Petitioner Valentin Sorto, convicted of murder and related offenses in New York state court, petitions for a federal writ of habeas corpus on the ground that the state courts unreasonably misapplied Batson v. Kentucky, 476 U.S. 79, 97-98, 106 S.Ct. 1712, 90 L.Ed.2d 69 (1986), and its progeny. During jury selection, Sorto twice asserted that the prosecution was discriminating against minority jurors in its exercise of peremptory strikes; both challenges were denied for failure to establish a prima facie ease of discrimination. Resolution of the Batson issue in this case requires more information about the possible jurors than the record discloses. Only limited portions of jury selection were recorded: This Court has not been presented with a full transcript of *167the voir dire, or with data describing the composition of the potential juror pool. Because Sorto bears the burden of demonstrating an unreasonable application of federal law, the insufficiency of the record defeats his petition, and we therefore affirm.
BACKGROUND
Valentin Sorto was arrested for the April 27, 1997 murder of Jose Alvarez and the severe beating of Lazaro Cruz. According to the prosecution, Sorto and another man retaliated for an attack on their fellow gang member by stabbing Alvarez in the neck and chest, leaving him to bleed to death in a stairwell; and Sorto punched Cruz and slashed his hands with a broken glass bottle. Sorto and his accomplice were indicted for murder in the second degree, assault in the second degree, and criminal possession of a weapon in the third degree. The accomplice pled guilty; Sorto went to trial and was convicted.
At Sorto’s trial, jury selection proceeded according to the “jury box” system, in which groups of fourteen prospective jurors are randomly called from the venire, interviewed, and then challenged by the attorneys. Following decision on the challenges for cause, the lawyers are afforded the opportunity to exercise one or more of their twenty peremptory challenges. A new set of potential jurors is then invited into the jury box, and the process repeated until a jury is empaneled. See generally People v. Webb, 187 Misc.2d 451, 722 N.Y.S.2d 349, 350-51 (N.Y.Sup.Ct.2001).
Round One
In the first round of jury selection, the prosecution challenged potential juror Vidal Martinez for cause, citing Martinez’s expressed sympathy for gang members, and his concession that he would have trouble deferring to the interpreter in the translations from Spanish. Sorto contested the challenge for cause, but allowed that the prosecution would be free to “us[e] one of his peremptories” to strike Martinez. The trial judge agreed and rejected the challenge for cause. Five more first-round jurors were dismissed for cause, all upon objection by the prosecutor.
Next, the prosecution exercised peremptory strikes against three jurors: [i] Martinez; [ii] Carlos Rivera, who is of Salvadoran descent; and [iii] and John Harper, an African American. Defendant then raised the first of his two Batson objections. Defendant argued: that Martinez was a peace officer who likely would be welcomed by the prosecution but for a discriminatory motive; that Rivera had filled out an unobjectionable jury questionnaire and that there was no basis for striking him other than his nationality, which was the same as the defendant’s; and that the use of three prosecutorial strikes against three minority potential jurors established (under the circumstances) a pattern of discrimination.1
*168The prosecution disputed the existence of a prima facie case of discrimination, and accordingly offered no further explanation for its strikes. However, the prosecution withdrew its objection to Martinez, thereby empaneling one of the two challenged Latino jurors.2 Defendant casts the prosecution’s about-face as a telling implicit admission; the court construed it as a token of the good faith.
The state court denied the Batson challenge for lack of a prima facie case, but agreed to remain seized of the issue, especially as related to the strike of Rivera: “the Court will keep it in mind as we proceed. So certainly we should keep both the questionnaire and the card of [Rivera].” Trial Tr. at 132.
Round Two
Only two jurors were successfully empaneled after round one; a second set of potential jurors were called to the jury box for voir dire. On this second round, the prosecution challenged Hazel Mays (an African American) for cause on the ground that Mays had hesitated before agreeing to be fair and impartial, and because she supposedly admitted that she “identifie[d] with the defendant because he is a member of a minority group.” When the challenge for cause was denied, the prosecution exercised a peremptory challenge to excuse her. After the, peremptory strike of Mays, the defendant interposed a second Batson challenge, claiming discrimination “in regards to the prosecution’s elimination of Mrs. Mays.” (emphasis added). The record does not clearly show what evidence was submitted to support the pri-ma facie case at this juncture. Defendant did not reprise the first-round eliminations of Harper and Rivera as evidence to support a prima facie case on this later motion, but the judge may have made that assumption, because he asked, with regard to this second Batson challenge, whether defendant placed “Hispanic and black in the same group.”
In response to the second Batson challenge, the prosecution spontaneously explained its strike of juror Harper — the African American dismissed in round one-even though Harper was not mentioned as the subject or basis of the second motion. The prosecutor defended that strike on the ground of Harper’s “sympathy” for his imprisoned nephew, and Harper’s prior hostile run-ins with the police. The prosecution did not attempt to explain its round-one strike of Rivera, nor was that strike ever mentioned by either party during round two.
Next, the prosecution explained that it challenged Mays because of her announced self-identification with the defendant. In any event, the prosecution asserted that no explanation was needed because no prima facie case had been stated.
The state court dismissed the second Batson challenge on the grounds that the defendant “hadn’t reached the threshold with respect to the particular juror,”3 and in the alternative (“in case another Court were to find differently”), that the prosecution had successfully offered non-pretextu*169al, race-neutral explanations for the dismissals of jurors Harper and Mays.
Sorto’s state appeal argued (inter alia) that the trial court (1) erroneously ruled that a prima facie case had not been established after the first round objections, (2) erroneously ruled that a prima facie case had not been established after the second round objection, and (3) erroneously found the prosecution’s proffered explanations for the Harper and Mays strikes were non-pretextual. The Appellate Division treated “defendant’s [Batson] contentions [as] either unpreserved for appellate review or without merit.” People v. Sorto, 274 A.D.2d 487, 487, 711 N.Y.S.2d 911 (N.Y.App.Div.2000). As to the existence of a Batson prima facie case, the parties agree that because the government offered no procedural default argument, the Appellate Division affirmance constitutes a ruling on the merits for purposes of the Antiterrorism and Effective Death Penalty Act of 1996 (“AEDPA”). The New York Court of Appeals denied leave to appeal. 95 N.Y.2d 893, 715 N.Y.S.2d 385, 738 N.E.2d 789.
Sorto next petitioned for federal habeas relief, challenging (inter alia) the Batson rulings. The district court denied the petition, but granted a certificate of appeala-bility as to the Batson claims.
DISCUSSION
Because the Appellate Division rendered a decision on the merits, our review of the prima facie rulings is governed by AEDPA. Torres v. Berbary, 340 F.3d 63, 68 (2d Cir.2003). Under AEDPA, a petition for a writ of habeas corpus claiming a state court error of law “shall not be granted ... unless the adjudication of the claim resulted in a decision that 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). See also Williams v. Taylor, 529 U.S. 362, 365, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). “[A] n unreasonable application of clearly established Supreme Court precedent occurs when a state court identifies the correct governing legal principle from the Supreme Court’s decisions but unreasonably applies that principle to the facts of the prisoner’s case.” Torres, 340 F.3d at 69 (internal citations omitted). While “[t] he precise method for distinguishing objectively unreasonable decisions from merely erroneous ones” is somewhat unclear, “it is well-established in this Circuit that the ‘objectively unreasonable’ standard of § 2254(d)(1) means that petitioner must identify some increment of incorrectness beyond error in order to obtain habeas relief.” Id. (internal citations omitted). This Court reviews a district court’s denial of petition for a writ of habe-as corpus de novo. Harris v. Kuhlmann, 346 F.3d 330, 342 (2d Cir.2003).
Round One
The Supreme Court’s decision in Batson v. Kentucky, 476 U.S. 79, 106 S.Ct. 1712, 90 L.Ed.2d 69 (1986), and its progeny limit the traditionally unfettered prerogative of exercising peremptory strikes by forbidding certain discrimination in jury selection. The Supreme Court has generally granted individual courts the leeway to adopt their own procedures to test for discriminatory strikes. See Howard v. Senkowski, 986 F.2d 24, 29 (2d Cir.1993) (“[T]he decisions ... recognize the role that remains for lower courts to work out the mechanics for implementing these requirements.”). That leeway is granted within a procedural framework:
The Batson Court ... established] a three-step burden-shifting framework for the evidentiary inquiry into whether a peremptory challenge is race-based: *170First, the moving party — i.e., the party challenging the other party’s attempted peremptory strike — must make a prima facie case that the nonmoving party’s peremptory is based on race. Second, the nonmoving party must assert a race-neutral reason for the peremptory challenge. The nonmoving party’s burden at step two is very low.... [Although a race-neutral reason must be given, it need not be persuasive or even plausible. Finally, the court must determine whether the moving party carried the burden of showing by a preponderance of the evidence that the peremptory challenge at issue was based on race.
McKinney v. Artuz, 326 F.3d 87, 97-98 (2d Cir.2003) (internal citations omitted).
The first step of the Batson analysis, requiring the showing of a prima facie case, is not meant to be onerous. Johnson v. California, 545 U.S. 162, 170, 125 S.Ct. 2410, 162 L.Ed.2d 129 (2005). However, this stage of the analysis still requires consideration of “all relevant circumstances.” Batson, 476 U.S. at 96, 106 S.Ct. 1712. As Batson explained:
[A] ‘pattern’ of strikes against black jurors included in the particular venire might give rise to an inference of discrimination. Similarly, the prosecutor’s questions and statements during voir dire examination and in exercising his challenges may support or refute an inference of discriminatory purpose. These examples are merely illustrative
Id. The prima facie inquiry is a hurdle that preserves the traditional confidentiality of a lawyer’s reason for peremptory strikes unless good reason is adduced to invade it: While litigants must now explain their motivations for certain strikes, courts must still be mindful of “each side’s historical prerogative to make a peremptory strike or challenge ... without a reason stated” if a prima facie case of discrimination has not been established. Miller-El v. Dretke, 545 U.S. 231, 125 S.Ct. 2317, 2324, 162 L.Ed.2d 196 (2005) (internal citation omitted).
To establish a prima facie case, “a defendant must show facts and circumstances that raise an inference that the prosecutor used the peremptory challenge to exclude potential jurors from the petit jury on account of their race.” Overton v. Newton, 295 F.3d 270, 276 (2d Cir.2002). The discharge of this burden may entail a review of prosecutorial strikes over the span of the selection process: Thus this Court has held, on habeas review, that a state court does not act unreasonably where it denies a Batson challenge early in the jury selection process. Id. at 279.
Where a litigant points to a pattern of strikes as evidence of discrimination, “statistical disparities are to be examined” as part of the Batson prima facie inquiry. United States v. Alvarado, 923 F.2d 253, 255 (2d Cir.1991). The need to examine statistical disparities may commend a wait-and-see approach. As we held in Overton (where the Batson challenge was brought after seven of ten potential African American jurors were dismissed through peremptory challenges), an early Batson challenge limits the state court’s ability to properly assess a prima facie case:
the trial judge never confronts], and the trial record does not reveal, what the statistics would [] show[] at the conclusion of jury selection. If those statistics sufficiently establish[ ] the inference that challenges [a]re based on race, the court could then [] implement] the Batson process to ensure that impermissible challenges [are] not [] allowed. If, on the other hand, the statistics at the conclusion fail[] to support a sufficient inference, there would be no need to engage in the 13 process.
*171Overton, 295 F.3d at 279 (emphasis added). Overton concluded that the state trial judge acted reasonably in “refusing] to implement Batson’s process for testing each questioned challenge midway in the process.” Id. at 280 (emphasis added).
Sorto raised his first Batson challenge after only three peremptory strikes. The state court acted reasonably in denying this challenge as premature, while remaining open to reevaluating these strikes as part of a later challenge. Accordingly, the district court did not err in denying the habeas petition challenging the denial of the first round Batson challenge.4
Round Two
The existence of a prima facie Batson case is a mixed question of law and fact. Overton, 295 F.3d at 276-77. On habeas review, then, we will disturb the state court ruling only if it “was contrary to, or involved an unreasonable application of, clearly established Federal law.” Id. at 277 (quoting 28 U.S.C. § 2254(d)(1)). Sor-to raises no argument that the state court identified the wrong legal standard; he therefore must show an unreasonable application.
“[A] state court decision fails the ‘unreasonable application’ prong of AEDPA analysis, ‘if the state court identifies the correct governing legal principle from [the Supreme Court’s] decisions but unreasonably applies that principle to the facts of the prisoner’s case.’ ” Id. (quoting Williams v. Taylor, 529 U.S. 362, 413, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000)). To challenge the application of law to fact, a petitioner must demonstrate the existence of a particular set of facts to which a legal rule was applied: We cannot say whether a properly identified rule of law was wrongly applied unless we know the set of facts to which the rule was applied. See generally Escalera v. Coombe, 826 F.2d 185, 193 (2d Cir.1987) (mixed questions of law and fact create “subsidiary questions of historical fact”). Facts on which a petitioner hopes to rely must be established by (at least) a preponderance of the evidence in the habeas court. Galarza v. Keane, 252 F.3d 630, 637 n. 5 (2d Cir.2001).
When, as here, a Batson prima facie case depends on a pattern of strikes, a petitioner cannot establish that the state court unreasonably concluded that the pattern was not sufficiently suspicious unless the petitioner can adduce a record of the baseline factual circumstances attending the Batson challenge. A sufficient record would likely include evidence such as the composition of the venire,5 the adversary’s *172use of peremptory challenges, the race of the potential jurors stricken, and a clear indication as to which strikes were challenged when and on what ground, and which strikes were cited to the trial court as evidence of a discriminatory intent. That information may be common knowledge in the courtroom based on the shared perceptions of the lawyers and the trial judge; but an appellate court does not have the benefit of what can be observed by those in the trial courtroom. Therefore, to the (appreciable) extent that information regarding the jury and the voir dire process bears upon establishing a pri-ma facie case, a sufficient appellate record may depend on a recitation of relevant information on the record in the trial court.
For example, in United States v. Alvarado we stated:
[T]he prosecution’s challenge rate against minorities was 50 percent (three of six) in the selection of the jury of 12, and 57 percent (four of seven) in the selection of the jury of 12 plus alternates. Whether this rate creates a statistical disparity would require knowing the minority percentage of the venire; for example, if the minority percentage of the venire was 50, it could be expected that a prosecutor, acting without discriminatory intent, would use 50 percent of his challenges against minorities.
923 F.2d 253, 255 (2d Cir.1991) (emphasis added). The analysis is thus driven by information regarding the prosecution’s strikes so that the federal court can usefully consider a prosecutorial strike pattern in the essential contexts.
The Alvarado Court met this need by taking judicial notice of the counties that compose the Eastern District of New York and the minority percentage of the populations of those counties, and then accepting that percentage as a “surrogate” for the minority population of the venire. Id. at 256. On direct appeal from a conviction in district court, a panel may, in a suitable case, supplement the record in such a manner (though that is a thin basis for assigning discriminatory motive to an officer of the court). But it is one thing to say that a panel may exercise that discretion on direct appeal in a suitable case, and quite another to hold on collateral review that it is contrary to or an unreasonable application of Batson for a state court to fail to take judicial notice of such “surrogate” facts and data, particularly where no such request appears to have been made, cf. N.Y. C.P.L.R. 4511, and where, for all we know, “surrogate” data in lieu of record facts may not have been at hand in the state court. In any event, we would in this case decline to exercise our discretion to take judicial notice of the counties from which Sorto’s venire was drawn in order to determine the minority percentages of those counties, or to assume that those percentages would reflect the ethnic or racial makeup of the venire in Sorto’s trial.
The record before us contains insufficient data as to the prosecution’s strike pattern to support a finding that the state court unreasonably applied Batson. For example, between the strikes of Rivera and Mays, the prosecution exercised peremptory challenges against potential jurors Mink and Burdonis. Petitioner’s brief states that Mink was not a minority; however, we have no particulars about Burdonis or about the prospective and em*173paneled jurors who were not challenged by the prosecution. Moreover, Sorto lacks any resource in the record to resolve in his favor conflicting reports as to the composition of the venire. At oral argument, petitioner suggested — without evidence — that the strikes of Rivera and Martinez removed all the Hispanic potential jurors from the venire. But petitioner’s brief acknowledges (at least) one additional Hispanic potential juror on the venire (potential juror Zate). Appellant’s Br. at 7. Beyond this, we have no information as to how many Hispanic and minority potential jurors remained on the venire after all of the challenged strikes. Absent this information, we cannot say that the state court acted unreasonably: The venire may have overwhelmingly consisted of minority jurors, rendering any individual peremptory strike of a minority juror less suspicious.6
The dissent illustrates what happens when insufficient care is taken to build a record of Batson discrimination: the case is made to depend on a labored piecing together of transcript fragments in an effort to intuit the race and ethnicity of jurors and to reconstruct and imagine what might have happened.
A well-crafted record in the state trial court is needed also to fix (1) the scope of a given Batson challenge and (2) the evidence adduced to support the motion. Sorto argues that the state “court’s failure to require a reason for the challenge to Rivera was an unreasonable disregard of its duty under Batson.” Petitioner’s Br. at 34. But the record is far too sketchy to support a conclusion that the state court acted unreasonably in refusing to demand an explanation for the Rivera strike. In reviewing the second-round Batson challenge, we are unable to identify (1) precisely which strikes were challenged, and (2) on what basis any challenge was made.
Seemingly, the round two challenge was limited to the strike of Mays: Sorto described the second round challenge as “regarding] the prosecution’s elimination of Mrs. Mays.” Trial Tr. at 208 (emphasis added). Petitioner argues that a question posed by the trial judge (whether the second round challenge grouped together strikes to African Americans and Hispanics) indicated that “the trial court understood that the scope of the renewed Batson challenge included all four challenged minority jurors.”7 Appellant’s Br. at 33. This may or may not have been the trial court’s thinking. But habeas may not be granted based on speculation as to the trial court’s thought process; the record limits the set of challenges under review. Petitioner explicitly limited the challenge to the strike of Mays; the record therefore does not command the conclusion that the strikes to Rivera and Harper were even in play in the second round;8 *174and consequently we cannot rule that the state court acted unreasonably in refusing to demand an explanation for the strike to Rivera. While unrecorded impressions may have given the trial judge certain clues as to the intended scope and basis of the round-two Batson challenge, we need a clear record.
The inadequacy of the record is one reason that the trial court’s rejection of Sorto’s second Batson challenge was not unreasonable; another independent reason is the preliminary stage at which the challenge was lodged. As discussed above, Sorto raised his first challenge .after only three peremptory strikes, and accordingly the state court’s denial of that challenge was reasonable. The same logic applies to Sorto’s second challenge, which came only after the prosecutor’s sixth peremptory challenge, four fewer than the number in Overton, 295 F.3d at 274. Between Sorto’s first and second Batson challenge, the prosecutor struck potential jurors Mink and Burdonis; neither of them, on the record before us, appears to have been black or Hispanic. It was the prosecutor’s sixth peremptory strike (of Mays) that precipitated Sorto’s second Batson challenge, at which point the prosecutor had used four of six peremptory strikes to remove black or Hispanic potential jurors. We cannot say that this stage of the voir dire was materially less preliminary than the stage at which Sorto made his first challenge. Accordingly, it was reasonable for the state court to conclude that a problematic pattern of strikes had not yet developed. Sorto did not renew his objection in later rounds of voir dire, and so we cannot say whether such a pattern ever developed. See id. at 279-80.
It is here that the dissent parts ways. Despite its agreement that “the state court acted reasonably in denying the first Bat-son challenge as premature,” the dissent concludes, leaning heavily on Green v. Travis, 414 F.3d 288 (2d Cir.2005), that the state court “unreasonably applied Batson” when it denied Sorto’s second challenge. Dissent Op. at 176. Between the first (premature) challenge and the second challenge, the government [i] withdrew its strike against a Hispanic juror (Martinez), [ii] struck a juror who was neither African American nor Hispanic (Mink), [iii] struck a juror who was African American (Mays), and [iv] struck a juror who was neither African American nor Hispanic (Burdonis). These intervening events furnish no appreciable support for a finding of discrimination beyond the showing that (we all agree) was insufficient and premature.
In any event, the dissent’s reliance on Green is misplaced. In Green, the “Appellate Division [had] not address[ed] whether the pattern of the prosecution’s peremptory strikes established a prima facie case of discrimination.” 414 F.3d at 299. So there was no state court determination on that issue to which the Green Court could give AEDPA deference: it is one thing to conclude that a pattern of strikes is prima facie evidence of discrimination; it is a very different thing to hold that the contrary conclusion would be an unreasonable application of Batson.
Accordingly, we hold that the record is insufficient to disturb the state court’s ruling on the existence of a prima facie case in support of the Batson challenge to the Mays strike. We similarly refuse to disturb the second round treatment of the Rivera strike, as the record does not even *175clearly indicate that that strike was at issue.9
We have considered petitioner’s remaining arguments and find each of them to be without merit. For the foregoing reasons, the judgment of the district court is affirmed.
. In the state court, the parties vigorously debated whether different minority groups should be aggregated-particularly African American and Latino groups-towards evaluating a Batson prima facie case. This Court has since held that "a defendant raising a Batson claim of purposeful racial discrimination does not have to demonstrate that all venireper-sons who were peremptorily excused belong to the same 'cognizable racial group.' ” Green v. Travis, 414 F.3d 288, 297 (2d Cir.2005) (internal citations omitted). The state court (not yet guided by our decision in Green) expressed reluctance to aggregate in discussing the second Batson challenge, but implied no view on the issue in denying the first Batson challenge. However, because the petitioner has not sufficiently established the factual circumstances giving rise to the second Batson challenge, the state court's erroneous view on aggregation is not implicated here.
. At trial, the parties disputed whether the "withdrawal” of a challenge has any impact for Batson purposes. For purposes of this appeal we will assume, arguendo, that the withdrawn strike still factors into a prima facie analysis.
. Use of the singular ("juror”) is suggestive: Even if the trial judge assumed at the onset that the second Batson challenge was supported by the Rivera and Martinez strikes, and even if the trial judge operated under this assumption while denying the challenge, reference to a "particular juror” indicates that the state court believed that only the Mays strike (and not the previous round’s strike of Rivera) had been challenged.
. Alternatively, petitioner argues that the prosecution's withdrawal of its peremptory challenge to juror Martinez was so irregular as to evince a prima facie case of discrimination. The state court interpreted this withdrawal as a gesture of good faith by the prosecution. This was a reasonable interpretation of the prosecution’s motive: The withdrawn challenge could reasonably be viewed as expressing a willingness to empanel one of two potential Hispanic jurors interviewed in round one. Though Sorto cannot understand why the prosecution would have withdrawn a peremptory from a juror previously challenged for cause, Sorto himself provides a possible answer: As part of his Batson challenge, Sorto reminded the prosecution that Martinez worked as a peace officer and would therefore likely be a favorable witness for the prosecution. Accordingly, the state court did not act unreasonably in ruling that the withdrawn challenge did not support a prima facie case.
. Here, "venire” refers to the jurors who were called to the jury box and subject to evaluation and strike. The term may also be used to reference the entire group of jurors eligible to be called into the jury box. Information regarding jurors who were eligible but not called may form part of a sufficient record insofar as that information assists a reviewing *172court in evaluating the pattern of strikes at issue or the strategy and motive of the lawyer exercising the strikes.
.Our analysis is naturally influenced by the context of this case: [i] a state court’s denial [ii] of a Batson motion that is premised on an allegedly pernicious pattern of strikes. Given our deferential habeas review, we cannot disturb a state court judgment as "unreasonable” unless we can consider the factual background that gave rise to a state court ruling. Background data as to the venire would seem less necessary when a Batson challenge is premised on evidence other than pattern, such as comments made during voir dire or during the exercise of challenges. See generally Batson, 476 U.S. at 96, 106 S.Ct. 1712.
. See our discussion at supra note 3.
. Sorto did not waive his ability to petition for habeas relief by his failure to restate his challenges to Rivera and Harper. It is true that a Batson objection is waived if not restated in the federal district court; but on a habeas petition challenging a state judgment, waiver is a matter of state procedure. DeBerry v. Portuondo, 403 F.3d 57, 66 (2d Cir.2005). The issue of restated challenges is not implicated here; our ruling is premised on the substantive insufficiency of the habeas claim. For example, we would be no more likely to *174find a Batson violation in Round Two even if petitioner had not challenged the Rivera and Harper strikes in Round One.
. Petitioner argues that the state court mooted the prima facie issue in addressing — for the sake of appellate review — the credibility of the prosecution's proffered explanations. Though that approach was taken in Hernandez v. New York, 500 U.S. 352, 359, 111 S.Ct. 1859, 114 L.Ed.2d 395 (1991), a habeas court remains free to affirm based on the prima facie rulings. See, e.g., United States v. Diaz, 176 F.3d 52, 77-78 (2d Cir.1999).