OPINION OF THE COURT
GIBBONS, Circuit Judge.Harold Oxman and William H. Pflaumer appeal from sentences imposed following their convictions of mail fraud, 18 U.S.C. § 1341 (1982), and conspiracy to commit mail fraud, 18 U.S.C. § 371 (1982). Their convictions arise from a scheme to defraud the states of Pennsylvania, Maryland, and New Jersey of fuel excise and road use taxes properly due from a corporation, Wm. H.P., Inc. (WHP), owned by Pflaumer. The defendants contend that certain errors in the conduct of the trial and the government’s deliberate nondisclosure of evidence favorable to the defense require a new trial. We conclude that Oxman’s conviction must be affirmed, but that in Pflaumer’s case an erroneous charge and the deliberate nondisclosure of evidence favorable to the defense, require a new trial.
I.
The Scheme
Pflaumer is the sole stockholder of WHP, a corporation engaged in the trucking business in Philadelphia, Pennsylvania. Charles Gillan was the President of WHP; Raymond Hill was its Philadelphia Terminal manager, Ralph Wille its comptroller. Pflaumer owns several other businesses, including a Philadelphia brewery, C., Schmidt & Sons, Inc., and another trucking firm, Burgmeyer Bros., Inc. Oxman was, at relevant times, a commission salesman for Park Oil Co., a fuel oil supplier in Newark, Delaware. Before it went out of business, Park Oil was owned by Oxman’s cousin, Frank Jock. United Fuel Oil and Burner Co. is a fuel oil supplier in Philadelphia, owned by John Luciano.
WHP’s diesel fuel is purchased in bulk, delivered to the WHP terminal in Philadelphia. As a bulk purchaser, WHP is entitled to file monthly returns and make monthly payments of the nine cents-per-gallon Pennsylvania excise tax on diesel fuel. Pennsylvania also charges a road use tax of nine cents per gallon. Taxpayers are permitted to offset any state fuel excise taxes against the road use tax. Maryland and New Jersey both have similar fuel excise and road use taxes, at rates of eight cents per gallon. The scheme charged in the indictment involved the alteration of invoices for diesel fuel supplied to WHP at its Philadelphia terminal in order to indicate, falsely, that the fuel was delivered to WHP in Maryland or New Jersey. This permitted the filing of false monthly excise tax returns with the Commonwealth of Pennsylvania which underreported the excise taxes due to that state. WHP filed no excise tax returns with Maryland and New Jersey but, on road use tax returns, claimed credit for excise taxes on bulk purchases allegedly delivered in those states. Mailings in furtherance of the scheme took place between April of 1978 and June of 1979. The indictment charges 21 substantive counts of mail fraud and a single count of conspiracy to commit mail fraud. Those indicted for mail fraud are Gillan, Oxman, Pflaumer and Hill. These four and Luciano are charged in the conspiracy count, which names Frank Jock as an unindicted co-conspirator. Willie is not named either as a defendant or as a co-conspirator.
*1301Prior to trial Gillan pleaded guilty to all counts. Pursuant to a plea agreement with the government, Luciano pleaded guilty to the conspiracy count. Oxman, Pflaumer and Hill stood trial. At trial the government sought, through the testimony of Luciano, Jock, Wille, and others, to establish that Oxman, Pflaumer and Hill all knew of the false invoice and tax fraud scheme, details of which are referred to hereafter. The theory of the defense was that while the mail fraud occurred, it was a matter about which the defendants had no knowledge and for which Gillan was solely responsible. Gillan did not testify.
II.
Pre-trial Discovery
Before trial Pflaumer tendered a number of discovery requests to the government. One series of requests sought all evidence having to do with criminal conduct on the part of any person to be called as a prosecution witness.1 A second series sought documents evidencing benefits conferred on, or agreements made with, potential government witnesses, including, specifically, immunity agreements.2
In response to these quite detailed requests, the government disclosed to counsel the prior criminal records of Luciano and Frank Jock, and disclosed that the government had entered into agreements with both men in exchange for their “truthful cooperation.” Luciano’s agreement involved his pleading guilty to the conspiracy count and to an independent charge of federal tax evasion. In exchange for his cooperation, the government agreed to forego further prosecution of Luciano and to report his cooperation at sentencing. Jock was already serving a term of incarceration for prior convictions. For his cooperation the government agreed to advise the Parole Board that he was cooperating and would be a good candidate for parole.
Although the requests were specific and covered all witnesses, the Luciano and Jock agreements were the only agreements disclosed to the defense. This, despite the fact that, as the United States Attorney well knew, the United States had entered into a written agreement, quoted in full in the margin,3 by which Wille was given use immunity with respect to information or evidence “relative to the federal investigation into certain activities of Charles Gillan and others during the period between June 2, 1978 and December, 1979.” App. at 1529. The Wille agreement was not dis*1302closed to the court or to counsel. Counsel for the defendants were unaware of the agreement until June 15, 1983, the day after the jury verdict, when Pflaumer's counsel heard about it from another attorney.
III.
Claimed Trial Error
A. Vouching for the Credibility of Jock and Luciano
Oxman and Pflaumer both contend that a new trial is warranted because the prosecuting attorney, during the trial and in closing argument, vouched for the truthfulness of the witnesses Frank Jock and John Luciano.
The alleged instances of vouching during the trial occurred when on direct examination the prosecuting attorney called to the jury’s attention that the plea agreements, which had been disclosed to defense counsel, obliged them to testify truthfully. The government could reasonably anticipate that the beneficial features *1303of the agreements would be used for impeachment purposes on cross-examination. If they were so used, reference to the condition requiring truthful testimony would be proper rehabilitation. See, e.g., United States v. Rohrer, 708 F.2d 429, 433 (9th Cir.1983); United States v. Edwards, 631 F.2d 1049,1051-52 (2d Cir.1980). Since the government could reasonably anticipate such impeachment, it was not improper to anticipate it on direct examination by disclosing the truthful-testimony condition. See United States v. Henderson, 717 F.2d 135, 137-38 (4th Cir.1983), cert. denied, — U.S. -, 104 S.Ct. 1006, 79 L.Ed.2d 238 (1984). Indeed, some courts have held that reference to the truthful-testimony condition is proper even when the witness’ credibility is not put in issue. See United States v. Winter, 663 F.2d 1120, 1133 (1st Cir.1981); United States v. Hedman, 630 F.2d 1184, 1198-99 (7th Cir.1980); United States v. Craig, 573 F.2d 513, 519 (7th Cir.), cert. denied, 439 U.S. 820, 99 S.Ct. 83, 58 L.Ed.2d 111 (1978).
The vouching during closing argument on which Oxman and Pflaumer rely occurred when the prosecuting attorney argued:
John Luciano’s testimony and the agreement that he entered into with the government is tied in to a condition that is written, and it is carved in stone, and there is no way around it. John Luciano must tell the truth. If he does not, he is going to be in far worse trouble than he already is.
App. at 1424-25. These two sentences, while hyperbolic, accurately describe the agreement with Luciano and the likely consequences of its breach. They were, therefore, appropriate responses to the defendants’ attack on Luciano’s credibility. Several sentences later, however, the prosecuting attorney continued:
If I made a mistake in entering into that deal with John Luciano, then I personally will have to be responsible for it.
App. at 1425. The reference to counsel’s personal responsibility in the event the jury believed him “mistake[n] in entering into that deal with John Luciano” was not an appropriate response. But this isolated reference to the possibility that the prosecuting attorney would be personally disadvantaged for entering injudiciously into a plea bargain with a witness, while in our view improper, is not ground for a new trial. Fed.R.Crim.P. 52(a).
B. Exclusion of Evidence
1. Other crimes evidence
Oxman contends that the trial court erred in excluding the testimony of an accountant that Luciano had defrauded Temple University of over $350,000. The trial court properly held that this evidence of specific instances of misconduct of a witness was inadmissible by virtue of Fed.R. Evid. 608(b).
2. Evidence of other tax payments
Pflaumer urges that the trial court erred in excluding evidence that during the time WHP was defrauding the states of Pennsylvania, Maryland, and New Jersey of about $200,000 in excise taxes, another company owned by him, C. Schmidt & Sons, Inc., was paying the federal government about $40 million a year in excise taxes. Pflaumer contends that this evidence would tend to negate any motive on his part to participate in the small fraud committed by WHP. The trial court excluded the testimony under Fed.R.Evid. 403 on the ground that any probative value it might have would be outweighed by the possibility of jury confusion. We find no abuse of discretion in that ruling.
C. Jury Instructions
Pflaumer contends that the trial court erred in refusing to give a proffered instruction as follows:
The indictment charges that the last tax return mailed as part of the alleged scheme and conspiracy was dated June 30, 1979. It was at that time, therefore, that the alleged conspiracy or scheme terminated. You may not find any defendant guilty merely on the basis of anything he did or said after June, 1979. *1304This is because a conspirator cannot join a conspiracy or scheme after its attempts to achieve illicit objectives [have] ended____ In short, if you find that the government has not proved beyond a reasonable doubt that Mr. Pflaumer knowingly and willfully became a member of the alleged conspiracy or scheme to defraud Pennsylvania, New Jersey and Maryland of diesel fuel taxes prior to June, 1979, you must acquit Mr. Pflaumer of all charges.
App. at 1498. To understand this request one must appreciate what was charged in the indictment.
The first twenty-one counts alleged twenty-one separate mailings, each a separate substantive violation of 18 U.S.C. § 1341 (1982). The first mailing took place on April 3, 1978, the last on June 30, 1979. These mailings were alleged to violate section 1341 because they were for the purpose of executing a scheme or artifice, which arguably extended over a longer period, to defraud the three states. The scheme itself does not, however, violate federal law; only the use of the mailings to effectuate the scheme does so. See Kann v. United, States, 323 U.S. 88, 94-95, 65 S.Ct. 148, 150-151, 89 L.Ed. 88 (1944); United States v. Tarnopol, 561 F.2d 466, 471 (3d Cir.1977).
The twenty-second count charges a conspiracy in violation of 18 U.S.C. § 371 (1982), which proscribes conspiracies “to commit any offense against the United States.” -The conspiracy charged in the indictment alleges:
a scheme and artifice to defraud the Commonwealth of Pennsylvania, the State of New Jersey, and the State of Maryland by means of false and fraudulent pretenses and representations, and for the purpose of executing such scheme and artifice to defraud and attempting so to do, to place in and cause to be placed in an authorized depository for mail matter, certain matter as described in Counts One through Twenty One, in violation of Title 18 United States Code, Section 1341____
App. at 17. As with the substantive section 1341 counts, the fraudulent scheme itself does not violate the federal conspiracy statute. The federal conspiracy must be to violate a federal substantive law — in this instance, the substantive prohibition against the use of the 21 mailings alleged in the first 21 counts.
Pflaumer’s point in making the quoted request for charge was straightforward. A conspiracy to violate a substantive prohibition in a federal statute ends when the unlawful object has been accomplished. Grunewald v. United States, 353 U.S. 391, 406-15, 77 S.Ct. 963, 974-79, 1 L.Ed.2d 931 (1957). The object of the conspiracy charged in the indictment was the use of the mails to effectuate a scheme to defraud three states of tax revenues. Although the scheme may have continued until 1981 — an issue on which we express no opinion — the use of the mails to effectuate the scheme ceased in 1979. Thus, the conspiracy charged in the indictment ended in June of 1979, and in order to convict the jury must have found that Pflaumer joined the conspiracy before that date.
From the viewpoint of Pflaumer’s defense, this issue was quite significant. There is evidence from which the jury could have found that when state agencies conducted audits of WHP after June of 1979, Pflaumer’s actions tended to conceal the scheme. This evidence was admitted at trial not to show Pflaumer’s membership in the conspiracy — that was prohibited by Grünewald — but to raise an inference of consciousness of guilt. See United States v. Mastropieri, 685 F.2d 776, 790-91 (2d Cir.), cert. denied, 459 U.S. 945, 103 S.Ct. 260, 74 L.Ed.2d 203 (1982); App. at 1665-66. The jury, however — not finely versed in these subtle distinctions — might have mistakenly applied evidence of concealment to show membership in the conspiracy. Hence Pflaumer requested an instruction expressly charging that the jury must, in order to convict, establish membership in the conspiracy based on evidence of acts before June of 1979.
*1305The court refused the requested instruction. The government defends that refusal on several grounds. One is that post-June 30, 1979 efforts at concealment “constitute a ratification of the co-conspirator’s antecedent conduct including any mailings made in furtherance of the scheme.” Appellee’s Br. at 50. That contention must be rejected because it conflicts with Grünewald v. United States, supra, which holds that acts of concealment after the principal object of the conspiracy (here use of the mails to defraud three states of tax revenues) has been completed do not comprise part of the conspiracy. See 353 U.S. at 403-06, 77 S.Ct. at 973-75.
Alternatively, the government contends that the charge as given was substantially in accordance with the request. The court charged in part:
In order to find any defendant guilty on Count 22 [conspiracy] you must find all the following elements beyond a reasonable doubt:
(1) That the conspiracy described in the Indictment was existing at or about the time alleged;
(2) That the particular defendant willfully became a member of the conspiracy with knowledge of its illegal objectives;
(3) That one of the conspirators thereafter knowingly committed at least one of the overt acts alleged in the indictment, at or about the time charged and during the period of the conspiracy; and
(4) That the overt act was knowingly done in furtherance of some object or purpose of the conspiracy as charged.
If you should find beyond a reasonable doubt from the evidence that the existence of the conspiracy charged in the Indictment has been proved and that during the existence of that conspiracy one of the overt acts alleged was knowingly done by one of the persons found by you to have been a conspirator, in furtherance of some object or purpose of the conspiracy, that proof of the conspiracy is complete; and it is complete as to every person found by you to have been willfully a member of that conspiracy at the time the overt act was committed, regardless of which of the conspirators did the overt act.
App. at 1458-59. Had the court stopped at this point, we could have accepted the government’s alternative argument. This much of the charge, while less specific in advancing Pflaumer’s theory than the requested instruction, is not inconsistent with the indictment and complies with Grünewald. But the court continued:
The Indictment alleges numerous overt acts in furtherance of each conspiracy. What must be shown is that after the defendant, whom you are considering, joined the conspiracy, some one of the conspirators performed an overt act in furtherance of the conspiracy.
Keep in mind that the Government must establish beyond a reasonable doubt that at least one of the overt acts as alleged in the Indictment occurred while the conspiracy was still in existence. That is, the overt act must have occurred between approximately November, 1976, and January, 1981. You’re not to consider any acts that occurred after the conspiracy was terminated.
App. at 1467-68 (emphasis added). Thus, the court charged that the jury may find that the defendants joined “each conspiracy” as late as January, 1981. Plainly the court here rejected Pflaumer’s theory that a section 371 conspiracy to violate the mail fraud statute ends with the last mailing, here in June of 1979. The reference to plural conspiracies, and to overt acts after June 30, 1979, suggests that the court focused on a conspiracy to defraud the states as itself a violation of section 371. Such a conspiracy would not be a violation of section 371, nor does the indictment so charge. In any event, because the charge permitted the jury to find that the defendants joined the conspiracy after its principal object had terminated in June of 1979, the charge as given is inconsistent both with the indictment and with Grünewald v. United States, supra.
*1306Finally, the government notes that, as the penultimate paragraph of the charge above (denoted [1]) indicates, the court charged that “[w]hat must be shown is that after the defendant ... joined the conspiracy, some one of the conspirators performed an overt act in furtherance of the conspiracy.” App. at 1467. The last overt act charged in the indictment occurred in March of 1978. The jury had a copy of the indictment. Therefore, the government maintains, although the charge permitted the jury to find an overt act as late as January, 1981, the jury must have found an overt act no later than March of 1978. Thus, the government concludes, the jury must have found that Pflaumer joined the conspiracy before that date. Appellee’s Br. at 47.
We cannot agree. We see no basis for believing that the jury would ignore the court’s instruction that it may find that the last overt act occurred in 1981. And we view with skepticism the suggestion that the jury would have understood which of the court’s instructions to obey and which to ignore. Moreover, the jury heard a significant amount of evidence suggesting possible concealment by Pflaumer after 1979, inviting it to do exactly what Pflaumer feared — find that he joined the conspiracy after the last mailing. Thus, far from suggesting that the jury may have ignored the court’s instructions, the evidence suggests that it may well have done precisely what the jury is expected to do: abide by them.
We must also determine whether the error is harmless. Fed.R.Crim.P. 52(a). If the evidence connecting Pflaumer to the conspiracy prior to June 30, 1979 were overwhelming, we could accept such an analysis. But as we observe in Part IV infra, that is not the case. Although some evidence tended to connect Pflaumer to the conspiracy before 1979, it is not sufficient to negate the possibility that the jury established his membership based in significant part on the post-1979 evidence. Because there is a reasonable likelihood that the jury may have considered acts of concealment as evidence of membership in the conspiracy alleged, we cannot find the error harmless.
Pflaumer’s counsel preserved his objection to this aspect of the charge at its completion. App. at 1486. Oxman neither requested a charge with respect to the termination date of the conspiracy nor objected to the charge, evidently for the reason that none of the evidence bearing on possible concealment by Pflaumer after June 30, 1979, implicated him. Thus the error in the charge affected only Pflaumer.
IV.
The Brady Violation
As we noted in Part II above, Oxman and Pflaumer made specific requests for information on agreements with prospective government witnesses, and the government knowingly elected not to reveal to the court or counsel the agreement with Wille quoted at note 3, supra. Both appellants rely on this concealment as a ground for a new trial. Under governing precedents a separate analysis of the impact of the concealment must be made for each defendant.
A. The Governing Law
In 1942, prior to the adoption of the Federal Rules of Criminal Procedure, a unanimous Supreme Court held that the deliberate suppression by state authorities of evidence favorable to a defendant violates due process. Pyle v. Kansas, 317 U.S. 213, 216, 63 S.Ct. 177, 178, 87 L.Ed. 214 (1942).4 The Pyle v. Kansas rule was *1307uncomplicated, and in this court it was applied in a relatively straightforward manner. It was not limited to the use of perjured testimony or to instances in which a request for exculpatory evidence was made. See United States ex rel. Almeida v. Baldi, 195 F.2d 815, 820 (3d Cir.1952) (state’s failure to disclose ballistics evidence helpful to defense), cert, denied, 345 U.S. 904 (1953); United States v. Rutkin, 212 F.2d 641, 644-45 (3d Cir.1954) (federal failure to disclose statement of a potentially useful defense witness); United States ex rel. Thompson v. Dye, 221 F.2d 763, 765 (3d Cir.) (state failure to disclose evidence of defendant’s intoxicated condition at time of arrest), cert, denied, 350 U.S. 875, 76 S.Ct. 120, 100 L.Ed. 773 (1955).
By the time these cases were decided, however, the Federal Rules of Criminal Procedure had been adopted. Rule 16, dealing with pretrial discovery in criminal proceedings, was drafted on the assumption that there was no right to discovery in criminal cases.5 That assumption, which is still the underlying predicate for Rule 16, produced inevitable tension with the due process rule of Pyle v. Kansas. That rule seemed to impose on prosecutors an affirmative obligation to make evidence available arguably favorable to a defendant, even though for the most part only a limited procedure existed for defendants to demand such discovery. Surprisingly, the Advisory Committee Note on the original Rule 16 made no reference to Pyle v. Kansas. Not surprisingly, this court’s opinion in United States v. Rutkin, our first occasion to apply the due process rule to a federal prosecutor’s failure to disclose, relied on Pyle v. Kansas and made no mention of Rule 16. Obviously the supposed general rule against discovery in criminal cases, at best a federal common law rule, yielded to a due process requirement.
In Brady v. State, 226 Md. 422, 174 A.2d 167 (1961), the Maryland Court of Appeals held that the state’s failure to disclose an arguably exculpatory statement violated the Pyle v. Kansas rule. The Maryland court made no mention of any request by the defendant. Id. at 427, 174 A.2d at 169 (relying on United States ex rel. Almeida v. Baldi, supra, and United States ex rel. Thompson v. Dye, supra). The statement, a confession by an accomplice named Bob-lit, asserted that Boblit had strangled the *1308victim. The court held that had the jury been informed that Boblit was the strangler, it might not have imposed the death penalty. Because the evidence did not bear on whether Brady was guilty of first degree murder, however, the court limited a new trial to the death-penalty issue alone. Id. at 430-31, 174 A.2d at 171. Only Brady petitioned for certiorari. 371 U.S. 812, 83 5. Ct. 56, 9 L.Ed.2d 54 (1962). Thus, the only issue presented to the Supreme Court was whether the state prosecutor’s concededly unconstitutional suppression of evidence required a new trial on guilt as well as punishment. The issue of whether the prosecutor’s suppression of evidence violated due process was neither briefed nor argued.6
The Supreme Court affirmed that the separate trial on punishment did not violate due process. Brady v. Maryland, 373 U.S. 83, 90, 83 S.Ct. 1194, 1198, 10 L.Ed.2d 215 (1963). Justice Douglas’ opinion is unfortunately unanalytical, in that it expresses approval of two of the Third Circuit cases mentioned above, of Pyle v. Kansas, and of the holding of the Maryland Court of Appeals that the Pyle v. Kansas rule had been violated, none of which had been briefed, argued, or was at issue in the appeal. Moreover, the opinion adds:
We now hold that the suppression by the prosecution of evidence favorable to an accused upon request violates due process where the evidence is material either to guilt or to punishment, irrespective of the good faith or bad faith of the prosecution.
373 U.S. at 87, 83 S.Ct. at 1196-97 (emphasis added). Of course, this “holding” of the Court was entirely unnecessary to its opinion; the Maryland Court of Appeals had held that “there was a duty on the State to produce the confession of Boblit that he did the actual strangling or at least to inform counsel for the accused of its existence,” 226 Md. at 427, 174 A.2d at 169, and Maryland had not appealed that holding. And for no reason discernible in the opinion, Justice Douglas’ formulation introduced the words “upon request.” Possibly he had in mind the federal common law rule against discovery in criminal cases, but if so he did not mention it. Neither Pyle v. Kansas, nor the Third Circuit cases applying the Pyle rule, nor the Maryland opinion following them, turned on the presence or absence of a request. Instead, these opinions focused on either the conduct of the prosecutor or the exculpatory quality of the evidence. The existence of a request is mentioned in none of the Maryland opinions addressing the due process issue, see Brady v. State, 222 Md. 442, 444-45, 160 A.2d 912, 914 (1960); Brady v. State, 226 Md. 422, 427-30, 174 A.2d 167, 169 (1961), and is discernible only by reference to the Supreme Court record.7
*1309Justice Douglas’ formulation referred to “materiality” as well as to the defendant’s request. The materiality requirement is consistent with the analysis in pre-Brady case law, e.g., United States ex rel. Thompson v. Dye, 221 F.2d at 765, and with the provision in Fed.R.Crim.P. 52(a) that “[a]ny error, defect, irregularity or variance which does not affect substantial rights shall be disregarded.” In Giglio v. United States, 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), the Court held that nondisclosure of a promise of leniency made to a key witness was material, since it could have affected the jury’s assessment of that witness’ testimony. “A new trial is required,” wrote Chief Justice Burger, if “ ‘the false testimony could ... in any reasonable likelihood have affected the judgment of the jury____’” Id. at 154, 92 S.Ct. at 766, quoting Napue v. Illinois, 360 U.S. 264, 271, 79 S.Ct. 1173, 1178, 3 L.Ed.2d 1217 (1959). Significantly, in Giglio the Supreme Court made no mention of the request feature of Justice Douglas’ Brady formulation.
Meanwhile, however, among prosecutors Justice Douglas’ choice of words was producing a reaction he probably neither intended nor anticipated. Prosecutors began to assume that except in the most egregious circumstances, such as the knowing use of perjured testimony, they had no obligation to disclose arguably exculpatory evidence in the absence of a request, and that they could decide for themselves the materiality of such evidence. These practices led most lower federal courts to hold that the disclosure obligation — now generally identified as a “Brady,” rather than a Pyle v. Kansas, obligation — existed even in the absence of a request. See United States v. Hibler, 463 F.2d 455, 459 (9th Cir.1972); Levin v. Clark, 408 F.2d 1209, 1210-12 (D.C.Cir.1968); United States v. Poole, 379 F.2d 645, 649 (7th Cir.1967); Levin v. Katzenbach, 363 F.2d 287, 290 (D.C.Cir.1966); Barbee v. Warden, 331 F.2d 842, 845-46 (4th Cir.1964); United States ex rel. Meers v. Wilkins, 326 F.2d 135,137 (2d Cir.1964); Simms v. Cupp, 354 F.Supp. 698, 700-01 (D.Ore.1972); Clements v. Coiner, 299 F.Supp. 752, 758 (S.D. W.Va.1969). In United States v. Keogh, 391 F.2d 138, 147 (2d Cir.1968), however, Judge Friendly suggested that the absence of a request was relevant to the determination of materiality.
In Moore v. Illinois, 408 U.S. 786, 794-95, 92 S.Ct. 2562, 2567-68, 33 L.Ed.2d 706 (1972), the Supreme Court reiterated Brady’s holding that when a request for material evidence is made, the suppression of such evidence favorable to the accused violates due process. Because in Moore a request had been made, the Court had no occasion to determine whether Justice Douglas’ reference to a request purported to narrow the rule of Pyle v. Kansas or to affect the materiality requirement when no such request is made.
There matters stood when the Supreme Court decided United States v. Agurs, 427 U.S. 97, 96 S.Ct. 2392, 49 L.Ed.2d 342 (1976). In Agurs, a federal prosecutor defended his failure to disclose the criminal record of the victim of a homicide on the ground that there had been no specific request for it. The Circuit Court for the District of Columbia Circuit, holding that the information was nevertheless relevant to the defendant’s self-defense theory, ordered a new trial. Before the Supreme Court, the United States took the position that Brady and Moore had narrowed Pyle by making a specific request a prerequisite to the prosecutor’s due process obligation to disclose material evidence favorable to the accused. Br. for Petitioner at 24-33, United States v. Agurs, 427 U.S. 97, 96 S.Ct. 2392, 49 L.Ed.2d 342 (1976). This position the Court rejected, thereby rehabilitating that portion of Pyle v. Kansas holding that the prosecutor has a duty to disclose material evidence favorable to the accused, even in the absence of a request. However, the Agurs Court refined the materiality requirement, developing for the first time a three-level test for materiality *1310depending on whether a request has been made.
Under Agurs, if the prosecutor knew or should have known of perjured testimony, whether or not a request has been made, then the disclosure of this fact is required by due process if there is “any reasonable likelihood that the false testimony could have affected the judgment of the jury.” Id. at 103, 96 S.Ct. at 2397 (footnote omitted). The standard for the use of perjured testimony, derived from Giglio, supra, is similar to, and evidently more lenient than, the well-known Rule 33 standard applicable to perjured testimony enunciated in Larrison v. United States, 24 F.2d 82, 87 (7th Cir.1928) (whether “the jury might have reached a different conclusion”).8
If a specific request has been made, then the non-disclosure of evidence favorable to the accused violates due process if a “substantial basis for claiming materiality exists.” 427 U.S. at 106, 96 S.Ct. at 2399; see United States v. McCrane, 547 F.2d 204, 205 (3d Cir.1976) (per curiam) (on remand from the Supreme Court in light of Agurs). As this court held in McCrane:
Agurs established a less stringent test to be applied when the defense requests specific evidence. In that situation, if ... a “substantial basis for claiming materiality exists, it is reasonable to require the prosecutor to respond either by furnishing the information or by submitting the problem to the trial judge.”
547 F.2d at 207 (quoting Agurs, 427 U.S. at 106, 96 S.Ct. at 2399). “When the prosecutor receives a specific and relevant request,” the Agurs Court added, “the failure to make any response is seldom, if ever, excusable.” 427 U.S. at 106, 96 S.Ct. at 2399.
If the defense made no request, or tendered one couched in general terms such as “all Brady materials,” the non-disclosure of evidence favorable to the accused violates due process if the undisclosed evidence suffices to establish a reasonable doubt. All three of these standards, of course, are subject to the over-arching constitutionally harmless error standard of Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967).
Since the Agurs Court “refined” the law with respect to the prosecutor’s duty to disclose exculpatory materials, replacing the rather clear test of Pyle with the foregoing materiality standards, this court has been faced with annoying frequency with gamesmanship by some prosecutors with respect to the duty to disclose. Our recent opinion in United States v. Starusko, 729 F.2d 256 (3d Cir.1984), outlines the problems we have encountered with the operation of the Agurs tests for materiality. It seems clear that those tests have a tendency to encourage unilateral decision-making by prosecutors with respect to disclosure. As we explain further below, the root of the problem is the prosecutor’s tendency to adopt a retrospective view of materiality. Before trial the prosecutor cannot know whether, after trial, particular evidence will prove to have been material. See Agurs, 427 U.S. at 108, 96 S.Ct. at 2399; Starusko, 729 F.2d at 261. Following their adversarial instincts, some prosecutors have determined unilaterally that evidence will not be material and, often in good faith, have disclosed it neither to defense counsel nor to the court. If and when the evidence emerges after trial, the prosecutor can always argue, with the benefit of hindsight, that it was not material.
It is equally clear that such self-help has the inevitable consequence of producing both appeals such as this and collateral attacks under 28 U.S.C. § 2255 (1982). Moreover, as we noted in Starusko, we are left with the nagging concern that material favorable to the defense may never emerge from secret government files. Starusko, *1311729 F.2d at 265. Our experience since Agurs suggests that its “refinement” of the due process requirements of Pyle v. Kansas as interpreted on occasion by prosecutors pays too much deference to the federal common law policy of discouraging discovery in criminal cases, and too little regard to due process of law for defendants in those cases.
This case is a good example of the conduct which the Agurs formulation encourages. The prosecutor, despite a specific request, withheld the Wille immunity agreement, well knowing that if the defendants should learn of it, he still could argue with the benefit of hindsight that it was insufficiently material. At oral argument the prosecutor conceded that before trial he simply satisfied himself that the evidence probably would not prove to be material. In cases where counsel has not been astute enough to make a specific request, the risk to the prosecution from nondisclosure is even less, indeed, nonexistent.
But while our superiors in this hierarchial system are free to reconsider the wisdom of the gamesmanship encouraged by the Agurs materiality standards, we are not. Thus we must turn to the task of deciding whether, in the words of that case, either defendant had a “substantial basis for claiming materiality.” 427 U.S. at 106, 96 S.Ct. at 2399. We approach that task mindful that, as the Agurs Court noted, “[w]hen the prosecutor receives a specific and relevant request, the failure to make any response is seldom, if ever, excusable.” Id. If we conclude that a “substantial basis for claiming materiality” existed, then we must determine whether the nondisclosure of the Wille immunity agreement was “harmless beyond a reasonable doubt.” Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967).
B. Application to this Case
1. Admissability of the Wille agreement.
In order to be material, evidence suppressed must have been admissible at trial. Brady, 373 U.S. at 89-90, 83 S.Ct. at 1197-1198. The government contends that the Wille immunity agreement is immaterial because it could not have been used for impeachment purposes. The government’s theory is that the agreement does not pertain to Wille’s testimony about the diesel fuel excise tax fraud, but to testimony about a separate but simultaneous check-kiting scheme.
Pflaumer hired Wille in June of 1978 as a comptroller for WHP. At that time, WHP had recently merged with a second trucking firm, C & R Transport Co., to form a third corporation, KMA Leasing, Inc. These three companies and a fourth, Burgmeyer Bros., Inc., all operated out of the WHP office. Charles Gillan served as President of WHP and Burgmeyer. In 1978, Burgmeyer had accumulated substantial federal tax liabilities. In order to satisfy cash flow needs, Pflaumer, Gillan and Wille allegedly operated an elaborate check-kiting scheme between KMA Leasing and Burgmeyer. That scheme was investigated by the United States Attorney’s Office. The government claims that the grant of immunity to Wille applied to that investigation alone, and was therefore irrelevant. Fed.R.Evid. 402.
We hold that the Wille agreement was admissible to impeach. The agreement is not specific on the subject matter of the government’s investigation. It refers simply to “the federal investigation into certain activities of Charles Gillan and others during the period between June 2, 1978 and December, 1979.” The government concedes that Pflaumer is one of the “others” to which the agreement refers. Both schemes arose out of cash flow deficiencies encountered by Pflaumer’s firms as a result of tax liabilities. Both were under investigation by the same persons at the same time. Wille was not privy to the government’s investigative files, and need not have known that in the government’s mind they were separate. Certainly the government’s artful phrasing of the agree*1312ment suggests that if Wille refused to cooperate in either of the investigations, he would be in breach of his undertaking for the immunity grant. Indeed, when the government advanced this argument, the trial court expressed skepticism:
MR. COLE: I believe the immunity he refers to had to do with the check cashing scheme.
THE COURT: In his own mind, in cross examination, if you were defense counsel wouldn’t you want to ask him the question: weren’t you immunized on a previous occasion. And the answer had to be, yes.
App. at 1600-01.
We share the trial court’s skepticism. Clearly defense counsel could have used the agreement in cross-examination of Wille. Any effort by the government to establish that the agreement meant less than it said would at best have gone to the weight that the jury might have given it for impeachment purposes. Thus we reject the government’s contention that it withheld inadmissible evidence.
2. Defense knowledge of the agreement
The government also urges that the Wille agreement is not Brady material because defense counsel “knew or should have known of it.” There is no record support for the contention that either the defendants or counsel knew of the agreement. During post-trial proceedings, defense counsel testified under oath that he first learned of the Wille agreement after trial from another attorney, and that testimony was accepted as a fact. Nothing in the record suggests that either defendant knew of the agreement and withheld the information from his own counsel. The government could have, but did not, produce Wille to testify whether he had in fact disclosed the immunity agreement to his employer. Thus, the testimony of Pflaumer’s counsel that he learned of the agreement after the verdict is uncontradicted.9
There is also no evidence that the defendants “should have known” of the Wille agreement.10 More importantly, however, it is not the law that exculpatory evidence in the government’s file is not Brady material if the defendant might have uncovered it through independent sources. That position was advanced by the United States and rejected by the Supreme Court in Agurs. There the government argued that certain prior convictions were not Brady material in part because they were public records, and therefore ascertainable by independent investigation. The Court rejected that view, applying the applicable materiality standard despite the government’s argument that defense coun*1313sel should have known of the records by virtue of their public status. See 427 U.S. at 111-14, 96 S.Ct. at 2401-03. Otherwise, the Court reasoned, “there would be no special significance to the prosecutor’s obligation to serve the cause of justice.” Id. at 111, 96 S.Ct. at 2401. It is simply not the policy of Brady to encourage pre-trial guessing games over whether the defendant might have an independent source for suppressed exculpatory evidence. The issue need not detain us, however, since there is no evidence in the record suggesting that defendants should have known of the secret immunity agreement.
3. Materiality
Finally, we must determine, whether “a substantial basis for claiming materiality exists.” 427 U.S. at 106, 96 S.Ct. at 2399. We hold that defense counsel has a substantial basis for claiming the materiality of evidence impeaching the truthfulness of a prosecution witness when, viewed prospectively as the prosecutor views the evidence before trial, the testimony of the witness incriminates the defendant, and the impeaching evidence significantly impairs the incriminatory quality of that testimony. We so hold because the impeachment of an incriminating witness with significant evidence attacking the truthfulness of his testimony “might affect” the jury’s assessment of reasonable doubt and thereby affect the outcome of the trial. See Agurs, 427 U.S. at 104, 96 S.Ct. at 2397-98. Thus, when the government has evidence in its files that serves to impeach a prosecution witness, and when, as here, a specific request for that evidence is made,11 then due process requires the government to disclose the evidence if, viewed prospectively, the witness incriminates the defendant at trial and the impeaching evidence significantly impairs the incriminatory quality of the testimony. A new trial must be granted if this standard is violated and if, viewed retrospectively against the totality of the circumstances, the error is not harmless beyond a reasonable doubt. Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967). Any doubt about whether impeaching evidence significantly impairs the believability of an incriminatory witness should be resolved by the trial court.
Thus, we reject the government’s invitation to speculate on whether or not, or in what particular fashion, trial counsel might have impeached the witness in question.12 The test for Brady material must be capable of application by the prosecutor before trial, and neither the prosecutor before trial nor we from this appellate perspective are capable of speculating on the particular manner and degree of impeachment. As we discuss below, the agreement was clearly usable in a significant manner to impeach the truthfulness of Wille’s testimony. Provided that Wille was an incriminating witness, which we also address below, that fact gives rise to a “substantial basis for claiming materiality.”
Similarly, we decline the invitation to speculate with hindsight on what the jury’s reaction to the agreement “might have *1314been” in assessing the prosecutor’s duty to disclose exculpatory evidence. Neither we from our appellate position nor the prosecutor before trial can decide these imponderables. If the Brady test is to work, it must be capable of prospective application by the prosecutor before trial. The standard to be applied by the prosecutor, therefore, cannot depend on considerations capable of ascertainment if at all only after the fact. Speculation of this kind simply invites the pre-trial gamesmanship with which this court is becoming all too familiar. Hence Agurs speaks in terms of “a substantial basis for claiming materiality,” a standard capable of prospective application. Any uncertainty about what evidence is covered will be narrowed by the specific request. If the prosecution has any doubts, it should turn the contested material over to the trial court for a pre-trial Brady ruling. What we can no longer tolerate is the prosecutor’s guess before trial that the evidence after trial will not prove to have been material, and the consequent decision to conceal it even from the trial court. As Judge Aldisert recently put it, “[t]he ‘game’ will go on, but justice will suffer.” Starusko, 729 F.2d at 265.
We do not agree with the suggestion in the dissent that the Agurs “test” for specific requests is whether the evidence “might have” affected the jury’s verdict, and that this “test” is to be analyzed retrospectively in light of the cumulative evidence adduced at trial, the possible intensity of impeachment, the possible impact on the jury, and so forth. The dissenting opinion misreads Agurs and this court’s decision in United States v. Higgs, 713 F.2d 39 (3d Cir.1983), and confuses the prospective Brady analysis which the prosecutor must apply before trial with the constitutionally harmless error determination which must be made after trial. The relevant passage of Agurs states:
A fair analysis of the holding in Brady [a specific-request case] indicates that implicit in the requirement of materiality is a concern that the suppressed evidence might have affected the outcome of the trial.
427 U.S. at 104, 96 S.Ct. at 2398. We agree that the concern in such cases is whether the suppressed evidence might have affected the outcome of the trial. But this observation does not propound a “test” for the specific-request case. Indeed, Agurs contains no such holding, since in that case there was no specific request. The metric articulated in Agurs for analyzing specific-request cases is the standard, adopted by this court in McCrane, of whether “a substantial basis for claiming materiality exists.” 427 U.S. at 106, 96 S.Ct. at 2399.13 That standard must be capable of reasonably certain application by the prosecutor before trial. Our holding gives content to the standard in a rule reasonably capable of prospective application by the prosecutor. Of course, a retrospective analysis of the totality of the case does enter the picture when, in this as in all cases, we determine whether any constitutional error was harmless beyond a reasonable doubt. Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967). We address that question in Part IV B 3 a infra.
Finally, we reject any per se standard of materiality. The standard is whether, on the facts of each case, “a substantial basis for claiming materiality exists.” Therefore, we turn to consider whether Wille’s testimony incriminated either Pflaumer or Oxman and whether the Wille immunity agreement significantly impaired its incriminatory character. We must further determine whether any error is harmless beyond a reasonable doubt. Chapman, 386 U.S. at 24, 87 S.Ct. at 828.
*1315 a. Pflaumer
i) Duty to disclose
(a) Incriminatory witness
In determining whether Wille’s testimony incriminated Pflaumer we must examine the evidence implicating Pflaumer in the excise tax fraud scheme. That scheme originated when Oxman, a commission salesman for Park Oil, arranged through John McCullough, the late president of a Philadelphia labor union, to meet with Pflaumer in an effort to obtain the fuel oil business of Pflaumer’s Philadelphia-based enterprises. Jock, Oxman and McCullough attended the initial meeting, but only Jock testified about it. According to Jock, he explained to Pflaumer that he could not offer better prices than those charged by Arco and Gulf “other than if we went with the tax-angle routine.” Jock agreed to report as delivery destinations “[w]hatever they told us to put on the invoice[s].” App. at 136. Jock also testified that Pflaumer instructed him to explain the “tax-angle routine” to Gillan. App. at 138-39. Jock’s testimony implicated Gillan and indicated that Oxman and McCullough were receiving “commissions” on WHP sales. While the scheme was in operation, WHP fell behind in payments. Jock called Pflaumer and told him he could not pay commissions to Oxman and McCullough because he was not getting paid for his product. App. at 150. Jock also stated that late in 1979, after the date of the last mailing, Pflaumer engaged in efforts to conceal the tax fraud from state auditors by destroying fuel invoices.
During cross-examination, questioning by defense counsel suggested that Jock had inculpated Pflaumer in order to obtain Jock’s release from prison. The examination established that the prosecution had orchestrated J'ock’s transfer to Eglin prison camp, a minimum security prison — characterized by Jock as “a country club,” App. at 270 — obtained a reduction in sentence making Jock eligible for parole immediately, App. at 272, and recommended favorable treatment from the Parole Board, which Jock received. App. at 274. As a consequence, Jock — who had at that time served only five months of a four-year prison term for wire fraud — was released on parole after only three additional months’ incarceration, two of which Jock served “at a half-way house in Palm Beach.” App. at 275. The import of Jock’s cross-examination was that the government had paid dearly for his testimony.
In order to bolster Jock’s testimony against Pflaumer, the government sought to establish that Pflaumer had a daily role in WHP’s financial affairs. Wille was one such prosecution witness. Wille testified to Pflaumer’s role in WHP as follows:
Q. What was Mr. Pflaumer’s involvement in Wm. H.P., Inc.?
A. He pretty much directed the company. His primary interest was in the operation of the brewery, Schmidt Brewery. He directed it from the financial end. From the part that I saw he directed the accounts payable, what was to be paid and the girl who wrote the checks, checked with him about what was to be paid.
Q. What would you say the extent of his involvement was in the financial end of the trucking company?
Could the check go out of there without Mr. Pflaumer’s knowledge?
A. No. That was his principal involvement on the financial end.
Q. Was there a cash flow problem with the company while you were there?
A. Yes.
Q. Did it require Mr. Pflaumer’s constant vigilance?
A. The office manager would check with him with respect to any check that was to be written if she did not think that he was aware that the check was going to be written that day.
Q. And did’ you personally talk to Mr. Pflaumer from time to time about the financial end of the trucking company?
A. Yes.
Q. On a daily basis?
A. Probably not on a daily basis but several times a week.
*1316Q. Did you observe the office manager talking to Mr. Pflaumer on a daily basis?
A. Oh, yes; usually several times a day.
App. at 859-61 (emphasis added). The government knew long before trial that Wille would testify to these matters by virtue of several FBI interviews with him.
Through Wille, therefore, the government established that WHP issued no checks without Pflaumer’s approval and that Pflaumer conferred several times daily with the WHP office manager. In contrast, Pflaumer’s defense was that Gillan and Hill ran WHP, while he spent most of his time at the brewery and had little involvement in the day to day financial affairs of WHP.
Thus, Wille’s testimony rebutted a principal defense on which Pflaumer relied. A significant portion of the government’s case consisted of establishing that Pflaumer had a daily role in WHP’s financial affairs. The government argued that Pflaumer’s assertion that he had no involvement in the affairs of WHP was false, and that the jury might infer that Pflaumer must have known of the scheme. Certainly, therefore, Wille’s testimony incriminated Pflaumer. If believed, it established that Pflaumer was sufficiently involved in the affairs of WHP to have direct control over every check that the company issued. The jury might infer, as the government argued, that Pflaumer must have known of the tax scheme. Consequently, we hold that Wille’s testimony incriminated Pflaumer in the tax fraud scheme and that the prosecutor before trial had ample basis for knowing that it would do so.14
In ruling that the Wille immunity agreement was immaterial, the trial court pointed out that the testimony of Jock and Luciano suggested Pflaumer’s involvement. The court then concluded:
Clearly, therefore, even if the defendants had been able to impeach Wille by virtue of the immunity agreement, there was sufficient other evidence implicating Mr. Pflaumer.
App. at 1673. This ruling is legal error. The test is not whether, viewed in retrospect, there was sufficient additional evidence such that, without Wille’s testimony, the case could have been submitted to the jury. Under Agurs, when a specific request for evidence is made, the standard is whether the defendant has a substantial basis for claiming materiality. Evidence significantly impeaching the truthfulness of an incriminating witness gives rise to a substantial basis for claiming materiality. United States v. Higgs, 713 F.2d at 43; United States v. McCrane, 547 F.2d at 206-08. Thus, we turn to whether the Wille immunity agreement was significant impeachment evidence.
(b) Significant impeachment evidence
We hold that the Wille immunity agreement was significant impeachment evidence, and that the prosecutor had a sufficient basis for appreciating this fact before and during trial. The government had already conferred immunity on two prosecution witnesses and extended substantial *1317benefits to them. The disclosure of yet a third immunity agreement, and of the extension of additional benefits, gave the prosecutor a sufficient basis for concluding that the Wille immunity agreement was significant impeachment evidence. The prosecutor should have appreciated that the disclosure of the existence of substantial benefits conferred on all of the government’s principal incriminatory witnesses might have led the jury to doubt their truthfulness. This gave rise to a substantial basis for claiming materiality. Therefore, the question whether the Wille immunity agreement was Brady material should at least have been submitted to the trial court for a pre-trial Brady ruling.
(ii) Harmless beyond a reasonable doubt
We have held that due process required that the government bring the Wille agreement to the attention of the trial court at least. A new trial is required, however, only if the non-disclosure of the Wille immunity agreement was not harmless beyond a reasonable doubt. Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967). This determination requires an overview of the totality of the case against Pflaumer.
Aside from Jock’s strong testimony about the initial meeting, his testimony about the commissions paid to Oxman and McCullough, and his testimony about Pflaumer’s interest in concealment, there is very little direct evidence of Pflaumer’s involvement. The testimony of Jock’s son Michael corroborated WHP’s involvement, but tended to inculpate Pflaumer only in an ambiguous passage referring to a price increase to cover federal excise taxes.15 Peter Cordua, Michael Jock’s accountant, also testified to an ambiguous conversation with Pflaumer concerning an audit of fuel deliveries in Pennsylvania. App. at 567. In closing, the government argued that because Pflaumer expressed no surprise at Cordua’s reference to “Pennsylvania” deliveries, the jury might infer that Pflaumer knew some deliveries had falsely been reported as having occurred outside of Pennsylvania. App. at 1324.16
John Luciano also testified to one conversation with Pflaumer. Luciano stated that he had falsified invoices at the direction of Ray Hill, whom the jury acquitted. Asked about a conversation with Pflaumer, Luciano responded:
A. He asked me if I worked everything out with Ray [Hill] and I said I did.
He said to make sure the stuff was good because he did not want any crap going into his trucks creating a lot of problems for him.
I told him we bought most of it from Exxon.
He said, keep your nose clean and don’t f— up. We have a good arrangement.
App. at 597. The jury might have inferred from Pflaumer’s allusion to Hill and to “a good arrangement” that Pflaumer was aware of the tax scheme. Luciano testified as well that Oxman once told him he had spoken to “Billy” — referring to Pflaumer— “and [that] everything looked fine.” App. at 591.
Lastly, a WHP accountant testified that Pflaumer’s signature appeared on several falsified tax returns. App. at 875-79. The *1318accountant did not, however, indicate that he was aware of Pflaumer’s knowledge of the scheme or his role in preparing the returns. And a Pennsylvania tax auditor testified to conversations with Pflaumer in 1980 in which Pflaumer had stated that the auditor had in his possession all of the available fuel invoices. App. at 951. In fact, many invoices had been destroyed. These conversations could be taken as evidence of concealment by Pflaumer.
This evidence, coupled with the foregoing testimony of Frank Jock and Wille, constituted the principal evidence of Pflaumer’s involvement.17 We do not agree that in light of this evidence, the error regarding Wille’s immunity agreement was harmless beyond a reasonable doubt. First, as we noted earlier, had Pflaumer’s defense counsel been in possession of the immunity letter, he would have been able to argue that Jock had received valuable benefits in exchange for testimony, that Luciano, who corroborated Jock, had also received benefits, and that Wille, who rebutted Pflaumer’s defense, had received benefits as well. Consequently, the jury would have perceived Wille as the third government witness whose testimony the prosecution had obtained in exchange for beneficial treatment. The jury, which in acquitting Hill had apparently discredited Luciano, who most directly incriminated Hill, might well have been influenced by the fact that the principal witnesses who incriminated Pflaumer all were the beneficiaries of prosecutorial largess.
Second, Pflaumer’s impeachment of Frank Jock invited the jury to seek corroboration from other witnesses. Wille was one such corroborating witness, and — in light of the weakness of the corroborating testimony of Michael Jock, Cordua, and Luciano — an important one. Wille was the only witness who testified that .Pflaumer exerted day-to-day control over the financial affairs of WHP, thereby affording a basis for inferring that Pflaumer must have known of the scheme. The government’s suggestion that Wille’s testimony “did not suggest that Pflaumer was aware of the fraudulent scheme,” Br. at 21, is simply erroneous. Effective impeachment of Wille might have undermined this impression.
Of course, the district court’s assessment of the likelihood of the impact of non-disclosure on the outcome of the trial “is entitled to great weight.” United States v. Provenzano, 615 F.2d 37, 49 (2d Cir.), cert, denied, 446 U.S. 953, 100 S.Ct. 2921, 64 L.Ed.2d 810 (1980). Agurs requires that we give deference to the trial court’s “firsthand appraisal of the record” when that appraisal is “thorough” and “reasonable.” 427 U.S.' at 114, 96 S.Ct. at 2402. The district court’s analysis, however, does not satisfy this standard. The entirety of the court’s assessment of the weight of the evidence consists of the following paragraph:
Frank Jock testified that he entered into a conspiracy with Mr. Pflaumer. John Luciano also testified as to Mr. Pflaumer’s involvement. Indeed, defendants’ memorandum in support of motion for judgment of acquittal acknowledges the inculpatory nature of their combined testimony. Defendants’ memorandum states “Mr. Luciano does not directly inculpate Mr. Pflaumer to the same degree that Mr. Jock does____” (Memorandum of Defendants, In support of Motion for Judgment of Acquittal, pp. 7 and 8). *1319Clearly, therefore, even if the defendants had been able to impeach Wille by virtue of the immunity agreement, there was sufficient other evidence implicating Mr. Pflaumer.
App. at 1672-73. This analysis is not adequate. First, unlike the trial court in Agurs, the district court did not “remain[ ] convinced of [Pflaumer’s] guilt beyond a reasonable doubt.” 427 U.S. at 114, 96 S.Ct. at 2402 (emphasis added). We cannot ascertain whether the court applied the Chapman standard of constitutionally harmless error. Second, the only testimony of Luciano inculpating Pflaumer was. the single oblique suggestion that Pflaumer might have known of Hill’s role and believed it “a good arrangement,” and that Oxman had spoken to Pflaumer and believed that “everything looked fine.” The court’s notation that “John Luciano testified to Mr. Pflaumer’s involvement” is not an informed assessment of the impact of this testimony. Rather than canvassing all of the relevant testimony and examining Wille’s role in light of it, the district court relied on an inference from a statement in the defendants’ post-trial memorandum. Such an inference does not substitute for an informed appraisal of the evidence in its totality. Because we cannot say that the trial court’s “firsthand appraisal of the record was thorough and entirely reasonable,” Agurs, 427 U.S. at 114, 96 S.Ct. at 2402, we are unable to defer to its assessment in this case. For these reasons, we cannot find the error harmless beyond a reasonable doubt.
Because the prosecutor withheld specifically requested information that might have been used to impeach significantly a witness who incriminated Pflaumer, and because the non-disclosure is not harmless beyond a reasonable doubt. Pflaumer is entitled to a new trial.
b. Oxman
Oxman’s case differs from Pflaumer’s in an important respect, for Wille’s testimony in no way incriminated Oxman. Wille added nothing to the government’s case against him. That case tied Oxman into the fraudulent scheme in which Gillan, Jock, and later Luciano were key participants. Because Wille in no way incriminated Oxman, the prosecutor did not breach a duty to disclose the Wille immunity agreement to him.
Oxman also contends that he was prejudiced by the tardy disclosure of material impeaching Luciano. We have frequently condemned the tardy disclosure of Brady material. See Starusko, 729 F.2d at 264. We repeat that admonition. But because Oxman’s counsel obtained the materials in time to conduct a vigorous cross-examination of Luciano, he was not prejudiced by this tardy disclosure. See Starusko, 729 F.2d at 262; United States v. Higgs, 713 F.2d 39, 44 (3d Cir.1983). Ox-man is not entitled to a new trial.
IV.
Conclusion
Because the trial court erred in rejecting Pflaumer’s request for a charge that the jury must find that he joined the conspiracy prior to the last mailing, and because the government withheld specifically requested Brady material, his conviction must be reversed, and the case remanded for a new trial. The judgment of sentence in Oxman’s case will be affirmed.
. Paragraph 26 of Pflaumer’s initial discovery request sought
Any information, in any form whatsoever, the existence of which is known, or by the exercise of due diligence may become known, to the government bearing upon the credibility of any person whom the government intends to call at trial, including but not limited to any prior criminal arrest or conviction, any pending criminal indictment or information, and any pending criminal or civil investigation related to any activity of such person.
App. at 1513. Paragraph 16 requested any and all evidence having to do with criminal conduct-local, state or federal — on the part of any person whom the prosecution intends to call as a witness at trial, of which the prosecution, its agents and representatives have become aware.
App. at 1511.
. Paragraph 18 sought
any and all promises, understandings, or agreements, formal or informal, between the prosecution, its agents and representatives and persons (including counsel for such persons) whom the government intends to call as witnesses at trial, together with copies of all documentation pertaining thereto. This request includes, but is not limited to, such promises, understandings, or agreements as may have been made in connection with other cases or investigations.
App. at 1512. Similarly, paragraph 30 requested [a]ny and all actions, promises, efforts or inducements — formal or informal — on the part of the government, its agents and representatives to aid, assist or obtain benefits of any kind at any time for any person whom the government now considers a potential witness at trial ____
App. at 1513.
.
September 26, 1980
Stephen A. Madva, Esquire
3 Parkway
20th Floor
Philadelphia, PA 19102
Re: Ralph P. Wille
Dear Mr. Madva:
This letter will memorialize our understanding and agreement regarding the cooperation and testimony of Ralph P. Wille relative to the federal investigation into certain activi*1302ties of Charles Gillan and others during the period between June 2, 1978 and December, 1979.
It is the understanding of the parties hereto that Ralph P. Wille is prepared to cooperate with the United States Government with respect to the above-mentioned investigation.
In exchange for Ralph P. Wille's complete cooperation and testimony at any hearings and/or trials pertaining to the above-mentioned investigation, the Government agrees that the information and testimony provided by Ralph P. Wille will not be used against him in any criminal prosecution, nor will any information or evidence derived from the information and testimony provided by Ralph P. Wille be utilized against him in any criminal prosecution in this district.
It is agreed by Ralph P. Wille that under this agreement he is obligating himself to provide truthful information and testimony, without reservation, regarding any and all of the matters relating to the above-mentioned investigation. As part of this understanding, Ralph P. Wille will hold himself available for interviews with the Government, in addition to testifying at any other hearing or trial proceedings when called upon to do so by the Government.
By the phrase "complete cooperation and testimony” the United States Attorney's Office and Ralph P. Wille understand that he will provide full and truthful responses to all questions asked, under oath or otherwise, by Government attorneys and their agents with respect to his complete knowledge and information regarding the investigation referred to above. Ralph P. Wille also agrees that, upon request by the Government or its agents, he will voluntarily take, and pass, a polygraph examination.
It is understood that the phrase "truthful information and testimony, without reservation” requires that Ralph P. Wille avoid falsely implicating any person as well as protecting any person or omitting any inculpatory testimony. In the event Ralph P. Wille is called upon by the Government to take a polygraph examination and his responses to that examination suggest a conscious intent to deceive or prevaricate, he will be afforded an opportunity to review and explain the deceptive responses to the Government. If the totality of circumstances convinces the Government that his statement is not complete and truthful, he will be so informed and any and all obligations under this agreement may be rendered null and void. Moreover, any withholding of information and/or the providing of false information by Ralph P. Wille will terminate the Government’s obligations under this agreement and cause this agreement to become null and void. In the event that Ralph P. Wille fails to comply with any of the terms of this agreement, and the agreement is thereby terminated by the Government, any and all information and testimony provided by Ralph P. Wille can be used against him in the event the Government chooses to proceed criminally against him.
It is also understood that if the Government discovers that Ralph P. Wille received any benefit from the activities described in paragraph 1 of this letter other than salary and reasonable and customary benefits related to his employment at Burgmeyer Bros., Inc., this agreement may be terminated by the Government and any and all information and testimony provided by Ralph Wille can be used against him in the event the Government chooses to proceed criminally against him.
Furthermore, any false information and/or testimony so provided by Ralph P. Wille under oath can be prosecuted under the perjury and false statement laws of the United States, and nothing in this agreement shall be construed, in any manner, to affect the Government's right to so prosecute.
Very truly yours,
Peter F. Vaira
United States Attorney
Stephen V. Wehner
Assistant United States
Attorney
App. at 1529-31 (emphasis added).
. See 317 U.S. at 215-16, 63 S.Ct. at 178 (emphasis added):
Petitioner’s papers ... set forth allegations that his imprisonment resulted from perjured testimony, knowingly used by the State authorities to obtain his conviction, and from the deliberate suppression by those same authorities of evidence favorable to him. These allegations sufficiently charge a deprivation of rights guaranteed by the Federal Constitution.
These holdings derived from Mooney v. Holohan, 294 U.S. 103, 110, 55 S.Ct. 340, 341, 79 L.Ed. 791 (1935) (per curiam), in which the *1307petitioner had urged "that the ‘knowing use’ by the State of perjured testimony to obtain the conviction and the deliberate suppression of evidence to impeach that testimony constituted a denial of due process of law.” The Mooney Court’s holding, however, was limited on its face to the use of "testimony known to be perjured." Id. at 112, 55 S.Ct. at 341. In Pyle the Supreme Court first held squarely that the deliberate suppression of favorable, non-perjured evidence violates due process. The fact that Pyle was not limited to the use of perjured testimony is occasionally overlooked. See United States v. Agurs, 427 U.S. 97, 103 n. 8, 96 S.Ct. 2392, 2397 n. 8, 49 L.Ed.2d 342 (1976).
. The original Rule 16 provided:
Upon motion of a defendant at any time after the filing of the indictment or information, the court may order the attorney for the government to permit the defendant to inspect and copy or photograph designated books, papers, documents or tangible objects, obtained from or belonging to the defendant or obtained from others by seizure or by process, upon a showing that the items sought may be material to the preparation of his defense and that the request is reasonable. The order shall specify the time, place and manner of making the inspection and of taking the copies or photographs and may prescribe such terms and conditions as are just. 327 U.S. 821, 846 (1946). The Notes of the Advisory Committee explained:
Whether under existing law discovery may be permitted in criminal cases is doubtful, United States v. Rosenfeld, 57 F.2d 74, C.C. A.2d, certiorari denied, 286 U.S. 556, 52 S.Ct. 642, 76 L.Ed. 1290. The courts have, however, made orders granting to the defendant an opportunity to inspect impounded documents belonging to him. United States v. B. Goedde and Co., 40 Fed.Supp. 523, 534, E.D. III. The rule is a restatement of this procedure. In addition, it permits the procedure to be invoked in cases of objects and documents obtained from others by seizure or by process, on the theory that such evidential matter would probably have been accessible to the defendant if it had not previously been seized by the prosecution. The entire matter is left within the discretion of the court.
Fed.R.Crim.P. 16 advisory comm. note. Rule 16 has since 1946 been modified from time to time so as to permit more discovery than the 1946 version authorized.
. The sole question presented by Brady for review was
[wjhether the restriction of the petitioner's new trial to the question of punishment and the denial of a re-trial of guilt denies the petitioner's constitutional rights.
Br. for Petitioner at 2, Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963). Brady theorized that the differential treatment of other defendants for whom suppressed evidence went to the jury, even though irrelevant to guilt, violated equal pro'ection, and that due process required a new trial on guilt either (1) because the constitutional violation vitiated all proceedings from the moment of its occurrence before trial, or (2) because it would be improper to speculate on whether the jury might have considered the evidence on guilt, even if improperly so. See Br. for Petitioner at 5-6. The Supreme Court characterized these arguments as espousing a “sporting theory of justice" — i.e., one assuming that the jury might have flouted the court’s instructions — -and rejected it. 373 U.S. at 90, 83 S.Ct. at 1198; see United States v. Agurs, 427 U.S. 97, 108-09, 96 S.Ct. 2392, 2399-2400, 49 L.Ed.2d 342 (1976).
. See Record at 24-25, Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963):
8. Had you ever requested from the State’s Attorney the copies of statements made by Boblit? A. Not in that phraseology. What I asked him for was — I asked him had he made any confessions.
9. You asked him who had made any confessions? A. Both of them [Brady and Bob-lit],
10. You did? A. And he said "yes” and I said that I’d like to have them. And I sat in his office one day and he gave me — I imagine three or four on each side____
13. ... [W]hen you asked [the prosecutor] for a — the statements by the various defendants, [did] he show[ ] you that one [of *1309Boblit, naming Boblit as the strangler]? A. No, he did not.
. See United States v. Meyers, 484 F.2d 113, 116 (3d Cir.1973); Newman v. United States, 238 F.2d 861, 862 n. 1 (5th Cir.1956); Gordon v. United States, 178 F.2d 896, 900 (6th Cir. 1949), cert, denied, 339 U.S. 935, 70 S.Ct. 664, 94 L.Ed. 1353 (1950); United States v. Flynn, 131 F.Supp. 742, 743 (S.D.N.Y.1955); United States v. Hiss, 107 F.Supp. 128, 136 (S.D.N.Y.1952), aff'd, 201 F.2d 372 (2d Cir.), cert, denied, 345 U.S. 942, 73 S.Ct. 830, 97 L.Ed. 1368 (1953). But see United States v. Stofsky, 527 F.2d 237, 245-47 (2d Cir. 1975), cert, denied, 429 U.S. 819, 97 S.Ct. 65, 50 L.Ed.2d 80 (1976).
. The district court suggested that because Pflaumer was present during post-trial hearings and chose not to testify — “thereby denying the government an opportunity to cross-examine him on [his knowledge],” App. at 1673 — an inference might be drawn from his refusal to testify that Pflaumer in fact knew of the Wille agreement. Pflaumer instead submitted an affidavit attesting that, until counsel had informed him of the fact after his conviction, he was unaware that Wille had entered into an immunity agreement with the United States. App. at 1654.
We need not decide the somewhat difficult questions whether, when the defendant submits an affidavit attesting to lack of knowledge, his refusal to testify during a post-trial Brady hearing can be taken as substantive evidence of knowledge, cf. Griffin v. California, 380 U.S. 609, 613— 15, 85 S.Ct. 1229, 1232-33, 14 L.Ed.2d 106 (1965) (comment on refusal to testify at trial burdens privilege against self-incrimination), or whether his submission of an affidavit constitutes a waiver of the privilege against self-incrimination, cf. Brown v. United States, 356 U.S. 148, 154-55, 78 S.Ct. 622, 626-27, 2 L.Ed.2d 589 (1958), or whether such a waiver was voluntary, see Minnesota v. Murphy, — U.S.-, 104 S.Ct. 1136, 1143, 79 L.Ed.2d 409 (1984), or which party has the burden of establishing knowledge or non-knowledge. Even if Pflaumer's post-trial decision to submit an affidavit rather than- to testify could be taken as substantive evidence — a point on which we express no view — that would simply be insufficiently probative to establish his knowledge.
. During post-trial proceedings, the government argued that Pflaumer must have known of the Wille agreement because he must have known that Wille had given testimony in other proceedings relating to Burgmeyer Brothers. App. at 1605-06. Even if true, such knowledge would not have alerted Pflaumer to an immunity agreement.
. A request for "all impeachment evidence,” of course, like a request "for all Brady materials," is not a specific request.
. The government argues that we should affirm because we can find that Pflaumer’s counsel would not have attempted to impeach Wille. The reason advanced in support of that proposed finding is that had Wille been confronted with the immunity agreement, the door would have been opened for the prosecutor to refer to the check-kiting scheme and Pflaumer’s involvement in it. Even if we were permitted to speculate in this fashion, that suggestion would not have appeal. The defendants could have exposed the immunity agreement and Wille’s favorable treatment under it without revealing the details of the underlying check-kiting scheme. Any mention by the prosecution of Pflaumer's role in the check-kiting scheme would have been inadmissible under Fed.R.Evid. 404(b) and 405(b), and would almost certainly have been excluded under Fed.R.Evid. 403. Only in the unlikely event that the defendants themselves had probed Pflaumer’s role in the check-kiting scheme would the risk suggested by the government have arisen. Of course, these difficulties highlight the hazards of hypothesizing about the particular course of events that may have ensued at trial had the prosecutor disclosed exculpatory evidence.
. Nor can we agree that the standard applied here is inconsistent with the language of Higgs. We apply here verbatim the standard for specific-request cases adopted by this court in United States v. McCrane, 547 F.2d 204, 205 (3d Cir. 1976) (per curiam) (on remand from Supreme Court in light of Agurs), on which this court in Higgs relied. The Higgs court made no effort to flesh out the materiality standard of McCrane because the government in Higgs had conceded that the requested information was Brady material. 713 F.2d at 43.
. The government maintains that Wille’s testimony was necessary only "so that he could name the person from whom he received his orders.” Br. at 20. This explanation is simply implausible. That person, as Wille testified, was Charles Gillan. App. at 861-66. Gillan, however, had already pleaded guilty. No purpose was served by calling Wille solely to inculpate Gillan. Nor was it necessary to establish that Wille took orders from Gillan "for completeness,” as the government suggests. If anything, such testimony detracted from the government’s case against Pflaumer, since it implied — as Pflaumer contended — that Gillan had orchestrated the scheme.
At oral argument, the government contended that Wille’s name had been included in the witness list before Gillan had pleaded guilty and, indeed, that its inclusion may have contributed to Gillan’s decision to plead guilty. While this explanation may make sense of the government’s assertion in its brief that Wille’s testimony was necessary to inculpate Gillan, it does not explain why Wille was called even after Gillan had pleaded guilty. And, of course, it ignores that portion of Wille’s testimony that directly inculpated Pflaumer.
. Michael Jock testified:
A. I told him [Pflaumer] [that] the [fuel] price would have to be increased.
Q. Why did you tell him that?
A. I would have to pay the four cents.
Q. Had you been charging him the four cents tax up to that point.
A. No.
Q. What did he say when you said that?
A. He said deduct my regular price four cents and add that to it and the price will come out the same.
App. at 421. The jury might have inferred from Pflaumer’s response that he knew that federal excise taxes were not being paid.
. This inference drawn from Pflaumer’s silence was a weak one. If all the deliveries were to Pennsylvania, Pflaumer may simply have observed nothing unusual in referring to them as "Pennsylvania deliveries.” It would have been more remarkable had the accountant referred to a list of Maryland or New Jersey deliveries and elicited no surprise from Pflaumer.
. The government maintains that additional witnesses further implicated Pflaumer. Br. at 21 n. 5. None of the testimony cited, however, added significantly to the evidence reviewed above. Pflaumer’s secretary, for example, Jacqueline Branson — a defense witness — testified that the meeting of Pflaumer, Oxman, Frank Jock, and McCullough had occurred. App. at 1251-52. The existence of the meeting, however, had already been established. Branson did not testify to the content of the conversations, nor could she have done so, for she had not been present. Delores Lyons, the WHP office manager, testified that from time to time she and Pflaumer had had discussions about the "financial end of the business.” App. at 734. Lyons did not, however, testify that these conversations inculpated Pflaumer. Wille’s testimony indicated that the conversations had occurred and established more strongly than did Lyons’ Pflaumer's financial role in the company.