt or the rule of lenity in their first appeal. Nor have they shown that our refusal to consider these issues now will affect their substantial rights. See Olano, 507 U.S. at 735, 113 S. Ct. at 1778. Accordingly, the defendants have waived their challenge to our prior decision.

The defendants maintain that, in disposing of their first appeal, we were wrong to let their convictions for violating § 924(c)(1) stand in the face of their acquittals on the predicate crime of conspiracy to murder federal agents. We based our decision to leave the convictions intact on several cases in which we had held that a conviction for the predicate offense is unnecessary to convict under § 924(c)(1). See United States v. Ruiz, 986 F.2d 905, 911 (5th Cir.1993); United States v. Munoz-Fabela, 896 F.2d 908, 910-11 (5th Cir.1990). The defendants argue that we should have ignored those cases and followed a more recent decision, United States v. Lucien, 61 F.3d 366, 377 (5th Cir.1995). They claim that Lucien contradicts the cases on which we relied and supports their argument.12

Even if Lucien conflicts with the cases on which we relied and supports the defendants' argument, we are unmoved. The law-of-the-case doctrine's progeny includes the rule that we must follow the first-in-time decision when faced with conflicting panel decisions. See In re Caddo Parish-Villas S., Ltd., 174 F.3d 624, 628-29 (5th Cir.1999); United States v. Iwegbu, 6 F.3d 272, 274 n. 1 (5th Cir.1993); 18 CHARLES ALAN WRIGHT ET AL., FEDERAL PRACTICE AND PROCEDURE § 4478 (2d ed.1981). Therefore, we must adhere to our earliest decision on whether or not a conviction on the predicate offense must occur to convict under § 924(c)(1). Our earliest decision rejects the position that the defendants take on the issue.13 See Munoz-Fabela, 896 F.2d at 910-11. Accordingly, we decline to reconsider our earlier holding that the absence of a finding of guilt on the predicate offense does not negate the defendants' convictions for violating § 924(c)(1).

The defendants assert that we should overturn their convictions for violating § 924(c)(1) because the district court's jury instruction on "use" failed to comport with Bailey. This argument implicates the waiver doctrine. The defendants could have raised their contention in their first appeal. Furthermore, they have not shown that district court's erroneous jury instruction affected their substantial rights and therefore amounted to plain error. See Olano, 507 U.S. at 735, 113 S. Ct. at 1778. In this regard, we previously have held that the evidence was sufficient to convict the defendants for using firearms in the sense contemplated by Bailey.14 See Branch, 91 F.3d at 735. Accordingly, the defendants have waived their challenge to the district court's jury instruction on "use."

The defendants criticize the district court for applying the Pinkerton doctrine for the purpose of determining the type of firearms that they had used. In our earlier decision, we stated:

[T]here is evidence from which it could be found that machine-guns and other enhancing weapons [e.g., destructive devices, firearms equipped with silencers] were used by one or more members of the conspiracy in the firefight of February 28[, 1993].... Should the district court find on remand that members of the conspiracy actively employed machine-guns, it is free to reimpose the 30-year sentence.

Id. at 740-41. The district court understood these comments to mean that on remand it was only to make findings on the active employment of machine guns, and that it was not to revisit the applicability of the Pinkerton doctrine.

The defendants' criticism implicates the mandate rule, a corollary of the law-of-the-case doctrine. See United States v. Becerra, 155 F.3d 740, 753 (5th Cir.1998). The mandate rule requires a district court on remand to effect our mandate and to do nothing else. See id.

We hold that the district court's decision not to reconsider the applicability of the Pinkerton doctrine on remand was consistent with our mandate. Nor do we see any reason to depart from the mandate rule in this case. The defendants claim that the Pinkerton doctrine was unavailable to the district court because the jury did not convict them of the predicate offense to the § 924(c)(1) charge. This argument ignores our earlier holding that the absence of a guilty verdict on the predicate offense does not invalidate a conviction for violating § 924(c)(1). See Branch, 91 F.3d at 732-33. The defendants also argue that the Pinkerton doctrine plays no role in determining a sentence for a violation of § 924(c)(1) because § 1B1.3 of the Sentencing Guidelines, an analogue to the Pinkerton doctrine, is inapplicable to § 2K2.4, which covers § 924(c)(1). However, we have endorsed the use of the Pinkerton doctrine to determine the sentence of a co-conspirator under 21 U.S.C. § 841(b), a statutory provision analogous to § 924(c)(1).15 See United States v. Ruiz, 43 F.3d 985, 992 (5th Cir.1995). For us to depart from the mandate rule, the defendants must show that we previously were "dead wrong." City Public Serv. Bd., 935 F.2d at 82. They have not done so. Accordingly, we decline to reconsider our prior approval of the district court's application of the Pinkerton doctrine.16

Finally, the defendants claim that the district court erred in making findings of fact for purposes of sentencing based on the preponderance of the evidence. They assert that the district court was obliged to apply a higher evidentiary standard because the increase in their sentences from 5 to 30 years--the consequence of the district court's findings of fact on active employment of machine guns--was dramatic and devastating. In support of their argument, they point to the following observation that we made in United States v. Mergerson, 4 F.3d 337, 343 (5th Cir.1993):

We recognize a growing number of cases decided by courts in other circuits in which a higher standard of proof has been suggested or required when a finding of a particular fact relevant to sentencing dramatically alters the sentencing options of the court to the disadvantage of the defendant. We also recognize dicta in the Supreme Court's decision in McMillan v. Pennsylvania, 477 U.S. 79, 106 S. Ct. 2411, 91 L. Ed. 2d 67 (1986), to the same extent.

The defendants' contention implicates the waiver doctrine. The district court initially imposed the defendants' sentences based on findings of fact made by a preponderance of the evidence. The defendants therefore could have attacked the correctness of the standard of proof in their first appeal. Moreover, the district court's error, if any, was not plain in light of several circumstances. See Olano, 507 U.S. at 734, 113 S.Ct. at 1777 (" 'Plain' is synonymous with 'clear' or equivalently, 'obvious.' "). We have noted that "[i]t is well-established that the preponderance of evidence standard is the applicable standard for sentencing purposes," United States v. Gaytan, 74 F.3d 545, 558 (5th Cir.1996), and have expressed "reluctan[ce] to part from the preponderance of evidence standard in a non-capital case," United States v. Millsaps, 157 F.3d 989, 995 (5th Cir.1998). Additionally, another circuit court has held that an increase in a sentence comparable to the increase in the sentences in this case is an insufficient reason to necessitate resort to a higher standard of proof. See United States v. Porter, 23 F.3d 1274, 1277-78 (7th Cir.1994) (citing an earlier case in which the sentence jumped from a Sentencing Guideline range of 33 to 41 months to consecutive 20-year terms). Accordingly, the defendants have waived their challenge to the standard of proof that the district court applied in sentencing them.

We agree with the Government that the law-of-the-case doctrine and the waiver doctrine determine the disposition of this appeal. Accordingly, we affirm.