POSNER, Circuit Judge.

In 1983 an Indiana jury found Mark Wisehart guilty of murder and robbery of an elderly woman, the burglary of her home, and the theft of her property. The jury recommended the death penalty, and the judge agreed and sentenced Wisehart accordingly. After exhausting state judicial remedies in Wisehart v. State, 484 N.E.2d 949 (Ind. 1985), 693 N.E.2d 23 (Ind. 1998), Wisehart unsuccessfully sought federal habeas corpus, and has now appealed to us.

The police discovered the body as the result of an anonymous phone call — by Wisehart himself, who disguised his voice. Wisehart lived in a homeless shelter called the "Christian Center" to which his victim had been a regular visitor. Another resident, a companion in crime to Wisehart named Johnson, testified that Wisehart had sent him a series of letters in which he talked about going to old people's houses and robbing them and killing anyone who got in the way; the letters were placed in evidence. Johnson also testified that after the murder Wisehart, realizing that Johnson would be a witness, told him: "Try to make it look like I'm crazy."

Wisehart gave the police a full and detailed confession a week after the murder. His defense at trial was that he was insane and his confession (which he admitted making) false.

He makes two arguments. (A third, that some of the jury instructions suggested he might be convicted on the basis of a mere preponderance of the evidence, was procedurally defaulted and is anyway completely without merit.) The first argument is that the state violated the Brady doctrine (see, e.g., Strickler v. Greene, 527 U.S. 263, 280-82, 119 S.Ct. 1936, 144 L.Ed.2d 286 (1999); Brady v. Maryland, 373 U.S. 83, 87, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963); United States v. Fallon, 348 F.3d 248, 251-52 (7th Cir. 2003)) by failing to disclose benefits that Johnson had received as a result of his agreeing to testify. The prosecution's giving a witness benefits — leniency, cash, or anything else — can be used by a cross-examining defense counsel to undermine the witness in two, possibly three, distinct ways. The first and most common is by showing that the benefits were given in return for the witness's providing testimony that would help the prosecution. He might have told the prosecutor what he would testify to if called and the prosecutor might have explicitly agreed to give him specified benefits if he testified consistently with his proffer. E.g., Giglio v. United States, 405 U.S. 150, 152-55, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972); Abbott v. United States, 195 F.3d 946, 948-50 (7th Cir. 1999); Shabazz v. Artuz, 336 F.3d 154, 161-62 (2d Cir. 2003). Or there might have been a tacit understanding that if his testimony was helpful to the prosecution, the state would give him a break on some pending criminal charge. Another example of an implicit agreement would be if the prosecutor promised a witness $100,000 contingent on the defendant's being convicted. Cf. United States v. Villafranca, 260 F.3d 374, 380 (5th Cir. 2001). Express or tacit, either way there would be an agreement, it would be usable for impeachment, and it would have to be disclosed to the defense. But the Indiana Supreme Court found that in this case there had been no agreement, express or implied, and as the finding has not been rebutted by "clear and convincing evidence," it binds us. 28 U.S.C. § 2254(e)(1); Miller-El v. Cockrell, 537 U.S. 322, 324, 123 S.Ct. 1029, 154 L.Ed.2d 931 (2003); Barrow v. Uchtman, 398 F.3d 597, 602-03 (7th Cir. 2005). The finding further implies that Johnson was not testifying falsely when he denied having any "kind of deal" with the prosecution regarding his other crimes; a deal is an agreement. And so there was no violation of the rule of Napue v. Illinois, 360 U.S. 264, 79 S.Ct. 1173, 3 L.Ed.2d 1217 (1959).

The second way of invoking Brady is by showing that although there was no quid pro quo, the state, as in our Boyd and Williams cases, United States v. Boyd, 55 F.3d 239, 243-45 (7th Cir. 1995); United States v. Williams, 81 F.3d 1434, 1438 (7th Cir. 1996); see also United States v. Sipe, 388 F.3d 471, 488-90 (5th Cir. 2004); United States v. Soto-Beniquez, 356 F.3d 1, 41 (1st Cir. 2004), had lavished benefits (sex, free long-distance calls, cash, or what have you) on its witnesses in the hope of making them feel part of the state's team and as a result inclined, out of gratitude, friendship, or loyalty, to testify in support of the prosecution. In Sipe, for example, the aliens who testified for the government "were given . . . significant benefits, including Social Security cards, witness fees, permits allowing travel to and from Mexico, travel expenses, living expenses, some phone expenses, and other benefits. They were essentially given all, and more, of the benefits they were arrested for trying to obtain illegally — benefits so valuable that they took great risks to obtain them by crossing the border illegally. . . . [Prosecutors stated] that the aliens needed to be `kept in orbit'; that the agents needed to maintain `close control' over the witnesses; that they must be kept `in pocket'; and that the aliens needed to be `re-commit[ted] to the cause.' This evidence, which was withheld from Sipe, reveals that the aliens were dependent upon the government for their most basic needs, such as visiting and communicating with their families." 388 F.3d at 488-90. There is nothing comparable here.

There may be a third category; it could be thought intermediate between the first two. This would be the definite benefit that is neither a quid pro quo nor lavish, yet permits an inference that the witness's testimony would be affected. Suppose the prosecutor had given Johnson $500, with no words exchanged, and later called him as a witness. Johnson might think either that his acceptance of the money had created an obligation to cooperate with the prosecution or that he should cooperate out of gratitude. There would be an argument for requiring disclosure of such a benefit, especially as the requirement would not create the problem of fuzzy boundaries that requiring disclosure of a mere forbearance to prosecute a witness for unrelated crimes, discussed next, would create. We need not decide how strong an argument; Johnson received no such definite benefit.

What is decisive for this case is this court's refusal to recognize a fourth Brady category, in which the state merely doesn't come down as hard on a witness as it could. "Todd cannot prove an agreement existed. He argues that at the very least Nielson had an `expectation' of benefit. But what one party might expect from another does not amount to an agreement between them. And Todd does not argue that the state knew of Nielson's expectation or that he could not have uncovered that expectation with reasonable diligence. This brings us back to the agreement, which Todd cannot show existed. Without an agreement, no evidence was suppressed, and the state's conduct, not disclosing something it did not have, cannot be considered a Brady violation." Todd v. Schomig, 283 F.3d 842, 849 (7th Cir. 2002). Or as the Second Circuit put it in Shabazz v. Artuz, supra, 336 F.3d at 165 (emphasis in original), "The government is free to reward witnesses for their cooperation with favorable treatment in pending criminal cases without disclosing to the defendant its intention to do so, provided that it does not promise anything to the witnesses prior to their testimony. . . . [T]he fact that a prosecutor afforded favorable treatment to a government witness, standing alone, does not establish the existence of an underlying promise of leniency in exchange for testimony."

The objections to the extension for which Wisehart contends are twofold. First, the category has no ascertainable boundaries. Rarely does the state end up charging a defendant with every possible crime that he may have committed. Because the state doesn't have the resources to do that, most criminal cases are disposed of pursuant to plea agreements that involve some concessions on its part. The implication of Wisehart's argument is that whenever the state uses a criminal as a witness, which it does very commonly in criminal cases, the entire history of the state's dealing with the individual must be excavated and displayed and inspected for intimations of leniency, and perhaps all his hypothetical future dealings as well, for he might think that cooperation now would yield benefits should he ever again become involved with the law. Any time the government had omitted to charge the witness with a crime, the omission would have to be disclosed to defense counsel and explained to the jury, unless the statute of limitations had run, since until then the government could punish the witness for unsatisfactory testimony by prosecuting him for the crime. But second, the impeachment value would be slight, once charging practices were explained to the jury.

Evidence presented in Wisehart's postconviction proceedings, and thus not available to the defense at trial, indicated that the state had not prosecuted Johnson for two burglaries that they suspected him of having committed, because they didn't want by doing so to dissuade him from testifying against Wisehart; they didn't want to antagonize him. But what would knowledge of this motive of the state's have added to the jury's consideration? Had the prosecutor testified that of course the state didn't want to risk losing a witness in a capital case merely to be able to convict the witness of burglary, this would not have helped Wisehart.

Yet "certainly Johnson was aware of the benefits he was receiving for providing what the police thought to be valuable testimony." 693 N.E.2d at 58 n. 50. And it's just a step from that to thinking that it would have been reasonable for Johnson to assume that if he failed to cooperate, the state might revive the charges against him. He thus had something to gain by testifying against Wisehart and much to lose if he stopped cooperating. But this just brings us back to the problem of indefinite boundaries. A criminal trial must not be allowed to turn into an inquiry into disparate treatment of criminals, with the witness being asked whether he'd received any benefit that he would not have received had the state not wanted his testimony and whether therefore he feared retaliation if he stopped playing ball. In sum, we cannot say that the Indiana Supreme Court's ruling "was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States." 28 U.S.C. § 2254(d)(1).

The other issue presented by the appeal is whether Wisehart was deprived of his right to trial by an impartial jury. In a postconviction hearing conducted more than a decade after his trial, Wisehart presented an affidavit from one of the jurors that stated: "I learned that Mark Wisehart had taken a polygraph test. The jury had been brought to the courthouse, and was preparing to begin court when we were told court would not be held that day. I learned the court session had been canceled because Mark Wisehart was to take a polygraph test. I do not recall who gave me the information about the polygraph. After the polygraph, the trial continued, and I never learned the results of [the] polygraph test." No effort was made to call the juror as a witness in an effort to obtain further detail of the incident, and no other factual inquiry was conducted; the quoted passage is the entire information we have about the incident.

Wisehart points to the statement in the Supreme Court's decision in Remmer v. United States, 347 U.S. 227, 229, 74 S.Ct. 450, 98 L.Ed. 654 (1954), that "in a criminal case, any private communication, contact, or tampering directly or indirectly, with a juror during a trial about the matter pending before the jury is, for obvious reasons, deemed presumptively prejudicial. . . . The presumption is not conclusive, but the burden rests heavily upon the Government to establish, after notice to and hearing of the defendant, that such contact with the jury was harmless to the defendant." Ripped from its context, the statement is difficult to take seriously, because it is so easy to imagine situations in which a "private communication . . . with a juror during a trial about the matter pending before the jury" would not create a rational presumption of prejudice. Suppose a juror's spouse said to the juror, "I saw you on television in the jury box, and you looked great." That would be a private communication concerning the case, but it would not be suggestive of jury tampering. General language does not decide particular cases, as Holmes liked to say. Judges expect their pronunciamentos to be read in context, which in Remmer, as the Court went on explain, involved "the sending of an F.B.I. agent in the midst of a trial to investigate a juror as to his conduct" — something that "is bound to impress the juror and is very apt to do so unduly. A juror must feel free to exercise his functions without the F.B.I. or anyone else looking over his shoulder." Id.

In short — and the subsequent case law is in accord with this interpretation of Remmer — the extraneous communication to the juror must be of a character that creates a reasonable suspicion that further inquiry is necessary to determine whether the defendant was deprived of his right to an impartial jury. How much inquiry is necessary (perhaps very little, or even none) depends on how likely was the extraneous communication to contaminate the jury's deliberations. Evans v. Young, 854 F.2d 1081, 1083-84 (7th Cir. 1988); Dyer v. Calderon, 151 F.3d 970, 974-75 (9th Cir. 1998) (en banc); United States v. Williams-Davis, 90 F.3d 490, 499-501 (D.C. Cir. 1996); see generally Oswald v. Bertrand, 374 F.3d 475, 477-78, 480 (7th Cir. 2004). The Indiana Supreme Court failed to apply this test or any reasonable variant of it. All the court said was that Wisehart had failed to present "live testimony" or any other evidence besides the affidavit, that the affidavit didn't indicate that the jury had been biased as a result of learning about the polygraph test, and that there is no indication that any other jurors learned about it. The first point is irrelevant; the affidavit was evidence. The second point shows only that the affidavit does not prove that the juror was biased but merely raises a suspicion. The third point is irrelevant, because a defendant is entitled to be tried by a jury no member of which has a bias induced by extraneous matter. Williams v. Bagley, 380 F.3d 932, 943-44 (6th Cir. 2004); United States v. Brande, 329 F.3d 1173, 1178 (9th Cir. 2003); see also Morgan v. Illinois, 504 U.S. 719, 729, 112 S.Ct. 2222, 119 L.Ed.2d 492 (1992); Smith v. Phillips, 455 U.S. 209, 102 S.Ct. 940, 71 L.Ed.2d 78 (1982); Remmer v. United States, supra, 347 U.S. at 229, 74 S.Ct. 450.

The affidavit was sufficient to necessitate a further inquiry at which the judge would have asked the juror how she had reacted to learning about the polygraph test. From the fact that the trial resumed after the test had she assumed that Wisehart had flunked it? If so, had she thought polygraph tests such reliable detectors of lies that she inferred that Wisehart must be guilty? The reliability of polygraph tests remains an open question (see references in United States v. Scheffer, 523 U.S. 303, 309-12, 118 S.Ct. 1261, 140 L.Ed.2d 413 (1998); id. at 333, 118 S.Ct. 1261 (dissenting opinion); see also United States v. Lea, 249 F.3d 632, 638 (7th Cir. 2001); King v. Trippett, 192 F.3d 517, 522 (6th Cir. 1999); Hubbard v. State, 742 N.E.2d 919, 924 (Ind. 2001)), but some laypeople may think them infallible. United States v. Scheffer, supra, 523 U.S. at 313-14, 118 S.Ct. 1261. Indiana worries about this; the prosecutor would have violated Indiana law had he told the jury that Wisehart had taken a polygraph test and flunked, since Wisehart hadn't stipulated to the admissibility of the test results. Willey v. State, 712 N.E.2d 434, 439-40 (Ind. 1999); Wisehart v. State, supra, 693 N.E.2d at 63 n. 72; Sanchez v. State, 675 N.E.2d 306, 308 (Ind. 1996).

It doesn't follow that permitting such evidence to be given in a criminal trial would violate the Constitution; a violation of a state's rule on the admissibility of evidence is not a violation of federal law. But it does follow that smuggling the defendant's test result into the jury room would have required a hearing under the Remmer line of cases. The concern with extraneous material in the jury room is not limited to material that would be inadmissible at trial; if news about a defendant's having taken a polygraph test reaches the jurors under the table as it were, the defendant is denied an ability to put the results in — to explain for example that he had passed the test, or at least not failed it; to describe the weaknesses of lie detectors; and in short to pull, or at least try to pull, the sting.

What happened in this case was not so egregious as telling the jury sub rosa that Wisehart had taken and flunked a polygraph test. But it was bad enough to require a hearing, however abbreviated, to determine what impact the news that he had taken the test had on the jury.

Back in 1994 it would have been relatively easy to call the juror as a witness and ask her to explain her reaction to learning about the polygraph test, though she might have forgotten because the trial had been conducted in 1983. It will be all the more difficult today to reconstruct an incident now more than twenty years in the past. But it was the state's burden, given the juror's affidavit, to present evidence that the jury's deliberations had not been poisoned by the reference to Wisehart's having been given a polygraph test.

The judgment must therefore be vacated with directions that the state release Wisehart, retry him, or conduct a further postconviction hearing addressed to the issue of jury bias.