OPINION

PELANDER, Justice.

¶ 1 This case is before us now for a fourth time on James Granvil Wallace's automatic appeal from two death sentences. See State v. Wallace (Wallace III), 219 Ariz. 1, 191 P.3d 164 (2008); State v. Wallace (Wallace II), 160 Ariz. 424, 773 P.2d 983 (1989); State v. Wallace (Wallace I), 151 Ariz. 362, 728 P.2d 232 (1986). For the reasons below, on independent review we find that the State has not established beyond a reasonable doubt that Wallace inflicted gratuitous violence on the two victims. The murders thus were not heinous or depraved under A.R.S. § 13-751(F)(6),1 the sole aggravating factor alleged in this case. Accordingly, we vacate Wallace's death sentences and sentence him to consecutive life terms of imprisonment.

I.

¶ 2 In early 1984, Wallace was living with his girlfriend, Susan Insalaco, and her two children, sixteen-year-old Anna and twelve-year-old Gabriel. When Wallace came home drunk on January 31, 1984, Susan told him to move out. The next day, Susan went to work, and Anna and Gabriel left for school while Wallace stayed home.

¶ 3 When Anna returned from school, Wallace was hiding behind the front door with a small wooden baseball bat. He struck Anna in the head with the bat from behind. She fell to the ground but continued breathing and moaning. Wallace struck her in the head at least ten times, eventually breaking the bat, but Anna was still alive. Wallace dragged her into the bathroom and drove the broken bat into her throat.

¶ 4 After killing Anna, Wallace found an 18-inch pipe wrench and decided to use it to kill Gabriel because he "didn't want him to suffer like [Anna]." Gabriel arrived home shortly thereafter and walked into his bedroom. Wallace followed and hit Gabriel in the head with the pipe wrench. Gabriel fell to the floor, and Wallace hit him in the head approximately ten more times until he had "crushed his skull."

¶ 5 When Susan arrived home from work a couple hours later, Wallace hit her in the head with the pipe wrench. She fell, and Wallace hit her several more times, ultimately killing her. Wallace reported the crimes to the police the next day. After waiving his Miranda rights, he confessed in detail to the murders but could not explain why he committed them.

¶ 6 Wallace pleaded guilty and was sentenced to death by the trial judge for all three murders. After this Court twice affirmed the convictions and death sentences on appeal, see Wallace I; Wallace II, the federal district court granted habeas corpus relief and ordered a new sentencing trial. That retrial before a jury in 2005 again resulted in the imposition of three death sentences.2

¶ 7 In 2008, we reduced the death sentence for Susan's murder to life in prison because the evidence was insufficient to prove an aggravating circumstance as to that murder. Wallace III, 219 Ariz. at 7-8 ¶¶ 36-39, 191 P.3d at 170-71. Finding error in the jury instructions, we also remanded the case for a new sentencing trial with respect to the murders of the children. Id. at 4-6 ¶¶ 18-25, 8 ¶ 39, 191 P.3d at 167-69, 171. In 2009, a jury found that Wallace murdered both children in an especially heinous or depraved manner through the use of gratuitous violence. The jury found that death was the appropriate sentence for each of those two murders.

II.

¶ 8 Because the murders occurred before August 1, 2002, this Court independently reviews the aggravation and mitigation findings, as well as the propriety of the death sentences. See A.R.S. § 13-755(A). In independent review, "we do not defer to the findings or decision of the jury, with respect to aggravation or mitigation, when determining the propriety of the death sentence." State v. Newell, 212 Ariz. 389, 405 ¶ 82, 132 P.3d 833, 849 (2006) (alterations and internal quotation marks omitted). The state must prove aggravating circumstances beyond a reasonable doubt. State v. Kayer, 194 Ariz. 423, 433 ¶ 28, 984 P.2d 31, 41 (1999).

¶ 9 The sole aggravating circumstance here, heinousness or depravity of the murders, requires proof beyond a reasonable doubt that Wallace inflicted gratuitous violence in murdering Anna or Gabriel.3 Wallace argues that the State failed to prove that aggravating factor beyond a reasonable doubt. "The term `heinous or depraved,'" as used in § 13-751(F)(6), describes "the defendant's state of mind." State v. Murdaugh, 209 Ariz. 19, 31 ¶ 59, 97 P.3d 844, 856 (2004). The state can prove heinousness or depravity by showing that a defendant inflicted gratuitous violence on his victim. State v. Womble, 225 Ariz. 91, 99 n. 7 ¶ 27, 235 P.3d 244, 252 n. 7 (2010).4

¶ 10 In State v. Bocharski, this Court clarified the standard for gratuitous violence, recognizing that our "prior cases ha[d] not been entirely consistent in describing the showing needed to establish" that factor. 218 Ariz. 476, 494 ¶ 85, 189 P.3d 403, 421 (2008); see also Wallace III, 219 Ariz. at 6 ¶ 28, 191 P.3d at 169 (noting that Bocharski "clarif[ied] the principles governing" gratuitous violence). Bocharski established a two-pronged test. First, the state must show that the defendant used "violence beyond that necessary to kill." 218 Ariz. at 494 ¶ 85, 189 P.3d at 421. Second, "[t]he State must also show that the defendant continued to inflict violence after he knew or should have known that a fatal action had occurred."5 Id. at ¶ 87.

¶ 11 In Bocharski, this Court found on independent review that the State failed to prove gratuitous violence beyond a reasonable doubt even though the defendant had stabbed the victim twenty-four times in the head, including eight wounds that penetrated deep into the victim's face and head. 218 Ariz. at 494 ¶¶ 85-86, 189 P.3d at 421. Although we could infer that the defendant used more violence than necessary to kill, we could not conclude that the State established the knowledge requirement. Id. at 495 ¶ 91, 189 P.3d at 422. A medical examiner testified that the immediately fatal wound "probably" occurred early in the sequence of wounds, but was uncertain precisely when. Id. at 494 ¶ 88, 189 P.3d at 421. The doctor also testified that the knife injuries occurred in "quick succession and that all the injuries were likely inflicted within a minute." Id. at 495 ¶ 89, 189 P.3d at 422. This Court concluded that "[b]ecause Bocharski only used a knife to inflict the wounds and completed his attack very rapidly, we find it unlikely he knew or should have known he had inflicted a fatal wound but continued nonetheless to inflict more violence." Id. at ¶ 90.

¶ 12 Even before Bocharski, this Court had made clear that multiple, rapid attacks on a victim, although reprehensible, do not necessarily establish gratuitous violence when the attacks were made in attempting to kill the victim. In State v. Cañez, for example, the defendant "attempted to strangle [the victim], stabbed him six times, and delivered 21 blunt force injuries, ten of them to the head." 202 Ariz. 133, 161 ¶ 106, 42 P.3d 564, 592 (2002). The defendant attacked the victim "with his fist, a frying pan, a laundry bag, and a knife." Id. Because the State asserted that the stabbing occurred last, however, the evidence demonstrated that the defendant "merely escalated his attacks until he succeeded in killing" the victim. Id. at 161-62, 42 P.3d at 592-93. Thus, this Court could not find beyond a reasonable doubt that the violence exceeded that necessary to kill. Id.

¶ 13 Similarly, in State v. Anderson, even though it was a "close[] question," we could not find gratuitous violence. 210 Ariz. 327, 355 ¶ 123, 111 P.3d 369, 397 (2005). This Court reasoned:

Id.

III.

¶ 14 Although this Court has previously considered whether Wallace inflicted gratuitous violence on his victims, the law, most notably in Bocharski, has evolved in this area. In Wallace III, we disavowed the reasoning used in Wallace I and Wallace II and held that the trial court erred in instructing the jury to consider "whether the defendant had available less violent alternatives to cause death" in evaluating gratuitous violence. 219 Ariz. at 4 ¶¶ 15, 18, 191 P.3d at 167. Without any showing that Wallace actually intended to inflict greater violence than that necessary to kill, the State had not proven the "killer's vile state of mind" as required to establish heinousness or depravity under § 13-751(F)(6). Id. at 4-5 ¶ 19, 191 P.3d at 167-68 (internal quotation omitted).

¶ 15 This Court also concluded in Wallace III that the issue of gratuitous violence should not have been submitted to the jury with respect to Susan's murder. Id. at 7 ¶ 36, 191 P.3d at 170. We noted that the attack on her involved four or five blows to the head "over a relatively brief period," and "the blows were apparently struck in rapid succession with the same implement that caused death." Id. Although the medical examiner had stated that each blow to Susan's head "might have" been fatal, he "was unable to opine as to which blow was fatal, let alone whether sufficient injury to kill had already been inflicted before the final blow." Id. at 7-8 ¶ 37, 191 P.3d at 170-71. In addition, the evidence did not support a reasonable conclusion that Wallace possessed the requisite mental state: "Although the assault on Susan was brutal and reprehensible, it `came in an attempt . . . to kill the victim, not to engage in violence beyond that necessary to kill.'" Id. (quoting Anderson, 210 Ariz. at 355 ¶ 123, 111 P.3d at 397).

¶ 16 This Court in Wallace III found sufficient evidence to submit the issue of gratuitous violence to the jury on remand with respect to the murders of Anna and Gabriel. Id. at 7 ¶¶ 32-35, 191 P.3d at 170. But because we remanded the case, we did not engage in independent review of the gratuitous-violence findings relating to those two victims. Id. at ¶ 35. That issue is squarely presented now and we turn to it, considering the evidence adduced on remand in the 2009 trial.

IV.

¶ 17 In the aggravation phase of the 2009 trial, the medical examiner, Dr. Parks, testified about Anna and Gabriel's injuries. Crime scene and autopsy photos depicting their wounds were admitted into evidence. Transcripts of the recorded statements Wallace made to police the day after the murders were also introduced. In our independent review, we must determine whether the State proved beyond a reasonable doubt that Wallace inflicted gratuitous violence on the children, analyzing the evidence separately as to each of those victims.

A. Anna's murder

1. More injury than necessary to kill

¶ 18 Dr. Parks testified that Anna suffered at least ten blows to the head, causing two skull fractures. But he was not able to determine the order of the blows, nor could he determine which blow caused her death. Describing the neck wound, Dr. Parks stated that the bat went through the skin of Anna's lower neck, into her left chest cavity, breaking a rib in her lower chest cavity, and pushed through the body to her back, leaving "a bulge in her back where the end of the bat came to rest."

¶ 19 Dr. Parks testified that the late Dr. Jones, who had conducted the autopsy of Anna, listed cerebral injuries as a cause of death, indicating that Dr. Jones did not consider the wound through the neck to be a cause of death. Dr. Parks stated that when the bat entered Anna's chest and neck area, it did not hit any major arteries or other blood vessels; thus, the neck wound alone would not have been fatal. But Dr. Parks could not determine whether Anna was alive when Wallace inserted the bat into her neck. When asked whether he thought the bat wound to Anna "finished the act of killing her," he stated "I don't exactly know." When asked whether it "hastened the act of killing her," he stated "it's possible it could have contributed."

¶ 20 Dr. Parks acknowledged that at the 2005 trial, he had testified that he could not conclude with certainty whether Anna sustained more injuries than were necessary to have caused her death. Based on the small amount of blood found in Anna's chest cavity, however, Dr. Parks opined, "[i]f [Anna] was alive when the bat was placed [in her neck], she didn't live much longer after that or she was in the process of dying when the bat was inserted." He explained "[s]he would not have been able to live too long and have such a small amount of blood accumulate."

¶ 21 Whether the evidence establishes that more injury than necessary to kill was inflicted on Anna is a close question. Dr. Parks never clearly expressed an opinion to that effect. But this Court can infer that Wallace did not have to stab Anna through the neck with the bat in order to kill her after already inflicting ten blows to her head. Although there is uncertainty about whether Anna was still alive when the neck wound was inflicted, the inquiry is not whether the victim was dead before further injury was inflicted, but rather whether more injury was inflicted than necessary to kill. See Bocharski, 218 Ariz. at 494 ¶ 86, 189 P.3d at 421. Dr. Parks indicated that the neck wound would not have been fatal, permitting an inference that the head injuries alone would have been. We find the evidence proved beyond a reasonable doubt that Wallace inflicted more injury on Anna than necessary to kill.

2. Knew or should have known requirement

¶ 22 The more difficult question is whether the State proved beyond a reasonable doubt that Wallace continued to inflict injury after he knew or should have known that he had inflicted a fatal wound. Wallace stated that he hit Anna on the head because "I thought she would die with one blow, that'd be it, like in the movies, it ain't that way, she looked me in the eye, she knew who was killing her." After Anna fell to the ground, Wallace said, she continued to moan and breathe. He stated, "she wouldn't die. I broke a f * * *ing bat on her head and she was still moaning. I don't know what people are like when they are dead. I wanted to put her out of her misery man, and she wouldn't f * * *ing die."

¶ 23 Wallace also said that even after he jammed the bat piece through Anna's throat, "she still wouldn't die." Before Gabriel's arrival, Wallace went to the shed to get a piece of steel to kill Gabriel because, he said, "I didn't know how hard it was to kill a human being, I didn't want Gabe to go through what I put Anna through, I wanted her to die quick and she didn't."

¶ 24 The State argues that Wallace's "self-serving" statements about Anna's moaning and breathing and his difficulty killing her should be given little weight. But Wallace made the statements the day after the murders when he turned himself in to the police and freely admitted his crimes, providing many incriminating details without attempting to escape blame or minimize his actions. Indeed, Wallace's statements were the only direct evidence about how the murders occurred.

¶ 25 Wallace's observations that Anna was "moaning" before he stabbed her through the neck were supported by Dr. Parks's testimony. Dr. Parks was not able to determine whether Anna was alive when Wallace drove the bat through her neck. In addition, Dr. Parks agreed that Anna might have still been conscious during the striking of the blows to her head, and that she might have still been moving such that "the person inflicting the blows would not realize that the person was, in fact, fatally injured." Referring to Wallace's next-day statement that Anna was breathing, moaning, making eye contact, and "just wouldn't die," the defense asked, "[i]s that medically logical and consistent with the injuries that you observed through your photos and [the autopsy] report?" Dr. Parks responded "[y]es."

¶ 26 On this record, we cannot find beyond a reasonable doubt that Bocharski's actual or constructive knowledge requirement was met. Viewed as a whole, the evidence casts reasonable doubt on whether Wallace knew or should have known a fatal wound had been inflicted when he stabbed Anna in the neck.

¶ 27 Anna's murder is more akin to the clumsy and escalating attacks in Cañez and Anderson than to murders in recent cases in which the Court has found gratuitous violence. See State v. Bearup, 221 Ariz. 163, 173 ¶¶ 50-53, 211 P.3d 684, 694 (2009) (upholding especially heinous or depraved finding when defendant cut off victim's finger an hour after beating victim with an aluminum baseball bat, reasoning that the removal of the finger constituted either gratuitous violence or mutilation);6 cf. State v. Gunches, 225 Ariz. 22, 26 ¶¶ 17-18, 20, 234 P.3d 590, 594 (2010) (finding insufficient evidence to establish beyond a reasonable doubt that the defendant, who shot victim four times, knew or should have known that he had already fired a fatal shot and yet continued to inflict violence). If Anna continued moaning and breathing before Wallace inflicted the neck wound, as the evidence suggests, a logical inference would be that Wallace's jamming of the piece of bat into her neck "came in an attempt . . . to kill the victim, not to engage in violence beyond that necessary to kill." Wallace III, 219 Ariz. at 8 ¶ 37, 191 P.3d at 171 (quotations omitted); see also Gunches, 225 Ariz. at 26 ¶ 21, 234 P.3d at 594 (evidence that victim was still breathing before final shots were fired supported finding that defendant did not knowingly inflict gratuitous violence by firing final shots).

¶ 28 In sum, the evidence does not establish beyond a reasonable doubt that Wallace knew or should have known that he already had inflicted fatal wounds upon Anna before committing his final assault. We therefore cannot find heinousness or depravity, the sole aggravating factor in this case.

B. Gabriel's murder

1. More injury than necessary to kill

¶ 29 Dr. Parks testified that there were a total of eleven lacerations on Gabriel's head, and that no more than eleven blows would have caused those lacerations. Dr. Parks was not able to determine the order of the blows or which particular one caused Gabriel's death, but opined that two separate wounds to Gabriel's head would alone "most likely [have been] fatal." One of those blows to the forehead exposed Gabriel's brain and caused a portion of brain tissue to separate from the rest of his brain and exit Gabriel's skull. Another fatal skull fracture near Gabriel's right ear protruded inward, causing a deep depression in Gabriel's head.

¶ 30 On cross examination, Dr. Parks acknowledged that he "could not say that the injuries [Gabriel] sustained were more . . . than were necessary to have caused his death." But later, when asked whether he thought either the "gaping hole" to Gabriel's forehead or the "caved in portion" of his head could by itself have caused Gabriel's death, Dr. Parks responded affirmatively. We therefore conclude that the State proved beyond a reasonable doubt that Wallace inflicted more injury than necessary to kill Gabriel.

2. Knew or should have known requirement

¶ 31 When asked by police whether he knew Gabriel was dead during the attack, Wallace said, "I just knew I had to kill him, I couldn't stand there and watch him go." When asked whether Gabriel was moving, Wallace said, "He might, he, body reflexes, he probably flinched or something, I smashed his skull in." As noted earlier, he also stated, "I never killed anybody before. . . . I tried to kill his sister and she wouldn't die. I broke a f* * *ing bat on her head and she was still moaning. I don't know what people are like when they're dead."

¶ 32 Dr. Parks testified that, if the two most severe blows to Gabriel's head were delivered in rapid succession, there would be "no time for the person to register the effect of Blow A versus Blow B." He further acknowledged that the potentially fatal blows could have been the final two impacts to Gabriel's skull. Dr. Parks agreed that Gabriel might have still been conscious as his head was struck and might have exhibited some bodily movement "to make one think the person was still alive." In that regard, Dr. Parks acknowledged that Wallace's statement that Gabriel might have been "flinching" during the blows was medically consistent with his observations. When asked about the blood found in Gabriel's lungs, Dr. Parks stated that this indicated Gabriel's "breathing following at least some of the injuries."

¶ 33 Whether Bocharski's knowledge requirement has been established with respect to Gabriel's murder is difficult to determine. In contrast to his recollection of Anna's murder, Wallace was less certain about whether Gabriel was flinching during the attacks, which suggests that he was not motivated by Gabriel's movements to continue attacking him. Given the heavy tool and force used to pummel Gabriel's head, the number of blows inflicted, and the nature and severity of the injuries Gabriel suffered, it is certainly arguable that Wallace should have known that striking the head of a 102-pound twelve-year-old with an 18-inch pipe wrench nine or fewer times, rather than ten or eleven times, would have been sufficient to kill. See Wallace III, 219 Ariz. at 7 ¶ 34, 191 P.3d at 170 ("[T]he nature of the attack [on Gabriel] and its results support an inference that Wallace either knew or should have known he had struck enough blows to kill yet continued his attack.").

¶ 34 But Wallace stated that he used the pipe wrench on Gabriel because he had such difficulty killing Anna and wanted Gabriel to die more quickly than her. As in Bocharski, the injuries occurred in quick succession, and all the blows were delivered with the means used to inflict death, facts that tend to cut against a finding of gratuitous violence. In addition, as Dr. Parks acknowledged, the most obviously fatal and gruesome wound that caused Gabriel's skull to split open could have been the final blow. Creating further doubt, Dr. Parks also acknowledged that in 2005 he could not say for certain whether more injury than necessary to kill was inflicted on Gabriel.

¶ 35 Although the issue is very close, on our independent review of this record, we cannot find beyond a reasonable doubt that Wallace "continued to inflict violence [on Gabriel] after he knew or should have known that a fatal action had occurred." Bocharski, 218 Ariz. at 494 ¶ 87, 189 P.3d at 421. Because the State did not meet its burden of proving beyond a reasonable doubt Wallace's intent to inflict gratuitous violence on Gabriel, the (F)(6) aggravating circumstance was not established.

V.

¶ 36 Even among capital cases, this case is atrocious. Wallace's premeditated, brutal murders of Anna and Gabriel clearly were senseless, and the unsuspecting, defenseless victims were helpless. Under Arizona's current statutes, Wallace most certainly would be eligible for the death sentence. See supra n. 3. But under the applicable law when Wallace murdered the victims, his eligibility for capital punishment requires a finding of gratuitous violence to establish heinousness or depravity under § 13-751(F)(6).

¶ 37 The United States Supreme Court determined that Arizona's (F)(6) aggravating circumstance is facially vague but "can be cured by a state appellate court applying a narrowed construction of the aggravator and determining de novo whether the evidence supported the finding of the aggravator." Anderson, 210 Ariz. at 352 ¶ 109, 111 P.3d at 394 (citing Walton v. Arizona, 497 U.S. 639, 654, 110 S.Ct. 3047, 111 L.Ed.2d 511 (1990), overruled on other grounds by Ring v. Arizona, 536 U.S. 584, 608-09, 122 S.Ct. 2428, 153 L.Ed.2d 556 (2002)). Bocharski's clarification and narrowing of the concept of gratuitous violence for establishing heinousness or depravity under (F)(6) were thus constitutionally required, as is our application of Bocharski's two requirements in this case.

¶ 38 In common parlance, Wallace's crimes undoubtedly would be characterized as heinous or depraved. But under the Supreme Court's jurisprudence, we must apply narrowing constructions of those words, not common understandings, to avoid "serious constitutional problems." State v. Carlson, 202 Ariz. 570, 585 ¶ 55, 48 P.3d 1180, 1195 (2002); see also Anderson, 210 Ariz. at 355 ¶¶ 123-24, 111 P.3d at 397.

¶ 39 Under Bocharski and our other (F)(6) narrowing jurisprudence, we conclude on independent review that the evidence does not show beyond a reasonable doubt that Wallace actually or constructively knew that he had delivered one or more fatal blows to Anna or Gabriel before he stopped striking them. We therefore vacate the death sentences imposed on the two convictions relating to Anna and Gabriel's murders and impose life sentences for each of those convictions.7 See State v. Snelling, 225 Ariz. 182, 190 ¶ 38, 236 P.3d 409, 417 (2010) (noting that a "death penalty may be imposed only if the state has proved the existence of at least one aggravating factor beyond a reasonable doubt") (citation and internal quotation marks omitted). These sentences will be served consecutively to the life sentence previously imposed for Susan's murder. See Wallace III, 219 Ariz. at 8 ¶ 38, 191 P.3d at 171.

CONCURRING: REBECCA WHITE BERCH, Chief Justice, ANDREW D. HURWITZ, Vice Chief Justice, and ROBERT M. BRUTINEL, Justice.*