GODBOLD, Chief Judge:
Brown is a Florida state prisoner convicted in 1974 of robbery, rape and murder. He received a death sentence on his murder conviction and two consecutive life sentences on the robbery and rape convictions. After pursuing state remedies 1 he filed a petition for habeas corpus in 1983 in the district court for M.D. Florida, raising 11 issues.2 The district court did not conduct a hearing but painstakingly considered the issues, found no merit in any, and denied the writ. Because the prosecution knowingly allowed material false testimony to be introduced at trial, failed to step forward and make the falsity known, and knowingly exploited the false testimony in its closing argument to the jury, in violation of the due process clause of the Fourteenth Amendment, Giglio v. U.S., 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), we reverse the district court and direct that the writ be granted.
I. Background
A. Trial
Brown was convicted for the robbery, rape and murder of Earlene Barksdale, co-owner of a small shop. The state’s case against Brown hinged on the testimony of Ronald Floyd, who was the only witness placing Brown at the scene of the crime and the only witness to testify to admis[1459] sions by Brown that he committed the murder and the rape. According to Floyd’s testimony, during the afternoon of the Barksdale murder he, Brown, and a man identified only as “Pooehie” (and never located) drove to the shop. Floyd waited in the car, parked across the street, while the other two entered the store. Floyd did not know what Brown and Pooehie intended to do in the store, and he did not see a gun, but he noticed that Brown had a bulge in his shirt that looked like a gun. About 15 minutes later Floyd went to the door of the shop to look in, heard a shot, entered the store and saw the foot of a body lying on the floor. Brown and Pooehie emerged from the shop, Brown bringing with him articles of new clothing, and all three men jumped in the car and drove away. While in the car Pooehie told Brown, “Man, you didn’t have to do that.”
Floyd also testified that the next day he, Brown, and Raymond Vinson were together and heard a radio broadcast concerning the Barksdale murder. Floyd said something like “People will do anything these days.” Brown responded, “Yes, she never should have done what she did.” Vinson’s testimony corroborated this conversation. Later that day, Floyd testified, he asked Brown outright whether he had killed “the woman” [Mrs. Barksdale].3 Brown responded “Yes,” and followed the answer with a crude sexual remark to the effect that he had had intercourse with her.
Floyd denied any prior knowledge that the crime was to take place.
Brown presented an alibi defense supported by the testimony of his girlfriend and her mother. Evidence was introduced tending to prove that Mrs. Barksdale had been sexually assaulted and raped and that money had been taken from the cash register. Several hours after her death Mrs. Barksdale’s nude body was found in the store, shot once in the head.
On the evening of the day of the murder, around midnight, Brown and Floyd committed a separate robbery of a man and woman at a motel. Testimony conflicted as to whether Vinson was present, but he admitted that his ear was used in this robbery. In this robbery Brown forced the female victim to remove her nightclothes, sexually abused her, and was just beginning the act of raping her when, according to Floyd’s testimony, Floyd persuaded Brown to desist. They tied up the victims and left. The next day Brown turned himself in to the police, confessed to the motel robbery, and implicated Floyd. Brown gave information that permitted the police to discover the gun used in the robbery. This gun, the property of Vinson, was introduced in the Bárksdale trial as the alleged murder weapon. Vinson testified, admitting that his car was used by the other two in the motel robbery and that he was implicated as an accomplice. All three were charged with the motel robbery.
At trial of the present case Brown’s defense counsel attempted to impeach Floyd’s testimony by casting doubt on his credibility. This attempt proceeded on three distinct grounds. First, he elicited from Floyd testimony as to his prior criminal record. Second, he brought out from Floyd that he had reason to seek revenge on Brown because Brown had told police that Floyd was involved with him in the motel robbery. Third, he inquired into the existence of a plea agreement with the state that was beneficial to Floyd. As to this third point Floyd testified as follows:
Q In this [motel] robbery, have you been sentenced?
A No, I have not.
Q You have not been sentenced?
A No.
Q When did you plead to it?
A October.
Q Of 1973?
A Yes.
[1460] Q Do you have any knowledge of why you haven’t been sentenced in this case?
A No, I haven’t. Just that I have been put on PSI.
Q PSI?
A Yes, presentence.
Q What does that stand for?
A Presentence investigation.
R. 895. Then, with respect to the Barks-dale case:
Q Right. Has the State made any promises or agreements with you in this case?
A Not to my knowledge they haven’t.
Q They haven’t?
A No.
Q Have you been charged with this case?
A No, I haven’t
Q Have you been given immunity in this crime?
A No, I have not, not as I know of.
Q You haven’t been charged and you haven’t been given immunity?
A Not as I know of.
Q Are you afraid that you might be charged with this crime?
A Yes, I am.
Q This is a first degree murder trial—
A Yes, I know that.
Q —isn’t it? And you are absolutely certain that you haven’t been given any immunity, is that correct?
A I’m certain.
R. 896. And later:
Q Has the State promised you anything in the sentencing in [the motel] case if you cooperated in this case?
A Well, like I told you before, I do not have any knowledge of it whatsoever.
Q Do you think it might be beneficial to you to testify in this case?
A I don’t know.
R. 909. And still later, with respect to the Barksdale case:
Q And, so, you’ve decided to just cleanse your soul and take the chances of whether the State of Florida is going to charge you with this murder?
A Yes.
R. 916-17.
In his closing argument to the jury the prosecutor used Floyd’s denial of any promises to bolster Floyd’s testimony.
There was an attempt made by defense counsel to impeach Ronald Floyd. You all heard that. Impeach means to reduce his credibility somehow by showing that for some reason or another he isn’t telling the truth. And I submit that there has been no promises made to Ronald Floyd for his testifying in this case. He testified to that from the witness stand. He has absolutely nothing to gain by testifying against this particular individual. As a matter of fact, if he wouldn’t testify at all, if he hadn’t said anything, you can see the kind of case we had. We had nothing linking him to this case.
R. 1199
Beyond Floyd’s testimony, the state introduced scant evidence linking Brown to the Barksdale crimes. The only other live testimony implicating Brown was that of Vinson. Vinson was not present at the scene of the murder, and his only testimony tending to implicate Brown was that he overheard the conversation between Floyd and Brown in which, after hearing the radio broadcast, Brown said, “Yes, she never should have done what she did.”4 There was no fingerprint evidence. The only physical evidence pointing to Brown was Vinson’s gun, to which Brown had access. The state introduced ballistic reports concerning the gun, but the evidence of whether the bullet that killed Mrs. Barksdale came from the gun was inconclusive (which the state concedes).
[1461] Floyd was never indicted for the Barks-dale murder.
We need not develop the facts concerning the plea agreement with Floyd relating to the motel robbery. There is evidence that an agreement was reached on this offense in October 1973, eight months before the Barksdale trial and before negotiations began for an agreement concerning the Barksdale case. At the time of the Barksdale trial Floyd had not been sentenced in the motel robbery case. As set out in the testimony quoted above, Floyd denied any knowledge of why he had not been sentenced. After the Barksdale trial, on pleas of guilty, Floyd and Brown were sentenced for the motel robbery. Floyd was given probation. Brown got 20 years. Possibly a'jury might infer that the agreement concerning the motel robbery was sufficiently linked to the agreement concerning the Barksdale murder that the existence of it should have been made known to the jury. But we do not base our decision upon the testimony concerning the agreement relating to the motel case, since the false testimony relating to the Barks-dale case agreement, and the prosecution’s misrepresentation of it, require granting the writ.5
B. Post-trial
After Brown’s trial, questions arose as to the veracity of Floyd’s testimony in which he denied that he was testifying pursuant to an agreement with the state. Some eight months after trial and while Floyd was in prison for his conviction on still another robbery, Floyd gave Brown’s attorney an affidavit in which he retracted his trial testimony concerning the crime and stated that in exchange for his testimony against Brown he had been offered “favorable consideration” in both the Barksdale case and the motel case. Brown’s conviction on the Barksdale case was then on appeal to the state supreme court. Brown moved for a new trial based on the affidavit, and the supreme court remanded for an evidentiary hearing on the issues raised by the affidavit.
At the evidentiary hearing before the trial court, in 1975, Floyd recanted his affidavit insofar as it stated that his trial testimony concerning the Barksdale murder had been untrue and testified that his trial testimony had accurately recounted the events of the murder. He reaffirmed, however, that part of the affidavit stating that he had entered into a plea agreement before he testified against Brown because of “favorable consideration” in both the Barksdale and the motel cases. At this point in the hearing the judge indicated that he intended to deny Brown’s motion for a new trial. The state, speaking through the prosecutor who had tried the case, then made a proffer of testimony it would have offered if the hearing had proceeded. The proffer included:
The state was then prepared to call Mr. Knight, Mr. Robert Knight, Attorney for Mr. Floyd at the time of these proceedings, to testify about the negotiations which were entered into by Mr. Floyd as far as plea negotiations.
And the State would proffer that his testimony would have been that upon entry of his plea to the robbery charge, there was a PSI with a cap of twenty years given, and that subsequent to that time, we discussed with him testifying before the Grand Jury and then testifying in the cause of this particular case, Joseph Green Brown. And that if he would testify and if he would tell us the truth and if he could pass a polygraph that he was not the true man in this offense, that we would not prosecute him [1462] for that offense. And that was the only plea negotiations we discussed with Mr. Floyd relative to his testimony. That there was no discussion made that if he testified in this cause, the murder case, that his penalty would be in any way lessened.
The State is also or was also ready and prepared today to present testimony from Detective Bebler regarding any plea negotiations which were entered into. And his testimony would have been essentially the same as that of Mr. Knight.
R. 1399-1400.
I was then prepared to testify myself as to the plea negotiations that were entered into by the State Attorney’s Office, which would have been essentially the same as those outlined that we proffered Mr. Knight’s testimony would have been.
R. 1401.
The trial court denied the motion for new trial without admitting the proffered testimony into evidence.6
In its brief filed with the Florida Supreme Court after the 1975 evidentiary hearing the state said this (p. 10):
No promise as to penalty was made based on whether [Floyd] testified. (R. 1399) Appellee feels compelled to note that the proffer included a promise not to prosecute Floyd if he testified truthfully, and “could pass a polygraph that he was not the true man in this case”. (R 1399) That may contradict Floyd’s testimony that he had no promises or agreements. (R 896)
In 1980 the supreme court decided the merits appeal including the remanded issues. 381 So.2d 690 (Fla.1980). The court appeared to hold that, absent the affidavit of Floyd, the state’s version of its dealings with Floyd, which version the record did not contradict, was consistent with Floyd’s trial testimony that he had been given no promise of favorable consideration. 381 So.2d at 693. The court noted that at the 1975 hearing Brown offered no testimony to substantiate the affidavit, therefore there was “nothing in the record” to suggest that the state in fact made promises as described in the affidavit. Id. The supreme court appeared to give no effect to the state’s proffer, which had set out that Brown’s attorney, Detective Bebler, and the prosecutor who tried the case (and made the proffer) all would testify to an agreement that was made, consisting of an agreement, upon described conditions, not to prosecute Floyd for the Barksdale murder. The decision appears therefore, to be based on failure of proof.
Any evidentiary reasons that may have led the supreme court to not consider the proffer have been outrun by events. The federal district court found as a fact that an agreement was made as described in the proffer. In its brief to this court the state now acknowledges that the agreement as found by the district court is the agreement that was made.
Here the state has steadfastly asserted throughout the trial and post-trial proceedings that Floyd was never given any favorable consideration in exchange for his testimony. The only agreement reached between the State and Floyd provided that if Floyd agreed to testify before the Grand Jury and in the trial of the case, and if he could pass a polygraph test while asserting his innocence in the present case, then the State would not prosecute him for this offense.
Brief of appellee, p. 23. This acknowledgement was reiterated at oral argument. These acknowledgements by the state drop [1463] out of the case the argument made earlier in the state’s brief that the finding by the supreme court — that the record did not support that a promise was made — is entitled to a presumption of correctness under 28 U.S.C. § 2254(d). Thus, we are squarely presented with an adequate record establishing that an agreement was made. With this point behind us, we address the issue of the effect of Floyd’s testimony that the state had made no promises or agreements with him and that he knew nothing about being given immunity for the murder, the state’s silence in the face of that testimony, and the prosecutor’s representation to the jury that no promises had been made to Floyd and that Floyd had nothing to gain by testifying.7
II. The district court and the test for materiality
The federal district court found that an agreement had been made as described in the proffer but went on to hold that Brown was not entitled to the writ because he had not demonstrated that Floyd’s false testimony was “material” to his conviction. In assessing the materiality of the testimony at issue the district court read U.S. v. Phillips, 664 F.2d 971 (5th Cir.1981) (Unit B), as requiring Brown to “demonstrate that [correction of the false testimony] probably would have resulted in an acquittal.” It then held that Brown had failed to meet this standard for “materiality.”
The starting point for analyzing whether a prosecutor’s knowing use of false testimony invalidates a conviction is Giglio v. U.S., 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), where the Supreme Court addressed a situation strikingly similar to the one in this case. The key government witness testified on cross-examination that he had not received any promises that he would not be indicted if he would testify against the defendant. In his closing argument the prosecutor told the jury that the witness did not receive any promises for his testimony. 405 U.S. 151-52, 92 S.Ct. at 764-65. After trial it was disclosed that the witness in fact had been testifying pursuant to a plea agreement he had negotiated with the government. In deciding that this use of false evidence by the government required Giglio’s conviction be set aside, the Court held that “[a] new trial is required 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). This reasonable likelihood standard has been reaffirmed by the Supreme Court as the appropriate test for a prosecutor’s knowing use of false testimony. U.S. v. Bagley, — U.S. -,-- -, 105 S.Ct. 3375, 3381-82, 87 L.Ed.2d 481, 491-93 (1985).8
[1464] The district court erred in requiring Brown to demonstrate that correction of the false testimony “probably would have resulted in an acquittal.” Phillips had announced such a standard for the situation where a prosecutor has failed to disclose purely impeaching evidence.9 This case does not involve mere nondisclosure of impeaching evidence but knowing introduction of false testimony and exploitation of that testimony in argument to the jury. The appropriate standard is that of Giglio and Bagley, brought forward into our en banc decision in McCleskey v. Kemp, 753 F.2d 877, 885 (11th Cir.1985) (en banc).10
III. The application of Giglio
Application of the correct standard requires that the writ be granted.
In this case there are several facets to the government’s action and inaction, only one or more of which might exist in other cases.11 The government has a duty to disclose evidence of any understanding or agreement as to prosecution of a key government witness. Haber v. Wainwright, 756 F.2d 1520 (11th Cir.1985); Williams v. Brown, 609 F.2d 216, 221 (5th Cir.1980); U.S. v. Tashman, 478 F.2d 129, 131 (5th Cir.1973). The government, in this case, did not disclose. The government has a duty not to present or use false testimony. Giglio; Williams v. Griswald, 743 F.2d 1533, 1541 (11th Cir.1984). It did use false testimony. If false testimony surfaces during a trial and the government has knowledge of it, as occurred here, the government has a duty to step forward and disclose. Smith v. Kemp, 715 F.2d 1459, 1463 (11th Cir.), cert. denied 464 U.S. 1003, 104 S.Ct. 510, 78 L.Ed.2d 699 (1983) (“The state must affirmatively correct testimony of a witness who fraudulently testifies that he has not received a promise of leniency in exchange for his testimony.”). It did not step forward and disclose when Floyd testified falsely. The government has a duty not to exploit false testimony by prosecutorial argument affirmatively urging to the jury the truth of what it knows to be false. See U.S. v. Sanfilippo, 564 F.2d 176, 179 (5th Cir.1977) (defendant’s conviction reversed because “The Government not only permitted false testimony of one of its witnesses to go to the jury, but argued it as a relevant matter for the jury to consider”). Here the government told the jury “there has been no promises made to Ronald Floyd for his testifying in this case” when it knew the contrary was true.
On the now-established record, the state urges several reasons why Giglio and related cases do not require granting the writ in this case. Its primary argument, described by it in argument as the “bottom line,” goes to the content of the agreement. The state insists that the agreement embraces no “promise of leniency.” Despite diligent inquiry by this court at oral argument, the basis for this contention has not been made clear. Certainly Giglio does not [1465] require that the word “promise” is a word of art that must be specifically employed. Nor can it rationally be contended that Giglio is triggered by a promise subject to acceptance but not by a bilateral agreement. Nor does it matter that the agreement contained conditions (that Floyd testify before the grand jury and trial jury and that he pass a polygraph test tending to show that he was not the “true man” in the Barksdale killing/rape.)6 *****12 The promise in Napue v. Illinois was conditional (that the witness testify and that his assertion that he was a “reluctant participant” in the robbery was “borne out.”). The agreement is no less an agreement because the conditions required that the witness “establish” that he was not culpable or was less culpable than the defendant. This was one of the conditions in Napue. Usually the prosecution will attach to any plea agreement with a co-defendant or putative co-defendant a condition that he will testify truthfully. We find no case suggesting that the existence of conditions such as existed in this case makes the constitutional principle inapplicable. The condition of passing a polygraph test would be for the jury to consider in determining whether in its judgment Floyd was testifying truthfully. It does not make an agreement any less an agreement.
The state’s argument misconceives the constitutional concerns addressed by Giglio. It is a constitution we deal with, not semantics. “The thrust of Giglio and its progeny has been to ensure that the jury knows the facts that might motivate a witness in giving testimony ...” Smith v. Kemp, 715 F.2d 1459, 1467 (11th Cir.), cert. denied 464 U.S. 1003, 104 S.Ct. 510, 78 L.Ed.2d 699 (1983), which testimony “could ... in any reasonable likelihood have affected the judgment of the jury.” Giglio, 405 U.S. at 154, 92 S.Ct. at 766, quoting Napue v. Illinois, 360 U.S. 264, 79 S.Ct. 1173, 3 L.Ed.2d 1217 (1959).
The constitutional concerns address the realities of what might induce a witness to testify falsely, and the jury is entitled to consider those realities in assessing credibility. The jury at Brown’s trial was entitled to know whether Floyd was testifying under an agreement that might make it possible for him to avoid prosecution for the Barksdale murder, and, if he was, to consider this in measuring his credibility. The agreement that we now know was struck gave Floyd immunity from prosecution in a capital case in which, by his own testimony, he had come to the scene of the crime with Brown, had been at the scene when the crime was committed, had fled with Brown, and later the same day had participated with Brown in another crime of violence with a similar modus operandi with respect to the female victim. It is not for the state to now say that the agreement was not important because Floyd really did not get very much in the trade, because it might not have been able to convict him of the Barksdale murder. Floyd got the benefit of the bargain for immunity. He was never even indicted for the Barksdale crimes.
The state makes an alternative argument that even if the agreement is within the sweep of Giglio the writ should not be granted because the false testimony only related to Floyd’s credibility. “It is of no consequence that the falsehood [bears] upon the witness’s credibility rather than directly upon [the] defendant’s guilt,” Williams v. Griswald, 743 F.2d 1533 (11th Cir.1984), quoting Napue v. Illinois.
Not every use of false evidence entitles a defendant to the writ. Giglio, 405 U.S. at 154, 92 S.Ct. at 766; Williams, 743 F.2d at 1542. The defendant must prove that the false evidence was “material” in obtaining his conviction or sentence or both. We have pointed out that the correct test in this case for whether evidence is “material” is whether there is “any reasonable likelihood that the false testimony [1466] could have affected the judgment of the jury.” U.S. v. Bagley, — U.S.-,-, 105 S.Ct. 3375, 3382, 87 L.Ed.2d 481, 492 (1985) (plurality opinion) quoting US. v. Agurs, 427 U.S. 97, 103, 96 S.Ct. 2392, 2397, 49 L.Ed.2d 342 (1976). See also Williams, 743 F.2d at 1543 (same). Brown meets this materiality standard. Floyd’s testimony was the keystone of the state’s case. Only he placed Brown at the scene. Only he testified that Brown admitted killing and raping Mrs. Barksdale. Absent his testimony the state was left with Vinson’s testimony of an arguably incriminating statement and inconclusive evidence concerning the pistol, possibly not even sufficient evidence to submit the case to the jury.13 In this case, as in Giglio and Napue, the prosecution’s case turned on the immunized witness’s testimony. Giglio, 405 U.S. at 154, 92 S.Ct. at 766. And the Giglio prosecutor, like the prosecutor in this case, argued that the key witness received no promises that he would not be prosecuted. Id. at 152, 92 S.Ct. at 765.
If knowledge of the agreement struck with Floyd for favorable treatment on the Barksdale case could reasonably have led a jury to disbelieve his testimony, Brown’s conviction and sentence were constitutionally invalid. There is a reasonable likelihood that disclosure to the jury that Floyd was testifying under an agreement that might save his skin could have affected the jury’s verdict and sentence.14 This is the type of incentive that existed in Giglio where non-disclosure of a plea agreement invalidated the conviction. The false testimony was used in an effort to rehabilitate Floyd’s credibility after Brown’s defense counsel had brought out two possible reasons Floyd might have had for implicating Brown. The evidence was material.
We reject the state's contention that the false testimony was not material because it was merely cumulative of Floyd’s possible bias. In the normal evidentiary sense cumulative evidence is excluded because it is repetitious. The testimony here did not merely reinforce a fact that the jury already knew; the truth would have introduced a new source of potential bias. See U.S. v. Sanfilippo, 564 F.2d 176, 178 (5th Cir.1977) (“The fact that the history of a witness shows that he might be dishonest does not render cumulative evidence that the prosecution promised immunity for testimony.”).
Both the state and the district court have made the point that disclosing the existence of the agreement could have damaged Brown’s case because the jury might have inferred that Floyd had passed the lie detector test, thus bolstering his testimony. Thus, it is contended, the false testimony was not “material” because any gain to Brown’s defense from revealing Floyd’s deal with the state would have been offset by possible harm to Brown’s defense from the jury’s knowing that Floyd had passed a lie detector test. The point has no force. During examination by the prosecutor, Floyd testified that initially he took a polygraph test concerning his knowledge of the Barksdale case and flunked it because he “did know something about [the case],” and that subsequently, after he told the police the details he later testified to at trial, he took “about three polygraphs” and passed all of them. Thus, one need not speculate whether revealing the agreement might have injured the defense by the jury’s inferring that Brown passed the polygraph, because Brown testified to having passed the polygraph “about three” times.
[1467] IV. Other issues
The following issues, as listed in footnote 2, have been abandoned because not raised on appeal: (3), (5), (6), (7), (8), (9), (10).
Because of our decision in the case we do not need to reach issues (1) and (2), ineffective counsel, and issue (11), denial of a fair hearing in state collateral proceedings, and a new issue raised on appeal, denial of a hearing by the federal district court.
The decision of the district court is REVERSED and the case is REMANDED to the district court with instructions to issue the writ of habeas corpus, subject to the right of the state to retry Brown.