United States v. Abdelaziz ( 2023 )


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  •           United States Court of Appeals
    For the First Circuit
    No. 22-1129
    UNITED STATES,
    Appellee,
    v.
    GAMAL ABDELAZIZ,
    Defendant, Appellant.
    No. 22-1138
    UNITED STATES,
    Appellee,
    v.
    JOHN WILSON,
    Defendant, Appellant.
    APPEALS FROM THE UNITED STATES DISTRICT COURT
    FOR THE DISTRICT OF MASSACHUSETTS
    [Hon. Nathaniel M. Gorton, U.S. District Judge]
    Before
    Barron, Chief Judge,
    Lynch and Lipez, Circuit Judges.
    Joshua C. Sharp, with whom Brian T. Kelly, Lauren A. Maynard,
    and Nixon Peabody LLP were on brief, for appellant Abdelaziz.
    Noel J. Francisco, with whom Yaakov M. Roth, Marco P. Basile,
    Harry S. Graver, Jones Day, Michael Kendall, Lauren M. Papenhausen,
    White & Case LLP, Andrew E. Tomback, and McLaughlin & Stern, LLP,
    were on brief, for appellant Wilson.
    Elliott M. Davis, Kateland R. Jackson, Scott A. Chesin, and
    Shook, Hardy & Bacon L.L.P. on brief for Former United States
    Attorneys, amici curiae.
    Michael J. Iacopino, Brennan Lenehan Iacopino & Hickey,
    Steven F. Molo, Leonid Grinberg, Justin V. Shur, Kenneth E. Notter
    III, and MoloLamken LLP on brief for the National Association of
    Criminal Defense Lawyers and the American Board of Criminal
    Lawyers, amici curiae.
    Robert T. Smith, Mary Fleming, Timothy H. Gray, and Katten
    Muchin Rosenman LLP on brief for Law Professors, amici curiae.
    Alexia R. De Vincentis, Assistant United States Attorney,
    with whom Rachael S. Rollins, United States Attorney, and Donald
    C. Lockhart, Ian J. Stearns, Stephen E. Frank, Leslie A. Wright,
    and Kristen A. Kearney, Assistant United States Attorneys, were on
    brief, for appellee.
    May 10, 2023
    LYNCH, Circuit Judge.      The convictions underlying this
    appeal arise from a government criminal prosecution of alleged
    misconduct related to college admissions.          The government alleged
    that Rick Singer -- a college admissions consultant -- and his
    clients engaged in various forms of bribery and fraud to help
    secure     those   clients'     children's     admission    to     competitive
    universities.      Singer, who pleaded guilty in a separate case to
    multiple     charges1    and     cooperated      with     the    government's
    investigation, is not a defendant here, and his culpability is
    well established.
    The defendants-appellants in this case are two parents,
    Gamal Abdelaziz and John Wilson, who hired Singer. Both men agreed
    with Singer to make payments purportedly to university accounts in
    exchange    for    university   employees'     securing    their    children's
    admission as athletic recruits -- a path to admission Singer
    referred to as the "side door."2            Their defense at trial and on
    appeal is that they believed Singer's services and the side door
    to be legitimate and that they acted in good faith.
    1    Singer   pleaded   guilty  to   conspiracy   to   commit
    racketeering, see 
    18 U.S.C. § 1962
    (d); conspiracy to commit money
    laundering, see 
    id.
     § 1956(h); obstruction of justice, see id.
    § 1512(c)(2); and conspiracy to defraud the United States, see id.
    § 371.
    2    Singer contrasted this side door with the "front door"
    (admission on merit) and the "back door" (admission through large
    "institutional advancement" donations).
    - 3 -
    The    government      charged       Abdelaziz    and     Wilson     with
    multiple offenses based on their work with Singer.                      It alleged
    that both defendants had participated in an overarching conspiracy
    not   only   with    Singer   but    also   with     other    Singer    clients    to
    corruptly    influence      university      employees     through      payments     to
    university accounts, in violation of the federal programs bribery
    statute.     See 
    18 U.S.C. § 666
    .        It further alleged that Abdelaziz
    and Wilson conspired with other parents to commit two types of
    mail and wire fraud: honest services fraud, by using their payments
    to deprive the universities of the honest services of their
    employees, and property fraud, by depriving the universities of
    property in the form of "admissions slots."              See 
    id.
     §§ 1341, 1343,
    1346, 1349. It also charged Wilson with several substantive counts
    of federal programs bribery and wire fraud, and with filing a false
    tax return in connection with his payments through Singer.                      See 
    26 U.S.C. § 7206
    (1).
    A   jury   convicted    both     Abdelaziz       and   Wilson   of   all
    charges.     The defendants challenge those convictions on a number
    of grounds.        They contend that payments to university accounts
    cannot violate § 666 or constitute honest services fraud because
    the   payments       were     intended      for     accounts        owned    by   the
    universities -- the alleged victims of the scheme.                      They argue
    that the property fraud theory is invalid because admissions slots
    are not property, or, in the alternative, that their convictions
    - 4 -
    must be vacated because the district court erred by instructing
    the jury that admissions slots are property as a matter of law.
    And   they     argue   that    the    government     proved     only   a   narrower
    conspiracy than the one alleged by the indictment and that this
    variance prejudiced them on all counts.              Wilson also asserts that
    various forms of trial error require us to vacate his conviction
    for filing a false tax return.                Our task in this appeal is to
    assess these arguments and determine whether the charged conduct
    falls within the specific crimes of which these defendants were
    convicted and whether the manner in which this case was charged
    and tried unacceptably deprived these two defendants of a fair
    trial on their own conduct, rather than the conduct of others.
    Nothing   in    this   opinion       should   be   taken   as   approval     of   the
    defendants'      conduct      in   seeking    college      admission   for    their
    children.
    We reject the defendants' argument that payments to
    accounts controlled by the alleged victim of a bribery scheme
    cannot violate § 666, which lacks any basis in the provision's
    text, and so deny their request for judgment of acquittal on that
    basis.    And we affirm Wilson's conviction for filing a false tax
    return.
    We do hold that the government's honest services theory
    is invalid as a matter of law under the Supreme Court's decision
    in Skilling v. United States, 
    561 U.S. 358
     (2010), and that, on
    - 5 -
    the arguments offered by the government, the district court erred
    in instructing the jury that admissions slots constitute property.
    Accordingly,   we   vacate   the   defendants'   mail   and   wire   fraud
    convictions.   We also hold that the government failed to prove
    that Abdelaziz or Wilson agreed to join the overarching conspiracy
    among Singer and his clients charged in the indictment, and that
    this variance prejudiced the defendants by allowing the government
    to introduce a significant amount of powerful evidence related to
    other parents' wrongdoing in which these defendants played no part,
    creating an unacceptable risk that the jury convicted Abdelaziz
    and Wilson based on others' conduct rather than their own.             On
    that basis, we vacate the conspiracy convictions and Wilson's
    substantive convictions under § 666.3
    I.    Background
    We begin by laying out the basic facts and procedural
    history.   We elaborate on this background information as necessary
    in our analysis of the legal issues.
    3    We acknowledge and thank the amici curiae for their
    submissions in this case.    Eleven former U.S. Attorneys, five
    criminal law professors, and the National Association of Criminal
    Defense Lawyers and the American Board of Criminal Lawyers filed
    briefs in support of Wilson.
    - 6 -
    A.
    1.
    The charges against Abdelaziz stem from his work with
    Singer in 2017 and 2018 to secure his daughter's admission to the
    University       of    Southern     California      ("USC").        Abdelaziz      had
    previously paid Singer to work with his two older children in 2012
    and 2013; the government does not argue or cite any evidence that
    these     2012   and    2013     services    for   Abdelaziz's      children      were
    improper.
    It is undisputed that Abdelaziz agreed with Singer in
    approximately June 2017 to pursue side-door admission to USC for
    his daughter.         Abdelaziz maintains that he believed this option to
    be   at    least      tacitly    approved    by    the   school     and    to   entail
    "preferential admissions treatment to students like [Abdelaziz's
    daughter] who could assist athletic teams as practice players or
    team      managers     and      whose   parents    donated     to    the    athletic
    department."          At the time, Abdelaziz's daughter had not played
    competitive basketball in over a year; she had played for her
    school's junior varsity team until January or February 2016 but
    had stopped playing after failing to make the varsity team.                      A USC
    admissions officer testified that Abdelaziz's daughter was also
    not an academically competitive applicant outside the athletic
    recruitment process.
    - 7 -
    On July 16, 2017, Singer sent Abdelaziz an email with
    the subject line "For Me to complete USC athletic profile" that
    requested information about Abdelaziz's daughter's scholastic and
    athletic    accomplishments,       including   "If   they   play   the   sport-
    Basketball," "Accolades if they have them," and "Action Picture."
    Abdelaziz forwarded the email to his wife, but the record does not
    otherwise show any response.            Eleven days later, Singer emailed
    Abdelaziz again to request "an action photo or two of [Abdelaziz's
    daughter] playing basketball."          Abdelaziz responded: "Got it."      He
    then sent Singer five photos of high school girls' basketball
    games, four of which -- as Abdelaziz represents in his brief and
    the government does not dispute -- contained his daughter, and one
    of which did not.         The file names for the photos used generic
    letters and numbers (for example, "DSC_0007.JPG") and differed
    only in the numbers.        Singer responded: "We will use this one."
    His email identified the chosen photo -- the one that did not
    contain Abdelaziz's daughter -- only by the file name.
    Singer instructed one of his associates to prepare the
    profile.     The associate did so; the result included the photo
    Singer     had   selected    and    various    basketball    statistics    and
    accolades    that   the     associate    invented    and    that   Abdelaziz's
    daughter had not earned.       Singer sent this profile to Abdelaziz in
    early August, together with a message from his associate: "Let me
    - 8 -
    know if you want me to add any other awards to her profile or if
    you think that is enough."
    As he did at trial, Abdelaziz disputes whether he saw
    this    message.             Singer    originally       sent        the     profile     to
    gamalaziz@cox.net.           This address generated an automatic reply that
    stated: "Please be advised that I have changed my e-mail address
    to gamalaziz797@gmail.com."                An FBI agent testified that he was
    not aware of any evidence that Abdelaziz opened or responded to
    the message, although Abdelaziz later responded to messages Singer
    sent to the same cox.net address, having apparently forwarded them
    to    the    gmail.com       account   mentioned      in     his    automatic     reply.
    The     next          day,    Singer        also      sent     the         profile      to
    amalaziz797@gmail.com -- the address mentioned in the automatic
    reply, but without the opening "g."                The same FBI agent testified
    that,       because    of    the   typo,    this   message         would    be   "off   in
    cyberspace."
    Singer then sent the profile to an administrator in the
    USC Athletics Department who had agreed with Singer to facilitate
    Abdelaziz's daughter's side-door admission.                        That administrator
    added additional falsehoods to the profile -- including a photo of
    a different girl.            This version of the profile became the basis
    for the profile presented to "Subco," a subcommittee of USC
    admissions       officers      responsible      for    overseeing          admission    of
    - 9 -
    athletic recruits.           Based on this profile, Subco considered and
    approved Abdelaziz's daughter's admission on October 5, 2017.
    On October 10, the same athletics administrator who
    revised Abdelaziz's daughter's profile sent Singer a letter from
    the     Dean    of    Admissions   conditionally       admitting    Abdelaziz's
    daughter as a recruited athlete, pending her submission of a full
    application packet and other administrative tasks.               Singer emailed
    this letter to Abdelaziz the same day.
    In early November, Abdelaziz forwarded the letter to a
    Singer employee, together with an email that stated: "[Singer]
    asked that we work with you to complete USC's application . . . ."
    Abdelaziz later exchanged emails with this employee, Singer, and
    another Singer associate about his daughter's application.                     In
    these    emails,      Singer    noted    that    it   was   important   for   the
    application to discuss "basketball as apassion [sic]."                  Abdelaziz
    later "reminde[d]" the others of this "direction" when it came
    time to edit his daughter's application essays.
    In    early   January    2018,    Abdelaziz's    daughter      sent
    Abdelaziz and the Singer employee an application essay in which
    she described "[t]he basketball court [as her] art studio" and
    wrote: "Whether I am playing alone or in a pickup game with friends
    or in front of a crowd of two hundred people at school, I feel an
    enormous release from my everyday life when I am on the court."
    Abdelaziz responded in the same email thread that he had "read
    - 10 -
    [his       daughter's]      essays,"   and   opined   that   "overall    [they
    were] . . . good."           His daughter had not played on the school
    basketball team for nearly two years at the time.                 The Singer
    employee submitted Abdelaziz's daughter's USC application the
    following day.
    Abdelaziz and the        government agree that Abdelaziz's
    daughter was formally admitted in March 2018, although they do not
    cite any exact date in the record.
    On March 16, 2018, a foundation run by Singer sent
    Abdelaziz an invoice for $300,000,4 purportedly for a "[p]rivate
    [c]ontribution."           Abdelaziz wired that sum to the foundation ten
    days later.         Where that money ultimately went is unclear from the
    record, but an FBI agent testified that Singer's "general pitch
    [to parents] was that [a side-door payment] was a donation to a
    program," and government counsel acknowledged at trial that Singer
    told Abdelaziz the payment would go to a university account.                The
    government does not argue on appeal that the jury could have found
    that Abdelaziz intended the payment to go to any USC employee
    personally.
    Abdelaziz's daughter enrolled at USC in fall 2018.            She
    never      played    for    or   otherwise   associated   with   the    women's
    basketball team.
    4  Singer ran both a for-profit business, The Key, and a
    nonprofit foundation, The Key Worldwide Foundation.
    - 11 -
    In September 2018, the FBI approached Singer, and he
    agreed to cooperate with the government's investigation of his
    clients and university insiders.        As part of this cooperation
    agreement, Singer made various recorded calls at the government's
    direction.
    The government cites two such calls involving Abdelaziz.
    The first occurred on October 25, 2018.      During the call, Singer
    told Abdelaziz that his foundation was being audited and that the
    IRS had "asked . . . about" Abdelaziz's payment.      Singer further
    stated that he was "not going to tell the IRS anything about the
    fact that your $300,000 . . . was paid to . . . [an athletics
    administrator] at USC to get [Abdelaziz's daughter] into school
    even though she wasn't a legitimate basketball player at that
    level."   Abdelaziz responded: "OK."    Singer asked: "You're OK with
    that, right?"    Abdelaziz answered: "Of course."    A moment later,
    he added:
    No, I -- I mean, I -- you know, I mean . . .
    my intention was to, uh, donate the money to
    the foundation and, uh, what -- you know, and
    then from there obviously, uh -- I don't
    think -- Uh, do they have the intention of
    reaching out to the people that sent those
    payments?
    Singer said he did not know and that he "wanted to make sure our
    stories are correct."     He told Abdelaziz that he was "going to
    essentially say that [the] $300,000 payment . . . was made to our
    - 12 -
    foundation to help underserved kids," and "wanted to make sure
    [Abdelaziz was] OK with that."    Abdelaziz replied: "I am."
    In the same exchange, Singer told Abdelaziz that the USC
    athletics administrator with whom Singer had arranged the side
    door had called Singer to say that she "loved" the profile created
    for Abdelaziz's daughter and that "going forward, anybody who isn't
    a real basketball player that's a female, [she] want[ed] [Singer]
    to use that profile."    Abdelaziz responded: "I love it."
    The second call took place on January 3, 2019.       Singer
    told Abdelaziz that the same athletics administrator had called
    him to "give [him] a heads up" that the Admissions Department had
    "asked . . . why [Abdelaziz's daughter] did not show up for Women's
    Basketball in the fall," and that the administrator had "told them
    that [Abdelaziz's daughter] had an injury."        Abdelaziz asked
    whether the Admissions Department would ask his daughter about the
    situation and whether he needed to "prepare her."    Singer stated
    that they would not contact Abdelaziz's daughter and that he had
    "wanted [Abdelaziz] to know what" the administrator had told the
    Admissions Department.    Abdelaziz responded: "I will answer the
    same, uh, should they call me."
    2.
    Wilson engaged Singer's services to facilitate side-door
    admission for his children on multiple occasions between 2013 and
    - 13 -
    2019: first for his son, and then, years later, for his twin
    daughters.
    Beginning in spring 2013, Wilson worked with Singer to
    secure his son's admission to USC through the side door as a
    purported water polo recruit.    Singer explained to Wilson by email
    that the USC men's water polo coach was "giving [him] 1 boys slot,"
    available on a "first come first [sic]" basis.          In response to
    Wilson's asking when payment was due, Singer responded: "No payment
    of money till [sic] [the USC men's water polo coach] gets a verbal
    and written [sic] from admissions . . . ."
    Wilson's son did play high school water polo, but his
    high school coach testified that he was not a player of the level
    ordinarily recruited by USC, a noted water polo powerhouse.           In
    emails sent at the time, Wilson expressed doubts about whether his
    son would fit in on the team, asking Singer whether "it w[ould] be
    known that [his son was] a bench warming candidate" and whether
    his son would "be so weak as to be a clear misfit at practice,"
    and stating that "[o]bviously his [son's] skill level m[ight] be
    below the other freshmen" and that he "want[ed] to be sure [his
    son would] not [be] a lepper [sic]."        Singer responded that "the
    commitment   is   to   be   on    the     roster[,]   not   attend   all
    practices[;] . . . he will have to attend drug tests and other
    mandatory functions for 1 year [but then can] walk away/ frankly
    after the 1st semester he can move on."
    - 14 -
    In August 2013, Singer noted in an email to Wilson that
    the water polo coach needed "a player profile so he [could] add
    [Wilson's son] to his recruit list and present him to admissions
    in October," and that Singer had the necessary materials to create
    the profile.    Wilson responded: "Great - let me know when [you]
    have verified [you] have it all completed and into [sic] [the water
    polo coach]."   In October, Singer updated Wilson: "[The water polo
    coach] has [Wilson's son's] stuff and asked me to embellish his
    profile more, which I am doing."    A few days later, Singer emailed
    the profile to Wilson.        It misrepresented several aspects of
    Wilson's   son's   athletic   qualifications   --   for   instance,   by
    erroneously describing him as a captain of his high school team
    and listing implausibly fast swim times.       Wilson's counsel argued
    both at trial and on appeal that the government did not prove
    Wilson was aware of the falsehoods in the profile.            Wilson's
    counsel emphasized that an FBI agent whose testimony the government
    used to introduce the email containing the profile acknowledged
    that he was not aware of emails or other evidence showing Wilson
    read or responded to Singer's message, and that the profile listed
    the wrong home address and used Wilson's son's SAT scores rather
    than his more impressive ACT scores, which Wilson purportedly would
    have wanted included.
    - 15 -
    An assistant water polo coach at USC relied on this
    profile to prepare the athletic profile used by Subco to consider
    whether to admit Wilson's son.
    Subco considered and approved Wilson's son's admission
    on February 28, 2014, relying in part on the falsified athletic
    profile.
    The day after Subco approved his son's admission, Wilson
    emailed Singer to "[t]hank[] [him] again for making this happen."
    He asked about "the options for the payment" and requested an
    invoice "for consulting or whatever from [Singer's business] so
    that [Wilson could] pay it from [his private equity firm's]
    corporate account."    After some further discussion of payment
    mechanics, Wilson's firm wired $220,000 to a combination of Singer,
    his business, and his foundation on April 7, 2014.   The government
    does not contend that any USC employee personally received a
    portion of this payment.   Singer passed $100,000 along to the USC
    men's water polo team and, so far as the record shows, retained
    the other $120,000.
    Wilson's son quit the water polo team after his first
    semester at USC.
    Several years later, in September 2018, Wilson called
    Singer about the possibility of helping Wilson's twin daughters,
    then juniors in high school, with their college applications.   The
    government recorded this call, which took place before Singer began
    - 16 -
    cooperating     with   the     government,   without   the    participants'
    knowledge.     Singer explained that for Wilson's daughters to gain
    admission to the highly competitive schools they were interested
    in "on their own" would require "essentially perfect grades" and
    excellent standardized test scores, but added that "if you said
    you wanted to . . . go through a different door you c[ould] do
    that."    Wilson inquired about "the other door," asking if it was
    "like . . . water polo and [a] donation," and Singer explained the
    price and availability of admission through the side door at
    various   universities.        Singer   informed   Wilson    that   side-door
    admission to Stanford or Harvard would cost "a minimum of [$]1.2
    million," since a coach would have to "giv[e] up his spot" to a
    purported recruit who is "not a good enough athlete[] to compete,"
    but would provide a "done deal.         Just like with [Wilson's son]."
    Singer explained that the sport "d[id]n't matter," saying he "would
    make them a sailor or something."            Wilson laughed in response.
    Wilson observed that they were discussing "big numbers," since
    "there's so many people that want to do th[is]," and asked if
    "there [was] any way to make [the payments] tax deductible as like
    donations to the school."         Singer stated that payments would be
    deductible as contributions to his foundation.
    Wilson and Singer continued to discuss the use of the
    side door for Wilson's daughters after Singer began cooperating
    with the government.         From this point on, Singer's references to
    - 17 -
    university insiders willing to facilitate side-door admission were
    part of a ruse created by investigators.
    In a September 29 call, Wilson confirmed that, while his
    daughters remained undecided on what schools they wanted to attend,
    he was "interested about the side door and that stuff" and asked
    what schools and sports were available.        Singer assured him that
    the side door "is gonna . . . happen where you want it to happen,"
    and that crew or sailing might be potential sports options. Wilson
    asked: "[W]hat if they're not really that good?" Singer responded:
    "[A]t the end of the day . . . I may be able to go to the sailing
    coach   and   say,   'Hey,   this   family's   willing   to   make   the
    contributions. . . . [The child] may not be up to the level you
    are, but . . . you're gonna get a benefit, and the family's gonna
    get [a] benefit.'"    Wilson also asked how the payment would work.
    Singer stated that "the money [went] into [his] foundation," and
    that he would then "split the money potentially to the coach or
    other . . . parties that are at that school that need the
    money[.] . . . Or it may go right to the coach, . . . depend[ing]
    on the school."
    In October, Wilson called Singer to discuss "making some
    donations now, whatever -- how that can work."      Singer stated that
    he worked with "a bunch of schools," including Harvard and Stanford,
    on a "first come, first served" basis, and that, "if [he] were to
    get a deposit . . . [of] like half a million dollars in the bank,"
    - 18 -
    they could "figure out where [Wilson's daughters] wanna go" later.
    "[H]aving the money already, in advance, [would] make[] it much
    easier" because coaches "don't want to give up a spot" unless "the
    family guarantee[s] . . . that they're gonna ante up and they're
    gonna   make   a   payment."   Wilson     asked   whether   his   daughters
    "actually ha[d] to do th[e] sport" or whether "[t]hey could just
    go in and      . . . be like . . . the scorekeeper[,] . . . water
    girl[,] . . . [or] manager."      Singer confirmed that they could be
    "[m]anager[s] or whatever you want to call 'em," and that Wilson's
    daughters were "athletic and . . . big" enough that he could "sell
    to anybody that they're athletic enough to be able to take 'em and
    there'll be no question."      Singer further assured Wilson that, if
    he used the side door, admission would be "a done deal."            Wilson
    requested that Singer send him wiring information, and confirmed
    with Singer that if he sent "[h]alf a million" he would be "locked
    in for 2." Wilson's private equity firm wired $500,000 to Singer's
    foundation two days later at Wilson's direction.
    Later that month, Singer told Wilson that he had spoken
    with Stanford's sailing coach, that the coach was willing to
    "guarantee[] a spot for next year," and that Wilson could "have
    first dibs" if Singer sent the coach the "[$]500,000 that [Wilson]
    wired into [Singer's] account to secure the spot for one of
    [Wilson's] girls."     Singer also mentioned that he had "asked [the
    coach] for a second spot in sailing and [the coach] said he
    - 19 -
    c[ould]n't do that because he ha[d] to actually recruit some real
    sailors so that Stanford d[id]n't . . . catch on."           Wilson laughed
    and said "[r]ight."       Wilson asked for more time for his daughters
    to decide where they wanted to go, and inquired whether there was
    "any news on the Harvard side."           Singer promised to get back to
    him.5
    On November 29, Singer informed Wilson that they had
    "got a spot if . . . [Wilson's daughter] want[ed] to go to Harvard."
    He claimed that "the senior women's administrator at Harvard [was]
    going to give [Wilson's daughter] a spot," in exchange for which
    Wilson would "have to give . . . her . . . $500,000" through
    Singer's foundation to "fund the senior women's administrator."
    He added: "I've already paid [the Stanford sailing coach] the
    [$]500      [thousand]    and   now    we'll   give   the   senior   women's
    administrator [$]500 [thousand] . . . . [Y]our total's going to be
    [$]1.5 [million].        [$]250 [thousand] will come in the spring for
    Stanford and [$]250 [thousand] for Harvard in the spring and
    we'll . . . be done."       Wilson responded: "OK, great."      When Wilson
    5 Singer and Wilson covered similar ground in a call a few
    days later, in early November. Singer told Wilson: "[W]e got the
    Stanford spot. They wanna know if you want it because I have to
    pay the coach, um, right away."      He reiterated that the coach
    "d[id]n't care if it's a sailor or not." He added that he was
    "working" on Harvard, but it would "not happen for several months."
    Wilson again asked for more time to figure out which schools his
    daughters wanted to attend. He also agreed with Singer that his
    daughters "weren't going to get into either" Harvard or Stanford
    without the side door.
    - 20 -
    asked what sport his daughter would need to play, Singer answered:
    "[The Harvard administrator will] figure it out. . . . [I]t doesn't
    matter the sport at this point.        She will . . . just get her in
    through . . . athletics in one of the sports but it won't matter."
    Singer also noted that Wilson's other daughter "w[ould]n't have to
    sail but we're going to put her through sailing" at Stanford.
    Wilson responded that "sailing is actually a logical thing.         She
    could be even the mascot, whatever, but she knows sailing."           He
    confirmed that the plan "sound[ed] fantastic" and was "great news."
    Wilson's private equity firm wired a further $500,000 to Singer's
    foundation on December 11, again at Wilson's direction.
    As with Abdelaziz, the government acknowledged at trial
    that Singer "told the parents," including Wilson, "that the money
    would go to the athletic program at the schools."        On appeal, it
    does not argue that the jury could have found that Wilson intended
    any of his payments to go to insiders' personal accounts, rather
    than   to   university-owned   accounts   related   to   the   insiders'
    positions.
    B.
    On March 5, 2019, a federal grand jury in the District
    of Massachusetts returned a single-count indictment charging David
    Sidoo -- a parent who had worked with Singer -- with conspiracy to
    commit mail and wire fraud in connection with Sidoo's allegedly
    having paid Singer to have one of his associates take various
    - 21 -
    standardized tests for Sidoo's children.   The indictment further
    alleged that Singer had also paid his associate "to secretly take
    the SAT and ACT for the children of other co-conspirators known
    and unknown to the Grand Jury."
    Six days later, the government filed a separate criminal
    complaint alleging that roughly thirty other parents, including
    Abdelaziz and Wilson, had conspired with Singer and others to
    commit mail fraud and honest services mail fraud.   The complaint
    alleged varying forms of misconduct by the named parents; some
    were alleged to have schemed to help their children cheat on
    standardized tests, while others -- like Abdelaziz and Wilson --
    were alleged to have conspired to bribe university employees to
    secure their children's admission.
    Many of the parents named in the complaint elected to
    enter plea agreements with the government.    For example, Gordon
    Caplan, Agustin Huneeus, and Bruce Isackson -- three parents whose
    interactions with Singer would later figure in the evidence at
    Abdelaziz's and Wilson's trial -- agreed to waive the requirement
    of indictment by a grand jury and plead guilty to various offenses
    pursuant to criminal informations.
    On April 9, 2019, a grand jury returned a superseding
    indictment in the government's case against Sidoo that named as
    codefendants eighteen parents from the complaint who had not
    entered plea agreements, including Abdelaziz and Wilson.      The
    - 22 -
    government superseded this indictment twice more in the following
    months.      During this period, four more parents pleaded guilty
    without written plea agreements.
    The operative          fourth superseding        indictment        in this
    case, returned on January 14, 2020, charged fifteen parents with
    an overlapping set of offenses.                     All fifteen defendants were
    charged with conspiracy to commit mail and wire fraud and honest
    services mail and wire fraud.               See 
    18 U.S.C. §§ 1341
    , 1343, 1346,
    1349.       This count alleged an overarching conspiracy among the
    defendants        and     others,     including       Singer,   to        defraud    two
    standardized       test    firms     and    five    universities     --    Georgetown;
    Harvard;     Stanford;      the     University      of   California,      Los   Angeles
    ("UCLA"); and USC -- in two alternative ways: first, by depriving
    them of property in the form of "standardized tests and test
    scores" (for the standardized test companies) and "admission to
    the   [u]niversities"         (for    the     universities),       and     second,    by
    depriving them of the honest services of their employees through
    the   use    of    "bribes    and     kickbacks."         Notably,       although    the
    indictment contained detailed allegations of fraud related to
    standardized testing with respect to several other defendants, it
    did not allege that either Abdelaziz or Wilson had engaged in or
    even been aware of that form of misconduct.
    Eleven parents, including both Abdelaziz and Wilson,
    were also charged with conspiracy to commit federal programs
    - 23 -
    bribery.    See 
    id.
     §§ 371, 666(a)(2).     The indictment alleged that
    the parents had "conspired . . . to bribe agents of USC to secure
    their children's admission to that university."
    Wilson -- but not Abdelaziz -- was charged with three
    substantive counts of wire fraud and honest services wire fraud,
    see id. §§ 1343, 1346, and two substantive counts of federal
    programs bribery, see id. § 666(a)(2), all in connection with his
    efforts to secure admission to Harvard and Stanford for his
    daughters.    In addition, Wilson alone was charged with filing a
    false tax return, see 
    26 U.S.C. § 7206
    (1), in connection with his
    treatment of his payments to secure his son's admission to USC on
    his 2014 tax return.6
    The defendants moved to dismiss on a number of grounds.
    Three of those grounds previewed arguments Abdelaziz and Wilson
    now make on appeal: First, the defendants moved to dismiss the
    federal    programs   bribery   and   honest   services   fraud   charges,
    arguing that payments made to the alleged victim of a bribery
    scheme -- here, the universities -- cannot constitute bribes.
    Second, they moved to dismiss the mail and wire fraud charges
    insofar as these charges alleged that the defendants had defrauded
    the universities of property, arguing that admissions slots do not
    6    All fifteen defendants were also charged with conspiracy
    to commit money laundering. See 
    18 U.S.C. § 1956
    (h). The district
    court dismissed this charge as to Abdelaziz and Wilson before trial
    on the government's motion.
    - 24 -
    constitute property.          Third, they moved to dismiss the conspiracy
    counts, arguing that the indictment alleged a "rimless wheel"
    conspiracy barred by the Supreme Court's decision in Kotteakos v.
    United States, 
    328 U.S. 750
     (1946), and to dismiss the entire
    indictment because it did not allege that the defendants had
    participated        in    a   single     act,    transaction,     or   series    of
    transactions, such that joinder was improper.
    The district court denied these motions in a memorandum
    opinion and order issued June 23, 2020.                United States v. Sidoo,
    
    468 F. Supp. 3d 428
    , 435 (D. Mass. 2020).                       It reasoned that
    "[p]ayments made to accounts controlled by university insiders,
    even       if   such     payments      were     ultimately     received    by   the
    universities," could support the § 666 and honest services fraud
    charges.        Id. at 445; see id. at 444-45.               It further concluded
    that       admissions    slots   "are    property    interests     owned   by   the
    university cognizable under the mail and wire fraud statutes."7
    Id. at 441; see id. at 440-42.                    And it determined that the
    indictment adequately alleged a single overarching conspiracy and
    that, as a result, joinder was appropriate.                  See id. at 437-39.
    7  The district court later incorporated this conclusion
    into its instructions to the jury on the mail and wire fraud
    counts, which stated: "For purposes of the mail and wire fraud
    statutes,   admission[s]  slots   are  the   property  of   the
    [u]niversities."
    - 25 -
    Before trial, twelve of the defendants reached plea
    agreements with the government, and one received a presidential
    pardon. After unsuccessfully moving to sever, Abdelaziz and Wilson
    were tried jointly before a jury in fall 2021.
    Singer himself did not testify at trial, despite having
    cooperated with the investigation.           The government did, however,
    introduce    a   substantial   amount   of    evidence    related   to   other
    parents' work with Singer without showing that Abdelaziz or Wilson
    were personally aware of those activities.               For example, Bruce
    Isackson -- one of the parents who worked with Singer who entered
    a plea agreement with the government -- was the government's first
    witness, and described his knowing misconduct in his dealings with
    Singer.     He testified, for instance, that he had paid to have his
    daughter's standardized test scores altered and that he "knew a
    good portion of th[e] money [he paid Singer to facilitate his
    children's college admission] was going into [Singer's] pockets
    and [to] the people who helped him" rather than to the university
    accounts for which it was purportedly intended.              The government
    also introduced recorded calls between Singer and other parents,
    but not between Singer and Abdelaziz or Wilson, in which Singer
    and those other parents discussed obviously wrongful activities,
    such   as   schemes   to   cheat   on   standardized     tests.     And   the
    prosecution drew the jury's attention to this evidence during its
    opening statement and closing argument.             For instance, during
    - 26 -
    closing argument, government counsel stated: "[One] thing that you
    need to find . . . [is] that the defendants . . . knew that what
    they were doing [was] wrong.   One way you know that [they did] is
    because Bruce Isackson told you that he knew it, from the witness
    stand."
    The defendants argued through counsel that they had
    believed Singer's services to be legitimate and had acted in good
    faith.
    The jury found both Abdelaziz and Wilson guilty on all
    counts.   Abdelaziz and Wilson moved for judgment of acquittal or
    a new trial, raising, as most relevant here, arguments similar to
    those made in the pretrial motions to dismiss.    The district court
    denied the motions, relying on the reasoning from its decision
    denying the motions to dismiss.    See United States v. Abdelaziz,
    
    578 F. Supp. 3d 110
    , 113-14, 116 (D. Mass. 2021).         The court
    sentenced Abdelaziz and Wilson to twelve and fifteen months'
    imprisonment, respectively.
    These timely appeals followed.
    II. Rejection of the Defendants' Theory that Their
    Convictions Under 
    18 U.S.C. § 666
     ("Theft or bribery concerning
    programs receiving Federal funds") Fail as a Matter of Law
    We begin with the defendants' argument that the charges
    under 
    18 U.S.C. § 666
     (as well as the related § 666 conspiracy
    counts) "fail as a matter of law."      We review these questions of
    law about the scope of § 666 de novo, see United States v.
    - 27 -
    Fernandez,      
    722 F.3d 1
    ,    8    (1st    Cir.   2013),   and   reject   the
    defendants' argument.
    The text of § 666 criminalizes "corruptly giv[ing],
    offer[ing], or agree[ing] to give anything of value to any person,
    with       intent     to        influence      or     reward    an    agent    of    an
    organization . . . in connection with any business, transaction,
    or series of transactions of such organization . . . involving
    anything of value of $5,000 or more."8                  
    18 U.S.C. § 666
    (a)(2).       The
    parties agree that the "agent[s]" in this case are the university
    employees who worked with Singer and that the "organization[s]"
    are the universities.              The defendants do not make any developed
    argument that their dealings with Singer and, through him, the
    university insiders were not "in connection with . . . business,
    transaction[s],            or     [a]       series    of    transactions      of    [the
    universities] involving anything of value of $5,000 or more."                        Nor
    do the defendants argue that the payments were not "inten[ded] to
    influence" the insiders in conducting that business or those
    transactions.
    8   Section 666 applies only if the "organization" in
    question "receives, in any one year period, benefits in excess of
    $10,000 under a Federal program involving a grant, contract,
    subsidy, loan, guarantee, insurance, or other form of Federal
    assistance." 
    18 U.S.C. § 666
    (b); see 
    id.
     § 666(a). On appeal,
    the defendants do not dispute that the universities satisfy this
    condition.
    - 28 -
    The focus of the defendants' arguments is instead on
    whether payments intended for university accounts -- which the
    government does not dispute the defendants' payments were -- can
    violate § 666.      That is, the defendants dispute that the phrase
    "any person" in § 666 can refer to the "organization" which is the
    agent's principal.      Here, the organization which is the agent's
    principal is a university.       The defendants contend that a payment
    to a university principal is not covered by § 666's text and does
    not align with common or historical understandings of the terms
    "bribe" and "bribery" or the purposes of "bribery" statutes.                In
    particular, the defendants emphasize that the government cannot
    produce a single case in the history of Anglo-American law in which
    a payment to an agent's principal was prosecuted as a bribe.             Thus,
    the defendants contend that construing the provision to proscribe
    such   payments     would   violate   several     canons   of    construction
    requiring    that    "ambiguous"      criminal    statutes      be   construed
    narrowly.
    A.
    We turn to the plain language of § 666.          See, e.g., Baker
    v. Smith & Wesson, Inc., 
    40 F.4th 43
    , 48 (1st Cir. 2022) ("We start
    with the [statutory] text . . . ."); see also Salinas v. United
    States, 
    522 U.S. 52
    , 55-57 (1997) (interpreting § 666 based on its
    "plain language").     That text refers to a thing of value given "to
    any person."      
    18 U.S.C. § 666
    (a)(2).         At oral argument, defense
    - 29 -
    counsel conceded that the "person" to whom a "[]thing of value" is
    given could be an organization.9           See 
    1 U.S.C. § 1
     (defining
    "person" to include "corporations, companies, associations, firms,
    partnerships, societies, and joint stock companies" "unless the
    context indicates otherwise").
    Given this concession, we see no textual reason to
    exclude the organizational principal from the set of entities that
    qualify as "any person" for purposes of § 666.         The Supreme Court
    has explained that courts should give effect to § 666's "expansive,
    unqualified language," including in its use of the word "any."
    Salinas, 
    522 U.S. at 56-57
    ; see also Sabri v. United States, 
    541 U.S. 600
    , 605 (2004) (declining to read § 666 to require proof
    that crime itself had nexus with federal money); Fischer v. United
    States, 
    529 U.S. 667
    , 677 (2000) (reading "benefit" in § 666 to
    include Medicare funds).         Salinas, for example, reasoned that
    Congress's   use   of   "any"   before   "the   business    or   transaction
    clause . . . undercut[] [a defendant's] attempt to impose [a]
    narrowing construction" that would limit § 666's application to
    bribes affecting federal funds.      
    522 U.S. at 57
    .       Here, the use of
    "any" before "person" militates against excluding principals from
    the set of eligible "person[s]."
    9    Similarly, in their briefing, the defendants stated that
    bribes could be directed to the agent's "political campaign,
    or . . . his favorite charity."
    - 30 -
    Section   666's    context    and   history   buttress   the
    conclusion that "any person" includes the agent's principal.          See
    Fernandez, 
    722 F.3d at 20-27
     (considering "statutory context,"
    including history of § 666 and related statutes, in interpreting
    § 666).    Congress amended § 666 into essentially its current form
    in 1986.     See id. at 21-22.    In so doing, Congress used the same
    operative language as used in 
    18 U.S.C. § 215
    , the bank bribery
    statute, which it had also revised earlier that year.        Indeed, the
    committee report on the § 666 amendment explained that "[t]he
    provision     parallels   the   bank   bribery   provision   (18   U.S.C.
    [§] 215)."     H.R. Rep. No. 99-797, at 30 n.9 (1986).        Compare 
    18 U.S.C. § 215
    (a)(1), with 
    id.
     § 666(a)(2).
    Before 1986, § 215 prohibited "directly or indirectly,
    giv[ing], offer[ing], or promis[ing] anything of value to any
    [agent] of any financial institution . . . or offer[ing] or
    promis[ing] any such [agent] to give anything of value to any
    person or entity, other than such financial institution, for or in
    connection with any transaction or business of such financial
    institution."    Act of Oct. 12, 1984, 
    Pub. L. No. 98-473, § 1107
    (a),
    
    98 Stat. 1837
    , 2146 (emphasis added).       The 1986 amendment to § 215
    revised the statute to bar "corruptly giv[ing], offer[ing], or
    promis[ing] anything of value to any person, with intent to
    influence or reward an officer, director, employee, agent, or
    attorney of a financial institution in connection with any business
    - 31 -
    or transaction of such institution."      Act of Aug. 4, 1986, 
    Pub. L. No. 99-370, § 2
    , 
    100 Stat. 779
    , 779.
    This history of § 215 and thus of § 666 shows that
    Congress knew how to exclude the agent's principal from the set of
    "person[s]" who could receive the thing of value, and it chose not
    to do so in revising § 215 or importing its language into § 666.
    Cf. Univ. of Tex. Sw. Med. Ctr. v. Nassar, 
    570 U.S. 338
    , 357 (2013)
    (comparing the text of antidiscrimination statutes and concluding
    that "when Congress elected to address [a concept] as part of a
    detailed statutory scheme, it did so in clear textual terms").
    Section 666 itself includes no express carveout -- the statute
    refers only to "any person."      That context further undercuts the
    defendants' effort to introduce a carveout for payments to the
    agent's principal.10
    The     defendants'     textual      counterarguments      are
    unpersuasive.    They first respond that the "person" and the
    "organization" must be distinct in order "to give each term
    independent   meaning."   Even    when    "person"   and   "organization"
    happen to refer to the same entity, however, each term does
    independent work in defining the offense: the former describes the
    recipient of the thing of value, while the latter identifies the
    10   The language of § 666 tracks the language of the revised
    version of § 215.       The defendants make no argument that,
    notwithstanding the revisions to the text of § 215, "any person"
    in that provision excludes the financial institution.
    - 32 -
    agent's principal.         See 
    18 U.S.C. § 666
    (a)(2); cf. Littlefield v.
    Mashpee Wampanoag Indian Tribe, 
    951 F.3d 30
    , 38 (1st Cir. 2020)
    (finding     no    surplusage        where     a    statute    contained       multiple
    definitions       of     potential     benefit       recipients    even    though    a
    particular     person      might     qualify       under   multiple     definitions).
    Further, the defendants do not explain -- and we can think of no
    explanation -- why their reasoning would not require concluding
    that the "agent" must also be distinct from the "person" who
    receives the thing of value.             Yet even the defendants acknowledge
    that a payment to the agent himself "in exchange for an exercise
    of his powers" is a paradigmatic form of bribery covered by § 666.
    The defendants turn for support to the Supreme Court's
    decision in Cedric Kushner Promotions, Ltd. v. King, 
    533 U.S. 158
    (2001), but that case does not help their cause.                      Cedric Kushner
    involved   a      suit    under    the   Racketeer         Influenced    and   Corrupt
    Organizations ("RICO") Act, which "makes it 'unlawful for any
    person employed by or associated with any enterprise'" to engage
    in certain conduct.          
    Id. at 160
     (quoting 
    18 U.S.C. § 1962
    (c)).
    The Court concluded that the RICO Act's text contemplates "two
    distinct entities: (1) a 'person'; and (2) an 'enterprise' that is
    not simply the same 'person' referred to by a different name,"
    because, by the statute's terms, the "person" must be "employed by
    or associated with" the "enterprise," and "[i]n ordinary English
    - 33 -
    one speaks of employing, being employed by, or associating with
    others, not oneself."      
    Id. at 161
    .
    The premise of the defendants' argument is that § 666
    "follows a similar structure" to that of the RICO statute and so,
    under Cedric Kushner, the "person" and the "organization" must be
    distinct.    The defendants' premise is wrong: § 666's language does
    not parallel the RICO statute in the relevant respect.             The RICO
    statute sets out a particular relationship between the "person"
    and   the   "enterprise"   --   the   former   must   be   "employed    by   or
    associated with" the latter -- that is incompatible with the
    "person" and the "enterprise" being synonymous.               In contrast,
    § 666's text does not require such a relationship between the
    "person" and the "organization."         Instead, it refers broadly to
    "any person" and, separately, to an "organization."
    At oral argument, Wilson's counsel also asserted that
    "the thing of value" required by § 666 cannot "be the type of
    professional benefit that the government has been relying on."
    See 
    18 U.S.C. § 666
    (a)(2) (requiring, inter alia, "giv[ing],
    offer[ing], or agree[ing] to give anything of value").                 We need
    not decide whether professional benefits can qualify as "anything
    of value" for purposes of § 666 because that phrase, as used in
    the statute, refers not to what the agent personally receives from
    the arrangement but to what the defendants "g[a]ve[], offer[ed],
    or agree[d] to give . . . to any person" -- here, the money the
    - 34 -
    defendants paid or agreed to pay to the universities.               Money is
    indisputably a thing of value.           See, e.g., Salinas, 
    522 U.S. at 57
    .   That is all § 666 requires.
    B.
    The   defendants      also    offer    nontextual   arguments    to
    support their view that § 666 cannot criminalize payments to the
    university principals.      They contend first that a payment to a
    university principal does not fall within ordinary or historical
    understandings of the terms "bribe" or "bribery" or implicate the
    purpose of antibribery provisions, and second that a series of
    statutory construction canons favor their reading: They assert
    that § 666 is "ambiguous," such that the rule of lenity applies.
    The government's reading, they argue, alters the balance of state
    and federal criminal jurisdiction, which should not be done without
    a clear statement by Congress.          They conclude by arguing there are
    vagueness concerns which require a narrow reading of § 666.
    Given   the   clear    meaning    of   the   language   chosen   by
    Congress, arguments about the meaning of "bribe" and "bribery" or
    the generalized purposes of "bribery" laws are beside the point.11
    As the Supreme Court said in Bostock v. Clayton County, 
    140 S. Ct. 11
      The section's caption does use the term "bribery," 
    18 U.S.C. § 666
    , but "[t]he caption of a statute . . . 'cannot undo
    or limit that which the [statute's] text makes plain,'" Intel Corp.
    v. Advanced Micro Devices, Inc., 
    542 U.S. 241
    , 256 (2004) (second
    alteration in original) (quoting Bhd. of R.R. Trainmen v. Balt. &
    Ohio R.R. Co., 
    331 U.S. 519
    , 529 (1947)).
    - 35 -
    1731 (2020), "[w]hen the express terms of a statute give us one
    answer and extratextual considerations suggest another, it's no
    contest."       Id. at 1737.   Nor does the fact that the government has
    not identified any historical bribery prosecution involving a
    payment to the agent's principal override § 666's clear text.                 Cf.
    id. at 1750-53 (rejecting narrow reading of Civil Rights Act of
    1964 based on historical applications).
    The     defendants'       arguments     based   on   various   canons
    calling for narrow constructions of ambiguous criminal statutes
    fail because the text of § 666 is not ambiguous with respect to
    whether it covers payments to the university principals.                      See
    Salinas, 
    522 U.S. at 66
     ("The rule [of lenity] does not apply when
    a statute is unambiguous or when invoked to engraft an illogical
    requirement to its text."); 
    id. at 59-60
     (holding that the canon
    of construction requiring a clear statement to alter the federal-
    state    balance    of     criminal    jurisdiction     "does    not   warrant   a
    departure from [§ 666's] terms" where the statute's "text . . . is
    unambiguous on the point under consideration"); cf. Skilling v.
    United States, 
    561 U.S. 358
    , 412 (2010) (finding no vagueness
    problem where it was "as plain as a pikestaff that" the conduct at
    issue would violate a statute (quoting Williams v. United States,
    
    341 U.S. 97
    , 101 (1951))).            The rule of lenity, for example, does
    not     apply    because    "there     is    [no]   '"grievous    ambiguity      or
    uncertainty" in the statute.'"              Muscarello v. United States, 524
    - 36 -
    U.S. 125, 139 (1998) (quoting Staples v. United States, 
    511 U.S. 600
    ,   619   n.17   (1994)).       The    Supreme       Court     has    consistently
    commanded that § 666 be interpreted in keeping with its "expansive,
    unqualified    language,"      which     "undercuts      .    .   .     attempt[s]   to
    impose . . . narrowing construction[s]."                Salinas, 
    522 U.S. at
    56-
    57; see also Sabri, 
    541 U.S. at 605
    ; Fischer, 
    529 U.S. at 677
    .
    Further,   the    defendants'           policy   argument      that     our
    interpretation of § 666 would upset the state-federal balance and
    "criminalize a large swath of ordinary transactions" suffers from
    several flaws.      That policy argument would arrogate to the federal
    judiciary choices which have been made by Congress.                            And the
    argument disregards that there are meaningful restrictions on
    § 666's scope.
    The statutory text of § 666 imposes several restrictions
    on the type of conduct proscribed by the provision.                        First, the
    reach of § 666 is limited by two dollar thresholds.                       Section 666
    applies only if the organization at issue "receives, in any one
    year   period,   benefits     in   excess       of    $10,000     under    a   Federal
    program."     
    18 U.S.C. § 666
    (b); see 
    id.
     § 666(a).                      And "to fall
    within the purview of § 666, [a bribe] must be made 'in connection
    with any business, transaction, or series of transactions of [the
    covered] organization, government, or agency involving anything of
    value of $5,000 or more.'"             Fernandez, 
    722 F.3d at 12
     (second
    - 37 -
    alteration in original) (quoting 
    18 U.S.C. § 666
    (a)(2)); see 
    id. at 12-13
     (discussing this "transactional element").
    Most importantly, § 666 requires that a defendant have
    acted "corruptly."      
    18 U.S.C. § 666
    .          This "corruptly" element
    provides a meaningful limit on the provision's sweep.               See, e.g.,
    Cooper Indus., Inc. v. Aviall Servs., Inc., 
    543 U.S. 157
    , 167
    (2004)   (describing    "the   settled    rule"    that   courts    "must,   if
    possible, construe a statute to give every word some operative
    effect").
    The government stresses that the requirement that the
    defendant    act   "corruptly"   restricts   the     scope   of    permissible
    prosecutions.      Its brief does not argue that any payment which
    violates any university policy could violate § 666.               Instead, the
    government    focuses   on   payments    intended    to   induce    university
    insiders to act contrary to the schools' underlying interests.
    But the definitions of "corruptly" that appear in the legislative
    histories of other federal bribery statutes point against the
    conclusion that the term somehow operates to exclude agents'
    principals from the set of "person[s]" who can receive the thing
    of value under § 666.    See H.R. Rep. No. 87-748, at 18 (1961) ("The
    word 'corruptly' [in 
    18 U.S.C. § 201
    , the federal officials bribery
    statute,] . . . means with wrongful or dishonest intent."); H.R.
    Rep. No. 99-335, at 6 n.24 (1985) ("The term 'corruptly' [in § 215,
    the bank bribery statute,] means that the act is done 'voluntantly
    - 38 -
    [sic] and intentionally, and with the bad purpose of accomplishing
    either an unlawful end or result, or a lawful end or result by
    some unlawful methods or means.       The motive to act corruptly is
    ordinarily a hope or expectation of either financial gain or other
    benefit to one's self, or some aid or profit or benefit to
    another.'" (quoting 2 E. Devitt & C. Blackmar,           Federal Jury
    Practice and Instructions § 34.08 (3d ed. 1977))).         Nothing in
    those definitions appears uniquely incompatible with payments made
    to agents' principals.
    Moreover,   because   the   defendants'   argument   for   the
    reversal of the § 666 counts does not squarely raise the meaning
    of "corruptly" in this context, we see no basis for reversing their
    convictions on any contention about the meaning of "corruptly."12
    We reject the defendants' argument that the charges
    under § 666 fail as a matter of law because the payments at issue
    were intended for university accounts.13
    12   Because, as we will explain, we vacate the § 666
    convictions on other grounds, we leave it to the district court to
    address the import of the meaning of "corruptly," if necessary, on
    remand after full briefing.
    13   The defendants also argue that they are entitled to a
    new trial due to alleged error in the jury instructions. Because
    we vacate the convictions under § 666 on other grounds, as
    discussed below, we do not address this argument.
    - 39 -
    We address the defendants' more successful argument that
    the § 666 convictions must be vacated for trial error in Section
    IV, set forth below.
    III. Acceptance of the Defendants' Defenses as to Convictions
    Under 
    18 U.S.C. §§ 1341
    , 1343, and 1346 ("Honest Services Fraud"
    and "Property Fraud")
    The defendants argue that the charges under the mail and
    wire fraud statutes, see 
    18 U.S.C. §§ 1341
    , 1343, 1346, fail as a
    matter of law.    These mail and wire fraud charges were based on
    two distinct legal theories: honest services fraud and property
    fraud.14   The defendants argue that the conduct charged in the
    indictment does not involve the core honest services doctrine
    identified in    Skilling.   They also argue that the statutory
    requirement that "property" be the subject of the alleged scheme
    or artifice to defraud cannot be met here.
    While the question is close, in the end we agree with
    the defendants. Considering each theory de novo, see United States
    v. Correia, 
    55 F.4th 12
    , 41 (1st Cir. 2022); Fernandez, 
    722 F.3d 14
       The defendants do not dispute that if either theory is
    legally viable and the jury instructions were proper, the evidence
    was sufficient. See United States v. Celestin, 
    612 F.3d 14
    , 24
    (1st Cir. 2010) ("[W]hen the government has advanced several
    alternate theories of guilt and the trial court has submitted the
    case to the jury on that basis, an ensuing conviction may stand as
    long as the evidence suffices to support any one of the submitted
    theories." (quoting United States v. Gobbi, 
    471 F.3d 302
    , 309 (1st
    Cir. 2006))). They argue separately that, even if the government's
    theory and the jury instructions were legally sound, they are
    nonetheless entitled to a new trial because of trial error; we
    address those contentions in Section IV below.
    - 40 -
    at 8, we conclude that the honest services fraud theory fails as
    a matter of law and that the government's arguments with respect
    to the property theory are not adequate to support the jury
    instructions given at trial.      We vacate the defendants' mail and
    wire   fraud    convictions,     including   the    related    conspiracy
    convictions (Counts One, Six, Eight, and Nine of the operative
    indictment).
    A.
    The     government's     honest     services    fraud   theory
    essentially    charges   the   defendants    with   a   non-traditionally
    recognized form of bribery.        Understanding this theory and the
    defendants' objections to it requires some background on the
    history of the mail and wire fraud statutes.        In particular, it is
    important to understand (1) the law in this area before the Supreme
    Court's decision in McNally v. United States, 
    483 U.S. 350
     (1987);
    (2) the congressional reaction to McNally; and (3) the Court's
    2010 decision in Skilling interpreting the current scope of the
    mail and wire fraud statutes after those developments and in light
    of constitutional concerns.      See generally Skilling, 
    561 U.S. at 399-402
     (recounting this history).
    1.
    Sections 1341 and 1343 both prohibit "any scheme or
    artifice to defraud, or for obtaining money or property by means
    - 41 -
    of false or fraudulent pretenses, representations, or promises."15
    
    18 U.S.C. §§ 1341
    , 1343 (emphasis added).      As the Supreme Court
    said in Skilling: "Emphasizing Congress' disjunctive phrasing, the
    Courts of Appeals, [beginning in the 1940s], interpreted the term
    'scheme or artifice to defraud' to include deprivations not only
    of money or property, but also of intangible rights."    
    561 U.S. at 400
    .    This "honest services doctrine" proscribed forms of fraud in
    which,
    [w]hile the offender profited, the betrayed
    party suffered no deprivation of money or
    property; instead, a third party, who had not
    been deceived, provided the enrichment. For
    example, if a city mayor (the offender)
    accepted a bribe from a third party in
    exchange for awarding that party a city
    contract, yet the contract terms were the same
    as any that could have been negotiated at
    arm's length, the city (the betrayed party)
    would suffer no tangible loss. Even if the
    scheme occasioned a money or property gain for
    the   betrayed    party,   courts    reasoned,
    actionable harm lay in the denial of that
    party's right to the offender's "honest
    services."
    
    Id.
     (citation omitted).      While honest services cases "[m]ost
    often . . . involved bribery of public officials," courts also
    applied the theory to the private sector.      
    Id. at 401
     (quoting
    15 The mail fraud statute applies to schemes involving use
    of the mails, 
    18 U.S.C. § 1341
    , while the wire fraud statute
    applies to those involving use of the wires, 
    id.
     § 1343. Apart
    from these elements, the Supreme Court has construed the statutes
    coextensively, see, e.g., Pasquantino v. United States, 
    544 U.S. 349
    , 355 n.2 (2005), and so we discuss them interchangeably.
    - 42 -
    United States v. Bohonus, 
    628 F.2d 1167
    , 1171 (9th Cir. 1980)).
    The defendants do not assert that the statutes cannot reach purely
    private actors.         "[B]y 1982, all Courts of Appeals had embraced
    the honest-services theory of fraud."                      
    Id.
     (citation omitted).
    But    the        Supreme       Court's    1987       decision   in    McNally
    "stopped [the theory] in its tracks."                       
    Id.
        "McNally involved a
    state   officer        who,    in     selecting       Kentucky's      insurance    agent,
    arranged    to    procure       a    share    of     the    agent's    commissions      via
    kickbacks paid to companies the official partially controlled."
    
    Id.
     at 401-02 (citing McNally, 
    483 U.S. at 360
    ).                        The prosecution
    did not allege that the scheme had cost the state money or resulted
    in worse insurance; rather, it argued that the scheme had deprived
    Kentucky of its right to honest services.                          
    Id.
     at 402 (citing
    McNally, 
    483 U.S. at 353, 360
    ).                The Court rejected this argument
    and the honest services doctrine, which, it reasoned, "le[ft] [the
    mail and wire fraud statutes'] outer boundaries ambiguous and
    involve[d]       the    Federal        Government       in    setting     standards     of
    disclosure and good government for local and state officials."
    
    Id.
     (quoting McNally, 
    483 U.S. at 360
    ).                           The Court "read the
    statute    'as    limited       in    scope    to     the    protection      of   property
    rights,'"    and       stated        that    "[i]f     Congress       desire[d]    to    go
    further, . . . it must speak more clearly."                       
    Id.
     (quoting McNally,
    
    483 U.S. at 360
    ).
    - 43 -
    Congress responded the following year by enacting 
    18 U.S.C. § 1346
    , which provides: "For the purposes of [the mail and
    wire fraud statutes], the term 'scheme or artifice to defraud'
    includes a scheme or artifice to deprive another of the intangible
    right of honest services."      See Skilling, 
    561 U.S. at 402
    .
    In   2010,   the   Supreme    Court    considered      a    vagueness
    challenge to § 1346 in Skilling.        See id.   The defendant, a private
    sector actor, was "charged . . . with conspiring to defraud [a
    company's] shareholders by misrepresenting the company's fiscal
    health,   thereby   artificially    inflating      its    stock       price"   and
    allowing the defendant to profit through his salary, bonuses, and
    stock sales.    Id. at 413.   He contended that § 1346 did not provide
    fair notice of the conduct it prohibits and that its "standardless
    sweep" would enable arbitrary prosecutions.              Id. at 403 (quoting
    defendant's brief); see also, e.g., Kolender v. Lawson, 
    461 U.S. 352
    , 357 (1983) ("As generally stated, the void-for-vagueness
    doctrine requires that a penal statute define the criminal offense
    with sufficient definiteness that ordinary people can understand
    what conduct is prohibited and in a manner that does not encourage
    arbitrary and discriminatory enforcement.").
    Recognizing the constitutional due process concerns,
    including fair notice and vagueness, raised by the statute, the
    Court chose to narrow the statute, rather than invalidate it, to
    "preserve what Congress certainly intended the statute to cover,"
    - 44 -
    that is, what the Court called the "core" of the pre-McNally honest
    services doctrine. Skilling, 
    561 U.S. at 404
    . This core "involved
    fraudulent schemes to deprive another of honest services through
    bribes or kickbacks supplied by a third party who had not been
    deceived."      Id.; see also 
    id. at 407
     ("Although some applications
    of      the     pre-McNally    honest-services      doctrine    occasioned
    disagreement among the Courts of Appeals, these cases do not cloud
    the doctrine's solid core: The 'vast majority' of the honest-
    services cases involved offenders who, in violation of a fiduciary
    duty,    participated    in   bribery   or   kickback   schemes."   (quoting
    United States v. Runnels, 
    833 F.2d 1183
    , 1187 (6th Cir. 1987))).
    "To preserve the statute without transgressing constitutional
    limitations, [the Court] . . . h[e]ld that § 1346 criminalizes
    only the bribe-and-kickback core of the pre-McNally case law."
    Id. at 408-09.      The defendants here argue that the conduct charged
    in the indictment does not involve the core honest services
    doctrine identified in Skilling.
    The Court explained that when the narrowed § 1346 is
    "[c]onfined to these paramount applications," it "presents no
    vagueness problem."       Id. at 404.    Turning to the issue of notice,
    the Court said that, "'whatever the school of thought concerning
    the scope and meaning of' § 1346, it has always been 'as plain as
    a pikestaff that' bribes and kickbacks constitute honest-services
    fraud."       Id. at 412 (quoting Williams, 
    341 U.S. at 101
    ).        And the
    - 45 -
    narrowing construction limited the risk of arbitrary prosecutions
    by the fact that § 1346's "prohibition on bribes and kickbacks
    draws content not only from the pre-McNally case law, but also
    from federal statutes proscribing -- and defining -- similar
    crimes," id. (citing, inter alia, 
    18 U.S.C. § 666
    (a)(2)), such
    that a "criminal defendant who participated in a bribery or
    kickback scheme . . . cannot tenably complain about prosecution
    under § 1346 on vagueness grounds," id. at 413.
    Because there was no allegation -- or plausible way of
    reading    the    facts   to   suggest   --     that    Skilling       himself   had
    participated in a bribery or kickback scheme, the Court concluded
    that he had not committed honest services fraud.                 Id.
    2.
    The    defendants    contend       that    their   payments     to   the
    universities,      the    parties     whose     interests      were    purportedly
    betrayed    by    their    agents,     cannot     constitute       bribes    under
    Skilling's interpretation of § 1346.                   There is no charge of
    kickbacks in the indictment.
    In    response,     the   government        relies   on     Skilling's
    statement that its narrow construction of § 1346's "prohibition on
    bribes . . . draws content not only from the pre-McNally case law,
    but also from federal statutes proscribing -- and defining --
    similar crimes.       See, e.g., 
    18 U.S.C. §§ 201
    (b), 666(a)(2); 
    41 U.S.C. § 52
    (2) . . . ."         
    561 U.S. at 412
    .         The government relies
    - 46 -
    on this language to argue that § 1346 effectively incorporates a
    version of § 666, such that -- borrowing the language of § 666 --
    § 1346 covers "[w]hoever . . . corruptly gives, offers, or agrees
    to give anything of value to any person, with intent to influence
    or reward an agent of [a principal]."         
    18 U.S.C. § 666
    .
    Although the question is a close one, we conclude that
    the government's reliance on this single statement in Skilling is
    misplaced.    The government's reading is, for several reasons,
    impossible to reconcile with Skilling's language and its core
    holding that § 1346 covers only "the bribe-and-kickback core of
    the pre-McNally case law."       
    561 U.S. at 409
    .    The government has
    not identified any pre-McNally case involving a purported bribe
    paid to the victim of an alleged bribery scheme.                Further, the
    statutes in force while courts developed the pre-McNally case law
    defining "bribery" do not support the conclusion that payments to
    the purportedly betrayed party constitute "bribes" as that term is
    traditionally understood or used in Skilling.             Nor is there any
    support for that view in other legal sources defining "bribery."
    Rather than interpreting the language the government cites to
    override    these    considerations,    we   understand    it    instead     to
    constrain    the    honest   services   doctrine's   sweep.        And     this
    understanding that Skilling's reference to other statutes does not
    mean that § 1346 is coextensive with these other statutes draws
    additional support from the facts that those statutes define
    - 47 -
    offenses broader than traditional bribery and that those statutes
    may vary from each other in their coverage.
    The      government's          reading    ignores     Skilling's     core
    "hold[ing] that § 1346 criminalizes only the bribe-and-kickback
    core of the pre-McNally case law."                   Id.; accord id. at 408
    (confining   the    scope    of      §     1346    "to   the   core   pre-McNally
    applications");     see   id.   at       404-09    (looking    to   "the   doctrine
    developed in pre-McNally cases in an endeavor to ascertain the
    meaning of the phrase 'the intangible right of honest services'"
    (quoting 
    18 U.S.C. § 1346
    )).               The government has not cited any
    pre-McNally honest services case involving a purported bribe paid
    to an agent's purportedly betrayed principal, and does not dispute
    that no pre-McNally       case involved            such a payment.         Skilling
    embodies a narrower understanding of the meaning of "bribery" for
    purposes of honest services fraud that cuts against concluding
    that the conduct involved here, which does not fall "[i]n the
    main . . . [of] the pre-McNally cases," is a "bribe" in the sense
    meant by Skilling.16      
    Id. at 404
    .
    The government emphasizes Skilling's characterization of
    pre-McNally case law as recognizing the potential for honest
    16   The defendants argue that the lack of pre-McNally
    precedent "is dispositive" and requires acquittal. We need not
    reach so far; even assuming that the lack of pre-McNally precedent
    is only a relevant but not a dispositive factor in our analysis,
    we reach the same result.
    - 48 -
    services fraud "[e]ven if [a] scheme occasioned a money or property
    gain for the betrayed party."       
    Id.
     at 400 (citing United States v.
    Dixon, 
    536 F.2d 1388
    , 1400 (2d Cir. 1976)).            But the government
    does not contend that the Court had in mind a case like this one,
    where the alleged bribe was paid directly to the purportedly
    betrayed party.    In fact, the cases to which the Court referred
    appear to have involved traditional bribery fact patterns that
    happened   incidentally     to      benefit   the      agent's    principal
    financially.     See 
    id.
     (citing, inter alia, Shushan v. United
    States, 
    117 F.2d 110
    , 115, 119 (5th Cir. 1941)).          Dixon, the case
    the Court cited for the proposition that schemes that financially
    benefitted the principal could still be actionable under pre-
    McNally doctrine, did not involve bribery at all; it simply stated
    in passing that honest services fraud could cover schemes that
    "enriched" the principal.        
    536 F.2d at
    1400 (citing United States
    v. Isaacs, 
    493 F.2d 1124
     (7th Cir. 1974) (per curiam)).           In support
    of that proposition, Dixon, in turn, cited a Seventh Circuit
    decision   in   which   racing    interests   bribed    certain    Illinois
    officials to allow additional racing events, which incidentally
    increased tax revenues.    See Isaacs, 
    493 F.2d at 1135, 1139
    , 1149-
    51.   That is a classic bribery fact pattern, distinct from the
    direct payments to the university principals involved here.
    Nor do statutes in effect during the pre-McNally period
    show that a payment to the purportedly betrayed party would have
    - 49 -
    been considered a "bribe." As discussed above, until its amendment
    in 1986, shortly before the Court decided McNally, the bank bribery
    statute, 
    18 U.S.C. § 215
    , expressly excluded payments to an agent's
    principal from its coverage.        See Act of Oct. 12, 1984, § 1107(a),
    98 Stat. at 2146 (prohibiting "directly or indirectly, giv[ing],
    offer[ing], or promis[ing] anything of value to any [agent] of any
    financial institution . . . or offer[ing] or promis[ing] any such
    [agent] to give anything of value to any person or entity, other
    than such financial institution, for or in connection with any
    transaction or business of such financial institution" (emphasis
    added)).     This limitation in scope undercuts any argument that it
    was clear before McNally that "bribery" would encompass a payment
    to the purportedly betrayed party.
    Other legal sources defining "bribery" either weigh
    against    the   government's     position   or   are   at   most   ambiguous.
    Black's Law Dictionary, for example, defines "bribery" as "[t]he
    corrupt payment, receipt, or solicitation of a private favor for
    official action."      Bribery, Black's Law Dictionary (11th ed. 2019)
    (emphasis    added).      While    the   government     contends    that    the
    university    insiders   stood    to   benefit    professionally     from   the
    defendants' payments, describing this type of indirect benefit
    from a payment to a university principal -- the alleged victim of
    the scheme -- as a "private favor" is at best a stretch.                    Cf.
    United States v. 
    Thompson, 484
     F.3d 877, 884 (7th Cir. 2007)
    - 50 -
    (observing, in a pre-Skilling prosecution for honest services
    fraud under § 1346, that "[t]he United States has not cited, and
    we have not found, any appellate decision holding that an increase
    in official salary, or a psychic benefit such as basking in a
    superior's approbation (and thinking one's job more secure), is
    the sort of 'private gain' that makes an act criminal" under
    § 1346, and rejecting the prosecution's theory (quoting United
    States v. Bloom, 
    149 F.3d 649
    , 655 (7th Cir. 1998))); see also,
    e.g., H. James, When Is a Bribe a Bribe?                     Teaching a Workable
    Definition of Bribery, 6 Teaching Bus. Ethics 199, 209–16 (2002)
    ("Any payment made to a principal, for any purpose, is not by
    definition a bribe.").
    The government's attempt to circumvent this lack of
    authority     by    relying       on    Skilling's          citation     to   other
    anticorruption statutes fails.           The government's reading of § 1346
    to   incorporate    a   version    of    §   666    eliminates     the   important
    limitations    on   liability      included        in   §   666,   including    the
    requirement that affected programs receive federal funds and the
    threshold dollar value involved, which we described above.17                    See
    
    18 U.S.C. § 666
    (a)-(b).         More importantly, the Supreme Court in
    17  The jury instructions that the government requested, and
    those that were ultimately given at trial, for example, did not
    require the jury to find that the alleged bribes satisfied either
    of those requirements in order to convict under an honest services
    theory.
    - 51 -
    Skilling emphasized these specific limitations in describing the
    statutes.    See 
    561 U.S. at
    413 n.45.
    Critically, the statement on which the government relies
    appears in the Court's discussion of why its construction of § 1346
    will prevent arbitrary prosecutions.           See id. at 412-13.       In that
    context, the citation is best read as constraining honest services
    prosecutions    by      referring     prosecutors         to    statutes   that
    collectively   offer    general     guidance    as   to   whether    particular
    conduct may be actionable, rather than as expanding the concept of
    "bribery" to incorporate even the outermost limits of the cited
    statutes' scopes.    Indeed, the government's argument would stretch
    criminal    liability     beyond     those     statutes'       context-specific
    limitations.    Skilling does not hold, as the government argues,
    that any conduct that might violate those other statutes also
    violates § 1346.
    Our reading properly accounts for the fact that Congress
    crafted § 666 and other federal anticorruption statutes to target
    particular classes of misconduct, and thus did not necessarily
    confine those statutes to criminalizing the classic crime of
    "bribery" in the sense described in Skilling and at the core of
    the pre-McNally case law.            The Supreme Court has repeatedly
    observed that § 666 uses "expansive, unqualified language" in
    service of Congress's unique interest in protecting federal funds
    from misuse.    Salinas, 
    522 U.S. at 56
    ; see 
    id. at 56-59
    ; see also
    - 52 -
    Sabri, 
    541 U.S. at 606-07
    .          That language sweeps beyond the type
    of "bribery" reflected in pre-McNally law and the other legal
    sources discussed above.            We do not think that while narrowly
    construing § 1346 to cover "the bribe-and-kickback core of the
    pre-McNally case law," Skilling, 
    561 U.S. at 409
    , the Court meant
    simultaneously to extend § 666's broad language to apply outside
    the particular context for which Congress designed it.
    Indeed,     the   government's   reading   would    threaten   to
    render    §   666   (and    other   specialized   anticorruption     statutes)
    superfluous, since § 1346 would cover the same ground while also
    extending to other contexts.             Cf. id. at 413 n.45 (addressing
    potential "superfluous[ness]" between § 1346 and more specialized
    anticorruption statutes).18          Our reading recognizes the statutes'
    distinct roles, with § 666 covering a broader set of types of
    conduct but applying only in a narrower context.                 Section 666's
    breadth is inseparable from its narrow focus.
    The government ignores the fact that Skilling itself
    recognized that other federal anticorruption statutes may vary in
    scope.        See   id.    at   412-13   (describing     these    statutes   as
    "defining . . . similar [but not necessarily identical] crimes").
    18   Skilling explained that construing § 1346 to overlap to
    some degree with specialized anticorruption statutes does not
    render § 1346 superfluous because it applies to a broader range of
    contexts. See 
    561 U.S. at
    413 n.45. It is a different question
    whether reading § 1346 to cover everything that § 666 covers, as
    well as other conduct, would render § 666 superfluous.
    - 53 -
    Given the number of potentially relevant statutes, the variation
    resulting from the government's reading would be problematic.                 For
    example, this variation would have existed in this case under the
    pre-1986   version    of   the   bank   bribery       statute,   §   215,    which
    explicitly excluded payments to an agent's principal.                Compare Act
    of Oct. 12, 1984, § 1107(a), 98 Stat. at 2146 (pre-1986 § 215),
    with, e.g., 
    18 U.S.C. § 666
    (a)(2), and Act of Oct. 23, 1962, 
    Pub. L. No. 87-849, 76
     Stat. 1119, 1119 (enacting 
    18 U.S.C. § 201
    , the
    federal officials bribery statute, which potentially applies to
    payments to "any public official" or "any other person").
    Construing § 1346 to cover conduct not covered by the
    core   pre-McNally    understanding     of    "bribes"    would      not   provide
    sufficient   notice    for   "ordinary       people    [to]   understand     what
    conduct is prohibited."          Skilling, 
    561 U.S. at 402
     (quoting
    Kolender, 
    461 U.S. at 357
    ).        An ordinary person would not be on
    notice that a payment to a purportedly betrayed party was bribery
    within the core of pre-McNally law, raising the same concern which
    motivated the Supreme Court in Skilling to construe honest services
    fraud as it did.     Our holding that § 666 may cover the defendants'
    conduct does not cure this concern: § 1346 does not have § 666's
    clear text, and, as we have explained, Skilling did not hold that
    liability under any other federal anticorruption statute suffices
    to render an act criminal under § 1346.
    - 54 -
    Various          canons   and    other     interpretive     methodologies
    employed by the Supreme Court reinforce our conclusion that, after
    Skilling, § 1346 does not cover the defendants' conduct as honest
    services fraud.          Unlike in our interpretation of § 666, these
    interpretive tools do apply here because the applicability of
    § 1346 to the charged conduct has little historical antecedent and
    would introduce ambiguity. And the Supreme Court "ha[s] instructed
    that   'ambiguity        concerning      the      ambit    of   criminal     statutes[,
    including the mail and wire fraud statutes,] should be resolved in
    favor of lenity.'"            Cleveland v. United States, 
    531 U.S. 12
    , 25
    (2000) (quoting Rewis v. United States, 
    401 U.S. 808
    , 812 (1971))
    (applying    rule       of    lenity    in    non-honest        services    mail     fraud
    prosecution). Similarly, in the honest services context, the Court
    has    repeatedly       "decline[d]          to   'construe      [federal        criminal]
    statute[s]   in     a    manner       that    leaves      [their]   outer    boundaries
    ambiguous    and        involves       the    Federal      Government       in    setting
    standards' of 'good government for local and state officials.'"
    McDonnell v. United States, 
    136 S. Ct. 2355
    , 2373 (2016) (quoting
    McNally, 
    483 U.S. at 360
    ); see, e.g., Skilling, 
    561 U.S. at
    408-
    12; McNally, 
    483 U.S. at 360
    .                Indeed, embracing the government's
    reading of § 1346 would go beyond "'setting standards' of 'good
    [state and local] government,'" McDonnell, 
    136 S. Ct. at 2373
    (quoting McNally, 
    483 U.S. at 360
    ), and stretch honest services
    bribery to potentially criminalize such parental actions as, for
    - 55 -
    example, donations to preschools by parents who hope to gain
    admission for their children.        Further, the contrast between the
    Court's repeated instruction to apply the honest services doctrine
    narrowly and its broad, textualist application of § 666 supplies
    another reason not to read Skilling as incorporating § 666 into
    § 1346.
    We should not be misunderstood.              We do not say the
    defendants' conduct is at all desirable.         That is far different
    from the issue we face of whether that conduct is in violation of
    § 1346's honest services language as interpreted by the Supreme
    Court in Skilling.   As Skilling explained, "[i]f Congress desires
    to go further, . . . it must speak more clearly than it has."            
    561 U.S. at 411
     (quoting McNally, 
    483 U.S. at 360
    ).          The government's
    honest services theory cannot support the defendants' mail and
    wire fraud convictions.
    B.
    Independently of honest services fraud, the government
    argues that we should affirm the defendants' mail and wire fraud
    convictions on the distinct property fraud theory.              The mail and
    wire   fraud   statutes   prohibit    use   of   the    mails     or   wires,
    respectively, to effect "any scheme or artifice to defraud, or for
    obtaining money or property by means of false or fraudulent
    pretenses, representations, or promises." 
    18 U.S.C. §§ 1341
    , 1343.
    A prosecution for property fraud under these statutes requires the
    - 56 -
    government to prove "that the 'object of the fraud . . . [was]
    [money or] property in the victim's hands.'" Pasquantino v. United
    States, 
    544 U.S. 349
    , 355 (2005) (second alteration and omission
    in   original)      (internal      quotation         marks     omitted)     (quoting
    Cleveland, 
    531 U.S. at 26
    ).
    The asserted "property" that the government argues was
    obtained here is "admissions slots."              Indeed, the district court
    instructed the jury that, "[f]or purposes of the mail and wire
    fraud   statutes,    admission[s]        slots    are    the    property     of   the
    [u]niversities."
    The defendants contend, however, that admissions slots
    can never qualify as property for purposes of the mail and wire
    fraud   statutes,    and    thus      that   their    convictions      under    these
    statutes must be reversed for that reason alone.                     As a fallback
    argument, they also contend that even if some admissions slots
    could be property for purposes of those statutes, we must vacate
    the convictions because, given the limitations of the government's
    arguments   and     evidence     in    this     case,    the    district     court's
    instruction that "admissions slots" are property was error.
    The government responds to these two different defense
    arguments   with    one    categorical        assertion.        It   contends     that
    "admissions   slots"       at   the    universities      supply      the   necessary
    - 57 -
    property    because      admissions    slots    by    their   nature     constitute
    property.19
    We reject the government's argument that admissions
    slots at any university always qualify as property for purposes of
    the mail and wire fraud statutes.               The government's categorical
    argument fails, for example, to recognize even the well-known
    variations in types of admissions slots offered at the university
    level;     for     instance,     early       admission,   rolling        admission,
    conditional       admission,     waiting-list        admission,    and     deferred
    admission.       Nor does the government's categorical approach account
    for the fact that admissions occur at all levels of education,
    from nursery school through postgraduate studies, and involve
    millions of students and parents.             We reject, too, the defendants'
    equally categorical contention to the contrary and so reject their
    argument     that    their     property-based        convictions    under     these
    statutes must be reversed on the ground that the government did
    not prove that property was involved in the commission of those
    offenses because "admissions slots" cannot be property.
    But     we   do   agree   with    the    defendants'   more     limited
    fallback argument that the jury instruction erred in stating, based
    on the arguments and record in this case, that "admission[s] slots
    19   The government does not develop any argument on appeal
    that the universities were defrauded of money or property, such as
    instructional   resources,   associated   with   the   defendants'
    children's enrollment.
    - 58 -
    are the property of the [u]niversities."         We see no basis for
    concluding that such a categorical statement is invariably true of
    any admissions slot, and the government has not identified any
    basis in the record that would indicate that the instruction could
    be upheld on the ground that there was evidence that the admissions
    slots in question in the charged offenses in this case qualified
    as property as a matter of law.
    We emphasize the narrowness of our holding: We do not
    hold that admissions slots cannot ever be property.          Nor do we
    hold that the jury instruction given by the district court could
    never be appropriate.     The resolution of these questions will
    require much more detail, both legal and factual, on the nature of
    the purported property interest at issue.         It may well be that
    there must be resolution of disputed facts by a jury and resolution
    of the ultimate legal question by the court.       A court may well be
    able to validly conclude on the evidence in a particular case that
    admissions slots constitute property.      Such increased detail would
    better position a district court to consider, for example, whether
    dictionaries,   case   law,   treatises,    or   other   legal   sources
    establish that similar interests are treated as property, see,
    e.g., 
    id.
     at 356 (citing such sources); Carpenter v. United States,
    
    484 U.S. 19
    , 26 (1987) (similar), and whether expert educational
    and/or economic evidence is warranted.       But here, the government
    does not identify from the record of this case adequate details
    - 59 -
    about the admissions slots at issue, or admissions slots generally,
    that would support the instruction given.                     Thus, we see no basis
    for concluding that the district court validly instructed the jury
    that,   "[f]or   purposes      of    the    mail       and    wire    fraud    statutes,
    admission[s] slots are the property of the [u]niversities."
    1.
    The fundamental problem with the government's argument
    as to why the instruction was not in error is that the government
    fails to describe the purported property interest in anything other
    than highly general, abstract terms, leaving us no firm basis on
    which   to   assess   whether       the    admissions         slots    at     issue   here
    constitute property.         The government's brief describes such slots
    as economically valuable and exclusively within the power of a
    university to issue, revoke, or prohibit transfer of, and on that
    basis alone it asks us to conclude that they are property.
    Under controlling Supreme Court precedent, we do not
    accept the government's argument that admissions slots always
    qualify as property for purposes of the mail and wire fraud
    statutes     merely    because      they        may    bear     some     hallmarks      of
    traditionally recognized forms of property.                     A series of Supreme
    Court   decisions     have    counseled         that    courts       should    resort   to
    traditional notions of property in construing the mail and wire
    fraud statutes. See Pasquantino, 
    544 U.S. at
    356 (citing treatises
    and case law in conducting property analysis); Cleveland, 531 U.S.
    - 60 -
    at 24 ("We reject the Government's theories of property rights [in
    part] because they stray from traditional concepts of property.");
    Carpenter, 
    484 U.S. at 26
    ; cf. Shaw v. United States, 
    137 S. Ct. 462
    , 466-67 (2016) (citing treatises and case law in analyzing
    whether   property      requirement   was    satisfied    under   bank   fraud
    statute, 
    18 U.S.C. § 1344
    ).           Carpenter, for example, held that
    "[c]onfidential business information" constitutes property under
    §§ 1341 and 1343 because it "has long been recognized as property";
    the Court cited in support of its conclusion an array of cases, a
    statute, and a treatise. 
    484 U.S. at 26
    . Based on these decisions,
    the   parties   agree    that   "[i]ntangible    rights    can    qualify   [as
    property] . . . if they have historically been treated as property
    or bear its traditional hallmarks."
    The Supreme Court's decision in Pasquantino explained
    that we must determine whether the alleged property interest
    constitutes "'property' as that term ordinarily is employed."               
    544 U.S. at 356
    ; see 
    id.
     ("When interpreting a statute, we must give
    words their ordinary or natural meaning." (quoting Leocal v.
    Ashcroft,     
    543 U.S. 1
    ,   9   (2004)    (internal    quotation     marks
    omitted))).     The Court's mail and wire fraud decisions offer
    several     potentially    relevant     guideposts   for     that    inquiry,
    including whether the purported property at issue falls within a
    dictionary definition of that term, whether it has been recognized
    as property in case law or other legal sources, and whether it
    - 61 -
    exhibits traditional attributes of property.             See, e.g., id. at
    355-57.    We consider these factors in turn, none of which support
    the   government's     categorical       position,      in   defending     the
    instruction, that admissions slots always constitute property.
    Pasquantino itself relied in large part on Black's Law
    Dictionary's definition of "property" for guidance on the term's
    ordinary meaning.      See id. at 356 (citing Property, Black's Law
    Dictionary (4th ed. 1951)).          Here, however, the government does
    not make any argument based on the dictionary definition of
    "property."
    The   government    does     offer    two    cases    recognizing
    interests purportedly analogous to admissions slots as property,
    but both are easily distinguishable.             The government cites the
    Supreme Court's decision in Bridge v. Phoenix Bond & Indemnity
    Co., 
    553 U.S. 639
     (2008), which allowed a group of plaintiffs who
    regularly sought to purchase tax liens at county auctions to pursue
    a RICO claim against a rival bidder based on allegations that the
    rival had committed mail fraud by fraudulently attesting that it
    had complied with a county rule regulating the number of bids that
    an entity could make.     See 
    id. at 642-44, 647-48, 661
    .             In fact,
    Bridge's    analysis   did     not    address    the    money    or   property
    requirement -- the Court accepted the case to answer a different
    question about the interaction between the RICO and mail fraud
    statutes.   
    Id. at 641-42
    .     Nonetheless, the government argues that
    - 62 -
    it held "that [the] false representation[s] to secure . . . extra
    bid[s] . . . , thereby depriving other bidders of the opportunity
    to obtain the liens, w[ere] . . .         'act[s] . . . indictable as
    mail fraud.'"    (Quoting 
    id. at 648
    .)    To the extent the government
    seeks to use this case, it misses the fact that the property the
    fraudulent bidder in Bridge sought to obtain, and of which the
    plaintiffs claimed to be defrauded, was not the bids, but the
    "valuable liens" available in the auctions.             
    Id. at 648
    ; cf.
    Pasquantino, 
    544 U.S. at 356
     (recognizing a "right to be paid
    money" as property under mail and wire fraud statutes).
    The    government's   other   authority,   United      States    v.
    Frost, 
    125 F.3d 346
     (6th Cir. 1997), held that a university "ha[d]
    a property right in [unissued] degree[s]."           
    Id. at 367
    .          The
    government    contends   that   admissions   slots   are    analogous     to
    unissued degrees.     We agree with the defendants, however, that
    unissued degrees are meaningfully different from admissions slots,
    at least insofar as the government has described such slots. Frost
    itself observed that a degree represents the culmination of the
    transaction between the university and the student, in which the
    university,     "in   return    for   tuition   money      and   scholarly
    effort, . . . agrees to provide an education and a degree."               
    Id.
    So far as the government's arguments show, an admissions slot, in
    contrast, involves an offer to participate in that transaction --
    one that a potential student may or may not accept.         Even assuming
    - 63 -
    Frost's correctness, it does not establish that admissions slots
    are a historically recognized form of property interest.
    The government falls back to an argument that admissions
    slots are necessarily property because they "bear the primary
    traditional hallmarks of property."20      It contends that an interest
    qualifies   as   property   if   it   exhibits   (1)   "exclusivity,"   see
    Carpenter, 
    484 U.S. at 26-27
     (observing that "exclusivity is an
    important aspect of confidential business information and most
    private property"), and (2) "economic value," see Pasquantino, 
    544 U.S. at 355-57
     (holding that "[v]aluable entitlement[]" to be paid
    taxes is property); Cleveland, 
    531 U.S. at 22
     (observing that
    unissued licenses had economic value to Louisiana, but concluding
    that this value alone did not make the licenses property in the
    State's hands).    It then asserts that admissions slots always have
    these features. We conclude that this purported test is too broad,
    as it would construe "property" to reach abstract interests that
    the Court, as well as several circuits, have concluded fall outside
    the statutes' scope.
    20   It may be that underlying the government's argument is
    an assumption that a contractual interest necessarily creates a
    property interest.    But this circuit has never so held, and
    Carpenter expressly rejected that argument as to honest services
    fraud. See 
    484 U.S. at
    25 (citing McNally, 
    483 U.S. at 355
    , 359
    n.8, 360) (explaining that a "contractual right to [an employee's]
    honest and faithful service" is not a cognizable property interest
    in this context).
    - 64 -
    Portions of the government's brief could be read to argue
    that economic value alone suffices to turn an intangible interest
    into a property right under the mail and wire fraud statutes.              In
    particular, the brief asserts that Pasquantino defined property as
    simply "something of value."        While the Court did use that phrase,
    Pasquantino, 
    544 U.S. at 355
     (quoting McNally, 
    483 U.S. at 358
    ),
    it did not state that economic value alone brought the purported
    property at issue within the scope of §§ 1341 and 1343; instead,
    it looked to a legal dictionary, treatises, and case law to
    determine whether the particular "[v]aluable entitlement[]" at
    issue constituted property,21 id. at 356; see id. at 355-57.
    But   even   to   the   extent   the   government   argues   that
    admissions slots are always property because they are, by their
    nature, both exclusive and economically valuable, its proposed
    test sweeps too broadly.        McNally illustrates the problem.           The
    purported   right   to   honest     services   that   McNally   declined   to
    recognize as property would satisfy both of the test's prongs: the
    right belonged exclusively to the entity to which honest services
    were owed, cf. Skilling, 
    561 U.S. at 401
     (characterizing pre-
    McNally honest services doctrine as proscribing "breache[s of an
    employee's] allegiance to his employer" (quoting United States v.
    21   A test looking only at whether the purported property
    interest has economic value would also sweep too broadly for all
    the same reasons as a test requiring both economic value and
    exclusivity, discussed below.
    - 65 -
    McNeive, 
    536 F.2d 1245
    , 1249 (8th Cir. 1976))), and plainly held
    economic value for that entity.                  The proposed test's failure to
    reach       the    right    result   on    McNally's    facts   demonstrates      its
    overinclusiveness.22
    The proposed test is also incompatible with multiple
    circuit decisions holding that various intangible interests that
    are both exclusive and valuable fall outside the scope of §§ 1341
    and 1343.         For example, the government's theory conflicts with the
    Sixth Circuit's decision in United States v. Sadler, 
    750 F.3d 585
    (6th    Cir.       2014).     There,      the   government   charged   one   of   the
    defendants, Nancy Sadler, with wire fraud in connection with the
    purchase of controlled substances.                   See 
    id. at 588-89
    .           The
    prosecution did not dispute that Sadler paid full price for the
    pills at issue; instead, to satisfy the property requirement, it
    argued that Sadler had lied to the sellers about the patients to
    22 The defendants argue that the government's test also
    conflicts with Cleveland, which held that unissued gaming licenses
    possessed by the State of Louisiana did not qualify as property
    for mail fraud purposes despite the State's exclusive control over
    the licenses, 
    531 U.S. at 23-24
    , and the fact that they could
    potentially generate revenue (in the form of application
    processing fees) even while in the State's possession, 
    id. at 22
    .
    But the Court reached that result because, in administering the
    licensing scheme, the State acted as a regulator, rather than as
    a property holder. See 
    id. at 20-25
    . That reasoning would not
    apply to entities, like the universities, without regulatory
    authority. Were Cleveland the only obstacle, then, the government
    could have proposed the same test but restricted its application
    to private parties.    There is no such easy fix for the test's
    incompatibility with McNally.
    - 66 -
    whom she would distribute the pills, "depriv[ing] the [sellers]
    of . . . a right to accurate information before selling the pills."
    Id. at 590-91.      The court rejected this argument, reasoning that
    "the ethereal right to accurate information" did not qualify as a
    property right, in part because it did not "amount[] to an interest
    that 'has long been recognized as property.'"            Id. at 591 (quoting
    Cleveland, 
    531 U.S. at 23
    ); accord United States v. Yates, 
    16 F.4th 256
    , 265 (9th Cir. 2021).         The government's proposed test would
    call for the opposite result, since a party's purported right to
    accurate   information    before    engaging    in   a    transaction   would
    presumably both have economic value for and belong exclusively to
    that party.      See also United States v. Bruchhausen, 
    977 F.2d 464
    ,
    467-68 (9th Cir. 1992) (declining to recognize as property a
    seller's right to control "the destination of [its] products after
    sale," even though such a right would, by hypothesis, be exclusive
    to the seller and potentially economically valuable).
    The     government's     highly     general      argument    would
    criminalize a wide swath of conduct.         Under the government's broad
    understanding of property applied to admissions slots as a class,
    embellishments in a kindergarten application could constitute
    property fraud proscribed by federal law.                Cleveland explained
    that it "reject[ed] the Government's theories of property rights
    not simply because they stray[ed] from traditional concepts of
    property," but also "because [they] invite[d the Court] to approve
    - 67 -
    a sweeping expansion of federal criminal jurisdiction in the
    absence of a clear statement by Congress."    
    531 U.S. at 24
    ; cf.
    Yates, 16 F.4th at 265 (rejecting property theory that "would
    transform all deception into fraud").
    Further, as the Court stated in Cleveland:
    [T]o the extent that the word "property" is
    ambiguous as placed in [the mail and wire
    fraud statutes], we have instructed that
    "ambiguity concerning the ambit of criminal
    statutes should be resolved in favor of
    lenity."      This   interpretive   guide   is
    especially appropriate in construing [the mail
    and wire fraud statutes] because . . . mail
    [and wire] fraud [are] predicate offense[s]
    under RICO . . . .       In deciding what is
    "property" [in this context], we think "it is
    appropriate, before we choose the harsher
    alternative, to require that Congress should
    have spoken in language that is clear and
    definite."
    
    531 U.S. at 25
     (citations omitted) (first quoting Rewis, 
    401 U.S. at 812
    ; and then quoting United States v. Universal C.I.T. Credit
    Corp., 
    344 U.S. 218
    , 222 (1952)); see 
    18 U.S.C. § 1961
    (1)(B)
    (defining mail and wire fraud as predicate offenses for purposes
    of RICO statute).
    2.
    At the same time, we must also reject the defendants'
    argument that no admissions slot at any university can qualify as
    property for purposes of the mail and wire fraud statutes and thus
    that their property-based convictions under those statutes must be
    reversed on this basis.   The defendants characterize admissions
    - 68 -
    slots as mere "offer[s] to engage in a transaction: The college is
    offering to provide educational services to a student in exchange
    for tuition payments."    However, the defendants do not address the
    complexities that would arise were there to be evidence that a
    particular admissions slot is more than a mere offer to transact.
    The same complexities which undercut the government's argument
    undercut this argument by the defense.            Cf. Tamboura v. Singer,
    No. 19-cv-03411, 
    2020 WL 2793371
    , at *1-5 (N.D. Cal. May 29, 2020)
    (dismissing for lack of standing two class action lawsuits to
    recover   application    fees    from      universities   at   which    Singer
    arranged side doors alleging that the plaintiffs "did not receive
    the 'fair' and 'objective' admissions process that they were
    promised" in a "bargain-for-exchange," only because the plaintiffs
    did not allege that they "applied for, were being considered for,
    or were denied . . . athletic spot[s]," which were the "focus[]"
    of "Singer's scheme").
    We thus cannot accept the defendants' contention that
    admissions slots can never be property, such that we could reverse
    their property-based convictions on that ground alone.
    3.
    There    remains   the    defendants'     argument    that    their
    property-based     convictions   must      be   vacated   because,     even   if
    admissions slots could constitute property in some circumstances,
    the jury instruction here was incorrect.            The defendants advance
    - 69 -
    two distinct arguments in this regard, one of which we cannot
    accept but the other of which we do.
    The defendants take aim at the instruction in part
    because they contend that the question whether admissions slots
    constitute property is -- as a matter of law -- a question of fact
    to be decided by the jury.          But neither party has provided any
    briefing on whether the question whether an interest constitutes
    property is, regardless of the facts of the case, a question of
    law to be decided by the judge.         Indeed, the parties' arguments
    are simply not clear as to what issues would present questions of
    fact to be determined by a jury, much less what are questions of
    law to be determined by a court.             Neither party has cited any
    Supreme Court case law resolving this issue, and, as best we can
    tell, the Supreme Court has not resolved the matter.            Perhaps it
    will in upcoming cases.       But we do not decide important issues of
    law based on vague, broad, and unsupported assertions by the
    parties in a case.   We need not resolve the issue here, and nothing
    in this opinion should be taken to suggest that the ultimate
    determination of whether admissions slots are property is an issue
    for the jury.
    That   said,   we    do    find    persuasive   the   defendants'
    separate contention that the jury instruction was erroneous here
    because it instructed that "admission[s] slots are the property of
    the [u]niversities."     There is some ambiguity as to whether, in
    - 70 -
    making   this       instruction,   the     district    court   accepted     the
    government's contention that any admissions slot at any university
    necessarily qualifies as property.            If so, then we have already
    explained why that conclusion is incorrect.
    If, however, the instruction was based on a more specific
    determination regarding the admissions slots at the universities
    at issue here, we fail to see the basis in the record for such a
    conclusion.      We do not understand admissions processes to be
    universally the same across universities, and the meaning of
    "admissions     slot"    may   differ      across    institutions,    yet   the
    government's argument treats them interchangeably and in sweeping
    terms.   Indeed, the government has cited no evidence and offered
    no argument specific to the admissions slots at the schools at
    which these two defendants sought admission for their children.
    Nor   does    the    government    offer     any    guidance   --    or   record
    citations -- for understanding the contours of these specific
    universities' admissions policies and processes or the rights,
    benefits, or obligations, if any, associated with obtaining an
    admissions slot at these universities.              It develops no argument,
    for example, that either a student's application or a university's
    offer of admission creates a contractual relationship between the
    applicant and the school.          Nor does it argue that every student
    awarded an admissions slot will eventually enroll -- in fact, it
    acknowledges, as it clearly must, that some will not.
    - 71 -
    With respect to what a proper jury instruction would
    say, or even whether one would be proper in this case given a more
    developed record on remand, we are not in a position to address
    the question, given the nature of the arguments that have been
    made to us.         We do emphasize, though, that the argument that
    admissions       slots   are   categorically      property   because    they    are
    exclusive and have economic value is insufficient.                   And, to the
    extent there are more case-specific arguments about the specific
    admissions slots involved in the charged offenses in a given case,
    we emphasize only that any argument that those admissions slots
    constitute property would have to show that, in light of what the
    record revealed about the nature of those particular slots, they
    would satisfy the standards that we have described above that the
    Supreme    Court    requires     us    to   apply   to   determine    whether    an
    intangible right is a species of property.
    We     recognize    that    our   analysis    leaves     considerable
    uncertainty as to how district courts should apply the mail and
    wire    fraud    statutes'     property     requirement    in   cases   involving
    admission to educational institutions.              There are sound reasons to
    be     prudent    and    cautious     about     criminalizing    conduct,      even
    unethical conduct, in this complicated area affecting so many
    students and parents.
    - 72 -
    4.
    We   hold   that,   based    on   the   arguments   made   by   the
    government, the district court's jury instruction was error.                 We
    therefore vacate the defendants' convictions on the mail and wire
    fraud charges, including the related conspiracy charges.
    IV. Acceptance of the Defendants' Argument that There Was a
    Prejudicial Variance with Respect to the Conspiracy Charges
    We turn to the defendants' core contention that under
    Supreme Court and First Circuit precedent the conspiracy charges
    are of an impermissible "rimless wheel" type forbidden by law,
    depriving them of fair trials.23              Count One of the indictment
    alleged an overarching nationwide conspiracy among Singer, his
    staff, university insiders, and parents to facilitate the parents'
    children's admission to Georgetown, Harvard, Stanford, UCLA, and
    USC by means of mail and wire fraud, in violation of §§ 1341, 1343,
    and 1346.    Count Two alleged an overarching nationwide conspiracy
    among a subset of the same individuals to secure children's
    admission to USC by means of federal programs bribery, in violation
    of § 666.    The defendants contend that the evidence is at most
    sufficient to show, however, that they agreed to join only a
    narrower conspiracy, which was to gain admission for each's own
    respective child or children (rather than to gain admission also
    23   The defendants are supported in this view by an amicus
    brief from eleven former U.S. Attorneys.
    - 73 -
    for other parents' children). As a result, they contend that there
    was a variance as to both counts, because that narrower conspiracy
    is not the broader one charged.24
    The    defendants'      characterization   of     the   charged
    conspiracy as a "rimless wheel" derives from the Supreme Court's
    decision in Kotteakos v. United States, 
    328 U.S. 750
     (1946).
    There, the government alleged that a single hub figure had assisted
    otherwise unrelated clients or groups of clients in fraudulently
    obtaining separate loans.        See 
    id. at 752-55
    .         The government
    indicted the hub figure and his clients as part of one overarching
    conspiracy.   
    Id. at 752-53
    .     The Court concluded that the evidence
    did   not   show   that   several    client-defendants      had   agreed   to
    participate in a single conspiracy with the other clients.25               See
    
    id. at 754-55
    .     Instead, "the pattern was that of separate spokes
    24  For convenience, we will refer to the two charged
    conspiracies as "the charged conspiracy" because the defendants
    contend that there was a variance because each charged conspiracy
    was broader than what they contend the evidence at most suffices
    to show -- their respective agreements to each join a conspiracy
    to gain admission for their own child or children.
    25  The government conceded in Kotteakos that the evidence
    did not support a finding of a single conspiracy. See 
    328 U.S. at 754-56, 768-69
    .     The Court endorsed that conclusion in both
    Kotteakos and Blumenthal v. United States, 
    332 U.S. 539
     (1947),
    and its reasoning in doing so informs our analysis. See Kotteakos,
    
    328 U.S. at 754-56, 768-69
    ; Blumenthal, 332 U.S. at 558; cf., e.g.,
    Brito v. Garland, 
    22 F.4th 240
    , 248 (1st Cir. 2021) (noting that
    this court is "bound to follow 'considered dicta' of the Supreme
    Court" (quoting United Nurses & Allied Pros. v. NLRB, 
    975 F.3d 34
    ,
    40 (1st Cir. 2020))).
    - 74 -
    meeting at a common center . . . without the rim of the wheel to
    enclose the spokes," which "made out a case, not of a single
    conspiracy, but of several."     
    Id. at 755
     (internal quotation marks
    omitted); see 
    id. at 754-55
    .         Further, the Court held that, while
    the failure to prove the single conspiracy charged might amount to
    harmless error in some cases, the defendants had been prejudiced
    by a defect in the jury instructions.             See 
    id. at 767-72
    .    More
    generally,    the   Court   warned    of   the    danger   of   prejudice   to
    defendants in cases where the government charges a broad conspiracy
    but proves only a collection of narrower ones, as the overbroad
    charge increases the risk that a jury will be exposed to and weigh
    against a defendant evidence that is actually relevant only to a
    separate conspiracy in which the defendant was not a participant.
    See 
    id. at 766-67
    .
    Abdelaziz and Wilson contend that the evidence fits
    Kotteakos's "rimless wheel" model, with Singer and his associates
    as the hub and parents as the spokes.            They assert that, whatever
    agreements might have existed among Singer and other parents, the
    evidence would not allow a reasonable jury to find that each of
    Abdelaziz and Wilson agreed to conspire with those parents.                 And
    they argue that this variance between the charges in the indictment
    and the proof at trial prejudiced them because the overarching
    conspiracy charges allowed the government to introduce evidence
    related to other parents' activities that undermined Abdelaziz's
    - 75 -
    and Wilson's defenses and led the jury to convict them for those
    other   parents'   conduct   and   not   for   their   own   actions.   In
    particular, the defendants contend that the overarching conspiracy
    charge enabled the prosecution to present to the jury inflammatory
    evidence, in the form of both witness testimony and recorded calls,
    of other parents' obviously culpable conduct in which Abdelaziz
    and Wilson played no part.
    This court determines whether convictions for conspiracy
    must be vacated on the ground that the scope of the conspiracy
    proved at trial varied from the conspiracy that was charged in the
    indictment by answering three questions:
    (1) Is the evidence sufficient to permit a
    jury to find the [conspiracy] that the
    indictment charges?     (2) If not, is it
    sufficient to permit a jury, under a proper
    set of instructions, to convict the defendant
    of a related, similar conspiracy [to violate
    the same statute]?     (3) If so, does the
    variance affect the defendant's substantial
    rights or does the difference between the
    charged conspiracy and the conspiracy proved
    amount to "harmless error?"
    United States v. Glenn, 
    828 F.2d 855
    , 858 (1st Cir. 1987) (Breyer,
    J.); see also United States v. Wihbey, 
    75 F.3d 761
    , 773 (1st Cir.
    1996) ("[S]o long as the statutory violation remains the same, the
    jury can convict even if the facts are somewhat different than
    [those] charged -- so long as the difference does not cause unfair
    prejudice." (quoting United States v. Twitty, 
    72 F.3d 228
    , 231
    (1st Cir. 1995))).
    - 76 -
    The answer to the second question is not in dispute: the
    government contends -- and the defendants do not make any developed
    argument to the contrary -- that the evidence was sufficient to
    permit a jury to convict each defendant of conspiring with Singer,
    his staff, and university insiders to secure his own child's or
    children's admission.     That leaves only the first and third
    questions at issue.
    We analyze first whether there was sufficient evidence
    to convict the defendants of the broader charged conspiracy, and
    second whether, if not, the resulting variance from the indictment
    prejudiced the defendants.      We conclude that the evidence was
    insufficient to prove that these defendants agreed to join the
    broader charged conspiracy and that the defendants were prejudiced
    by the variance, and so we vacate the defendants' conspiracy
    convictions.26    We   also   vacate    Wilson's   substantive   §   666
    convictions.
    26   We have already vacated the defendants' mail and wire
    fraud conspiracy convictions in Section III.        This variance
    analysis provides an alternative ground for that holding, in
    addition to providing the sole ground for vacating the convictions
    for conspiracy to commit federal programs bribery under § 666.
    Because we have already vacated the mail and wire fraud conspiracy
    convictions, we do not address Abdelaziz's argument that his
    conviction for conspiracy to commit mail and wire fraud must be
    vacated because of alleged error in the jury instructions'
    description of the scope of the charged conspiracy.
    - 77 -
    A.
    To assess whether the claimed variance occurred, we must
    determine whether the evidence sufficed for a rational juror to
    find beyond a reasonable doubt that the defendants agreed to join
    the broader charged conspiracy.      Glenn, 
    828 F.2d at 858
    .       As with
    all sufficiency challenges, our review is de novo, and we must
    review the evidence in the light most favorable to the verdict.
    United States v. Dellosantos, 
    649 F.3d 109
    , 115, 117 (1st Cir.
    2011).    The evidence cannot suffice to support the verdict through
    the kind of inference-stacking that "would require impermissible
    speculation on the jury's part."     Glenn, 
    828 F.2d at 860
    .      We begin
    this inquiry by providing the relevant legal background, which
    reveals the relevance to the inquiry of three specific factors.
    We then turn to the record in this case regarding each of those
    factors.
    1.
    The three factors that we have found to be helpful in
    guiding the inquiry into whether the evidence suffices to show
    that a defendant agreed to join a conspiracy as broad as the one
    charged rather than only a smaller, narrower one are "(1) the
    existence of a common goal [among the alleged participants in the
    charged    conspiracy],   (2)   interdependence    among   [the   alleged]
    participants [in the charged conspiracy], and (3) overlap among
    the   [alleged]   participants     [in     the   charged   conspiracy].'"
    - 78 -
    Dellosantos,     
    649 F.3d at 117
          (quoting   United   States   v.
    Mangual-Santiago, 
    562 F.3d 411
    , 421 (1st Cir. 2009)). The analysis
    is "pragmatic," United States v. Fenton, 
    367 F.3d 14
    , 19 (1st Cir.
    2004), and no single factor "is necessarily determinative," United
    States v. Díaz-Arias, 
    717 F.3d 1
    , 21 (1st Cir. 2013); see, e.g.,
    Dellosantos,     
    649 F.3d at 120-21
       (considering     factors
    collectively).
    It is particularly important in this inquiry to look not
    only to how these three factors bear on individuals alleged to
    have performed a similar role in the charged conspiracy to the
    role allegedly played by the specific defendants before us in these
    appeals, but also to whether these specific defendants agreed to
    join that broader conspiracy rather than at most only a narrower
    one.   As then-Judge Breyer described:
    [W]e recognize that conspiracy law, like most
    criminal law, focuses upon the activities of
    an individual defendant.      It is therefore
    dangerous to think of a conspiracy as a kind
    of "club" that one joins or a "business" in
    which one works.     Those metaphors falsely
    suggest that the "member" or "employee"
    automatically becomes legally responsible for
    the entire enterprise. Instead, "the gist of
    the   [conspiracy]    offense    remains  the
    agreement, and it is therefore essential to
    determine   what   kind   of    agreement  or
    understanding existed as to each defendant."
    Glenn, 
    828 F.2d at 857
     (second alteration in original) (citation
    omitted) (quoting United States v. Borelli, 
    336 F.2d 376
    , 384 (2d
    Cir. 1964)).   And while an agreement to conspire may be express or
    - 79 -
    tacit and can be proven using direct or circumstantial evidence,
    see id. at 857-58, "[the government] can prove only the agreement
    or   understanding    that   the   evidence     .    .   .    implies    beyond    a
    reasonable doubt," id. at 858.
    In Glenn, the court held that the evidence did not
    suffice to show that a defendant, Glenn, had joined the single
    conspiracy charged by the government to import marijuana from
    Thailand and hashish from Pakistan.         See id. at 858-60.            Instead,
    the court held that the evidence sufficed to show only that Glenn
    had joined a narrower conspiracy to import hashish, although that
    conspiracy was with some of those alleged to be part of the broader
    conspiracy described in the indictment.              See id.
    Glenn     emphasized    that   the       inquiry    into     whether    a
    defendant has agreed to join the conspiracy charged focuses on the
    scope of the activity in which the defendant agreed to join.                      See
    id. at 857.   It explained that the record might have sufficed to
    show that several other individuals in what it referred to as "the
    core group" had conspired to import both marijuana and hashish.
    Id. at 859.   But Glenn further explained that while the evidence
    showed that Glenn had dealings with that core group with respect
    to the distribution of hashish and was aware that the core group
    was involved in a broader conspiracy than one just to distribute
    hashish, that did not mean that the evidence sufficed to show that
    he had agreed to join that marijuana-hashish conspiracy.                   See id.
    - 80 -
    The court explained that, while the evidence showed that Glenn was
    aware of efforts to import marijuana by those in the core group,
    there was no evidence that he understood himself to have a stake
    in the success of those efforts or saw them as interdependent with
    his efforts to import hashish.         See id.      As a result, the court
    determined   that    the     government     had       not    proven      Glenn's
    participation in the broader multidrug conspiracy -- just his
    participation in a narrower, hashish-only scheme.                Id.
    2.
    With   this     legal   framework     in    mind,     we    begin   by
    considering what the evidence shows with respect to whether the
    defendants shared a common goal with the other alleged participants
    in the broader charged conspiracy.             We do so because if the
    evidence does show as much, then it would point in favor of finding
    that the defendants had agreed to join in the charged conspiracy.
    We acknowledge that, as the government emphasizes, the
    common goal factor "is given wide breadth."           Dellosantos, 
    649 F.3d at 117
         (internal       quotation      marks      omitted)         (quoting
    Mangual-Santiago, 
    562 F.3d at 421
    ).              In the context of drug
    distribution rings, this court has repeatedly recognized that
    "selling   cocaine   for    profit"   can   qualify     as   a    common   goal.
    Mangual-Santiago, 
    562 F.3d at 422
    ; accord, e.g., United States v.
    Portela, 
    167 F.3d 687
    , 695 (1st Cir. 1999).
    - 81 -
    But this is not a drug distribution case, and the Supreme
    Court has distinguished between fact patterns in which members of
    a broader conspiracy seek to "achiev[e] a single unlawful end" and
    those in which the alleged coconspirators each pursue "an end in
    itself, separate from all others, although all [a]re alike in
    having similar illegal objects."   Blumenthal v. United States, 
    332 U.S. 539
    , 558 (1947).   In the latter class of cases, there is no
    common goal shared by the alleged participants in the single
    broader charged conspiracy, even though each alleged participant
    may have a "similar illegal object[]" as the other participants.
    
    Id.
    Several decisions from this circuit have also found that
    alleged coconspirators lacked a common goal where they pursued
    similar but distinct ends or acted based on different motives from
    those common to the charged conspiracy.      See United States v.
    Monserrate-Valentín, 
    729 F.3d 31
    , 44 (1st Cir. 2013) (finding no
    common goal between alleged coconspirators who aimed to commit one
    robbery and others who sought to commit a series of robberies, and
    noting that the former group was motivated by a desire to seek
    revenge against the victim while the latter's objective was "purely
    pecuniary"); United States v. Franco-Santiago, 
    681 F.3d 1
    , 9-10
    (1st Cir. 2012) (contrasting goal of committing one robbery with
    goal of committing a series of robberies), abrogated on other
    grounds by Musacchio v. United States, 
    577 U.S. 237
     (2016); cf.
    - 82 -
    Glenn, 
    828 F.2d at 859-60
     (distinguishing between agreement to
    import marijuana, agreement to import hashish, and agreement to
    import both).       Other circuits have held the same.               See, e.g.,
    United States v. Swafford, 
    512 F.3d 833
    , 842 (6th Cir. 2008)
    (finding no "common goal" where alleged coconspirators engaged in
    similar   conduct      but   were   unaware   of    and   indifferent   to   one
    another's activities); United States v. Rosnow, 
    977 F.2d 399
    , 406
    (8th   Cir.    1992)   (finding     no   common    purpose   where   defendants
    "engaged in similar acts for similar reasons . . . in order to
    benefit themselves individually[ or] to gain revenge on their
    individual perceived enemies, and not to benefit the group as a
    whole"); United States v. Harrison, 
    942 F.2d 751
    , 757 (10th Cir.
    1991) (distinguishing between an overarching conspiracy with a
    "common" goal and multiple conspiracies with "identical" -- but
    not common -- goals).
    We ask then whether the government's evidence as to each
    defendant was sufficient to show that that defendant falls into
    the former class of cases.           The government offers two kinds of
    arguments to show that the cases at hand fall into the former
    class -- the first of which concerns what the evidence shows about
    the nature of the alleged scheme and the second of which concerns
    what the evidence shows as to more specific conduct by each
    defendant.
    - 83 -
    a.
    The   government's   contention   that    the   nature   of   the
    alleged scheme here itself provides a basis for concluding that
    these defendants shared a goal in common with the other alleged
    participants faces an immediate difficulty: the alleged scheme has
    the hallmarks of a hub-and-spoke conspiracy.        On the government's
    own account, the evidence shows that a hub figure or figures
    (Singer and others working directly with him to assist parents in
    gaining admission for their children) had dealings with many
    separate spokes (the individual parents who obtained services from
    Singer and his group).     We consider whether the hub-and-spoke
    nature of the scheme charged would, in and of itself, support a
    reasonable inference that any "spoke" shared a common goal with
    the other "spokes," and reject the government's argument.
    Blumenthal is a case of a hub-and-spoke conspiracy in
    which the evidence was deemed sufficient to show that the spokes
    shared a common goal due to the nature of the scheme.              There,
    several individuals were charged with conspiring to sell whiskey
    at rates above a government price ceiling.         See 332 U.S. at 541.
    The Court held that the evidence was sufficient to show that the
    defendants, each of whom was charged with purchasing whiskey from
    a single supplier to distribute and then selling it to others, had
    a common goal -- "to sell . . . whiskey unlawfully [at an above-
    market rate]" -- with the other defendants, which included not
    - 84 -
    only the supplier but other distributors.           Id. at 559.        That was
    so because the other potential explanations for those defendants'
    conduct in purchasing the whiskey at an above-market price from
    the conspiracy's hub were so economically irrational as to be
    "scarcely conceivable."      Id. at 550.     And the Court concluded that
    the   fact   that   they   shared   that   goal   with   the   other    alleged
    conspirators supported a finding that they had tacitly agreed to
    join the single, charged conspiracy.          See id. at 550, 559.
    Blumenthal explicitly contrasted the scenario involved
    in that case with the contrary outcome in the scenario at issue in
    Kotteakos, where a hub figure had helped otherwise unconnected
    clients or groups of clients fraudulently obtain loans.                See id.
    at 558; Kotteakos, 
    328 U.S. at 753-55
    .            The Court explained that
    although each client (or group of clients) in Kotteakos pursued a
    "similar illegal object[]," none "was interested in whether any
    loan except his own went through," and that this lack of common
    purpose cut against treating the clients as participants in a
    single overarching conspiracy.             Blumenthal, 332 U.S. at 558.
    Rather, the evidence "made out a case, not of a single conspiracy,
    but of several."     Kotteakos, 
    328 U.S. at 755
    ; see 
    id. at 754-55
    .
    The defendants here do not dispute that the evidence
    suffices to show that all parents alleged to have conspired with
    Singer and his core group had similar unlawful goals in one sense:
    getting their own children into particular universities through
    - 85 -
    illicit means.       And the evidence does suffice to show that Singer
    and   others    in   his   core   group    shared   a   goal   of   facilitating
    admissions into universities for the children of parents who sought
    the group's services, as the business model of the alleged scheme
    depended on their ability to secure those side doors.
    The relevant question, though, is whether the nature of
    the alleged scheme is such that it would be reasonable to infer
    that any parents who sought the assistance of the core group shared
    a goal of getting children other than their own into any university
    just because they sought such assistance for their own children.
    We do not see how the nature of the alleged scheme would support
    such an inference.
    The defendants were purchasing a service from the core
    group in the way that any consumer of a service would purchase it
    from a service provider.          We do not commonly infer, however, that
    a buyer shares a common goal with a seller just because the two
    transact with one another.          See United States v. Bedini, 
    861 F.3d 10
    , 15 (1st Cir. 2017) (looking for "more than a mere buyer-seller
    relationship" (quoting United States v. Ortiz-Islas, 
    829 F.3d 19
    ,
    25 (1st Cir. 2016) (Souter, J.))); Ortiz-Islas, 
    829 F.3d at 25
    (finding "more than a mere buyer-seller relationship" due to
    evidence that "seller" fronted wholesale quantities of cocaine to
    "buyer," showing "act of trust that assumed an ongoing enterprise
    with a standing objective");              cf.   Kotteakos, 
    328 U.S. at
    753
    - 86 -
    (explaining   that   hub   figure's   relationship   to   each   alleged
    coconspirator was that of a "broker . . . charging a five per cent
    commission for his services").
    Moreover, this is not a case like Blumenthal, in which
    there is some straightforward reason to draw an inference that the
    defendants had a goal beyond benefiting themselves.         Here, unlike
    in Blumenthal, the two defendants had a clear self-interest in
    dealing with the hub figures: obtaining their own children's
    admission in a discrete buyer-seller transaction.         Thus, there is
    a quite "conceivable" explanation for their willingness to seek
    Singer's assistance that by no means entails their having a broader
    goal of ensuring that other parents could obtain similar assistance
    from Singer for their children and thus a common goal with other
    spokes.   Cf. Blumenthal, 332 U.S. at 550 (describing alternative
    explanation for defendants' conduct other than participation in a
    broader conspiracy as "scarcely conceivable").
    Indeed, the nature of the defendants' status as buyers
    in this scheme much more easily leads to the opposite inference:
    that the defendants were indifferent or even adverse to whether
    other parents' children were admitted to the schools to which they
    sought admission, and had no interest in what happened to parents
    seeking admission at other universities.       It is commonplace that
    universities'   admissions   processes   are   competitive    and   often
    highly competitive.    The defendants argue that, far from proving
    - 87 -
    pursuit of a common goal, the evidence showed that "Singer's
    clients were at times led to believe they were competitors."                   They
    cite as an example an email exchange involving Singer, Wilson, and
    Wilson's wife in which Singer stated that USC's water polo coach
    was "giving [him] 1 boys [sic] slot" and that he had "5 + wanting
    in that are boys [--] 2 polo[,] 3 others."
    Of course, as the government points out, this court and
    others         have    explained      that      competition      among      alleged
    coconspirators does not itself preclude a finding of an overarching
    conspiracy where there is other evidence supporting a finding of
    an overarching conspiracy notwithstanding that competition.                    See,
    e.g., United States v. Rivera Calderón, 
    578 F.3d 78
    , 92 & n.2 (1st
    Cir. 2009) (concluding that "even if there was some competition
    [among coconspirators in a drug ring], that alone d[id] not detract
    from     the     various   ways    the   appellants       conspired      together,"
    particularly          because   the   evidence      did    not   show      "serious
    competition," such as undercutting one another's sales).                   However,
    that does not make competition irrelevant; indeed, competition
    cuts against the reasonableness of inferring that the defendants
    shared a common goal with all the alleged coconspirators here.
    See United States v. Townsend, 
    924 F.2d 1385
    , 1397 (7th Cir. 1991)
    ("The evidence does not suggest that [a defendant conspired with
    other individuals]; rather, it shows that he was competing with
    them.").
    - 88 -
    We do not say that, on different evidence, it would be
    impossible for any parent who sought out services from Singer and
    the core group to have adopted the common goal of advancing the
    success of children seeking admission though side doors.      Some
    parents may have had an interest in the broader success of the
    venture.    Here, the government cites to testimony from Bruce
    Isackson, an alleged unindicted coconspirator parent who pleaded
    guilty and cooperated with the government as its lead evidence,27
    that he thought it was "good" that lots of parents worked with
    Singer because "[m]ost of these people have very complicated [tax]
    returns," which "would [make it] pretty hard [for the IRS] to
    figure things out."
    As Glenn instructs, however, we must keep our attention
    focused on whether each individual defendant agreed to join the
    broader conspiracy that was charged.    There is no evidence that
    either defendant before us on appeal ever spoke with or even was
    aware of Isackson's dealing with the core group, let alone that
    they shared his view that the participation of other parents was
    "good" for the success of the core group's venture.28   Isackson's
    27   Isackson was named alongside Abdelaziz and Wilson in the
    government's original criminal complaint.      He entered a plea
    agreement with the government pursuant to a criminal information,
    waiving the right to indictment, and so was not indicted with the
    defendants.
    28   Isackson testified that he had never met or spoken with
    Abdelaziz and that he had met Wilson at school or charitable events
    - 89 -
    testimony about his own understanding of how the breadth of
    participation   by   other     parents   mattered   does    not   support   a
    reasonable inference that any parent who worked with Singer was
    similarly    interested   in     ensuring   that    other    parents     were
    participating, too.
    The government also points to evidence that "a stock
    part of Singer's pitch" included describing the benefits of broad
    participation, and urges us to conclude that it would be reasonable
    to infer that each defendant heard that pitch.              While the pitch
    may help clarify the nature of the scheme, the evidence that the
    government cites at most shows that Singer told at least some
    parents, including Wilson, that his operation worked with a large
    number of parents and schools.        That is the usual assertion of any
    successful venture.     The idea that a larger venture is more likely
    to succeed than a smaller one is not necessarily true.
    Moreover,   while   the   evidence   suffices    to   show   that
    Singer and his core group had a financial interest in whether
    children of parents other than the defendants obtained admission,
    no parent had any similar financial stake in how successful other
    children were in getting admitted through the services of the core
    group.   Glenn and Kotteakos do not permit us to conclude that the
    defendants' mere awareness that Singer and the core group had other
    but had never had any "substantive conversation[s]" with him.
    - 90 -
    parents enrolled suffices to permit a rational juror to infer that
    the defendants shared the goal of advancing the success of that
    broader conspiracy.       Glenn, 
    828 F.2d at 859
    ; Kotteakos, 
    328 U.S. at 755
    .
    b.
    The government also points to evidence that is more
    defendant specific to show that the common goal factor favors its
    position.    For example, the government points to evidence that the
    Wilson family referred other parents to Singer, and that Abdelaziz
    responded "I love it" when told that Singer would be using his
    daughter's profile as a model when creating profiles for children
    of other parents seeking admission to USC.              The government thus
    contends that even if the nature of the scheme -- even as fleshed
    out through the Isackson testimony and the evidence of Singer's
    pitch   --   does   not   in   and   of    itself   suffice   to   support   the
    reasonable inference that the defendants had the common goal of
    advancing the success of the broader venture, this defendant-
    specific evidence does when considered in the context of the
    evidence as a whole.
    In pressing this point, the government asks us to accept
    that the common goal of the charged conspiracy was merely to
    advance the conspiracy's success.              By defining a common goal at
    that high level of generality, the government's argument threatens
    to drain the common goal factor of any independent significance in
    - 91 -
    the inquiry into whether the evidence suffices to show that the
    scope of the conspiracy that the defendants joined is the same as
    the one charged.       Nor does the government identify any prior
    precedent of ours that treats as the common goal of the conspiracy
    charged    merely   advancing   the   conspiracy's   success.   Even   in
    Blumenthal, the "common end" was described not merely as ensuring
    the conspiracy's success, but more specifically as "to sell the
    whiskey unlawfully" and "to aid in disposing of the whiskey."          332
    U.S. at 559.
    The defendant-specific evidence that the government puts
    forward to show that these defendants shared a common goal is more
    relevant to the factor that we discuss in the next section --
    interdependence.      Interdependence, after all, "concerns whether
    'the activities of one aspect of the scheme are necessary or
    advantageous to the success of another aspect of the scheme.'"
    Dellosantos, 
    649 F.3d at 117
     (quoting Mangual-Santiago, 
    562 F.3d at 422
    ).
    Indeed, in connection with the interdependence factor,
    the government does point to the same evidence about the specific
    conduct of the defendants in allegedly aiding other parents in
    obtaining admission for their children through Singer's venture.
    We discuss this body of evidence -- and the weakness of it -- in
    the next section.     The same weaknesses which lead us to conclude,
    as we next explain, that this evidence does not supportably show
    - 92 -
    based on the interdependence factor that the defendants tacitly
    agreed to join the broader charged conspiracy also leads us to
    conclude that it fails to do so based on the common goal factor.
    3.
    We turn to the other factor in dispute: interdependence.
    Glenn is instructive once again.           As then-Judge Breyer explained
    in the context of an alleged drug-distribution conspiracy:
    [K]nown interdependence . . . makes it
    reasonable to speak of a tacit understanding
    between the distributor and others upon whose
    unlawful acts the distributor knows his own
    success    likely  depends.       When   such
    interdependence   is   missing,    when   the
    distributor is indifferent to the purposes of
    others in the enterprise -- say, other
    distributors -- the tacit understanding does
    not exist.
    Glenn, 
    828 F.2d at 857-58
     (citation omitted).              Thus, as we have
    explained, "[e]ach individual must think the aspects of the venture
    interdependent, and each defendant's state of mind, and not his
    mere   participation    in   some   branch    of   the   venture,    is   key."
    Dellosantos, 
    649 F.3d at 117
     (quoting Mangual-Santiago, 
    562 F.3d at 422
    ).
    The indictment here alleged that the defendants agreed
    to conspire not only with Singer and the core group but also with
    the purported coconspirator parents.          See Glenn, 
    828 F.2d at 857
    (observing     that      conspiracy        requires      agreement        among
    coconspirators).      It is therefore insufficient for the government
    - 93 -
    to show only that each of the defendants conspired individually
    with Singer and the core group to secure admission for their own
    children to prove the broad overarching conspiracy charged.                    The
    government must show that each of these two defendants agreed to
    conspire with the other parents charged as coconspirators in the
    larger conspiracy.          See Kotteakos, 
    328 U.S. at 755
     (recognizing
    that no single conspiracy existed where the evidence showed only
    "separate spokes meeting at a common center . . . without the rim
    of the wheel to enclose the[m]").                 For, as other circuits have
    explained,       where     the   government    has    charged   a   hub-and-spoke
    conspiracy like that alleged here, interdependence must exist
    between the spokes, and not simply between the hub and each spoke,
    for the interdependence factor to support a finding of a single
    conspiracy.        See, e.g., United States v. Chandler, 
    388 F.3d 796
    ,
    811      (11th      Cir.     2004)       (evaluating     whether     there      was
    "interdependence of the spokes"); United States v. Mathis, 
    216 F.3d 18
    , 24 (D.C. Cir. 2000) (rejecting an argument that the
    government must "show interdependence only among the hub . . . ,
    not among the spoke[s]" because the "spoke[s] . . . in a hub
    conspiracy must not only have a connection to the hub . . . but
    must also have interdependence among each other in order to form
    a     rim    and    constitute       a    single     conspiracy").        Without
    interdependence between the defendants and other parents, the
    "tacit      understanding"       necessary    for    these   defendants   to   have
    - 94 -
    agreed to conspire with the other parents does not exist.                     Glenn,
    
    828 F.2d at 858
    ; see 
    id. at 857-58
    .
    We look at whether the evidence of the conduct of each
    of these two defendants shows interdependence with the other
    parents in the broader charged conspiracy. See, e.g., Dellosantos,
    
    649 F.3d at 119-20
     (assessing interdependence of two branches of
    alleged drug conspiracy, one of which distributed cocaine and one
    of which distributed both cocaine and marijuana).
    a.
    Here,       too,   the    government    argues,   relying      on   drug
    distribution cases, that the jury could infer from "the nature of
    the scheme" that the defendants must have understood themselves to
    be interdependent with other parents.                We cannot agree.
    This court has found that form of inference appropriate
    in the context of defendants selling "wholesale quantit[ies]" of
    drugs   to   drug        distribution     rings     for   resale    to    individual
    buyers/users, e.g., Portela, 
    167 F.3d at 697
    ; see 
    id. at 697-98
    ,
    but that fact pattern is not at all analogous to this case.                      It is
    clearly reasonable to infer that a drug supplier must understand
    that "[t]he success of [his] transaction [i]s dependent on [the
    existence    of]     a    conspiratorial        network   capable    of    disposing
    profitably of the [drugs], and the very existence of such a network
    [i]s necessarily dependent on the existence of other wholesale
    suppliers."     
    Id. at 697
    .           The wholesale supplier knows there must
    - 95 -
    be a further distribution chain for the distribution of his
    wholesale quantities of drugs.     See 
    id.
    Singer, though, is not a wholesaler of any good, and
    neither defendant is a distributor.        The evidence here is that
    Singer brokered such arrangements as he made for the admission of
    these two defendants' children on an individual basis, with each
    "an end in itself" rather than an integrated part of a larger
    conspiracy.   Blumenthal, 332 U.S. at 558.    Indeed, as with "common
    goal," the competitive nature of college admissions would, if
    anything, cut against a finding of interdependence.         See United
    States v. Carnagie, 
    533 F.3d 1231
    , 1240 (10th Cir. 2008) (noting
    that   "direct   competition"   between   alleged   coconspirators   cut
    against finding interdependence).     Nor is there any evidence here
    of anything akin to coconspirators' "fronting" each other money or
    drugs, which we have looked to even in drug conspiracy cases to
    substantiate the notion that coconspirators were interdependent.
    See, e.g., Bedini, 
    861 F.3d at 16
    .
    The government also argues that a rational jury could
    infer interdependence between the defendants and other parents
    from just the nature of the scheme because "[b]road participation
    allowed . . . parents to rely on . . . the scheme's success," as
    "Singer [could] recruit more coaches and . . . offer parents . . .
    more options at more schools."       To support this "nature of the
    scheme"-based method of proving interdependence, the government
    - 96 -
    cites a phone call between Singer and parent Agustin Huneeus, an
    alleged unindicted coconspirator.29     In that call, according to the
    government, Huneeus "motivat[ed] himself to participate in the
    scheme based on the experience of" another parent who worked with
    Singer and "the scheme's track record."
    Even if a parent chose to work with Singer based on his
    past success with other parents, the government must prove as to
    interdependence that "the activities of one aspect of the scheme
    are necessary or advantageous to the success of another aspect of
    the    scheme."      Dellosantos,   
    649 F.3d at 117
       (quoting
    Mangual-Santiago, 
    562 F.3d at 422
    ).     A track record of success may
    have made parents more confident that Singer could help their
    children, but it does not mean that those parents necessarily
    viewed their children's admission as in some way dependent on
    Singer's work     with other parents.      Thus, even accepting the
    government's interpretation of the inferences to be drawn from the
    call about Huneeus's understanding of the scheme, they do not
    supportably show that Abdelaziz and Wilson had the same view.     See
    
    id.
     (noting focus on each defendant's own state of mind).      To draw
    such a conclusion would require stacking inference upon inference
    29  Like Isackson, Huneeus was named in the government's
    original criminal complaint alongside Abdelaziz and Wilson, but
    pleaded guilty pursuant to a criminal information before Abdelaziz
    and Wilson were indicted.
    - 97 -
    through "impermissible speculation on the jury's part."                 Glenn,
    
    828 F.2d at 860
    .
    The government separately argues that a rational jury
    could find the required interdependence from the nature of the
    scheme because the inclusion of additional parents in the scheme
    benefited the defendants by making "the interconnected web of
    relationships and finances . . . more difficult to unravel."                In
    support of this "nature of the scheme"-based theory for finding
    that the interdependence factor points in favor of the convictions,
    the government again cites Isackson's testimony that he thought it
    was "good" that lots of parents worked with Singer because "[m]ost
    of these people have very complicated [tax] returns," which "would
    [make it] pretty hard [for the IRS] to figure things out."                 The
    government also asserts that "Wilson and Abdelaziz . . . join[ed]
    the   other    parents   in   channeling    millions    of   dollars   through
    [Singer's operation]," that they discussed "the mechanics of money
    flow" with Singer, and that "both were sophisticated and successful
    businessmen" who would understand "the advantages of this feature
    of the scheme."
    We disagree with the government: this evidence does not
    support   a    finding   of   interdependence.         Isackson's   testimony
    reflected his own personal view that it was "good" that other
    parents of means had also engaged Singer because in his view that
    made a coverup easier.         Neither Abdelaziz nor Wilson discussed
    - 98 -
    Singer's services with Isackson, and there was no evidence they
    shared his views or even would have thought those views plausible.
    See Portela, 
    167 F.3d at 695
     (emphasizing that interdependence
    depends on defendant's own state of mind); Glenn, 
    828 F.2d at 859
    (evaluating       evidence    of   defendant's    state     of   mind).     While
    sometimes a defendant's efforts to cover up a conspiracy may be
    probative of his agreement to join the conspiracy, in which case
    a cover-up effort interdependent with other coconspirators might
    suffice to show interdependence, the proof the government offered
    here was of a different actor's guilt.
    Further, the discussions about "the mechanics of money
    flow"     that    the    government      cites   involved    basic   logistical
    questions about payment due dates and wiring instructions; at no
    point during those exchanges did the defendants allude to any
    perceived benefit from intermingling their payments with other
    parents'.        That parents made their payments to the universities
    through    Singer       is   naturally    explained   by    Singer's      role   in
    coordinating the transactions.            The government stretches too far
    in arguing that the payments through Singer necessarily show that
    any parent who hired Singer intended to allow Singer to commingle
    funds for the purpose of making it more difficult for prosecutors
    to show Singer was engaged in a criminal conspiracy with other
    parents by means of transmission of funds to the universities.
    - 99 -
    We conclude that the nature of the alleged scheme is not
    such    that,    because    the   evidence     suffices   to   show   that   the
    defendants sought Singer's services in connection with their own
    children, the evidence also suffices to show that the defendants
    were interdependent with other parents for whom Singer coordinated
    side-door deals.
    b.
    As with the "common goal" factor, the government also
    contends     that   there   is    more   defendant-specific     evidence     that
    suffices to show that the interdependence factor supports the
    conclusion that each of the defendants before us agreed to join
    the broader charged conspiracy,               even if the evidence of the
    scheme's nature in and of itself does not.            We conclude, however,
    that this evidence, when considered in the context of the record
    as a whole, does not so suffice without the sort of inference-
    stacking that "would require impermissible speculation on the
    jury's part."       Glenn, 
    828 F.2d at 860
    .
    The parties do not dispute that both Abdelaziz and Wilson
    were at least aware that Singer conducted side doors for other
    parents.30      However, as the defendants point out, mere awareness
    30 Abdelaziz argues that he "did not know that the 'side
    door' was broader than USC," but the government presented evidence
    that "a stock part of Singer's pitch" was the "wide[] variety of
    school options" that he was able to offer parents, which could
    allow a jury to conclude that Abdelaziz learned the same from
    Singer.
    - 100 -
    of a common figure's involvement in similar dealings with similarly
    situated    people     is     far    from   enough    to   show   interdependence.
    Indeed, in Glenn we concluded that even though Glenn had "attended
    one meeting where the core conspirators discussed the smuggling of
    both marijuana and hashish," this showed "at most that Glenn knew
    about   the       marijuana    venture,"      not    "that   [he]      thought   [it]
    interdependent" with the hashish venture in which he was involved.
    
    Id. at 859
    .
    As evidence that Abdelaziz thought his work with Singer
    interdependent with other parents', the government relies on a
    phone call between Abdelaziz and Singer in October 2018 -- after
    Abdelaziz's daughter had been admitted to and enrolled at USC.
    During the call, which Singer initiated at the government's request
    after   agreeing       to    cooperate      with    investigators,      Singer    told
    Abdelaziz that a USC athletics administrator had "loved" the
    basketball profile they had used to facilitate his daughter's
    admission and wanted Singer to use it for other applicants who
    "[are]n't     .    .   .    real     basketball     player[s]"    in    the   future.
    Abdelaziz responded: "I love it."
    That response by a father about his daughter, ambiguous
    as it is, does not bear the weight the government posits.                     Further,
    Abdelaziz's daughter had already been admitted to USC well before
    Singer made that call.              It is not an admission by Abdelaziz that
    he joined an interdependent conspiracy with other parents in
    - 101 -
    advancement of his own interests.               Construing this conversation as
    evidence that Abdelaziz considered himself interdependent with
    other parents would again require impermissible inference-stacking
    on the jury's part.
    As evidence of interdependence related to Wilson, the
    government relies on evidence that Wilson as well as other parents
    "referred and recruited other parents."                  And the government adds
    that Wilson had a motive to make such referrals because he had
    other teenage children for whom he might seek Singer's services in
    the future, and thus he was likely to be a repeat player.
    The government points to four such referrals in its
    brief, only one of which involved Wilson as the speaker: First, a
    2013    email       from    Wilson's     wife    to    Singer    asking   about    the
    possibility of two of Wilson's son's friends' taking part in a
    "UCLA       workshop/internship"         and    "college       counseling"   run     by
    Singer.31      Second, a 2017 email, also from Wilson's wife, to Marci
    Palatella      --    a     codefendant    in    this   case,    including    in    both
    conspiracy counts -- which states: "I had a few thoughts about
    [Singer] & USC -- easier to talk on the phone."                    The third is not
    from Wilson or his wife, but from Palatella -- a 2018 phone call
    31 One of the friends was later admitted to USC with
    Singer's assistance; his family made a $100,000 contribution for
    "USC baseball" through Singer's foundation. The friend's parents
    do not appear to have been charged in this case, and the
    government's brief does not describe them as codefendants.
    - 102 -
    between Palatella and Singer in which Palatella mentioned telling
    a neighbor about Singer's services and assured Singer that she
    "d[id]n't say much to anybody unless [she] th[ought] they'd be a
    good candidate" and that she would "hand [him] the right people."
    And fourth, a 2018 text exchange and phone call between Singer and
    Wilson    in    which   Wilson   stated    that    he   had   a    "good    --   very
    wealthy -- friend [with] a daughter applying to Brown."                       Wilson
    described the friend as "willin[g] to pay a million, 2 million" to
    secure his daughter's admission, and said he would "connect" Singer
    and the friend by email.32
    Because Wilson's own "state of mind . . . is key" to the
    interdependence inquiry, the actions of his alleged coconspirators
    are of limited relevance. Dellosantos, 
    649 F.3d at 117
    ; see, e.g.,
    Glenn, 
    828 F.2d at 859
     (distinguishing between views of defendant
    and alleged coconspirators).              Further, the referral by Wilson
    involving Brown University does not fall squarely within the
    conspiracy alleged by the indictment, which did not include Brown
    among the set of universities allegedly targeted by the parents.
    Regardless, this one referral involving Wilson does not
    support    an     inference   that   Wilson       viewed   his     activities     as
    interdependent with Singer's work with other clients.                       Wilson's
    (or another parent's) referring a friend to Singer does not
    32   Wilson's friend was not indicted                    in   the     operative
    indictment for his interactions with Singer.
    - 103 -
    necessarily provide an "indication that [Wilson] thought that [his
    work with Singer to obtain admission for his own children] was
    'necessary or advantageous to the success'" of Singer's work with
    other parents, or vice versa.         Monserrate-Valentín, 
    729 F.3d at 44
    (quoting Dellosantos, 
    649 F.3d at 117
    ).
    A jury could plainly infer that Wilson wanted Singer to
    work with the friend and that he hoped that that work would be
    successful.     But to conclude based on this evidence that Wilson
    thought that Singer's work with other parents would be beneficial
    to his own success, even if he was likely to be a repeat player,
    would again require stacking inference upon inference.33             For, as
    the defendants point out, Kotteakos itself contained evidence that
    both the defendant and other alleged coconspirators, many of whom
    were repeat players, had referred others to the "hub" figure, but
    held that such evidence did not suffice to show that either the
    defendant or the other alleged coconspirators who made referrals
    had   thereby   joined   with   the    "hub"   figure   in   an   overarching
    conspiracy.     See 
    328 U.S. at 754
     ("Kotteakos . . . sent Brown [(the
    hub figure)] applications on behalf of other persons.").
    33  At oral argument, the government contended that the
    record shows more than "bare referrals" and that parents
    effectively vetted potential new participants in the scheme to
    ensure they would benefit Singer's network. The only evidence to
    this effect is Palatella's statement to Singer, not made by Wilson
    or even his wife.
    - 104 -
    We conclude that there was insufficient evidence from
    which a rational jury could find interdependence with respect to
    these two defendants.            This factor, too,          points against the
    conclusion     that    the   defendants     agreed     to    join    the     broader
    conspiracy that was charged rather than merely the narrower ones
    to ensure admission for their own children.
    4.
    The defendants do not contest that the final factor,
    overlap among the participants, is satisfied by Singer's and his
    associates' interactions with the parents.              See Dellosantos, 
    649 F.3d at 118
    . Rather, they assert, and we agree, that the "evidence
    of   overlap   .   .   .   was   insufficient     to   outweigh      the    lack   of
    interdependence" and common goal.           
    Id. at 120
    .
    In an abundance of caution before we turn to the issue
    of prejudice, we discuss the totality of the evidence.                 See United
    States v. Canty, 
    37 F.4th 775
    , 796 (1st Cir. 2022) (considering
    totality     of    circumstances      in    addition        to      common     goal,
    interdependence, and overlap).          The government does not cite any
    additional     considerations      beyond   the    three     factors       discussed
    above; the defendants offer one.
    As the defendants point out, the dissimilarity in the
    conduct of the parents alleged to have conspired with Singer
    undercuts the reasonableness of finding a single conspiracy.                       See
    Franco-Santiago, 
    681 F.3d at 10
    .            Franco-Santiago held that the
    - 105 -
    evidence   was   insufficient    to     show   that   a   defendant   who   had
    conspired to commit one robbery had agreed to join a broader
    conspiracy to commit a series of robberies in part because "the
    robbery in which [the defendant] did participate [was] notably
    different from the other robberies encompassed by the . . .
    overarching conspiracy."       Id.; see 
    id. at 11-12
    .         In particular,
    the robbery in which the defendant had participated "was one of
    cash from a person, whereas the other four robberies were all
    robberies of places of business."          
    Id. at 10
    .      In this case, the
    evidence showed significant differences between the conduct of the
    defendants   and   that   of    their    alleged      coconspirators.       The
    government introduced evidence that other parents who purportedly
    participated in the alleged overarching conspiracy knowingly made
    payments to university insiders' personal accounts and paid to
    alter standardized test scores or have third parties take online
    classes for their children.       The evidence does not show, and the
    government does not argue, that Abdelaziz or Wilson engaged in
    those practices.
    Evaluating the record as a whole, we conclude that there
    was insufficient evidence from which a rational jury could find
    beyond a reasonable doubt that the defendants joined the broader
    conspiracy charged in the indictment.
    - 106 -
    B.
    Our conclusion that the proof varied from the indictment
    does not, on its own, "upset [the defendants'] conviction[s]."
    Glenn, 
    828 F.2d at 858
    .       After all, "[a]s long as administrative
    convenience leads the government to prosecute many, or all, members
    of a large criminal enterprise at a single trial, variances between
    the scope of the conspiracy charged and that proved may, at least
    as to some defendants, be fairly common," 
    id.,
     and "a defendant
    'can hardly . . . complain when the government's proof at trial
    establishes a scheme similar to but somewhat narrower in breadth
    and   malignity     than   that     charged     in   the     indictment,'"
    Monserrate-Valentín, 
    729 F.3d at 49
     (internal quotation marks
    omitted) (quoting United States v. Mubayyid, 
    658 F.3d 35
    , 48-49
    (1st Cir. 2011)).
    To succeed on appeal, then, the defendants must also
    show that the variance "prejudiced [them -- that] it 'affect[ed]
    [their] substantial rights.'"        Glenn, 
    828 F.2d at 858
     (second
    alteration   in   original)    (internal      quotation    marks   omitted)
    (quoting Berger v. United States, 
    295 U.S. 78
    , 82 (1935)); accord,
    e.g., Dellosantos, 
    649 F.3d at 124
    .         This circuit has "recognized
    at least three" possible forms of prejudice in this context,
    Dellosantos, 
    649 F.3d at
    124:
    First, a defendant may receive inadequate
    notice of the charge against him and thus be
    taken by surprise at trial.       Second, a
    - 107 -
    defendant may be twice subject to prosecution
    for the same offense. Third, a defendant may
    be prejudiced by "evidentiary spillover": the
    "transference of guilt" to a defendant
    involved in one conspiracy from evidence
    incriminating     defendants    in    another
    conspiracy in which the particular defendant
    was not involved.
    
    Id. at 125
     (quoting Wihbey, 
    75 F.3d at 774
     (citations omitted)).
    Abdelaziz and Wilson focus exclusively on the third form
    of prejudice -- evidentiary spillover.                   They argue that because
    the   government    charged,      but    failed     to    prove,   an   overarching
    conspiracy,    it   "was    able    to     admit    mountains      of   inflammatory
    evidence    about   markedly      different        conduct   by    other   parents,"
    including    evidence      that    other    parents      were   aware    that   their
    payments would go to university officials personally, that other
    parents paid to alter standardized test answers and scores, and
    that other parents paid Singer's staff to take courses for their
    children.     This evidence involved parental knowledge and conduct
    markedly different in kind from Abdelaziz's and Wilson's alleged
    activities and focused on parents with whom the defendants did not
    interact.     Meanwhile, the government chose not to put Singer on
    the stand, where he would have been subject to cross-examination,
    despite the facts that Singer was the person with whom these two
    defendants did interact and that he had cooperated with the
    prosecution.     The defendants argue that all of this "created an
    - 108 -
    impermissibly high risk that the jury . . . could not fairly
    evaluate [the defendants'] own knowledge or intent."
    In   response,   the   government   emphasizes    that   "[t]o
    prevail on a claim of prejudicial spillover, a defendant must prove
    prejudice so pervasive that a miscarriage of justice looms."
    Wihbey, 
    75 F.3d at 776
     (internal quotation marks omitted) (quoting
    United States v. Levy-Cordero, 
    67 F.3d 1002
    , 1008 (1st Cir. 1995)).
    The government contends that the defendants cannot meet that
    standard because, it asserts, (1) some evidence related to the
    other parents may have been admissible even to prove the narrower
    conspiracies   involving   the    defendants,   (2)   the   government
    compartmentalized its presentation of the evidence in a way that
    would prevent evidentiary spillover, (3) the trial court issued
    appropriate limiting instructions to the jury, and (4) the proof
    of the defendants' participation in those narrower conspiracies
    was overwhelming.
    The record does not support the government's assertions.
    Rather, applying de novo review, Dellosantos, 
    649 F.3d at 124
    , we
    agree with the defendants: "The dangers for transference of guilt
    [in this case were] . . . so great that no one really can say
    prejudice to substantial right has not taken place," Kotteakos,
    
    328 U.S. at 774
    .
    - 109 -
    1.
    The   government   does    not        dispute   that   to   convict
    Abdelaziz and Wilson of conspiring to violate either § 666 or the
    mail and wire fraud statutes, it had to show that the defendants
    possessed the requisite mental state to commit the underlying
    offense.   Violation of § 666 requires acting "corruptly . . . with
    intent to influence or reward an agent of an organization."                 
    18 U.S.C. § 666
    (a)(2).   Violation of the mail and wire fraud statutes
    requires acting "with the specific intent to defraud."                  United
    States v. Martínez, 
    994 F.3d 1
    , 7 (1st Cir. 2021) (quoting United
    States v. Woodward, 
    149 F.3d 46
    , 54 (1st Cir. 1998)).                     Both
    conspiracy convictions thus required the government to prove that
    Abdelaziz and Wilson acted with some culpable state of mind.               And
    at trial both defendants argued that they lacked this mental state
    and had instead acted in good faith, believing Singer's side door
    to be a path to admission of which the universities at least
    tacitly approved.
    Notably,   the   government      at    trial    acknowledged   that
    Singer provided some "totally legitimate" services, including
    "assistance with college applications," and did not contend that,
    even when he was committing fraud with some parents, he was
    committing fraud with all parents who engaged him.                Indeed, the
    evidence at trial showed that Abdelaziz paid Singer for work with
    his two older children in 2012 and 2013 -- years before Abdelaziz
    - 110 -
    allegedly began conspiring with Singer to facilitate his youngest
    daughter's admission, sometime in 2017.      The government has not
    argued or cited any evidence that those earlier transactions were
    in any way improper.
    Given this context, "there was a pervasive risk" that
    "the jury might have unfairly transferred to [Abdelaziz and Wilson]
    the guilt relating to" other parents who worked with Singer.
    Dellosantos, 
    649 F.3d at 125
    .    Based on the overarching conspiracy
    charge, the government introduced powerful evidence of culpable
    intent on the part of other parents that presented a pervasive
    risk of prejudicing the jury's assessment of each defendant's own
    intent.
    For   example,   in    contrast   with   the   government's
    acknowledgment that Singer told Abdelaziz and Wilson that their
    payments would go to the universities, the government's first
    witness, Isackson, testified that he "knew a good portion of [the
    money he paid Singer] was going into [Singer's] pockets and [to]
    the people who helped him."      He further described his concerns
    about shielding the transactions from IRS scrutiny, again evincing
    a consciousness of guilt.       In addition, a government auditor
    testified about payments from other parents through Singer to the
    personal accounts of university insiders at Georgetown, Yale, and
    UCLA, while a USC soccer coach testified to accepting bribes from
    Singer for facilitating the admission of other parents' children.
    - 111 -
    Further, through recordings of calls between Singer and
    other parents, as well as the testimony of Isackson and a Singer
    associate, the government introduced evidence of other parents'
    paying Singer to facilitate clearly fraudulent conduct that was
    both plainly wrongful and dissimilar in kind from Abdelaziz's and
    Wilson's actions.        Isackson testified that he "paid to have one of
    [his] daughter's test scores altered."                 In a recorded call, parent
    Gordon Caplan, an alleged unindicted coconspirator,34 discussed
    with    Singer    a    scheme    to     have   his    daughter   fake   a   learning
    disability in order to secure extra time on a standardized test
    and to bribe a proctor to correct her answers.                   A Singer employee
    testified    to       having    taken    online      courses   with   the   parents'
    knowledge for the children of Singer clients other than Abdelaziz
    and Wilson.       The government thus "subjected the [d]efendants to
    voluminous testimony relating to unconnected crimes in which they
    took no part."         
    Id.
    The government, citing case law holding that the risk of
    "prejudice [is] minimized [where] . . . transactions not directly
    involving [a defendant are] of the same character as the ones that
    did involve him," United States v. Levine, 
    569 F.2d 1175
    , 1177
    (1st Cir. 1978), argues that at least some other parents engaged
    34 Like Huneeus and Isackson, Caplan was named in the
    government's original criminal complaint alongside Abdelaziz and
    Wilson, but pleaded guilty pursuant to a criminal information
    before Abdelaziz and Wilson were indicted.
    - 112 -
    in the same kind of conduct as Abdelaziz and Wilson by "pay[ing]
    Singer to present [their] child[ren] as . . . Division I athletic
    recruit[s] based on falsified credentials."
    The defendants dispute the degree to which they were
    aware of any falsified credentials.               But, even accepting that some
    other    parents'    conduct     was        similar   in   some    respects   to   the
    defendants', much of the evidence introduced by means of charging
    the broader conspiracy nonetheless involved forms of conduct that
    were    different    in   kind     from      Abdelaziz's     and   Wilson's.       Cf.
    Dellosantos, 
    649 F.3d at 125
     (noting risk of evidentiary spillover
    where    government,       based       on     overarching     conspiracy      charge,
    presented evidence of marijuana transactions against defendants
    who had participated only in cocaine distribution). All this other
    evidence threatened to influence the jury on the core issue of the
    defendants' state of mind.
    Our   precedent     in    Martínez      supports     our    conclusion.
    There, a defendant and former public official, López, was charged
    with    and   convicted    of    receiving        bribes    from    a    codefendant,
    Hernández, in violation of § 666 and the mail and wire fraud
    statutes, and argued that her being tried jointly with Hernández
    created an unacceptable risk of evidentiary spillover.                        See 994
    F.3d at 4-5, 11.          López's "primary defense . . . was that she
    merely accepted gifts from [Hernández] without any sort of quid
    pro quo."      Id. at 15.        Yet, because she was tried jointly with
    - 113 -
    Hernández, who was also charged with a series of unrelated bribes
    involving other public officials, "[t]he jury before which López
    was tried was exposed to days of detailed evidence regarding
    Hernández's    role    in"     those    other    bribery       schemes,     including
    "direct evidence of the corrupt intentions of those alleged to
    have been involved."           Id. at 14.            This court vacated López's
    conviction because "the evidence about how Hernández corruptly
    schemed with others . . . to which her jury . . . was exposed . . .
    create[d] a grave risk of spillover prejudice."                    Id. at 15; see
    id. at 15-16.        In particular, "that evidence risked leading the
    jury in considering her charges to impute the states of mind of
    [the individuals who had conspired with Hernández in the separate
    bribery schemes] . . . to López."             Id. at 15.
    The   same      reasoning     applies       here.     The   overarching
    conspiracy charge enabled the government to introduce evidence of
    other parents' corrupt intent and actions in working with Singer,
    which we have described.            Just as Martínez recognized a "grave
    risk"   that   the    jury    imputed    to   López      the   state   of    mind   of
    Hernández's other collaborators, id., we see an unacceptable risk
    that the jury in this case may have imputed other parents' culpable
    mental states to the defendants.
    Finally, in addition to the specific evidence of other
    parents' intent and dissimilar conduct, the sheer number of alleged
    coconspirators       and     the   breadth      of    the   alleged    overarching
    - 114 -
    conspiracy      further    substantiates        that    there       was   prejudicial
    evidentiary spillover in this case.              Kotteakos explained that the
    risk of prejudice increases with the number of defendants, the
    number    of    conspiracies      proven,       and    the    number      of   alleged
    coconspirators.        See 
    328 U.S. at 766-67
    ; see also United States v.
    Kemp, 
    500 F.3d 257
    , 292 (3d Cir. 2007).                 Although only Abdelaziz
    and Wilson went to trial, the operative indictment charged fifteen
    parents   from    twelve       families    in    addition      to    alleged     named
    coconspirators like Caplan, Huneeus, and Isackson who were not
    charged    in    the    same    indictment       but    who     were      treated   as
    coconspirators for evidentiary purposes at trial.                     Cf. Kotteakos,
    
    328 U.S. at 766
     (noting that "only one conspiracy was charged, but
    eight separate ones were proved, involving at the outset thirty-
    two defendants"); Dellosantos, 
    649 F.3d at 110
    , 111 n.2, 125
    (finding spillover prejudice where government charged eighteen
    individuals, of whom three went to trial, and evidence proved two
    distinct conspiracies).
    For those reasons, we agree with the defendants that
    there was a significant risk that the evidentiary spillover in
    this case prejudiced them and affected their substantial rights.
    See Monserrate-Valentín, 
    729 F.3d at 49-50
    .
    2.
    The    government's       responses        are    unpersuasive.          The
    government first contends that at least some of the evidence
    - 115 -
    related to other parents may have been admissible against Abdelaziz
    or Wilson as proof of their participation in narrower conspiracies.
    But "[w]e . . . cannot see how evidence of such depth and quality
    about the nature of the allegedly corrupt scheme[s]" in which
    Singer engaged with other parents "could have been admitted at a
    trial against" Abdelaziz or Wilson on narrower conspiracy charges,
    and "the admission of that evidence in a trial of [Abdelaziz or
    Wilson]    still       would      have    been    limited   by    Federal    Rule    of
    Evidence . . . 403."35            Martínez, 994 F.3d at 14.
    Next,   the     government        argues   that   the   record   shows
    distinct treatment of evidence related to other parents sufficient
    to mitigate the risk of prejudice resulting from evidentiary
    spillover.       We disagree.        The prosecution did, at various times
    during its opening statement and closing argument, remind the jury
    that "[t]his trial is about [Abdelaziz and Wilson], what they knew,
    what they intended and what they agreed to do," and encourage
    jurors    to    "[l]ook      at    what   the     defendants     did   and   what   the
    defendants said."         But at other times it invited the jury to use
    evidence related to other parents to prove the guilt of Abdelaziz
    and Wilson; for example, it chose to call as its first witness
    35   Rule 403 provides that "[t]he court may exclude relevant
    evidence if its probative value is substantially outweighed by a
    danger of one or more of the following: unfair prejudice, confusing
    the issues, misleading the jury, undue delay, wasting time, or
    needlessly presenting cumulative evidence."
    - 116 -
    Isackson -- a parent who confirmed that he had never met Abdelaziz
    and that, while he had met Wilson at school or charitable events,
    he had never had any "substantive conversation[s]" with him, and
    whose    own    scheme    was    obviously         wrongful.       It   then   built   on
    Isackson's testimony and used it during its closing argument.                          Its
    description during closing argument of "the evidence of how the
    scheme worked" drew almost exclusively on Isackson's testimony and
    recordings of calls between Singer and other parents.                       Also in its
    closing, the government told jurors: "You know . . . that these
    defendants      joined     in    .     .    .   that     conspiracy      knowingly     and
    intentionally[,] . . . that the defendants knew what they were
    doing and . . . intended to do it, and that they knew that what
    they were doing [was] wrong . . . because Bruce Isackson told you
    that he knew it . . . ."               (Emphasis added.)          These statements by
    the     government     itself        seriously        undermine    the     government's
    contention      that     the    jury       would   not   have   considered     evidence
    regarding other parents in determining what the defendants had
    done.
    The government also argues that the district court's
    instructions to the jury were sufficient to prevent the risk of
    spillover.       The district court instructed the jury that it must
    base its verdict as to each defendant "upon evidence of his own
    words and actions" and "assess the evidence against each defendant
    individually."         However, these limiting instructions "did not
    - 117 -
    suffice to mitigate th[e] risk of spillover prejudice here."             Id.
    at 15.   Martínez explained that a similar limiting instruction was
    inadequate to address the risk of evidentiary spillover where
    trying a defendant jointly with her codefendants had "enabled the
    government to put forth direct evidence of the corrupt intent of
    [an alleged coconspirator's] collaborators in a distinct scheme,
    even though the government had only circumstantial evidence as to
    [the defendant's] state of mind and the trial . . . implicated a
    number of players and . . . complicated charges."            Id.   The same
    concerns apply here.        "[W]e do not readily assume that a jury
    disregards clear directions," Wihbey, 
    75 F.3d at 775
    , but, given
    the pervasive risk of evidentiary spillover in this case, the
    limiting instructions cannot save the convictions.
    The government lastly asserts that the convictions can
    stand because there was "overwhelming evidence that [Abdelaziz and
    Wilson] participated in smaller conspiracies that advanced their
    own side-door deals."      See, e.g., United States v. Morrow, 
    39 F.3d 1228
    ,    1235-36   (1st    Cir.   1994)   (finding     no   prejudice   from
    evidentiary spillover where "[t]he admissible evidence against
    each appellant amply proved his complicity in [a] narrow[er]
    conspiracy").      We     think   the   government's    evidence   of   each
    defendant's smaller conspiracy with Singer (which the defendants
    admit was sufficient) was not so strong as to be overwhelming in
    relation to the significant risk of prejudice posed by the evidence
    - 118 -
    regarding other parents.          Cf. Martínez, 994 F.3d at 11, 15-17
    (vacating conviction due to spillover prejudice where there was
    sufficient evidence to support conviction but government's case
    depended      on   circumstantial       evidence    from    which      competing
    inferences were possible).
    We reject the government's leading argument relying on
    the transcripts of calls between Singer and Wilson in which Wilson
    verified that his daughters did not actually need to play the
    sports for which they would purportedly be recruited; agreed with
    Singer's statement that the girls were "athletic enough" for Singer
    to   "sell"    them     without   raising   any    "question"        and   without
    "Stanford['s] . . . catch[ing] on"; and suggested that they could
    be   nonplayers       such   as    "scorekeeper[s],"       "water      girl[s],"
    "manager[s],"      or    "mascot[s]."       The    reference    to    Stanford's
    "catch[ing] on" was made by Singer, not Wilson, and record evidence
    supports the notion that Singer had for years prior to this
    conversation represented to Wilson that               the side door was a
    longstanding path to admission, the proceeds of which went to the
    university itself.       This statement, coming as late as it did, must
    be weighed against the impression Wilson had gained from years of
    conversations with Singer and from Wilson's experience arranging
    a side door for his son years earlier.            The government's view also
    requires a further inference by the jury that Wilson did not
    believe that the university and upper-level administrators tacitly
    - 119 -
    approved of side-door admission even if that policy was not known
    to everyone in the university, and that he did not think Singer
    was    referring    to   such    a   dynamic   when    he   spoke       about
    "Stanford['s] . . . catch[ing] on."
    The references to other nonplayer roles for Wilson's
    daughters (for instance, as managers) at least equally support the
    defendants' argument that they understood side-door admission
    through athletics was a practice the universities had approved, at
    least tacitly.     Indeed, testimonials on Singer's website described
    his assisting other clients in obtaining both college admission
    and positions as "managers" for college sports teams.             Thus, to
    construe this evidence as showing Wilson's guilty intent, the jury
    would have to make at least one inference, even if a reasonable
    one:   that   Wilson     made   these   statements    knowing    that     the
    universities in question did not condone the practice, such that,
    even if he believed his payments to be part of a quid pro quo, he
    did not in good faith believe that such a quid pro quo was welcomed
    by the universities and so did not amount to bribery.           And such an
    inference regarding Wilson's good or bad faith intent based on his
    understanding of Singer's scheme is precisely the sort that stands
    to be prejudiced by the evidence of the bad faith intent of other
    parents when committing similar conduct, as well as evidence of
    dissimilar conduct that could not plausibly be accompanied by good
    faith intent.
    - 120 -
    As to Abdelaziz, the government points chiefly to a phone
    call that took place between him and Singer after his daughter had
    already matriculated at USC, in which Singer told him that USC's
    Admissions Department was investigating why his daughter "did not
    show up for Women's Basketball in the fall," and that the USC
    administrator with whom Singer had negotiated the side door had
    told the Admissions Department that it was because Abdelaziz's
    daughter had suffered an injury.           Singer then told Abdelaziz: "And
    I doubt that Admissions will call you regarding [your daughter],
    you know, getting in through the side door and . . . not showing
    up for practice. . . . But they may ask you . . . So I just wanted
    you to know in case they call . . . ."               Abdelaziz responded by
    asking Singer whether the Admissions Department would ask his
    daughter and whether he should "prepare her."              Singer said they
    would    not    ask   her,   but   might   call   Abdelaziz,   and   Abdelaziz
    responded: "That's fine.           I will answer the same [regarding the
    purported injury], uh, should they call me."
    The government argues that this phone call is strong
    evidence that Abdelaziz understood that the side door was "at odds
    with the Admissions Department's expectations."                But, while it
    certainly is relevant evidence on that score, the jury -- as with
    Wilson's phone call -- would have to make multiple inferences to
    arrive at such a conclusion, such as that this later conversation
    (which     occurred      after     Abdelaziz's     daughter    had     already
    - 121 -
    matriculated at USC) was also reflective of his earlier intent,
    and that he did not believe that other actors and administrators
    at USC, including potentially higher-up administrators, tacitly
    approved of and welcomed these side doors even if other USC
    administrators were not aware of that.
    We again point out that the more sweeping the charged
    conspiracy, the higher the bar for showing that the evidence was
    "overwhelming," and consequently for showing that an error was
    harmless.    As Kotteakos itself explained, "it is one thing to hold
    harmless the admission of evidence [in a case] where only two
    conspiracies involving four persons all told were proved, and an
    entirely different thing to apply the same rule where, as here,
    only one conspiracy was charged, but eight separate ones were
    proved."    
    328 U.S. at 766
    ; see also Kemp, 
    500 F.3d at 292
    .   Indeed,
    the Court in Kotteakos itself commented that each defendant "was
    clearly shown to have shared in the fraudulent phase of the
    conspiracy in which he participated," but nonetheless reversed the
    lower court's finding that the error was harmless because it "d[id]
    not understand how it can be concluded, in the face of the
    instruction, that the jury considered and was influenced by nothing
    else."   
    328 U.S. at 771
    .   And since that was the case in Kotteakos,
    where the Court found that the indictment alleged at least eight
    separate conspiracies, we do not see how that would not be the
    case here, where the indictment alleges at least fifteen.
    - 122 -
    For these reasons, we agree with the defendants that the
    risk of evidentiary spillover in this case rendered prejudicial
    the   variance    between    the   broader    conspiracy    charged     in   the
    indictment and the narrower ones shown at trial.                 We therefore
    vacate the conspiracy convictions (Counts One and Two of the
    operative indictment).36
    C.
    Because we have already vacated Wilson's conspiracy and
    substantive wire fraud convictions and address his tax conviction
    in the next section, we focus here on whether our holding vacating
    the conspiracy convictions due to a prejudicial variance requires
    us also to vacate his substantive convictions for federal programs
    bribery under § 666.        Wilson's brief repeatedly asserts that the
    prejudicial      variance   requires    a    new   trial   on   "all   counts."
    Although the government's brief explicitly acknowledges Wilson's
    contention that the prejudicial variance requires us to vacate his
    36  We do not reach Abdelaziz's argument that the variance
    was prejudicial because there was not venue in the District of
    Massachusetts for the smaller conspiracy. Abdelaziz has not argued
    that we must conduct a venue analysis even where we conclude that
    a variance is prejudicial on alternative grounds. Nor did he seek
    distinct relief based on his venue argument in his opening brief,
    which sought only vacatur of his conviction.       To the extent
    Abdelaziz claims an entitlement to different relief in his reply
    brief, that argument has been waived.     See, e.g., Green Earth
    Energy Photovoltaic Corp. v. KeyBank Nat'l Ass'n, 
    51 F.4th 383
    ,
    391 n.15 (1st Cir. 2022).
    We do not address any Double Jeopardy Clause issues as
    they are not before us, and we express no opinion on the matter.
    - 123 -
    substantive     convictions,      it     does    not    develop    any    distinct
    counterargument against this claim, instead relying on its general
    argument    that   there   was    insufficient         spillover   prejudice     to
    warrant vacating any of the defendants' convictions.37
    Wilson's argument for vacating the § 666 convictions
    effectively amounts to a retroactive-misjoinder claim.                   See, e.g.,
    Mubayyid, 
    658 F.3d at 72-73
    , 72 n.39; United States v. Hamilton,
    
    334 F.3d 170
    , 181-82 (2d Cir. 2003).                   "Retroactive misjoinder
    occurs where joinder was proper initially because of a conspiracy
    allegation, but where later developments, such as [a] . . . court's
    decision . . . to set aside a defendant's conspiracy conviction,
    appear to render the initial joinder improper."                    Mubayyid, 
    658 F.3d at
    72 n.39 (quoting United States v. Deitz, 
    577 F.3d 672
    , 693
    (6th    Cir.   2009)   (alterations        and    internal      quotation    marks
    omitted)); accord Hamilton, 
    334 F.3d at 181
    .               To win "a new trial
    on the ground of retroactive misjoinder, a defendant 'must show
    compelling     prejudice,'"      which    "may   be     found   where    there   is
    '[p]rejudicial spillover from evidence used to obtain a conviction
    37 In particular, the government has not argued that Wilson
    failed to preserve his claim that the prejudicial variance requires
    vacating his substantive convictions, and so we need not decide
    whether plain error review would apply, had the government argued
    for it. See United States v. Encarnación-Ruiz, 
    787 F.3d 581
    , 586
    (1st Cir. 2015) ("When the government fails to request plain error
    review, we, and many of our sister circuits, review the claim under
    the standard of review that is applied when the issue is properly
    preserved below.").
    - 124 -
    subsequently reversed on appeal.'"        Hamilton, 
    334 F.3d at 181-82
    (alteration   in    original)   (first      quoting   United    States   v.
    Vebeliunas, 
    76 F.3d 1283
    , 1293 (2d Cir. 1996); and then quoting
    United States v. Jones, 
    16 F.3d 487
    , 493 (2d Cir. 1994)).              As in
    the variance context, succeeding on a claim of spillover prejudice
    requires Wilson "to show prejudice so pervasive that a miscarriage
    of justice looms."    United States v. Correia, 
    55 F.4th 12
    , 36-37
    (1st Cir. 2022) (internal quotation marks omitted) (quoting United
    States v. Simon, 
    12 F.4th 1
    , 43-44 (1st Cir. 2021)).
    For all the reasons just discussed in the variance
    analysis, we conclude that this case meets that standard.          As with
    the conspiracy charges, Wilson's intent in his dealings with Singer
    was a key issue with respect to the substantive § 666 counts.            See
    
    18 U.S.C. § 666
    (a)(2)    (covering     only    parties   that     act
    "corruptly . . . with intent to influence or reward an agent of an
    organization").    And the government offers no reason -- and we can
    think of none -- why the risk of spillover from evidence of other
    parents' dealings with Singer would be less acute with respect to
    Wilson's substantive § 666 counts than with respect to the related
    conspiracy charges.
    Nor is this "a case in which the results of the trial
    might be thought to undermine any claim of prejudice."           Martínez,
    994 F.3d at 16.    We have observed that "a discriminating verdict,"
    in which the jury convicts on some charges but not others, "is an
    - 125 -
    indication that spillover prejudice did not infect the jury's
    decisional calculus."     Correia, 55 F.4th at 38; see id. at 38-39.
    The jury in this case returned guilty verdicts on all counts,
    offering    no     reassurance       that   jurors    were    "[]able    to
    compartmentalize the evidence of each offense." Id. at 39 (quoting
    Mubayyid, 
    658 F.3d at 74
    ).
    Case law from other circuits supports our conclusion.
    See, e.g., United States v. Tellier, 
    83 F.3d 578
    , 581-82 (2d Cir.
    1996) (finding retroactive misjoinder where a RICO count on which
    the court found there had been insufficient evidence to convict
    had allowed the government to introduce "enormous amount[s] of
    prejudicial spillover evidence" related to "criminal activities in
    which [the] defendant did not participate"); Jones, 
    16 F.3d at 492-93
     (applying retroactive misjoinder where count on which court
    had   vacated    conviction    had    allowed   government   to   introduce
    inflammatory evidence of defendant's criminal history); United
    States v. Aldrich, 
    169 F.3d 526
    , 528-29 (8th Cir. 1999) (similar).
    The Second Circuit, for example, has developed a "three-
    part test" for assessing retroactive-misjoinder claims based on
    prejudicial spillover.        Hamilton, 
    334 F.3d at 182
    .     It examines:
    (1) whether the evidence introduced in support
    of the vacated count "was of such an
    inflammatory nature that it would have tended
    to incite or arouse the jury into convicting
    the defendant on the remaining counts,"
    (2) whether the dismissed count and the
    remaining counts were similar, and (3) whether
    - 126 -
    the government's evidence on the remaining
    counts was weak or strong.
    
    Id.
     (quoting Vebeliunas, 
    76 F.3d at 1294
     (internal quotation marks
    omitted)).     This test is satisfied here.     As to the first prong,
    the powerful evidence of other parents' obviously culpable conduct
    was potentially inflammatory -- far more so than the evidence of
    Wilson's own acts.      As to the second prong, the Second Circuit has
    explained that "prejudicial spillover is unlikely if the dismissed
    count and the remaining counts were either quite similar," such
    that the evidence relevant to the invalidated count would be
    independently admissible in connection with the other charges, "or
    quite dissimilar," such that a jury could readily compartmentalize
    the evidence.     Id. at 182; see id. at 182-83.           For the reasons
    described in the variance analysis, this case falls in the middle
    ground between these extremes,        where a retroactive-misjoinder
    finding is appropriate. At least the vast majority of the evidence
    of   other   parents'    activities   would   not   have    been   properly
    admissible in a prosecution of Wilson alone.        But the evidence was
    not so disconnected from Wilson's activities -- bearing, as it
    did, on other parents' understanding of their work with Singer --
    as to limit the risk that the jury would impute those parents'
    mental states to Wilson.      Cf. Martínez, 994 F.3d at 14.        And, as
    to the third prong, we have already explained that the government's
    - 127 -
    evidence related to Wilson himself was insufficiently strong to
    counteract the pervasive risk of prejudice.
    We vacate Wilson's convictions on the substantive § 666
    charges (Counts Eleven and Twelve of the operative indictment).
    This result makes it unnecessary to address -- outside the context
    of Wilson's tax conviction -- various evidentiary arguments raised
    by the defendants as grounds for vacating their convictions.           Many
    of these rulings may well become moot on remand, and we express no
    view on the merits of the defendants' evidentiary arguments with
    respect to the conspiracy, mail and wire fraud, and § 666 charges.
    We note, however, that the government does not attempt to defend
    on appeal many of the bases on which the district court relied in
    excluding various exhibits.
    V.    Affirmance of Wilson's Conviction for Filing a
    False Tax Return Under 
    26 U.S.C. § 7206
    (1)
    Finally, we turn to Wilson's conviction for filing a
    false tax return under 
    26 U.S.C. § 7206
    (1), which makes it a felony
    to "[w]illfully make[] and subscribe[] any return, statement, or
    other   document,   which   contains   or   is   verified   by   a   written
    declaration that it is made under the penalties of perjury, and
    which [the individual] does not believe to be true and correct as
    to every material matter."      We first lay out the relevant facts,
    then analyze and reject Wilson's challenges to his conviction.
    - 128 -
    A.
    The   tax    count    arises    from   Wilson's    designation   of
    payments he made in 2014 to secure his son's admission to USC as
    business expenses and charitable contributions on his 2014 income
    taxes.
    At the time he worked with Singer to obtain his son's
    admission, Wilson was the sole shareholder of Hyannis Port Capital,
    an S corporation.         An S corporation is a "pass-through entity"
    which    does   not     separately   pay    federal   taxes;    instead,     the
    corporation's     "income,       losses,    deductions,   and    credits     are
    attributed to individual shareholders," Bufferd v. Comm'r, 
    506 U.S. 523
    , 524-25 (1993); accord Benenson v. Comm'r, 
    887 F.3d 511
    ,
    513 n.1 (1st Cir. 2018), such that Hyannis Port Capital's profits,
    losses, and deductions directly affected Wilson's personal tax
    liability.
    On March 1, 2014, the day after Subco considered and
    approved his son's admission, Wilson sent Singer an email with the
    subject line "USC fees" that read: "Thanks again for making this
    happen!    Pls give me the invoice.            What are the options for the
    payment?   Can we make it for consulting or whatever from the [K]ey
    [(Singer's business)] so that I can pay it from the corporate
    account ? [sic]"        Singer replied: "Yes we can send you an invoice
    for business consulting fees and you may write off as an expense."
    He also requested "the name address etc you want the invoice to be
    - 129 -
    made out to."      Wilson responded: "Awesome!"           He provided billing
    information for Hyannis Port Capital.
    On March 29, 2014, Wilson emailed Hyannis Port Capital's
    office manager, Debbie Rogers, that "Monday we will get an invoice
    and wiring instructions for $250k.           To be paid by hpc inc."          When
    Rogers asked what account the invoice should be charged to, Wilson
    replied:     "Business   Consulting      -    the     invoice   will     be       for
    consulting - pls work with [Singer] to get invoice correct."                      The
    following day, Wilson emailed Rogers again, stating that "the
    amount is $200,000 not [$]250,000."
    Two days later, Rogers told Wilson by email that she had
    consulted Singer and one of his business associates, and that they
    had   said   the   payment   structure   would       be   "[$]100k   .   .    .    to
    [Singer's] foundation, [$]100k an invoice from the Key [(Singer's
    business),] and [$]20k to Rick Singer."               Rogers noted that this
    totaled $220,000, rather than the $200,000 Wilson had previously
    mentioned, and Wilson explained that he had "added $20 k for
    [Singer's] expenses." He also confirmed that the entire sum should
    be invoiced to Hyannis Port Capital.
    Consistent with this plan, Hyannis Port Capital wired
    $100,000 to Singer's business, $100,000 to Singer's foundation,
    and $20,000 to Singer on April 7, 2014.             Nine days later, Singer's
    business issued a $100,000 check made out to "USC Men's Water Polo"
    on behalf of the "Wilson family."
    - 130 -
    Singer's foundation sent Wilson a letter dated April 7,
    2014, acknowledging his "contribution of $100,000" and stating
    that "no goods or services were exchanged" for the payment. Singer
    and his business also provided invoices, both dated April 1, to
    Hyannis Port Capital.        The business's invoice purported to be for
    "Business    Consulting,"        and     Singer's      referred   to     "Special
    Consulting     .    .   .   Concept,      design,      and   implementation    of
    Professional       Development    program        for   Hyannis    Port    Capital
    associates."       Wilson offered no evidence and does not argue on
    appeal that Singer or his business actually provided any consulting
    services to Hyannis Port Capital.            In July, USC also sent Wilson
    and his wife a thank-you letter acknowledging a $100,000 gift,
    apparently in connection with the $100,000 check from Singer's
    business.
    On his 2014 tax return, Wilson deducted the $120,000
    Hyannis Port Capital paid to Singer and Singer's business as
    business expenses and the $100,000 payment to Singer's foundation
    as a charitable contribution.38           Wilson does not dispute that the
    payment could not properly be deducted as a charitable contribution
    if he received goods or services in exchange for the payment.
    38   Wilson originally filed a 2014 return in December 2015.
    He later filed two amended returns, neither of which made any
    changes   to   his  claimed  business   expenses  or   charitable
    contributions.
    - 131 -
    An   IRS   agent   testified    that   these   deductions    saved
    Wilson $88,546 in taxes. On cross-examination by Wilson's counsel,
    the same agent testified that treating the $120,000 in payments to
    Singer and his business as business expenses, rather than deducting
    the   full   $220,000     as    charitable    contributions,   saved     Wilson
    roughly $1,425.         Wilson's tax preparer testified that the exact
    effect of deducting the payments as business expenses, rather than
    charitable contributions, would have been uncertain until all of
    Wilson's tax information was available at the end of the year.
    The indictment charged Wilson with filing a false tax
    return based on allegations that both the $100,000 charitable
    contribution deduction and the $120,000 business expense deduction
    were improper.     During closing argument, Wilson's counsel asserted
    with respect to the charitable deduction that Wilson was "not real
    attentive to his taxes" and may have made some unintentional
    errors, but "thought he made a donation" and "could get a deduction
    for it."     He further argued that, given that belief, Wilson would
    have had little motive to willfully fraudulently deduct the payment
    as a business expense, since the deduction made only a small
    difference in his tax liability compared to deducting the entire
    sum as a charitable contribution.
    The jury returned a general verdict of guilty on the tax
    count without specifying the theory on which it relied.             Wilson's
    - 132 -
    proposed verdict form had not requested a special verdict giving
    the basis for the jury's decision.
    B.
    Wilson challenges his tax conviction on three grounds.
    First, he contends that the conviction must be vacated pursuant to
    the Supreme Court's decision in Yates v. United States, 
    354 U.S. 298
     (1957), and its progeny because the verdict may rest on an
    invalid legal theory.   Second, he argues that spillover prejudice
    from the variance in the conspiracy counts also tainted this
    conviction.   Third, he asserts that the district court prejudiced
    his defense through erroneous evidentiary rulings.     We consider
    these arguments in turn.
    1.
    We first reject Wilson's argument that we must vacate
    his conviction under the Supreme Court's decision in Yates.
    The defendants in Yates were charged with a single count
    of conspiring (1) "to advocate and teach the duty and necessity of
    overthrowing the Government of the United States by force and
    violence" and (2) "to organize, as the Communist Party of the
    United States, a society of persons who so advocate and teach."
    
    Id. at 300
    .   The Court concluded that the conduct underlying the
    latter "organiz[ing]" object had occurred outside the relevant
    statute of limitations, such that the defendants could not lawfully
    have been convicted of conspiracy on that basis.       See 
    id.
     at
    - 133 -
    303-11.      The Court then rejected the government's argument that
    the convictions could nonetheless stand based on the alternative
    "advoca[cy]" object, holding that "a verdict [must] be set aside
    in cases where the verdict is supportable on one ground, but not
    on another, and it is impossible to tell which ground the jury
    selected."     
    Id. at 312
    ; see 
    id. at 311-12
    .           In later cases, the
    Court has reaffirmed this rule while clarifying that it applies
    where "a particular theory of conviction submitted to [the jury]
    is contrary to law," and not where one of several alternative bases
    for conviction is legally sound but supported by insufficient
    evidence.     Griffin v. United States, 
    502 U.S. 46
    , 59 (1991).
    Here, the indictment did offer two types of allegedly
    false statements on which the jury could have convicted: Wilson's
    deductions for purported charitable contributions and business
    expenses.      Wilson does not, however, develop an argument that
    either of those theories of conviction set forth in the indictment
    is legally unsound.       He does not disagree with the government's
    assertion that the jury could convict based on a "find[ing] that
    Wilson willfully made a false statement as to a material matter on
    his tax return by falsely claiming the payments were business
    expenses and/or that he received no goods or services in exchange
    for   them    [as   necessary   to   claim     a   payment   as   a   charitable
    contribution]."      On the contrary, he acknowledges that "perhaps a
    jury could [were it not for the alleged Yates and other trial
    - 134 -
    errors] convict on the [business expense deduction] theory," and
    appears to concede that "a quid pro quo . . . vitiates any
    charitable deduction even if the exchange was otherwise lawful."
    He also does not argue that the jury instructions on the tax count
    were improper.39
    Instead,      Wilson   posits        that    the     jury   might   have
    convicted   him    of    tax   fraud    based    on     its    conclusion   that   he
    committed bribery or property fraud, on a theory that unlawful or
    fraudulent payments are not deductible.                 And because, he argues,
    his bribery and property fraud convictions were based on legally
    flawed premises, the "[d]erivative" tax conviction cannot stand
    under Yates.
    We reject this argument because the record does not
    support Wilson's claim that the jury could have convicted him on
    the tax count based on its verdicts on the bribery or property
    fraud charges.40        He has not cited anything in the indictment or
    the jury instructions that would lead a juror to conclude that a
    39    The district court instructed the jury that this count
    required the government to prove that (1) "Wilson made or caused
    to be made a [2014] federal income tax return . . . that he verified
    to be true," (2) the "return was false as to a material matter,"
    (3) "Wilson signed the return willfully and knowingly -- knowing
    that it was false," and (4) the "return contained a written
    declaration that it was made under the penalty of perjury."
    40   We have also partially rejected Wilson's premise that
    all of the bribery and property fraud charges are legally
    unsupportable. See supra Sections II-III.
    - 135 -
    guilty verdict on other charges would necessarily affect the
    outcome   of    the   tax   count.41   See   Skilling,    
    561 U.S. at 414
    (characterizing Yates as holding that "constitutional error occurs
    when a jury is instructed on alternative theories of guilt and
    returns a general verdict that may rest on a legally invalid
    theory" (emphasis added)); Griffin, 
    502 U.S. at 59
     (explaining
    that the Yates rule reflects the fact that "[j]urors are not
    generally equipped to determine whether a particular theory of
    conviction submitted to them is contrary to law" (emphasis added)).
    The jury instructions for the tax count, for example, did not
    cross-reference the other counts or otherwise suggest that the
    jury's findings on the bribery or property fraud charges were
    relevant to the tax count.         Cf. United States v. Foley, 
    73 F.3d 484
    , 493-94 (2d Cir. 1996) (vacating tax fraud conviction where
    jury had convicted defendant of bribery under legally invalid
    theory    and   was   instructed   that   bribes   were   not   deductible),
    abrogated in part on other grounds by Salinas, 
    522 U.S. 52
    .
    Certainly, a juror likely would -- and should -- have viewed some
    41   As we discuss below, to the extent Wilson argues that,
    even if the jury did not treat its verdict on the other counts as
    dictating the outcome of the tax count, the presence of such
    inflammatory charges might have influenced its verdict on the tax
    count, we note that Wilson has not challenged on appeal the
    district court's denial of his motion to sever the tax count from
    the other charges before trial, and the argument that we must
    vacate his tax conviction due to retroactive misjoinder lacks
    merit.
    - 136 -
    factual findings as relevant to multiple counts; for instance,
    whether Wilson understood himself to be entering a quid pro quo
    guaranteeing his son's admission in exchange for payment would
    bear both on the § 666 charge and on the legitimacy of his
    charitable deductions.       But Wilson offers no support for the
    assertion that a juror would have viewed the legal status of his
    payments as bribes as controlling the legitimacy of his deducting
    those payments.
    Wilson's strongest argument is that the instruction to
    the jury that, "[f]or purposes of the mail and wire fraud statutes,
    admission[s] slots are . . . property" -- which we have already
    held was erroneous based on the theory presented by the government
    at trial and on appeal, see supra Section III.B -- may have
    influenced the verdict on the tax count.     He contends that, having
    been told that an admissions slot is property as a matter of law,
    the jury might have concluded that Wilson's payment could not
    constitute a valid charitable contribution, since it would have
    been made in exchange for goods or services.           Taken in context,
    however, this erroneous instruction on different counts does not
    undermine   the   jury's   verdict.   The   property    instruction   was
    specifically limited to "the mail and wire fraud statutes," and
    Wilson cites nothing else in the instructions that would indicate
    to a juror that the verdict on the fraud count should influence
    - 137 -
    the tax count.42   "[We] presum[e] that jurors, conscious of the
    gravity of their task, attend closely the particular language of
    the trial court's instructions in a criminal case and strive to
    understand, make sense of, and follow the instructions given them."
    United States v. Olano, 
    507 U.S. 725
    , 740 (1993) (alterations in
    original) (quoting Francis v. Franklin, 
    471 U.S. 307
    , 324 n.9
    (1985)).   And, although Wilson did move below to sever the tax
    count from the other counts due in part to the confusion that might
    be caused by the theory that admissions slots are property, he
    does not now style his argument as an objection to the district
    court's denial of any motion to sever, nor to a failure by the
    district court to issue a clarifying instruction on the tax count
    to dispel that potential confusion.    In the absence of any such
    argument, the alleged confusion caused by the district court's
    42   The specific limiting language in the property
    instruction and the distinctness of the instructions relevant to
    the different counts undermines Wilson's reliance on United States
    v. Lindberg, 
    39 F.4th 151
     (4th Cir. 2022).     Lindberg vacated a
    defendant's conviction under § 666 because it concluded that a
    jury instruction incorrectly defining the term "official act" for
    purposes of a separate bribery charge had "infected" the jury's
    consideration of the § 666 count. Id. at 164; see id. at 164-65.
    Although § 666 does not contain an "official act" requirement (it
    refers instead to payments intended to influence an agent in
    connection with "any business, transaction, or series of
    transactions" of an organization), the court recognized that the
    district court had nonetheless "provided instructions on both
    counts at the same time using the term 'official act.'" Id. at
    164. Wilson does not cite any such overlapping instructions in
    this case.
    - 138 -
    instructions on a separate count cannot by itself provide a basis
    for vacating his conviction on this count.
    Wilson also argues that post-trial statements by the
    government and district court show that the jury may have convicted
    him on the tax count based on its conclusions about the other
    charges, but neither statement provides a basis for vacating the
    conviction.    First, he highlights that the government, in a
    memorandum opposing Wilson's post-trial motion for a new trial on
    Yates grounds, stated that "the jury could convict [either on the
    theory that] Wilson deducted an illegal bribe as business and
    charitable deductions, or because he deducted payments . . . that
    were not in fact charitable contributions or business expenses."
    Although this statement appears to accept Wilson's premise that
    the jury could have convicted on this basis, neither Wilson's
    motion nor the government's response cited any material presented
    to the jury that would have led jurors to believe that the verdict
    on the bribery and fraud counts should control the verdict on the
    tax count.    We decline to vacate the jury's verdict based on a
    single sentence in a post-trial filing.      Cf. United States v.
    McGregor, 
    650 F.3d 813
    , 824 n.4 (1st Cir. 2011) (explaining that
    concession by government in district court does not bind this
    court).
    Second, Wilson relies on a ruling by the district court
    during sentencing that, for purposes of the tax loss amount
    - 139 -
    applicable during sentencing, Wilson could not claim any of the
    $220,000 payment as a legitimate charitable contribution "because
    the entire payment was fraudulent."          Wilson cites no authority for
    the proposition that a statement by the court during sentencing is
    relevant to determining the basis for a jury's verdict.                  This
    statement also does not require vacating Wilson's conviction.
    Without some clearer grounding in the record, Wilson's
    broad argument that the jury might, on its own initiative, have
    based its resolution of the tax count on the bribery or property
    fraud charges does not create a Yates problem.
    2.
    We turn to Wilson's argument that the same spillover
    prejudice that led us to vacate his conspiracy and substantive
    § 666 convictions also requires us to vacate his tax conviction,
    which we understand to be a retroactive-misjoinder argument, even
    though Wilson has not characterized it as such.            See United States
    v. Cadden, 
    965 F.3d 1
    , 21 n.7 (1st Cir. 2020) ("'[R]etroactive
    misjoinder' arises where joinder of multiple counts was proper
    initially, but later developments . . . render the initial joinder
    improper." (quoting Jones, 
    16 F.3d at 493
    )).              In discussing this
    alleged evidentiary spillover, Wilson does not draw a distinction
    between   the   charitable   and   business     expense    deductions.     We
    conclude that Wilson has not met his burden to "prove prejudice so
    pervasive that a miscarriage of justice looms."             Wihbey, 75 F.3d
    - 140 -
    at 776 (quoting Levy-Cordero, 
    67 F.3d at 1008
    ).                    Our analysis
    applies to both the business expense and charitable contribution
    theories.
    The government's case on the tax count relied largely on
    Wilson's own emails discussing how to structure the payments
    related to his son's admission, limiting the risk that Wilson was
    convicted based on other parents' conduct.                    See id. at 775-76
    (noting      that    distinctness      of   evidence     related    to    charged
    coconspirators' conduct reduced danger of evidentiary spillover or
    jury confusion); United States v. Moran, 
    984 F.2d 1299
    , 1304 (1st
    Cir. 1993) (similar).           The distinctness of this evidence from
    evidence related to other parents also increases the probability
    that   the    jury   was   able   to    comply   with   the    district   court's
    instruction to determine Wilson's "guilt or innocence . . . on an
    individual basis."         See Wihbey, 
    75 F.3d at 775
     (observing that
    similar instruction limited risk of spillover prejudice).
    Further, the evidence against Wilson on the tax count,
    based on his own words and conduct, was very strong.                See Morrow,
    
    39 F.3d at 1235-36
     (concluding variance did not warrant vacating
    convictions where evidence properly admissible against defendants
    "amply proved" their guilt).           Importantly, Wilson does not develop
    on appeal any argument that, as a matter of substantive tax law,
    the statements on his return were true -- that is, that the
    payments     were    properly     deductible     as     business   expenses    or
    - 141 -
    charitable contributions.     He does not meaningfully dispute the
    government's theory that the payments did not qualify as business
    expenses (because of their personal nature) or as charitable
    contributions (because they were his side of a quid pro quo to
    secure his son's admission).43        His argument as to spillover
    prejudice, and as to the alleged evidentiary errors discussed
    below, focuses instead on the required mental state: that the false
    return be "ma[de] and subscribe[d]" "[w]illfully."        
    26 U.S.C. § 7206
    (1) (emphasis added).      We thus have no cause to reexamine
    the conviction on the basis of substantive tax law, and instead
    focus on the evidence of Wilson's intent.
    That evidence, with respect to both the business and
    charitable   deductions,   was   powerful.   After   having   already
    discussed the cost of the side door with Singer, Wilson, apparently
    43   In particular, with respect to the business deduction,
    Wilson does not argue on appeal that the fact that his business
    was a pass-through S corporation is relevant to the payments'
    deductibility, thereby waiving any such argument.      See United
    States v. Zannino, 
    895 F.2d 1
    , 17 (1st Cir. 1990). With respect
    to the charitable deduction, as discussed below, Wilson argues
    that the district court improperly excluded evidence related to
    USC's practices in acknowledging donations for tax purposes. He
    asserts that this evidence would have shown that "back-door and
    non-Singer side-door donations were . . . fully deductible." But
    Wilson styles this argument as involving an "evidentiary error[]"
    bearing on his mental state, and, in his opening brief, does not
    cite any legal authority explaining why this USC practice would
    bear on the deductibility of his payments.       We thus deem any
    challenge to the legal deductibility of the payments waived. See
    id.; United States v. Diggins, 
    36 F.4th 302
    , 320 (1st Cir.)
    (explaining that arguments not developed in an appellant's opening
    brief are waived), cert. denied, 
    143 S. Ct. 383 (2022)
    .
    - 142 -
    on his own initiative, asked Singer if the payment could be
    designated "for consulting or whatever from [Singer's business] so
    that [Wilson] c[ould] pay it from the corporate account."                    When
    Singer responded that he could send an invoice for consulting that
    Wilson    could    "write    off   as   an   expense,"   Wilson      responded,
    "Awesome!"       Wilson then directed his office manager to charge the
    invoice to "Business Consulting" and to expect invoices from
    Singer.     Wilson offered no evidence and does not argue on appeal
    that any such consulting actually took place.             Indeed, any notion
    that Wilson deducted these expenses because he believed that
    Singer's    college    counseling       services    qualified   as     "Business
    Consulting" is severely undermined by the fact that Wilson deducted
    the $20,000 that went to Singer personally on the basis that it
    was   for   the    "[c]oncept,     design,    and   implementation      of    [a]
    [p]rofessional [d]evelopment program for Hyannis Port Capital
    associates," a service completely unrelated to college counseling,
    and one that Singer clearly never performed.
    Wilson does not argue that he could plausibly have relied
    in good faith on Singer's representation that the payment could
    legitimately be "writ[ten] off as an expense" so long as it was
    falsely described as for business consulting services, nor would
    such an argument be tenable.              Instead, Wilson's enthusiastic
    response    to    Singer's   statement    linking    payment    from    Wilson's
    corporate account with a "write off" provides evidence that Wilson
    - 143 -
    was aware of the tax benefits of falsely characterizing the
    payments as business expenses and acted willfully to obtain those
    benefits.
    As    for   Wilson's        deducting       the    $100,000       payment    to
    Singer's nonprofit foundation as a charitable contribution, it is
    first worth noting that, although Wilson paid $100,000 to Singer's
    foundation, it was Singer's for-profit business that then paid
    $100,000 to USC.        Without some link between the two payments, the
    payment to Singer's nonprofit would clearly be in exchange for
    goods     and    services   --     the        same     services       that    Wilson    was
    simultaneously attempting to deduct as a business expense.
    However, even assuming that Wilson deducted the $100,000
    payment    to    Singer's   foundation           intending       that    it    reflect    a
    charitable contribution that would in turn be made to USC, the
    government offered strong evidence that Wilson understood this
    payment to be part of an explicit quid pro quo to secure his son's
    admission, rather than a gift that might also curry some intangible
    amount of favor with the university.                   Singer told Wilson that the
    USC men's water polo coach was "giving [him] 1 boys slot" on a
    "first come first [sic]" basis.                Later, Wilson asked Singer: "When
    is [the USC men's water polo coach] going to be able to give us
    decision [sic] on USC?             And when do I pay u [sic]?"                        Singer
    responded:       "No   payment     of       money     till    [sic]    [the    USC     men's
    water     polo    coach]    gets        a     verbal    and     written       [sic]    from
    - 144 -
    admissions . . . ."        The day after Subco considered and approved
    his son's admission, Wilson thanked Singer "for making this happen"
    and asked for an "invoice" with no differentiation between payment
    for   Singer's     services   and     any   contribution     to   USC,   further
    suggesting that he viewed the payment as his side of an exchange
    to secure his family a concrete benefit, rather than as a gift.
    That the deductions ultimately saved Wilson tens of
    thousands of dollars on his taxes would give the jury further
    reason to conclude that Wilson acted willfully.
    Wilson's    brief   on    appeal    downplays    this   evidence,
    arguing that Wilson was living overseas at the time of filing;
    could not have foreseen "whether or how" classifying the payments
    as business expenses, rather than charitable donations, would
    affect     his   tax   liability;    and    ultimately   saved    only   $1,425,
    relative to his liability had he deducted the entire $220,000 as
    a charitable contribution.44          This argument rests on an assumption
    that Wilson believed he could legitimately deduct the entire sum
    as a charitable contribution, but the government produced strong
    evidence to the contrary.           Further, even if this assumption were
    correct, the argument offers no plausible good faith explanation
    of why Wilson himself sought to classify the payments as consulting
    44  Notably, this argument echoes the position Wilson's
    counsel unsuccessfully presented to the jury during closing
    argument.
    - 145 -
    fees, and the unpredictability of the resulting tax benefits cuts
    both ways -- Wilson may have hoped to gain more through the
    maneuver.
    We   conclude   that    Wilson    has   not   shown   a    risk   of
    evidentiary spillover that warrants vacating his tax conviction.
    3.
    Finally,   Wilson      contends    that   allegedly       erroneous
    evidentiary rulings by the district court require us to vacate the
    tax conviction.     The excluded material he cites falls into three
    categories: (1) statements by Singer, (2) statements by Wilson,
    and (3) evidence related to USC's practices in acknowledging
    donations.45     We conclude that, even assuming the district court
    erred in excluding this evidence -- a question we need not and do
    not decide -- those errors were harmless with respect to the tax
    conviction.46
    45   Wilson also argues that the district court's exclusion
    of various evidence related to USC's admissions practices
    prejudiced his tax count defense by "hobbl[ing] the defense to the
    bribery and fraud counts." This claim repackages Wilson's Yates
    argument, and we reject it for the same reasons.      In addition,
    Wilson asserts elsewhere in his brief that the district court erred
    in declining to suppress certain call recordings the government
    captured beginning in September 2018. He does not reference this
    issue in discussing the tax count, and so we do not consider it
    here.
    46   We emphasize that our holding that any error in the
    exclusion of this evidence was harmless is limited to Wilson's tax
    conviction. We express no view as to whether the exclusion may
    have been harmful with respect to any other count.
    - 146 -
    "An error will be treated as harmless only if it is
    highly probable that the error did not contribute to the verdict."
    United States v. Kilmartin, 
    944 F.3d 315
    , 338 (1st Cir. 2019)
    (internal   quotation   marks   omitted)   (quoting   United   States   v.
    Fulmer, 
    108 F.3d 1486
    , 1498 (1st Cir. 1997)).          "To sustain the
    verdict, the reviewing court must be able to say with a fair degree
    of assurance that the erroneous ruling did not substantially sway
    the jury." 
    Id.
     (quoting Ruiz-Troche v. Pepsi Cola of P.R. Bottling
    Co., 
    161 F.3d 77
    , 87 (1st Cir. 1998)); see also 
    id.
     (requiring "a
    panoramic, case-specific inquiry" (quoting United States v. Piper,
    
    298 F.3d 47
    , 57 (1st Cir. 2002))).
    The first category of excluded evidence cited by Wilson
    encompasses a variety of statements by Singer to listeners other
    than Wilson, including public presentations and communications
    with other parents. In these statements, Singer generally depicted
    himself as well connected in the realm of college admissions and
    experienced in using the side door to help clients' children gain
    admission. For example, in a corporate presentation, Singer stated
    that he "read applications at two schools every year," including
    some of "the . . . most prominent universities in America," and
    had "d[one] 761 side doors into the best schools in America."           He
    asserted to one parent that the "side door is not improper" but is
    part of "how all schools fund their special programs or needs,"
    - 147 -
    and    told      another    that    he    would     soon    be   meeting     with    "the
    President[s] of Harvard and Tufts for lunch."47
    Wilson does not contend that he was aware of these
    statements during his dealings with Singer.                      Instead, he argues
    that "[a] jury could have reasonably inferred that Singer pitched
    the side-door to Wilson in the same way he pitched his advice to
    other parents," and that such a pitch would tend to show that
    Wilson plausibly could have believed that Singer's business was
    legitimate.         For several reasons, we conclude that any error in
    excluding this evidence was harmless with respect to the tax count.
    First,    there    was    other     evidence      before    the     jury
    regarding Singer's pitch.            See, e.g., United States v. Veloz, 
    948 F.3d 418
    , 434 (1st Cir. 2020) (finding erroneous exclusion of
    evidence harmless where record contained other evidence "to the
    same effect"); United States v. Wilkerson, 
    251 F.3d 273
    , 280 (1st
    Cir.    2001)      (similar).       An    IRS   agent      who   assisted    the    FBI's
    investigation of Singer and his clients testified at trial that
    "Singer's        general    pitch    was    that     [side-door      payments       were]
    donation[s] to . . . program[s]"; that "Singer did not use the
    word 'bribe' when he was discussing the side door"; and that even
    after       he   began    cooperating      with     the    government,      Singer    "at
    times . . . would go back to his whole pitch and go back to the
    47 The district court excluded these statements                               on   a
    combination of relevance, hearsay, and Rule 403 grounds.
    - 148 -
    old cover story as part of his just being used to saying 'donation'
    when we asked him to say payment."                   The agent even acknowledged
    that Singer's framing of the side door as a donation "does not
    sound bad."         The jury evidently           found this similar evidence
    unpersuasive.
    The   probative    value     of    the    excluded    material   with
    respect to the tax count was also quite low.                    See, e.g., United
    States   v.    Brown,   
    805 F.3d 13
    ,     17    (1st   Cir.   2015)   (finding
    evidentiary error harmless where jury likely gave little weight to
    improperly admitted material).                Singer's statements have little
    apparent relevance to the business deduction theory of tax fraud,
    as a belief on Wilson's part in Singer's legitimacy would not
    provide a basis for classifying the payments as business expenses,
    and Wilson does not offer any explanation for how that could be
    otherwise.      With respect to the charitable deduction, Wilson's
    understanding of Singer's legitimacy would at most be tangentially
    relevant to his understanding of whether the payment was pursuant
    to a quid pro quo, which Wilson does not dispute would "vitiate[]
    any charitable deduction."             And in order to rely on the evidence
    at all, the jury would need to infer that Singer made similar
    statements to Wilson himself and that Wilson relied on those
    statements -- a possible, but hardly inevitable, conclusion.
    Most importantly, as discussed above, the government's
    evidence on the tax count was powerful with respect to both the
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    charitable and business expense deductions.    See, e.g., Kilmartin,
    944 F.3d at 338-39 (noting that "the strength or weakness of the
    government's evidence of guilt is normally the most important
    integer in the harmlessness equation").     Considering the totality
    of the circumstances, we conclude that any error in the exclusion
    of Singer's statements was harmless.
    The second category of excluded evidence consists of
    statements by Wilson himself.       Wilson's brief identifies six
    excluded exhibits of this type.   The first three are email threads
    referring other parents to Singer.       In one thread, for example,
    Wilson told another parent: "There is a company and [a] CEO I know
    well who was a great help to us [with Wilson's son] that might be
    very helpful for you too.    I would be happy to connect you and pay
    for it . . . if you and your daughter would find it useful."
    Wilson asserts that a jury might infer "that Wilson would refer
    Singer's services to [other parents] only if he believed those
    services were legitimate."
    The other three excluded statements appear in emails
    from Wilson to Singer about his son's admission and enrollment at
    USC.   In one, sent at the beginning of his son's senior year of
    high school, Wilson asked whether it was important for his son to
    retake the ACT and SAT, saying that it "[s]eem[ed] like a lot [to
    do] in the fall."   The other two, sent during the summer before
    his son's enrollment at USC, inquired about the start date for his
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    son's water polo practices.48       Wilson argues that these emails
    might lead a jury to conclude that Wilson believed that "Singer
    was legitimate, that his son needed to do well on his tests, and
    that his son was going to play water-polo."
    We   conclude,   for   similar   reasons   as   with   Singer's
    statements, that, even assuming the exclusion of this evidence was
    erroneous, it was harmless with respect to the tax count.
    Again, evidence similar to the excluded material was
    before the jury.   See Veloz, 948 F.3d at 434; Wilkerson, 
    251 F.3d at 280
    .   In a recorded call introduced into evidence by the
    government, Wilson told Singer that he planned to refer a friend
    to him -- apparently the same friend who received one of the three
    excluded referrals.49   Another government exhibit included emails
    from Wilson to Singer in which Wilson asked whether "it w[ould] be
    known that [his son was] a bench warming candidate," noted that
    "[o]bviously his skill level m[ight] be below the other freshmen,"
    inquired whether his son would "be so weak as to be a clear misfit
    at practice," and stated that he "want[ed] to be sure [his son
    48   As with the Singer statements, the district court
    excluded these exhibits on a combination of relevance, hearsay,
    and Rule 403 grounds.
    49   During the call, Wilson identified the friend by name,
    identified his employer, and said the friend had a daughter who
    "want[ed] to go to Brown."     One of the excluded referrals was
    addressed to a parent with the same name whose email address
    corresponded to the friend's employer, and noted that the parent's
    daughter "ha[d] a strong interest in Brown."
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    would] not [be] a lepper [sic]."         And, although neither party
    mentions as much in its briefing, one of Wilson's water polo
    inquiries appears to have been admitted as part of a government
    exhibit; indeed, when introducing the exhibit, the government had
    a witness read Wilson's email aloud to the jury.
    The   probative   value   of   the   excluded   material   with
    respect to the tax count was also low.     See, e.g., Brown, 
    805 F.3d at 17
    .   For the same reasons as with Singer's statements, whether
    Wilson believed that Singer's services were legitimate has limited
    relevance to his state of mind for purposes of the tax count.        And,
    in any event, the excluded statements provide only weak evidence
    about Wilson's understanding of Singer's legitimacy.        Wilson does
    not dispute that Singer provided at least some legitimate services
    in addition to admission through the side door, and the referral
    emails -- which do not mention the side door by name -- could be
    for those services, shedding little light on Wilson's perception
    of the legitimacy of the services Singer provided to him. Further,
    the email regarding exam scores is, as the government argues,
    "consistent with questioning whether [Wilson's son] had to worry
    more about test scores at all given the side-door arrangement in
    place," and the emails about water polo would be consistent with
    Wilson's believing that, even if his son was not a legitimate
    athlete, he needed to at least pretend to be a team member for the
    side-door scheme to work.    Cf. United States v. Sabean, 885 F.3d
    - 152 -
    27, 42 (1st Cir. 2018) (finding any error harmless where excluded
    "statements had as much of a tendency to inculpate the defendant
    as to exonerate him," because "the net impact of the evidence was
    likely a wash").
    Given the government's strong evidence on the tax count,
    we conclude it is highly probable the exclusion of these exhibits
    did not taint Wilson's conviction.             See Kilmartin, 944 F.3d at
    338-39.
    Finally,   Wilson       argues     that    the    district   court
    erroneously   excluded     evidence    related    to     USC's   practices     in
    acknowledging   donations     for    tax     purposes.       Federal   tax   law
    generally   requires   a   charitable      organization      that   receives    a
    payment made "partly as a contribution and partly in consideration
    for goods or services" to (1) inform the donor that only the excess
    value of the payment beyond the value of those goods or services
    is deductible for federal tax purposes, and (2) provide an estimate
    of the value of the goods or services for the donor to use in
    preparing her taxes.       
    26 U.S.C. § 6115
    (b); see 
    id.
     § 6115(a).
    Wilson sought to introduce evidence that USC did not advise other
    donors that they were required to offset the value of their
    charitable deductions by the value of any admissions benefits
    received in exchange for their donations.                 In particular, in
    response to a subpoena, USC offered "to provide a declaration
    explaining that . . . it ha[d] no responsive documents reflecting
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    that USC instructed a donor to reduce his or her tax deductability
    [sic] in connection with an admission decision because USC does
    not offer admission to any applicant in exchange for quid pro quo
    payments."   Wilson argues that this evidence, which the district
    court excluded on relevance grounds, "would have supported the
    good-faith legitimacy of Wilson's deductions" by showing that he
    "reasonably believed that no offset was required."   The government
    does not develop any argument that this evidence was properly
    excluded; it instead contends that the exclusion was harmless.    We
    agree.
    The evidence's probative value was limited.     See, e.g.,
    Brown, 
    805 F.3d at 17
    .    It had no bearing on the legitimacy of
    Wilson's deductions for business expenses.    And it would provide
    only indirect support for Wilson's claim that he believed the
    charitable deductions to be legitimate.    Wilson did not contend
    below that he was aware of USC's donation acknowledgement practices
    at the time he filed his tax return; indeed, the district court
    excluded the evidence because Wilson did not show such awareness.
    Instead, we understand Wilson's brief to argue that evidence that
    USC did not instruct other donors to offset charitable deductions
    by the value of any admissions boost received would have made
    Wilson's belief that such an offset was not required appear more
    reasonable and, as a result, more likely to be sincere.   See United
    States v. Lachman, 
    521 F.3d 12
    , 19 (1st Cir. 2008).       But to the
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    extent     Wilson     asserts    that     this     evidence    confirms       the
    reasonableness of his belief in the deduction's legitimacy because
    it shows that USC shared that belief, the language of USC's
    proposed    declaration    undercuts      his    argument.    The    university
    claims not to have instructed donors to offset the value of their
    contributions only because it "does not offer admission to any
    applicant in exchange for quid pro quo payments."                   But we have
    already shown why the evidence is very strong that Wilson could
    not have believed in good faith that he was not engaging in a quid
    pro quo.    And to the extent Wilson argues that he was not aware
    that the benefit he received in that quid pro quo was the kind of
    valuable    benefit    that    would   preclude    his   payment    from    being
    deductible,    nothing    in    the    disputed    document   bears    on   that
    question, since the document at most establishes the ordinary point
    that a donation not given pursuant to a quid pro quo is fully
    deductible.    In truth, then, the proposed declaration does not
    support Wilson's position, meaning that the exclusion of that
    declaration was not prejudicial.
    Given this context, the government's strong evidence on
    both theories of the tax count again enables us to "say with a
    fair degree of assurance that the [exclusion of this evidence] did
    not substantially sway the jury," rendering any error harmless.
    Kilmartin, 944 F.3d at 338 (quoting Ruiz-Troche, 
    161 F.3d at 87
    ).
    We affirm Wilson's tax conviction.
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    VI.   Conclusion
    For the foregoing reasons, we affirm Wilson's conviction
    for filing a false tax return in violation of 
    26 U.S.C. § 7206
    (1)
    (Count   Thirteen   of   the   operative   indictment).     We   vacate
    Abdelaziz's   and   Wilson's   other   convictions,   and   remand   for
    proceedings consistent with this opinion.
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