BOUDIN, Circuit Judge.

William Walsh was charged with various offenses growing out of a bank fraud scheme and convicted on a majority of the counts. His present appeal is primarily directed at procedural issues. We affirm.

I.

Walsh was indicted in 1992, together with four co-defendants, and charged with conspiracy, twenty-nine counts of bank fraud, and twenty-nine counts of false statements. 18 U.S.C. § 2, 371, 1344, 1014. The substance of the indictment was that Walsh carried out a scheme to defraud Dime Savings Bank of New York ("Dime-NY"). He did so, according to the charge, by directing his employees to obtain 29 specific loans through the use of deceptions so that customers could purchase condominiums from Walsh and his associates.

Walsh's trial occurred in February and March 1994. Taken in the light most favorable to the verdict, United States v. Tuesta-Toro, 29 F.3d 771, 773 (1st Cir. 1994), cert. denied, ___ U.S. ___, 115 S. Ct. 947 (1995), the evidence permitted the jury to find the following. Walsh was a Cambridge, Massachusetts, city councillor, lawyer, and real estate developer. With a group of investors, he purchased apartment buildings or complexes, converted the property into condominiums, and sold the condominiums to customers, using the unit sales to pay off the acquisition financing.

Walsh ordinarily served as a trustee of the realty trust that acquired the building, acted as legal counsel to the trust, and usually served as the trust's representative in the sale of the individual condominium units. During 1986, sales of units in one of the projects started to fall behind schedule and the trust began to have difficulty repaying its acquisition loan. Walsh then discovered that Dime-NY had recently created a wholly owned subsidiary, called Dime Real Estate Services of Massachusetts, Inc. ("Dime-MA"), to originate mortgage loans in Massachusetts. Dime-MA made mortgage loans available rapidly — with no verification of income, assets or down payments — but the loans required a twenty percent down payment and secondary financing was prohibited.

On this failing project, and then on two others, Walsh directed his employees to arrange loans from Dime-MA for unit purchasers and to falsify documents submitted to Dime-MA to conceal the existence of secondary financing (and in some cases third mortgages as well). In the three projects, approximately half the customers defaulted and Dime-MA incurred substantial losses. An investigation ensued, followed by the indictment already described. Three of the four co-defendants pleaded guilty; charges against the fourth co-defendant were abandoned.

Jury deliberations began on March 22, 1994. Following the dismissal of a juror during deliberations, the jury (now reduced to 11 members) continued deliberations, and on March 28, 1994, it returned 41 guilty and 18 not guilty verdicts. Walsh was thereafter sentenced and now appeals. Most of the claims of error concern the dismissal of the juror and its aftermath, so we begin with that subject, starting with a description of the pertinent events.

II.

On March 23, 1994, a note was received from the jury indicating that one of the jurors wished to meet with the judge, adding: "He has several questions and we cannot relate to him in any way, shape, or form." The judge declined to meet with an individual juror, but the following day a court security officer reported that the foreperson was concerned that one of the jurors had become "mentally unstable." After consulting with counsel, the trial judge interviewed the foreperson, and learned of constant interruptions by "juror X", irrelevant statements by juror X about events in his past life, and juror X's efforts to show other jurors written materials consisting of a campaign brochure and a newspaper clipping from his prior efforts to win elective office.

After consulting further with counsel, the trial judge interviewed juror X; as in the judge's interview with the foreperson, counsel and Walsh himself were present. The judge cautioned juror X not to indicate his views on the merits of the case. The interview, which began by focusing on the material that the juror brought into the jury room, involved disjointed and rambling comments by juror X. Juror X also mentioned a self-described "nervous problem" and his general discharge from the military. Some of the questioning was based on questions that had been suggested by the government and defense counsel.

Finally, out of the presence of juror X, the court asked both sides for their position as to whether juror X should be excused, and defense counsel after consultation with Walsh indicated that he "would not object if the Court decided to keep him or eliminate him. . . . [Either way] we would move for a mistrial." The government said that it thought the juror was disabled and should be excused. The trial judge then excused the juror, agreeing that he was "not a person capable of engaging in rational discussions based upon the evidence."

Thereafter, the remaining jurors were sent home for the rest of the day. The following morning Walsh filed a motion for mistrial, arguing that the ability of the remaining jurors to be impartial and open-minded had been undermined by their exposure to juror X. No one at this point knew or claimed to know how juror X had proposed to vote. The court agreed to question the remaining jurors and solicited and received proposed questions from counsel.

Then the district judge, in the presence of counsel and the defendant, questioned each of the 11 jurors individually as to whether juror X had discussed the merits prior to the jury's deliberations, had brought material into the jury room, and had discussed his own personal experiences — and whether the juror being questioned could, to the extent that these events had occurred, put them aside and decide the case impartially based on the evidence presented. Eight of the jurors had been exposed to a campaign brochure and an old newspaper article about one of juror X's campaigns; all of the jurors had heard juror X discuss his personal experiences; and three jurors had heard comments from juror X about the merits of the case prior to the start of deliberations.

Each juror affirmed his or her ability to put aside the campaign material, the personal experiences of juror X, and any comments made by him before deliberations began. Defense counsel challenged three jurors who had heard comments by juror X before deliberations began, the substance of the comments not being revealed. With respect to each of the three jurors, the trial judge made findings that the juror was credible in saying that the pre-deliberation comments of juror X would have no effect. The trial judge then denied a mistrial.

At the request of defense counsel, the trial judge told the jury that it could begin its deliberations from the beginning if it wished; the court also told the jury not to discount a position taken "just because [juror X] took it." The jurors then deliberated for the rest of the day. Returning after a weekend break, they continued deliberations and asked for reinstruction on substantive issues. Late in the same day, they returned the 41 guilty and 18 not guilty verdicts.

1. Walsh's first claim of error is that the trial court erred in dismissing juror X. Walsh argues that there was no psychological testing or psychiatric examination of juror X, and the evidence did not show that he was either mentally incompetent or otherwise incapable of engaging in rational decision-making. In substance, Walsh says that juror X was simply an unpopular, perhaps irritating participant who probably sided with the defendant and whose removal led to a prompt agreement to convict.

Walsh did not make a timely objection on this ground. At the time of the dismissal, his counsel did not object to excusing juror X, or argue for psychiatric testing, or suggest that juror X could be dismissed only if a higher degree of irrationality were shown. Instead, Walsh made clear his intention to move for a mistrial; and when the mistrial motion was filed, the ground — inconsistent with the contention now made — was that juror X was someone whose "psychiatric problems" had been "clearly demonstrated" and whose "negative influence" on other jurors was apparent.

Although, for these reasons, the objection now made is reviewable only for plain error, the dismissal of juror X was not error at all. Federal Rule of Criminal Procedure 23(b) permits the judge to excuse a juror "for just cause" during deliberations and to allow the remaining 11 jurors to reach a verdict. The trial judge has substantial discretion in exercising this responsibility and may remove the juror when "convinced that the juror's abilities to perform his duties [have] become impaired." United States v. Huntress, 956 F.2d 1309, 1312 (5th Cir. 1992), cert. denied, 113 S.Ct. 2330 (1993). Similarly, in United States v. Molinares Charris, 822 F.2d 1213, 1223 (1st Cir. 1987), we permitted a judge to excuse a juror who had taken a tranquilizer pill and appeared somewhat unstable.

The trial judge carefully and repeatedly consulted with counsel in determining the course of the inquiry and the questions to be put to juror X. See United States v. Chorney, 63 F.3d 78, 81 (1st Cir. 1995). The transcript of juror X's voir dire, which need not be repeated in detail, gave the trial judge ample basis for concluding that the juror was not able to perform his duties. Whether or not juror X was incompetent as a juror under 28 U.S.C. § 1865(b)(4), "just cause" existed under Rule 23(b) for his removal in this case. See United States v. Reese, 33 F.3d 166, 172-73 (2d Cir. 1994), cert. denied, ___ U.S. ___, 115 S.Ct. 756 (1995) (just cause not limited to incompetence).

There is no evidence that the trial judge knew that juror X favored acquittal, if indeed juror X did. Nor did defense counsel make any such suggestion when he acquiesced in the dismissal of juror X. If anything, Walsh's mistrial motion suggested that juror X might be hostile to Walsh because Walsh was a lawyer and politician. Dismissal of a known holdout juror raises an entirely different question. Compare United States v. Hernandez, 862 F.2d 17 (2d Cir. 1988), cert. denied, 489 U.S. 1032 (1989).

2. Walsh's second objection is that even if juror X was properly removed, the court erred in permitting the 11 remaining jurors to return a verdict. Walsh's broadest grounds for this objection are legal: he claims that the Constitution does not permit a jury with only 11 jurors to decide a federal criminal case over the objection of a defendant. He also asserts that Rule 23(b) — which does permit this course — could not be lawfully enacted through the Rules Enabling Act procedures. 18 U.S.C. § 3771, 3772 (1982).

Neither of these objections was made at the time that the district court was determining whether to permit the 11 remaining jurors to deliberate and, accordingly, both objections are subject to review only for plain error. It is true that both issues were raised in the trial court after the verdict by a post-verdict motion for dismissal or a new trial. But the usual rule is that an objection must be made known at the time that the court is making its decision to act, e.g., United States v. Gonzalez-Torres, 980 F.2d 788, 791 (1st Cir. 1992), and here the proper time to raise the objections was when the court was deciding whether to continue with 11 jurors.

In this case, in any event, the standard of review does not matter as to the constitutional claim because in Williams v. Florida, 399 U.S. 78, 103 (1970), the Supreme Court said that the 12-member jury was not required by the Constitution and that Congress and the states could select a different number. We think that Williams effectively answers the claim that 11 jurors are too few. A number of circuits have held that a jury of 11 can constitutionally decide a federal criminal case, without consent of the parties, where a juror has been removed for cause. E.g., United States v. Ahmad, 974 F.2d 1163, 1164 (9th Cir. 1992).

Williams directly rejects the argument that the historical number of jurors is binding — how many would be too few is not an issue in this case — and we think that this conclusion is not altered by Walsh's attempt to rephrase the challenge as a concern for a "unanimous" jury. The Supreme Court has not said whether a less-than-unanimous verdict is acceptable. Compare Apodaca v. Oregon, 406 U.S. 404 (1972). But whether it is or not, we think that rendition of a verdict agreed to by all jurors, after one juror with unknown views has been dismissed for cause, is a unanimous verdict.

The gist of Walsh's claim under the Rules Enabling Act is that Congress might be able to alter the requisite number from 12 to 11 but that Rule 23(b) was adopted — under the procedures specified by the Rules Enabling Act — by judicial action coupled with Congress' failure to veto the change. Inaction, says Walsh, is not enough for a fundamental change. The Second Circuit has concluded, however, that this change can be accomplished through the enabling procedures. United States v. Stratton, 779 F.2d 820, 831 (2d Cir. 1985), cert. denied, 476 U.S. 1162 (1986).

Rules that are "strictly procedural" can be adopted through the Rules Enabling Act without an affirmative vote by Congress, Burlington Northern Railroad Company v. Woods, 480 U.S. 1, 5 (1987), and this extends to rules that fall "within the uncertain area between substance and procedure, [but] are rationally capable of classification as either." Id. (quoting Hanna v. Plumer, 380 U.S. 460, 472 (1965)). In view of the defendant's failure to make a timely objection, we need not decide this claim outright but are satisfied that the use of the 11-member jury did not constitute "clear error" based on the Rules Enabling Act claim.

3. Walsh did make in timely fashion an objection that this jury was not capable after juror X's discharge of rendering a fair and impartial verdict. When this issue was raised by Walsh immediately after the discharge, the district court properly undertook "an adequate inquiry to determine [what had happened and] . . . whether it was prejudicial." United States v. Ortiz-Arrigoitia, 996 F.2d 436, 442 (1st Cir. 1993), cert. denied, ___ U.S. ___, 114 S.Ct. 1368 (1994). A trial judge enjoys discretion to determine the scope of the inquiry in deciding whether the jury has been tainted. United States v. Boylan, 898 F.2d 230, 258 (1st Cir.), cert. denied, 498 U.S. 849 (1990).

As already noted, the district court judge separately examined each of the jurors, asking his own questions as well as various questions suggested by counsel. In each instance, the judge received a forthright declaration that the juror was not going to be affected by the personal comments made by juror X, by materials he had brought into the jury room, or — in the case of three jurors — by the comments that juror X made about the merits before deliberations began. The judgment of the trial judge, who can appraise the jurors face to face, deserves great weight.

Although Walsh now complains that the district judge limited his own questioning unduly — in an effort to avoid learning how the jurors were leaning — Walsh did not press for more detailed inquiry at the time. The trial judge treads a delicate line in this kind of inquiry. Assuming arguendo that Walsh is right in saying that Fed.R.Evid. 606(b) does not apply prior to the verdict, there are still obvious good reasons for a trial judge to avoid learning how an individual juror is leaning. United States v. Rengifo, 789 F.2d 975, 985 (1st Cir. 1986).

Although Walsh now argues that there is a substantial chance that the jurors were prejudiced by juror X, nothing in the record makes this at all likely. In addition to the jurors' own denials, we note that the brochure had nothing directly to do with the trial; there is no reason to believe that a newspaper article brought in by a juror regarding his prior political campaign contained anything material; and judging by the voir dire of juror X, his personal experiences were also not germane to the trial.

Walsh now argues that juror X was hostile to lawyers and politicians (Walsh was both) and that this view may have been passed on to the other jurors. In fact, juror X's brochure was more qualified, expressing (in a description of X's "positions") objections to "the [unspecified] unethical ethics practiced by certain members of the bar" and "machine controlled politics and . . . [unspecified] dirty tricks." The jurors said that they paid little attention to the pamphlet. Further, it is Walsh who now takes the position that juror X favored Walsh, which hardly suggests that juror X was denigrating Walsh.

Finally, Walsh now complains that by discharging juror X the court led the jury to think that juror X's views should be disregarded. In fact, the judge expressly cautioned the jury not to discount views simply because they were earlier expressed by juror X. Walsh also says that the jury should have been directed to start its deliberations anew. The judge told the jury that it was entitled to start anew. We think that this was all that was either useful or necessary. At the time, Walsh raised no objection to the instructions given.

4. Walsh's last claim of error based on the juror X episode relates to a post-trial event. According to an affidavit from Walsh's secretary, she received a call a week or so after the verdict from someone identifying himself as juror X who said he had been on the defendant's side, that the defendant had been "railroaded," and that she (the secretary) "would not believe what went on in the jury room." About a month later, Walsh submitted this affidavit in support of a request that the jury and juror X be subject to further voir dire or authorized inquiry by counsel.

The district court declined to hold such a post-verdict inquiry or to authorize discussions with the jurors or to grant a new trial based on the affidavit. Walsh now argues that because the parties were barred from unsupervised contact with the jurors after the verdict, United States v. Kepreos, 759 F.2d 961, 967 (1st Cir.), cert. denied, 474 U.S. 901 (1985), the trial court had an obligation to conduct an investigation itself. The abuse of discretion standard governs this claim, see Boylan, 898 F.2d at 258, and we think that there was no such abuse in this case.

The restrictions on post-verdict contact and the limitations on juror testimony about deliberations, Fed.R.Evid. 606(b), exist to protect important interests in the finality of the verdict and the privacy of the deliberations. See Tanner v. United States, 483 U.S. 107, 120 (1987). The affidavit contains only general rhetoric from juror X and no specific allegations of misconduct. Given what the district court already knew about juror X, the telephone call — assuming (as we do) that it came from juror X — did not require any further inquiry.

III.

Walsh's brief raises two further issues, both unrelated to juror X. The first claim relates to the government's admitted failure to turn over certain documents in a timely fashion. The documents related to Frances Schwartz, a senior attorney working for Walsh who was assigned to the three development projects involved in this case. Schwartz was indicted with Walsh and was one of the co-defendants who pled guilty to the conspiracy count and testified against Walsh at trial.

On direct examination, Schwartz gave damaging testimony against Walsh. In addition to identifying a number of documents and describing the operations of Walsh's office, Schwartz testified to discussions and correspondence with Walsh that — as recounted and interpreted by Schwartz — confirmed Walsh's knowing participation in and direction of the fraud. Schwartz' testimony was thus quite damaging, although another co-defendant who pled guilty also testified that Walsh knowingly directed the concealment of the secondary financing.

Early in her cross-examination, Schwartz mentioned that she had "daytimers" or calendars that she had used to refresh her recollection. Later, on re-cross, she mentioned that she had allowed the government to review the daytimers and make copies of them. The defense immediately objected that it had never received the daytimers. The government said that these daytimers should have been disclosed earlier but had been overlooked when other materials from Schwartz had been made available to Walsh's counsel. Copies of the daytimers were provided to the defendant later that day.

Following a timely motion by Walsh to dismiss the case because of this delay, the trial court denied the motion, finding that Walsh's strategy would not have been substantially different if the daytimers had been disclosed earlier. The court instructed the jury that the government had failed in its discovery obligation, and it allowed Walsh to recall Schwartz to continue her examination, using the daytimers to try to establish inconsistencies between Schwartz' prior testimony and the daytimers. Walsh now complains that this was inadequate.

This court previously considered the issue of delayed disclosure of impeachment material required to be disclosed under the Jencks Act. United States v. Arboleda, 929 F.2d 858, 862-65 (1st Cir. 1991). We said that the critical question was whether the delay had "prevented [the material's] effective use by the defense," id. at 862, and that some showing of prejudice was required beyond mere assertions that the defendant would have conducted cross-examination differently. Id. at 864. Cf. United States v. Lanoue, No. 95-1140, slip op. at 34 (1st Cir. 1995). Delayed disclosure of Brady material is subject to the same rule. See United States v. Osorio, 929 F.2d 753, 758 (1st Cir. 1991).

On this appeal, Walsh argues that if his trial counsel had received the daytimers earlier, he would have focused at the outset on the alleged inconsistencies between Schwartz' testimony and the daytimers instead of attempting to cast doubt on the reliability of her memory. In fact, the initial cross-examination did not focus on Schwartz' memory but rather on her veracity, which the defense counsel sought to undermine by emphasizing her prior drug use and her desire for a lenient sentence. And when Schwartz was subject to further cross after the daytimers had been produced, Walsh's counsel paid minimal attention to the supposed inconsistencies.

Walsh says that when Schwartz was recalled for further cross after the daytimers had been produced, it was too late to cross-examine effectively on inconsistencies because she had been "well prepared by the government to explain away any inconsistencies." As it happens, there is no evidence of any such discussion after the daytimers first became an issue. As to preparation prior to the original direct examination, the government was entitled to prepare the witness, and the risk of facing an initially prepared witness would have existed whether or not the daytimers had been produced.

Walsh's final claim of error, a claim raised in the district court and rejected there, is that the evidence failed to show that the victim was a federally insured financial institution. At the time of the fraudulent filings, 18 U.S.C. § 1344 aimed at schemes to defraud "a federally chartered or insured financial institution" or to obtain property owned by, or under the custody or control of such an institution through falsehoods. See United States v. Brandon, 17 F.3d 409, 424 n. 11 (1st Cir.), cert. denied, ___ U.S. ___, 115 S.Ct. 81 (1994). Walsh's argument turns on the fact that Dime-NY was a federally insured bank, but Dime-MA — the immediate maker of the loans — was not.

The government quotes to us in response a statement from Brandon, 17 F.3d at 426, that

the government does not have to show the alleged scheme was directed solely toward a particular institution; it is sufficient to show that defendant knowingly executed a fraudulent scheme that exposed a federally insured bank to a risk of loss.

That language, however, was directed to the scienter requirement, and not to the nexus claim made here. As it happens, the Brandon court also rejected a nexus argument somewhat similar to Walsh's argument here but on different facts. The intermediaries with whom defendants in Brandon dealt were mortgage brokers who forwarded the fraudulent applications to the federally insured bank which individually approved the loans and forwarded the money back to the mortgage brokers. See id. at 423, 426-27 n. 16.

While the nexus in Brandon was different — one can argue about whether it was closer or more remote — Brandon does confirm that a defendant can violate section 1344 by submitting the dishonest loan application to an entity which is not itself a federally insured institution. Here, Dime-MA was practically an alter ego of Dime-NY: it was a wholly owned subsidiary of Dime-NY; all of the subsidiary's directors and principal officers were officers of the parent; and Dime-MA was subject to examination by the same federal bank examiners as Dime-NY and reported its result on a consolidated basis.

Further, focusing on the loan process, the connection between the defendant and the federally insured victim is even tighter. Dime-NY provided all of the funds for Dime-MA both for its operating expenses and to fund mortgage closings. Dime-NY determined what loan products should be offered and, on the closing of a loan by Dime-MA, the mortgage was immediately assigned to Dime-NY, which then serviced the loan. For most practical purposes, and certainly for the purposes underlying section 1344, the mortgage fraud perpetrated against Dime-MA was effectively a fraud against Dime-NY.

We agree that there must be some outer limits to section 1344. For example, ruinous fraud directed against a major bank customer, but unrelated to a customer's deposits in or loans from the bank, might ultimately harm the bank itself, if only through loss of a valued customer. But here, as in Brandon, "this case presents a situation of direct harm to [a federally insured bank] resulting from a scheme specifically designed to fraudulently avoid the requirements of that federally insured bank in order to obtain funds originating directly from [that bank]." 17 F.3d at 427 n. 16. As in Brandon, we confine our affirmance to the present facts and decline to contrive general rules to govern myriad variations.

Affirmed.