HOWARD, Circuit Judge.

In April 1999, a Massachusetts jury convicted Stanley Donald of rape, kidnaping, assault and battery with a dangerous weapon, and carjacking. After a series of unsuccessful attempts at post-trial relief in state court, Donald filed a petition for habeas corpus in federal district court in Massachusetts.1 His petition set forth two general categories of alleged constitutional infirmities in the state court proceedings. First, that he was denied effective assistance of counsel both at trial and during post-conviction proceedings; and second, that he was denied post-conviction discovery in the state courts and that said denial took place without a hearing. During the pendency of his habeas petition, Donald proffered a motion for discovery, seeking access to various pieces of physical evidence which Donald wanted to subject to DNA testing more advanced than that performed prior to his trial. The district court denied the discovery motion without comment and subsequently denied Donald's petition in its entirety. Donald v. Spencer, 685 F.Supp.2d 250 (2010).

On appeal, Donald challenges only the district court's denial of his discovery motion.2 He also argues that the Antiterrorism and Effective Death Penalty Act of 1996 ("AEDPA"), pursuant to which the district court evaluated his petition, unconstitutionally restricts review to existing Supreme Court precedent. We affirm.

I.

While habeas corpus proceedings under 28 U.S.C. § 2254 are civil in nature, they are governed by "a discrete set of procedural rules." Teti v. Bender, 507 F.3d 50, 60 (1st Cir.2007) (citing Mayle v. Felix, 545 U.S. 644, 654 & 655 n. 4, 125 S.Ct. 2562, 162 L.Ed.2d 582 (2005)). Accordingly, "`[a] habeas petitioner, unlike the usual civil litigant in federal court, is not entitled to discovery as a matter of ordinary course.'" Bader v. Warden, 488 F.3d 483, 488 (1st Cir.2007) (quoting Bracy v. Gramley, 520 U.S. 899, 904, 117 S.Ct. 1793, 138 L.Ed.2d 97 (1997)). Instead, Rule 6(a) of the Rules Governing Section 2254 Cases allows the federal habeas judge, "for good cause, [to] authorize a party to conduct discovery under the Federal Rules of Civil Procedure and may limit the extent of discovery." Under Rule 6(b), the requesting party must provide reasons for the request. We review the district court's denial of discovery for abuse of discretion. Teti, 507 F.3d at 60 (citing Bader, 488 F.3d at 488).3

To demonstrate "good cause" Donald must present "specific allegations that give a court reason to believe that the petitioner may, if the facts are fully developed, be able to demonstrate that he is ... entitled to relief." Id. (citing Bracy, 520 U.S. at 908-09, 117 S.Ct. 1793). Undergirding this construct is our admonition that "[a] habeas proceeding is not a fishing expedition." Id. Here, Donald argues that he provided the district court with "a reason to believe" that more refined DNA testing would demonstrate that he was not the perpetrator of the rape. This necessarily requires us to recount the evidence presented against him at trial.

II.

We summarize the relevant facts as described in the Massachusetts Appeals Court's decision affirming Donald's conviction. Commonwealth v. Donald, 56 Mass.App.Ct. 1102, 775 N.E.2d 1283, 2002 WL 31246493 (Mass.App.Ct.2002) (unpublished table decision); see Teti, 507 F.3d at 58 (noting § 2254(e)(1)'s "presumption of correctness" applicable to state appellate courts' fact-finding).

On Tuesday, October 21, 1997, a woman was accosted at 9:40 AM in the garage of her apartment in Watertown, Massachusetts. The perpetrator smashed her face into the cement floor of the garage, breaking the victim's nose. The man said he needed money for drugs. When the victim said she didn't have sufficient cash, the man ordered her into her own car, shoved her into the passenger seat and drove out of the garage. He subsequently took her driver's license and demanded her automated teller machine (ATM) card and password.

The perpetrator, who was threatening the victim throughout the ride and had become increasingly agitated, eventually stopped the car. He pulled the victim out the passenger-side door, her head still covered with the jacket he had placed over it. Then he hit her on the side of the head, perhaps with a rock. She tried to play dead but he ordered her to walk. He commanded her to remove her hosiery and underwear, and to lay on the ground with her legs in the air. He then raped her several times.

Finally, she heard him walk to the car and drive away. After getting dressed, she ran to the road and flagged down a passing motorist, who took her to the nearest medical facility — a nearby animal clinic — where the staff called police. Responding officers found the bloodied victim with an obviously broken nose, numerous cuts and abrasions, and disheveled hair and clothing. An ambulance transported her to a hospital. From the victim's description, police quickly located the crime scene, about one-quarter mile from the animal clinic.

At roughly the same time that the victim was arriving at the animal clinic, a $300 withdrawal was made from her bank account at an ATM four miles from the rape scene. The videotape of the withdrawal was secured, and still photographs of the transaction were made from the taped footage.

At 8:00 PM on the night of the attack, police discovered the victim's abandoned car about twenty miles from the scene of the rape, partially obscured by brush. At 5:30 AM the following day, a handyman sweeping the area found a driver's license on the ground a short distance from the location of the victim's car. The name on the license was "Stanley Donald." The man returned it by mail to the address on the license.

Having discovered this information, investigators tracked down Donald's employer. In November 1997, they showed the still images from the ATM transaction to the company president and to Donald's supervisor. Both identified the man depicted in the image as Donald. Both also identified him at trial. At the time of the attack, the defendant had been working steadily at the company for more than five months. He did not show up for work as scheduled on October 21, the day of the attack. He never showed up for work again and he never picked up his final paycheck.

On the afternoon of the attack, a chemist from the Massachusetts State Police collected the clothing that the victim wore when she was raped. He later took a cutting from her underpants and submitted it for DNA testing, along with known samples of Donald's and the victim's blood. The result: the frequency of a match between the sample and Donald's profile occurred in one in 7,800 African-Americans.

On November 11, 1997, the defendant's mother gave police an envelope addressed to her, containing a one-page handwritten letter and a handwritten prayer. Experts opined that all three documents were written by the defendant. The envelope was postmarked from New York within three weeks of the attack. The letter read:

On November 20, 1997, the victim viewed a photographic array at a police station. She selected a photograph of the defendant and said that it looked "very, very much like the person who assaulted me." She also identified him at trial. Donald was arrested in Florida in June 1998. Donald, 2002 WL 31246493, at *1-4.

III.

Donald's claim of "good cause" for discovery combines two approaches. First, he tries to minimize the evidence arrayed against him at trial, suggesting that the victim's eyewitness testimony was faulty and that the photograph of him using the victim's ATM card to withdraw money from her bank account minutes after the rape and near the crime scene could be explained by other "possible circumstances" that would render him guilty of theft, but not rape. To recount his theory is to highlight its weakness. Although conceding that he was the person photographed at the ATM, Donald posits that "another man" (possibly his relative) raped the victim, and took her car, ATM card and password. "A short time later" that person "encountered" Donald and gave him the items. The two then disposed of the victim's car, during which time Donald lost his license. Whatever iota of theoretical plausibility this theory might hold, it is bereft of any record support.

Donald's second tack is based on advancements in DNA testing, which he claims would inure to his benefit by establishing his innocence. To be sure, "[m]odern DNA testing can provide powerful new evidence unlike anything known before." Dist. Attorney's Office v. Osborne, ___ U.S. ___, 129 S.Ct. 2308, 2316, 174 L.Ed.2d 38 (2009). But given the evidence amassed against him, Donald offers us nothing other than pure speculation to suggest that he would ultimately be entitled to relief. He cites no facts upon which to base a conclusion that new DNA evidence will exonerate him. By contrast, in Bracy, the Supreme Court reversed the denial of discovery under Rule 6 because of evidence that the petitioner's trial judge was later convicted of accepting bribes from other defendants, and that his own appointed counsel had been an associate of the crooked jurist. 520 U.S. at 906-09, 117 S.Ct. 1793. The petitioner in Bracy suggested that his lawyer accepted appointment to the case and — to the petitioner's detriment — helped bring it to a quick conclusion to help deflect suspicion from rigged cases that came before and after his own. Id. at 909, 117 S.Ct. 1793. The Court said that these "specific allegations" entitled him to discovery. Id. Donald's speculative theories and baseless allegations come nowhere close to meeting the standard articulated in Bracy. Accordingly, we find that the district court did not abuse its discretion in denying his motion for discovery.4

IV.

As for Donald's claim that AEDPA unconstitutionally restricts review to Supreme Court precedent, we continue to reject it. See Abrante v. St. Amand, 595 F.3d 11, 20 (1st Cir.2010) ("`[T]he Constitution is not offended when lower federal courts are prevented from substituting for that of a state court their judgment as to reasonable application of Supreme Court precedent.'" (quoting Evans v. Thompson, 518 F.3d 1, 8 (1st Cir.2008))) cert. denied, ___ U.S. ___, 131 S.Ct. 168, 178 L.Ed.2d 99 (2010).

The judgment of the district court is affirmed.