*794A trial court in a criminal case is charged with properly instructing the jury on the elements of all crimes and enhancements. ( People v. Tidwell (1970) 3 Cal.3d 82, 87, 89 Cal.Rptr. 58, 473 P.2d 762.) Does the trial court violate its duty to instruct-and potentially entitle the defendant to a retrial-if the court (1) tells the jury that the People have to prove an element that the law does not require, and (2) fails to properly define that element? We conclude that a mistake pertaining to a superfluous element does not constitute instructional error. In light of this conclusion, as well as our determinations that the trial court did not otherwise commit instructional error and that the prosecution did not commit misconduct, we affirm the murder conviction in this case. We nevertheless conclude that there are several sentencing errors that must be corrected, and remand for the trial court to consider whether the defendant has the ability to pay a restitution fine.
FACTS AND PROCEDURAL BACKGROUND
I. Facts
A. The crime itself
On December 3, 1978, 20-year-old Leslie Long (Long) was working at a full-service gas station in Palmdale, California. Long was a mother of three, *795including a baby she was still nursing. As she finished her shift alone that evening, two men arrived at the station, demanded that she open the *333safe, and took between $600 and $1,200. The men abducted Long and drove her 10 to 12 miles outside of town to a secluded spot in the high desert surrounding Palmdale. Each man vaginally raped Long, and one man anally raped her. Long was shot five times in the back of the head, "execution-style," and left for dead.
B. The "cold case" investigation
More than 30 years later, law enforcement ran DNA tests on the sperm samples collected from Long's body and ran the results through its DNA database. There was a hit. Some of the DNA in Long's vagina belonged to Terry Moses (Moses), a long-time gang member and a repeat killer. When confronted with the DNA match, Moses initially refused to cooperate. Moses later changed his mind. In exchange for the People's promise not to seek the death penalty, Moses pled guilty to the murder of Long and several others, to be sentenced to five life sentences (three of which were without the possibility of parole), and to name the second man who was with him in December 1978. Moses first fingered a since-deceased man. But when law enforcement indicated its intent to exhume the deceased man's body to test for DNA, Moses admitted that his cohort was Neal Matthews (defendant). DNA tests confirmed that it was defendant's sperm in Long's vaginal and anal cavities.
II. Procedural Background
A. Charges
In the operative information, the People charged defendant with the first degree murder ( Pen. Code, § 187 )1 of Long. The People alleged three special circumstances that, if true, would dictate a mandatory sentence of life without the possibility of parole (LWOP)-namely, that Long's murder was committed in the commission of the crimes of robbery, kidnapping and rape (§ 190.2, subds. (c)(3)(i), (c)(3)(ii) & (c)(3)(iii) ). The People further alleged that defendant "personally used a firearm" (§ 12022.5, subd. (a) ) and that a principal was armed with a firearm (§ 12022, subd. (a)(1) ). The People additionally alleged that defendant had three prior felony convictions that qualified as "strikes" under our Three Strikes Law (§§ 667, subds. (b)-(j), 1170.12, subds. (a)-(d) ).
B. First trial
The matter proceeded to trial, but the jury hung 11-1 in favor of guilt after the jurors reported the 12th juror refused to deliberate.
*796C. Retrial
The matter proceeded to a second jury trial.
The prosecutor read into the record defendant's testimony from the first trial. In that testimony, defendant stated that (i) Long had voluntarily accompanied Moses on the 75 mile trip from Palmdale to Los Angeles so that Moses could buy angel dust at the place where defendant was staying; (ii) defendant raped Long while she was alone with him in the kitchen; and (iii) Long thereafter left with Moses, which was the last time defendant ever saw her. Defendant acknowledged that the statute of limitations had expired on the 1978 rape, so admitting the rape subjected him to no criminal liability.
The court instructed the jury on two theories of criminal liability for Long's murder-namely, that (1) defendant himself committed the killing with "malice aforethought," and (2) defendant was liable under a felony-murder theory because Long was killed during the commission of *334the robbery or rape defendant aided and abetted. The jury was also instructed on the special circumstance.
The jury found defendant guilty of Long's murder, found all three special circumstances to be true, and found that a principal was armed with a firearm. The jury rejected the allegation that defendant personally used a firearm.
Defendant waived his right to a jury trial on his prior convictions, and the trial court subsequently found them to be true.
D. Sentencing
The trial court sentenced defendant to prison for life without the possibility of parole. The court also imposed a $10,000 restitution fine, a $10,000 parole revocation fine, and ordered that a hearing be set for restitution to the victim and her family. The court awarded 901 days of actual pre-sentence custody credit, and no conduct credits.
E. Appeal
Defendant filed a timely notice of appeal.
DISCUSSION
In this appeal, defendant argues that the special circumstance finding must be overturned due to three instructional errors; that his underlying murder *797conviction is invalid due to prosecutorial misconduct; and that the trial court committed a number of sentencing errors.
III. Instructional Errors
Defendant launches three separate attacks on the correctness of the special circumstance jury instruction. We independently review such challenges. ( People v. Hamilton (2009) 45 Cal.4th 863, 948, 89 Cal.Rptr.3d 286, 200 P.3d 898.)
A. Failure to define "physically aided"
Defendant first contends that the special circumstance instruction is invalid because (1) the jury was instructed under the 1977 version of the special circumstance statute, which requires that the People prove, among other things, that "the defendant physically aided or committed [the] act or acts causing [Long's death]" (former § 190.2, subd. (c), italics added); (2) the jury was not instructed that a defendant "physically aid[s] or commit[s] [the] act or acts causing death," under the 1977 version, only if "his conduct constitutes an assault or battery upon the victim or if by word or conduct he orders, initiates, or coerces the actual killing of the victim" (former § 190.2, subd. (d), as set forth in Carlos v. Superior Court (1983) 35 Cal.3d 131, 139, 197 Cal.Rptr. 79, 672 P.2d 862 ( Carlos ), overruled on other grounds in People v. Anderson (1987) 43 Cal.3d 1104, 1138-1147, 240 Cal.Rptr. 585, 742 P.2d 1306 ); and (3) the trial court's failure to use the statutory definition is error because it differs significantly from the "commonly understood" meaning of the phrase "physically aided" ( People v. Johnson (2015) 234 Cal.App.4th 1432, 1456, 184 Cal.Rptr.3d 850 [court should instruct on "particular meaning" of phrase defined by statute]; People v. Griffin (2004) 33 Cal.4th 1015, 1022-1023, 16 Cal.Rptr.3d 891, 94 P.3d 1089 ). The People respond that the trial court's failure to define "physically aided" is irrelevant because the 1978 version of the statute-which was the version in effect on the date of Long's murder-eliminated the requirement of "physical aiding." (Stats. 1977, ch. 316, §§ 1-26, pp. 1255-1266; Carlos , at pp. 140, 143, 197 Cal.Rptr. 79, 672 P.2d 862 ; see People v. Murtishaw (2011) 51 Cal.4th 574, 586, 121 Cal.Rptr.3d 586, 247 P.3d 941 [noting that 1978 version of special circumstances statute *335became effective on November 8, 1978].)
This case accordingly tees up the question: If a trial court mistakenly instructs the jury that the People must prove a fact as an element of a crime but does not properly define that fact, does that failure constitute instructional error when that fact is not-in actuality-an element of the crime? In other words, does a mistake in instructing the jury on a superfluous "element" of a crime constitute instructional error?
*798We conclude the answer is "no," and do so for three reasons.
First and foremost, what matters to the validity of a conviction is whether the jury is correctly instructed on the elements of a crime-that is, on those "fact[s] that, by law, increase[ ] the penalty for a crime." ( Alleyne v. United States (2013) 570 U.S. 99, 103, 133 S.Ct. 2151, 186 L.Ed.2d 314.) Those elements are defined by the statute(s) in effect on the date of the charged crime. ( People v. Anderson (2009) 47 Cal.4th 92, 101, 97 Cal.Rptr.3d 77, 211 P.3d 584 [" 'Every crime consists of a group of elements laid down by the statute or law defining the offense ...' "]; People v. Gardeley (1996) 14 Cal.4th 605, 615, 59 Cal.Rptr.2d 356, 927 P.2d 713 ["consider[ing] the version of the statutory provisions ... in effect" at the time the "crimes ... took place"], overruled on other grounds in People v. Sanchez (2016) 63 Cal.4th 665, 204 Cal.Rptr.3d 102, 374 P.3d 320.) As a result, mistakes in instructing on facts that are not elements do not undermine the validity of a conviction. If, for instance, a trial court mistakenly instructed a jury that the elements of murder are (1) the unlawful killing of a human being, (2) with malice aforethought, (3) when the moon is made of cheese, no court in California would overturn that murder conviction merely because the court failed to define what kind of cheese the moon had to be. The same result obtains here.
Second, a rule providing that defects in superfluous "elements" invalidate a conviction leads to further problems that are best avoided. Such a rule effectively allows a judge's mistake in listing the elements of a crime to redefine those elements. This ostensibly transgresses the separation of powers, which ascribes the task of defining crimes solely to the legislative branch. ( In re Brown (1973) 9 Cal.3d 612, 624, 108 Cal.Rptr. 465, 510 P.2d 1017 ["In California all crimes are statutory and there are no common law crimes. Only the Legislature and not the courts may make conduct criminal."].) Such a rule effectively creates a "greater included" offense comprised of the statutory elements plus the superfluous element, and then puts the jury to an all-or-nothing choice of finding defendant guilty of this newly created offense or acquitting him of all criminal conduct. This ostensibly transgresses the rule that eschews "forc[ing] the jury to make an 'all or nothing' choice between conviction of [a greater crime] or complete acquittal." ( People v. Barton (1995) 12 Cal.4th 186, 196, 47 Cal.Rptr.2d 569, 906 P.2d 531.) Just as a jury must be permitted to convict a defendant of a lesser included offense, so must a "lesser" conviction for the crime defined by its actual statutory elements be permitted to stand. Such a rule also effectively erects a game of "procedural gotcha": Mess up the elements by adding an extra one, and-"gotcha"-now you have to properly instruct on that superfluous element on pains of reversal, even though reversal undermines the People's "legitimate interest" in upholding a conviction for which all statutorily required elements are found beyond a reasonable doubt (see *799People v. Breverman (1998) 19 Cal.4th 142, 155, 77 Cal.Rptr.2d 870, 960 P.2d 1094 ; *336People v. St. Martin (1970) 1 Cal.3d 524, 533, 83 Cal.Rptr. 166, 463 P.2d 390 ).
Third, the cases that have considered related issues are consistent with the holding we reach today. In People v. Dayan (1995) 34 Cal.App.4th 707, 40 Cal.Rptr.2d 391 ( Dayan ), the appellate court upheld a conviction for misdemeanor sexual battery notwithstanding the trial court's mistake in instructing the jury with the narrow definition of "touching" applicable to felony sexual battery. The court squarely rejected the "startling proposition" advanced by the defendant that "a conviction must be reversed" "if a court's instruction erroneously adds an element to an offense." ( Id. at p. 717, 40 Cal.Rptr.2d 391.) Our Supreme Court cited Dayan ' s analysis on this point favorably in People v. Santana (2013) 56 Cal.4th 999, 157 Cal.Rptr.3d 547, 301 P.3d 1157, when it ruled that the mistaken addition of a "serious bodily injury" requirement to the crime of mayhem "did not prejudice defendant." ( Id. at p. 1011, 157 Cal.Rptr.3d 547, 301 P.3d 1157.) And the United States Supreme Court in Musacchio v. United States (2016) --- U.S. ----, 136 S.Ct. 709, 193 L.Ed.2d 639 ( Musacchio ) unanimously held that "when a jury instruction sets forth all the elements of the charged crime but incorrectly adds one more element, a sufficiency [of the evidence] challenge should be assessed against the elements of the charged crime, not against the erroneously heightened command in the jury instruction." ( Id. at p. 715.) Defendant urges that Dayan is not directly on point because the instructional mistake in that case was not due to a mix-up as to which version of a statute to apply, and that Musacchio analyzed the sufficiency of the evidence, not an instructional error. But we do not see how the reason why a superfluous "element" is added matters. And whether or not Dayan , Santana and Musacchio dictate the result we reach, they are undoubtedly consistent with it.
Defendant makes three arguments in response.
First, he argues that the People, in proposing the version of the special circumstance statute with the "physically aided" element, elected between the 1977 and 1978 versions of the statute and must be held to that election. To be sure, when the People elect between one of two statutory methods of proving a crime ( People v. Centers (1999) 73 Cal.App.4th 84, 91-92, 86 Cal.Rptr.2d 151 ) or between one of several factual theories for proving an element of a crime ( People v. Mabini (2001) 92 Cal.App.4th 654, 657-658, 112 Cal.Rptr.2d 159 ; People v. Nunez & Satele (2013) 57 Cal.4th 1, 37, 158 Cal.Rptr.3d 585, 302 P.3d 981 ), the People will be held to that election. But this principle applies when the People are choosing among statutorily authorized ways of proving a crime. This principle does not by its terms or rationale extend to the choice between proving a statutorily authorized crime and something other than the statutorily authorized crime. Defendant also cites *800In re Stankewitz (1985) 40 Cal.3d 391, 220 Cal.Rptr. 382, 708 P.2d 1260 in support of this argument, but that case dealt with a juror 's injection of outside information into jury deliberations. ( Id. at pp. 399-400, 220 Cal.Rptr. 382, 708 P.2d 1260.)
Second, defendant contends that the state of Washington's "law of the case" doctrine obligates the People to prove a superfluous "element" if the People mistakenly submit an instruction that contains that element. ( State v. Johnson (2017) 188 Wash. 2d 742, 756, 399 P.3d 507 ; State v. Hickman (1998) 135 Wash. 2d 97, 102, 954 P.2d 900.) Because California also has a "law of the case" doctrine, defendant reasons, we should adopt the same rule as Washington. We reject this argument because *337its central premise-that California's law of the case doctrine is the same as Washington's-is wrong. Washington applies its law of the case doctrine whenever the People include an "element" in an instruction without objection. ( Johnson , at p. 756, 399 P.3d 507.) California's law of the case doctrine, however, only applies when a higher appellate court has passed on an issue ( People v. Gray (2005) 37 Cal.4th 168, 196-197, 33 Cal.Rptr.3d 451, 118 P.3d 496 ; accord, Musacchio , supra , 136 S.Ct. at p. 716 [federal law of the case doctrine does not apply when "an appellate court assesses a lower court's rulings"].) More to the point, California's doctrine-unlike Washington's-does not bind an appellate court to follow an "erroneous [jury] instruction." ( O'Neill v. Thomas Day Co. (1907) 152 Cal. 357, 361-362, 92 P. 856.) The closest analogue to Washington's law of the case doctrine is California's "invited error" doctrine, but that binds a litigant only when that litigant "mislead[s] the trial court" or otherwise engages in "affirmative conduct demonstrating a deliberate tactical choice." ( Norgart v. Upjohn Co. (1999) 21 Cal.4th 383, 403, 87 Cal.Rptr.2d 453, 981 P.2d 79 ; Huffman v. Interstate Brands Corp. (2004) 121 Cal.App.4th 679, 706, 17 Cal.Rptr.3d 397.) Contrary to what defendant suggests, nothing indicates that the People's mistaken reliance on the wrong version of the special circumstance statute was meant to mislead the court or evinced a deliberate tactical choice; nor would it, as there was no advantage to be gained from using that version.
Lastly, defendant asserts for the first time in his reply brief that the trial court's failure to define the superfluous "element" somehow reduced the People's burden of proof. As explained above, any reduction in the burden of proving a fact the People were not statutorily required to prove is not a cognizable statutory or constitutional error.
B.-C.**
DISPOSITION
The judgment is modified to reflect an award of presentence conduct credit of 450 days in addition to defendant's 901 days of actual credits for a total of 1,351 days. The case is remanded for resentencing to allow the trial court to (1) conduct a hearing to determine the defendant's ability to pay the restitution fine of $10,000 it previously imposed; (2) strike the parole revocation fine; and (3) strike the Three Strike findings. Upon resentencing, the trial court is directed to prepare an amended abstract of judgment and forward a certified copy of it to the Department of Corrections and Rehabilitation. The judgment is otherwise affirmed.
We concur:
LUI, P. J.
ASHMANN-GERST, J.