This appeal concerns a settlement reached in 2002 between F.A.C., Inc. (“FAC”) and Cooperativa de Seguros de Vida de Puerto Rico (“COSVI”). FAC is a Consulting firm that was hired by the Puerto Rico Department of Health (“the Department”) to review certain past Medicare cost reimbursement claims submitted by the Department to the federal government. COSVI is a so-called “fiscal intermediary” acting for the federal government’s Medicare program in evaluating claims submitted by claimants.
It appears that FAC contracted with the Department to look for unreimbursed (or under-reimbursed) Medicare claims from past years and to obtain reopening of such claims, receiving in exchange the promise of a portion of any new recovery. In July 1995 FAC did submit a package of claims to COSVI, seeking enlarged reimbursement for the Department. COSVI rejected the reimbursement claims, ultimately leading the Department to terminate its contract with FAC.
In 1998 FAC sued COSVI in the federal district court in Puerto Rico. In this suit, FAC charged COSVI with violations of RICO, 18 U.S.C. § 1962(a), (d) (2000), based on alleged extortion attempts relating to the reimbursement requests FAC had made on behalf of the Department. On April 17, 2002, on the third day of a jury trial, FAC and COSVI reached a settlement agreement. They subsequently informed the district court that they had settled, and, at the parties’ request, the court on April 23, 2002, dismissed the case with prejudice pursuant to Fed.R.Civ.P. 41(a)(2).
Unfortunately, the settlement agreement was oral, which is permissible, see Quint v. A.E. Staley Mfg. Co., 246 F.3d 11, 15 (1st Cir.2001), cert. denied, 535 U.S. 1023, 122 S.Ct. 1618, 152 L.Ed.2d 631 (2002), although often unwise. Thereafter, the district court was on two different occasions asked to clarify and enforce the settlement. Importantly, both parties sought (and benefitted from) the district court’s intervention after the settlement was reached.
As early as May 2002, FAC began a succession of court filings and protest letters to COSVI, claiming that COSVI was not living up to the settlement. Most per*188tinent to the present appeal, FAC complained that COSVI had not sent a promised letter to the Centers for Medicare and Medicaid (“CMS”), the federal agency that had power to approve the reopening of certain of the past reimbursement claims identified by FAC.1
Both sides agreed that a letter was promised, but they differed as to its terms. FAC construed COSVI’s obligation as one of admitting to CMS that fraud had occurred within the COSVI organization. COSVI, writing to CMS in May 2002 to request reopening of the claims, was apparently anxious to minimize the scope of any admissions and went no further than to explain that “[t]his request is based on facts learned by COSVI as a result of its litigation with FAC, Inc., and pursuant to 42 CFR Section 405.1885(d).” The cited regulation permits an “intermediary determination” to be “reopened and revised at any time if it is established that such determination ... was procured by fraud or similar fault of any party to the determination.” 42 C.F.R. § 405.1885(d) (2005).
CMS refused to reopen, saying only that the exceptions to the otherwise applicable three-year time limit on reopening requests — which include subsection (d) — “do not fit the facts of this case.” COSVI wrote a second letter in July 2002 that went no further in admissions but offered to seek a court order to provide certain “documentary evidence” for CMS to consider, possibly an oblique reference to FBI reports. CMS still took no action to reopen the claims.
In the meantime, COSVI, in response to filings by FAC in the district court seeking enforcement of the settlement agreement, not only defended its own actions but in turn sought district court action to compel FAC to dismiss claims it had earlier brought against COSVI in the local Puerto Rico trial court. These local claims encompassed the same underlying conduct as FAC’s federal court RICO claims, but sought relief on behalf of individual plaintiffs under Puerto Rico defamation law. COSVI argued to the district court, and FAC denied, that dismissal of such claims was part of the federal court settlement.
In August 2002, the district court resolved the matter by amending its original April 2002 judgment of dismissal with prejudice. The amended August 2002 judgment began by summarizing the court’s understanding of the settlement as follows:
This Court actively participated in the settlement discussions held by the parties before and during the trial of the case. The Court understood the agreements reached to be summarized as follows:
a. The payment to Plaintiffs officials by Defendant COSVI of an amount that the parties agreed not to disclose publicly-
b. The payment is to be in consideration of Plaintiffs claims under 31 P.R. Laws Ann. §§ 1802 and 1803, as all allegations under RICO have been dismissed.
c. The sending of a letter by COSVI to [CMS], on behalf of FAC, Inc., requesting the reopening of the Medicare Part A reimbursement claims.
d. That any payments made as a result of the audit of the Medicare claims would be deposited in this Court for distribution purposes.
*189e. The dismissal with prejudice of the instant case and of [FAC’s local court action].
The amended judgment then discussed the dispute between the parties as to whether the settlement agreement required FAC to dismiss its local court action, and determined that it did. The judgment concluded by saying: “As per the terms of the settlement agreement and Fed.R.Civ.P. 41(a)(2), the Amended Complaint is hereby DISMISSED with prejudice. ...” Neither side appealed from this amended judgment; FAC dismissed the state claims and the district court subsequently disbursed certain funds that had been deposited with the court by COSVI to implement the settlement.
During 2003, CMS was apparently given access to further information relating to an FBI investigation, but it still did not authorize the reopening of the disputed claims. In late 2003, FAC filed a motion asking the district court to order COSVI to comply with various obligations which FAC claimed to be part of the settlement, including reopening the claims in question on its own authority. COSVI denied that it could reopen the claims on its own authority and said it had complied with all its obligations, including the required request to CMS to reopen the claims.
In September and October 2004, the district court denied FAC’s new motion and then, on FAC’s reconsideration motion, reopened the matter. In August 2005, the district court entered a new order discussing at length the procedural history of the case and concluding that COSVI’s earlier letter to CMS had not complied with its settlement obligations. The court ordered COSVI to send CMS a new request for reopening, which was to include the following sentence: “It has come to our attention that fraudulent activities took place within our organization with regard to the claims at issue, and, pursuant to 42 CFR Section 405.1885(a) and (d), we request that you reopen and reassess the claims in question due to a finding of fraud or similar fault.”
COSVI then filed the present appeal to this court and obtained a stay pendente lite of the direction that it send the new letter. On this appeal, COSVI argues that the district court had not retained jurisdiction to construe and enforce the settlement. In the alternative, it says that the district judge improperly modified the terms of the settlement and erred in concluding that COSVI had failed to comply with it. We begin, of course, with the jurisdictional objection. See Bolduc v. United States, 402 F.3d 50, 55 (1st Cir.2005).
In the district court, COSVI did not object to the court’s authority to construe and enforce the settlement, but the objection to subject matter jurisdiction is not waivable and may be raised for the first time on appeal. Am. Fiber & Finishing, Inc. v. Tyco Healthcare Group, LP, 362 F.3d 136, 138-39 (1st Cir.2004). If the district court’s authority rested solely on the original judgment, we would agree that the settlement could not be enforced, save by an independent law suit; but, as we will see, the entry of the amended judgment alters the situation adversely to COSVI.
The law is now settled that a federal court does not have inherent jurisdiction to enforce a settlement merely because it presided over the law suit that led to the settlement. Kokkonen v. Guardian Life Ins. Co. of Am., 511 U.S. 375, 379-80, 114 S.Ct. 1673, 128 L.Ed.2d 391 (1994). The federal court has “ancillary” jurisdiction to enforce only if
the parties’ obligation to comply with the terms of the settlement agreement had been made part of the order of *190dismissal—either by separate provision (such as a provision “retaining jurisdiction” over the settlement agreement) or by incorporating the terms of the settlement agreement in the order.
Id. at 381, 114 S.Ct. 1673. Otherwise, “enforcement of the settlement agreement is for state courts, unless there is some independent basis for federal jurisdiction,” id. at 382, 114 S.Ct. 1673, such as diversity of parties.
Here the original judgment recited that “[a]ll parties have come to a settlement agreement” and then said that “[pjursuant to that agreement and Federal Rule of Civil Procedure 41(a)(2),” the case was dismissed with prejudice. In our view, this bare reference to “a settlement agreement” does not satisfy Kokkonen, and, although the question is closer, even the “pursuant to” language is not enough to “incorporate the terms” of the unwritten settlement into the judgment.
The circuits appear to be in accord on the first point.2 On the second, there are two holdings in accord with our own, see RE/Max, 271 F.3d at 642; Phar-Mor, 172 F.3d at 274, with one judge in another circuit believing that language such as “pursuant to” or “in accordance with” is enough. Lucille v. City of Chicago, 31 F.3d 546, 549 (7th Cir.1994) (Cudahy, J., concurring), cert. denied, 513 U.S. 1154, 115 S.Ct. 1109, 130 L.Ed.2d 1074 (1995). Hard and fast rules may be unwise because of variations in language and context; but in the present case we think that Kokkonen’s requirements for retained jurisdiction were not satisfied by the first judgment.
However, in August 2002, four months after the original judgment, the district court entered an amended judgment without any objection from the parties as to its authority to do so. In the amended judgment, the court set forth its understanding of the main terms of the settlement, proceeded to “adjudicate” the dispute as to whether FAC had to dismiss its claims in the local-court action, “ordered” that claims be dismissed, and gave some further directions before reentering the dismissal with prejudice.
This new “amended final judgment” does go far enough to satisfy Kokkonen: it incorporates the terms of the settlement and, even more plainly, expresses by its very action in adjudicating a dispute about those terms an intention to retain jurisdiction to resolve disputes about the settlement. Neither party appealed from this new judgment. It therefore constitutes a final judgment asserting ongoing authority over the case and itself provides the baseline for evaluating any further action by the district court.
COSVI counters by asserting that the district court had no authority to enter the amended judgment. Because the amended judgment granted relief sought by COSVI (namely, the direction to FAC to dismiss the pending local suit), one might ask whether COSVI should be permitted to question the court’s authority. But in any event the court did not lack subject matter jurisdiction to determine whether to amend its judgment, and the merits of its decision to amend are no longer open to review.
Rule 60 of the Federal Rules of Civil Procedure sets out a series of grounds on which a district court may amend a final *191judgment. These include “errors” in the judgment “arising from oversight or omission,” Fed.R.Civ.P. 60(a); “mistake, inadvertence, surprise, or excusable neglect” (only available within the first year after judgment has issued), Fed.R.Civ.P. 60(b)(1); and “any other reason justifying relief from the operation of the judgment,” Fed.R.Civ.P. 60(b)(6).
Quite possibly it was a mistake to amend the judgment in August 2002, but this is not because of Kokkonen; that decision restricts enforcement of existing judgments and says nothing about whether and when a judgment may lawfully be amended. See 511 U.S. at 378, 114 S.Ct. 1673. A plausible objection would start with the notion that, although permissible grounds for amendment are broadly stated in Rule 60, they have been construed somewhat stringently because they compromise the finality of judgments. See, e.g., Morgan Gty. Tr. Co. of N.Y. v. Third Nat’l Bank of Hampden County, 545 F.2d 758, 760 (1st Cir.1976); cf. Roger Edwards, LLC v. Fiddes & Son, Ltd., 427 F.3d 129, 134 (1st Cir.2005).
The judge’s authority to amend the judgment is perhaps clearest where the judgment as entered merely failed to express the court’s intention. See Fed.R.Civ.P. 60(a); Boston Car Co. v. Acura Auto. Div., Am. Honda Motor Co., 971 F.2d 811, 814-15 (1st Cir.1992). Here, within a few months of the original April 2002 judgment, both the judge and parties acted as if the court had originally intended to reserve authority to superintend enforcement. The inference that this was the judge’s original intention is not especially strong, but it is at least colorable.
That is all that matters. The district judge had subject matter jurisdiction to determine whether to find that the circumstances permitted and warranted amendment. This is the message of Bell v. Hood, 327 U.S. 678, 682-83, 66 S.Ct. 773, 90 L.Ed. 939 (1946), which remains settled law, see, e.g., Arhaugh v. Y & H Corp., — U.S. -, - & n. 10, 126 S.Ct. 1235, 1244 & n. 10, 163 L.Ed.2d 1097 (2006). If his decision to amend was mistaken, it was up to the party objecting to amendment to file a timely appeal from the amended judgment, which COSVI failed to do.
There is no doubt that the amended judgment was appealable at the time it was entered. It substituted a new judgment which expressed the court’s understanding of certain terms of the settlement, established implicitly the court’s continuing enforcement authority and directed compliance by FAC with an unmet obligation. No further proceedings having been ordered, it was unquestionably a final judgment that either side could have chosen to appeal. See United States v. Metropolitan Dish Comm’n, 847 F.2d 12, 14 (1st Cir.1988).
The amended judgment also disposes of COSVI’s argument that the original judgment of dismissal under Rule 41(a) did not allow the court to retain jurisdiction because such a retention requires the consent of both parties. The case is now governed by the amended judgment, not the original dismissal. As it happens, COSVI’s premise even as to the original dismissal is mistaken; that dismissal was under Rule 41(a)(2), which permits the court to condition dismissals.3
*192We turn now to the merits of the present appeal. The question is whether the district court permissibly construed the settlement agreement to require COS-VI to write a letter to CMS containing the language mandated by the order under review, namely, an admission that “fraudulent activities took place within our [ie., COSVI’s] organization with regards to the claims at issue.” To sum up at the outset, the district court handled a very odd situation in a practical and (we conclude) defensible manner.
That a major case should be settled on complex terms without anything in writing is itself peculiar. A further curiosity is that neither side offered or proffered any testimony as to what the lawyers or principals actually said in the settlement negotiations. All we know is that the judge was involved in the settlement negotiations and asserts that his new order implements the parties’ intentions.
The law is a shade unsettled as to the standard of review we should apply — specifically, as to what weight, if any, is to be given to the district judge’s construction of a settlement agreement or consent decree. Our own precedent, perhaps surprisingly, suggests that in this circuit review of the interpretation of settlement agreements (as well as consent decrees) is ordinarily de novo 4 But our precedents also recognize that this cannot be a hard and fast rule. See Navarro-Ayala v. Hernandez-Colon, 951 F.2d 1325, 1337-38 (1st Cir.1991) (exception for “public interest” consent decrees).
The present situation- — no evidentiary hearing but personal knowledge by the judge based on his judicial participation in negotiations — argues for some deference. Indeed, some of our cases say that an exception may exist where the district judge has “special knowledge concerning the parties’ intentions.” Navarro-Ayala, 951 F.2d at 1339 n. 17; cf. Malave v. Carney Hosp., 170 F.3d 217, 221 (1st Cir.1999). We think this is only common sense.
Showing even the mildest deference, it is easy to sustain the district court’s construction of the settlement. Apart from relying on its own knowledge, the district court pointed out that COSVI’s May 2002 letter to CMS reflected an understanding that the reopening it was committed to support was to be based on a regulation that permitted reopenings after the ordinary three-year deadline “if it is established that such determination or decision [on a claim] was procured by fraud or similar fault,” 42 C.F.R. § 405.1885(d). The court continued:
The Court FINDS that a letter that is silent on the reasons for the request for reopening the reimbursement claims would clearly fail to satisfy the terms of the settlement. Without direct acknowledgment to CMS of the fact that fraud or similar fault occurred in relation to the reimbursement claims in question, CMS would have no compelling reason to reopen the claims.
The logic is not invincible, but it is pretty strong. In theory, COSVI might have bargained for the extremely guarded admission it actually made in its May 2002 letter, which said that the request to reopen was based on “facts learned by COS-*193VI as a result of its litigation with FAC” and was made pursuant to the regulation. In the real world, it is not easy to imagine FAC settling for anything so ambiguous. Or at least that seems unlikely in the context of the litigation to which the letter refers.
FAC says in its brief (and the trial transcript reflects) that at the outset of the trial COSVI’s own counsel said to the jury about COSVI’s vice president: “Andres Rodriguez ... is the gentleman who may have been involved in bribes, may have been involved in illegal activities and I think the evidence will show it.” In other words, the settlement terminated litigation in which it appears quite likely that FAC was about to establish the facts pretty close to what COSVI now refuses to put in writing to CMS.
There is one other piece of strong evidence in favor of FAC’s construction. The settlement was reached on April 17, 2002. Five days later, on April 22, FAC’s counsel sent a letter to COSVI’s counsel. The letter set forth, among the points that COSVI’s letter to CMS “should include,” the following: “COSVI’s acknowledgment that such .claims must be reopened by [CMS] pursuant to regulation (42 CFR Section 405.1885(d)) as a result of the ‘fraud or similar fault’ of Andres Rodriguez, a party to the prior determination.”
FAC told the district judge that its April 22 letter “was sent immediately upon returning ... from trial in Puerto Rico, was based upon notes taken at trial and during settlement negotiations and was the correspondence most contemporaneous to the trial which was exchanged between the parties after trial.” There is no indication that COSVI wrote back, in the usual fashion of fencing lawyers, to dispute FAC’s version of events.
Instead, COSVI wrote a letter in May 2002 to CMS, bearing out its understanding that a letter had to be written along the lines demanded by FAC but varying the punch line to avoid any direct repetition of what its own counsel appeared to have admitted in open court. COSVI can surely argue that its letter expressed its own understanding of its obligation under the settlement. But the absence of an immediate corrective response to FAC’s April 22 letter is still telling.
FAC’s case for its reading is hardly air tight. A “term sheet” that both parties agree was circulated during settlement negotiations says nothing about any letter, but that itself is not surprising. FAC’s initial position appears to have been that COSVI itself could reopen the claims, and the term sheet took no position on this issue but dealt with how the reopened claims were to be processed and with other elements of the settlement. Conceivably, the letter to CMS was introduced late in the day to deal with COSVI’s asserted inability to reopen the claims on its own.
What is harder for FAC to explain is why it did not immediately protest as inadequate the obscure wording (“facts learned by COSVI”) used when COSVI sent its May 2002 letter to CMS. Yet at that time no one likely knew exactly how CMS would respond to the letter which, after all, made the basic request to reopen, cited the pertinent regulation and could be read to imply the predicate admission of fraud or like fault. Possibly FAC hoped that, without starting a separate quarrel, it would all work out.
On this appeal, as in the district court, COSVI focuses on the language in the district court’s August 2002 amended judgment, which says that the settlement agreement included “[t]he sending of a letter by COSVI to [CMS], on behalf of FAC, Inc., requesting the reopening of the Medicare Part A reimbursement claims.” According to COSVI, the failure of the *194amended judgment to require COSVI to admit to any fraud in its letter to CMS is dispositive of the issue. This is quite unpersuasive.
The amended judgment merely purported to “summarize! ]” the agreement and it did not describe in detail the contents of the required letter. The judge’s subsequent interpretation of the agreement as requiring COSVI to admit fraud in its letter to CMS is not inconsistent with this earlier summary. Indeed, the focus of the August 2002 decision was not on the letter but on the dispute as to whether the local court claims had to be dismissed.
In the end, the district judge’s assessment of COSVI’s obligation is reasonable and more likely right than wrong. Because the parties did not put their precise understanding in writing, anything done to reduce to precise language what was probably only a concept has to be an approximation. Parties who settle cases in the fashion done here create the problem for themselves. The district judge is to be admired for grasping the nettle.
About the only real concern one might have is the absence of an evidentiary hearing. This case appears tailor-made for direct testimony as to what the parties said to one another at the time of the settlement. Like other courts, we have regularly endorsed this course “if there is a genuinely disputed question of material fact regarding the existence or terms of [a settlement] agreement.” Malave, 170 F.3d at 220; see also Kinan v. Cohen, 268 F.3d 27, 32-33 (1st Cir.2001); accord, e.g., Wilson v. Wilson, 46 F.3d 660, 664 (7th Cir.1995).
Yet this assumes that the parties, or at least one of them, requested such a hearing and had testimony worth presenting. Although COSVI claimed at oral argument before us that it did request an evidentiary hearing in the district court, it is decisive that COSVI made no such claim in its briefs here and has not argued that the failure to hold an evidentiary hearing was error. See Lopes v. Met. Life Ins. Co., 332 F.3d 1, 6 n. 8 (1st Cir.2003) (claims made in this court for the first time at oral argument are waived).
Under these circumstances, we can hardly fault the district court for relying upon its own familiarity with the background events and .with inferences that are adequately based on the record. Parties are perfectly free to submit issues for resolution on whatever limited evidence they choose to present. The courts decide disputes on the records before them and do not warrant their findings sub species ae-temitatis.
Whether CMS will or should do anything as a result of the new sentence in the new letter required of COSVI is plainly beyond the scope of this opinion. The parties have left in obscurity the relationship of the original Department claims to FAC’s later ones and the relationship to either of any fraud that occurred or was attempted. All we decide is that FAC is entitled to the benefit of its bargain with COSVI — for whatever that may be worth.
Affirmed.
. The Centers are an entity within the federal Department of Health and Human Services. They are the successor entity to the Health Care Financing Administration ("HCFA”), which was the relevant entity at the time the complaint was filed. We use the term "CMS” throughout, and substitute it where quoted material from the record referred to HCFA.
. See RE/MaxInt’l, Inc. v. Realty One, Inc., 271 F.3d 633, 642 (6th Cir.2001), cert. denied, 535 U.S. 987, 122 S.Ct. 1539, 152 L.Ed.2d 466 (2002); In re Phar-Mor, Inc. Sec. Litig., 172 F.3d 270, 274 (3d Cir.1999); Scelsa v. City Univ. of New York, 76 F.3d 37, 41 (2d Cir.1996); Miener v. Mo. Dep’t of Mental Health, 62 F.3d 1126, 1128 (8th Cir.1995).
. Compare Kokkonen, 511 U.S. at 381, 114 S.Ct. 1673 (decision to retain jurisdiction to enforce settlement agreement after Rule 41(a)(2) dismissal is "in the court's discretion''), with Mimic, of San Juan v. Rallan, 318 F.3d 26, 30 (1st Cir.2003) (district court may exercise continuing jurisdiction after Rule 41(a)(l)(ii) dismissal only if the parties consent); Metro-Goldwyn Mayer, Inc. v. 007 Safe*192ty Prods., Inc., 183 F.3d 10, 14 (1st Cir.1999) (same).
. Whitehouse v. LaRoche, 277 F.3d 568, 573 (1st Cir.2002); The Korman Co. v. Cumberland Farms, Inc., 140 F.3d 331, 333 (1st Cir.1998). Compare, e.g., Washington Hosp. v. White, 889 F.2d 1294, 1299 (3d Cir.1989) (de novo review), with Therma-Scan, Inc. v. Thermoscan, Inc., 217 F.3d 414, 419 (6th Cir.2000) (abuse-of-discretion standard).