Luis Amaya Manzanares (“Amaya”), a native of El Salvador, apparently entered the United States without authorization; the Immigration and Naturalization Service (“INS”), as it was then called, says that it has no record of a lawful entry by Amaya. In 1990, Amaya applied for asylum and was granted an “employment authorization card” by the INS, renewed annually while his asylum claim was being *41considered. See 8 C.F.R. § 264.1(b) (2004). The card differs from the alien registration card — the so-called “green card” — available to aliens who are legal permanent residents. See id.
According to the government, Amaya sought employment in late 2001 and early 2002 with two different service companies doing business in secure areas at Logan Airport in Boston. Under federal regulations, such employment requires an access badge granted after review of an application to the airport operator and a criminal background check. In support of his applications, the government says that Ama-ya submitted both his genuine employment authorization card and a green card bearing his name, photo, signature and actual “alien number” issued incident to his asylum application.
Thereafter, the government says that it determined that the green card was a forgery. INS records, according to the government, show that no green card was ever issued to Amaya. In April -2002, a grand jury indicted Amaya for use or attempted use of a forged, counterfeited, altered or falsely made green card. 18 U.S.C. § 1546(a) (2000). Thereafter, in October 2002, the district court scheduled trial for January 6, 2003.
On December 31, 2002, the prosecutor told defense counsel that the prosecution had just learned that an INS agent not involved in the case had spoken to Amaya on or about January 24, 2002, and that Amaya had made inculpatory statements concerning his acquisition of the green card. On January 2, 2003, Amaya moved to dismiss the indictment because of the “late disclosure.” After an evidentiary hearing held the next day, the district court refused to dismiss but ordered that any statements by Amaya to the agent be suppressed.1
On January 2, 2003, the government also filed a trial brief and list of trial exhibits, including two exhibits provided to Amaya’s counsel on December 31, 2002. One exhibit, entitled “certificate of nonexistence of record,” was prepared by the chief of the INS Records Services Branch on December 30, 2002, and stated in part that the INS maintained records of entry and that a diligent search showed no “lawful admission” into the United States by Amaya. A second exhibit, similarly titled, said that there was no record of issuance of the green card used by Amaya.
On January 6, 2003, the date scheduled for trial, Amaya moved to exclude both exhibits on the ground that they had been disclosed unduly late, allegedly in violation of applicable disclosure rules. Amaya also argued that the certificate indicating Ama-ya’s unlawful entry was irrelevant and, if relevant, should be excluded as unduly prejudicial. Fed.R.Evid. 401, 403. Amaya also moved in limine to suppress, as irrelevant and unduly prejudicial, any testimony that Amaya had entered the country unlawfully.
When the district judge took the bench, he announced that he was granting both motions. He said, perhaps referring mainly to first of the two certificates but logically encompassing both, that the evidence had not been turned over in timely fashion. And, apparently referring to the testimony as to unlawful entry, the judge said that he would , exclude it too because “I don’t think it is relevant at all to the issue in this case.”
*42The prosecutor then asked to address the court, defended the disclosure of the certificates as timely and urged that evidence of Amaya’s unlawful entry was being offered “simply to prove the crime that he is charged with.” The district court reiterated its ruling, noting that it was not suggesting bad faith by the government. Further argument by a supervising prosecutor was also unavailing and the government then filed this interlocutory appeal from the January 6 ruling. 18 U.S.C. § 3731 (2000).2
The controlling issue is whether evidence that Amaya entered the country unlawfully is admissible to show an element of the crime charged. Although the district court excluded proposed documentary evidence to that effect — the certificate indicating lack of lawful entry — on a procedural ground, the court also ruled that any testimonial reference to Amaya’s unlawful entry was barred as irrelevant. If the latter ruling is sound, the certificate insofar as it relates to unlawful entry would be inadmissible even without the procedural bar.
Nevertheless, the procedural ruling has substantial implications for the government’s conduct of future prosecutions and so it is best to begin by addressing that ruling. The Federal Rules of Criminal Procedure provide that, on request, the government in a criminal case must disclose to the defense
books, papers, documents, data, photographs, tangible objects, buildings or places ... if the item is within the government’s possession, custody, or control and ... the government intends to use the item in its case-in-chief at trial.
Fed.R.Crim.P. 16(a)(1)(E). By local rule, the district court requires such production within 28 days of arraignment. D. Mass. R. 116.1(C)(1)(a).3 Whether Rule 16(a)(1)(E) is intended to apply to substitutes for testimony, such as a deposition or certificate as to what government records do or do not contain, may be open to question; but we will assume arguendo that the rules do apply.
Nevertheless, the rule by its terms is directed to materials that the government actually possesses. By contrast, the document in question, which was a short-cut to avoid having a record-keeper testify, was created for trial purposes well after the arraignment (and then promptly turned over to the defense); indeed, it was created — perhaps on a precautionary basis— just before the district court ruled, shortly in advance of the scheduled trial, that the government could not offer Amaya’s admission that he had bought the green card.
On the face of the matter, Rule 16(a)(1)(E) did not apply to the document until it was created. See United States v. Kahl, 583 F.2d 1351, 1354 (5th Cir.1978) (upholding a district court’s refusal to grant discovery of government statistical *43compilations, when such compilations did not exist); United States v. Schembari, 484 F.2d 931, 935 (4th Cir.1973) (“[T]he government cannot disclose what it does not have .... ”); cf. United States v. Harper, 432 F.2d 100, 102 (5th Cir.1970) (holding that “the failure to produce non-required records when they do not exist” is not a violation of the Jencks Act).
Conceivably a remedy could be found if the government were manipulating the timing for its creation of evidence to prejudice the defense; but nothing of the kind is evident in this case. Nor is there any reason to think that the defense was prejudiced by the timing of the disclosure. Whatever objections the defense might have as to admissibility, it had good reason to fear that the government would offer testimony to show that Amaya was an unlawful entrant who knowingly used a forged green card because he could not get a valid one.
This brings us to the broader question of the relevance of any showing of unlawful entry. In order to show that Amaya was guilty of the crime charged — use of a false green card — the government had to show (1) that Amaya used the card or attempted to do so (which is probably not in dispute), (2) that the card was false and (3) that he knew it was false. 18 U.S.C. § 1546(a). The question is whether the proof that the government wants to offer — that Amaya entered the country as an illegal alien — is relevant to either of these last two elements.
The government asserts that Amaya’s unlawful entry is relevant to both falsity and knowledge, but the first of the two elements — falsity—is something of a red herring. Strictly speaking, Amaya’s unlawful entry (if proved) makes it more likely that the card is false than it would be without evidence of such entry; after all, an unlawful entrant would have use for a false green card, while a lawful entrant would have a far better chance of qualifying for a valid card.
But proof that the card was false could be furnished straightforwardly by a certification or other testimony that no such card had been issued to Amaya. Indeed, in this case the government apparently has the false card in hand and could prove its falsity directly. By contrast, a certificate showing Amaya’s unlawful entry (and perhaps his continued lack of eligibility for a green card) has only circumstantial bearing on the question whether the card is a fake.
No sensible judge would be likely to let in the unlawful entry evidence to show falsity. Rule 403 does not give judges a free hand to refashion the government’s proof to make cases less dramatic or more even-handed. Yet a judge who allowed in prejudicial evidence — and here we mean unfairly prejudicial — to prove something that was so easily and definitively proved without such prejudice and by more straightforward means would be courting reversal. This argument for relevance thus tends more to taint than to assist the government’s position.
The other reason given for the evidence — -to show that Amaya knew that the card was false — has much more substance. At first blush one might think, after proof that the card was false, that further independent evidence of Amaya’s knowledge would be unnecessary for conviction. After all, if the card were false, then Amaya could not have obtained it from the INS. The most natural alternative explanation would be that he purchased it from someone who deals in green card forgeries, knowing that he was purchasing a fake.
Yet conceivably some juror might think that there are means by which Amaya might have acquired a false green card *44without necessarily knowing that it was false. Perhaps a lawyer or other intermediary had, for a price, offered to procure a valid green card and then delivered it to Amaya representing that it was valid. True, a judge might limit argument by defense counsel about such possibilities in the absence of some proof, but the government was not obliged to gamble and could reasonably look for relevant, evidence of knowledge. So relevant evidence to show knowledge was hardly a waste of time.
Arguing for relevance, the government says that because illegal aliens cannot get green cards until their status is adjusted, Amaya must therefore have known that his own card was necessarily false.4 The government does not fully explain its reasoning and certainly did not do so to the district judge; but the government’s conclusion is correct for at least one reason and possibly for two, although the second poses a trickier issue that need not be resolved.
The most straightforward reason for relevance is that Amaya’s status as an unlávvful entrant who has not upgraded his status makes it more likely that he acquired the forged green card by buying it from someone who was obviously not a government. agent authorized to issue it. This might be conceived of as motive evidence (one who is an unlawful entrant has a motive to buy a forged green card) or mechanically (someone with Amaya’s sta: tus would have been rebuffed if he had sought a green card from the INS and so had to resort to the black market).
Either way, if Amaya bought the green card on the black market, this practically assures that Amaya knew the card he acquired was forged rather than valid. - It is not simply the possession of a forged document that creates a strong inference of knowledge that it is false; someone with a single forged $20 bill in his wallet may as easily be a victim rather than a perpetrator. Rather, it is the added proof (that Amaya’s status prevented him from getting a valid green card from the INS) which greatly increases the chances that he engaged in a transaction that by its nature would also have led him to believe that the document he had obtained was not valid.
Thus, we cannot agree that the unlawful entry evidence was not “relevant” to an element of the crime charged. Yes, the standard of review on appeal of evidentia-ry rulings is often said to be for “abuse of discretion” (although the phrase is somewhat misleading, see Invessys, Inc. v. McGraw-Hill Cos., 369 F.3d 16, 19 (1st Cir.2004)). But whether a fact tends to make another fact more or less likely depends heavily on the logic of the connection, and there is no “discretion” to ignore a logical relationship. Cf. Tiller v. Baghdady, 244 F.3d 9, 14 (1st Cir.2001).
There is an alternative argument for relevance. Conceivably, a jury might be able to infer without independent proof that someone who is an unlawful entrant in the United States is quite likely to know that he or she cannot get a valid green card and therefore that Amaya, since he was found with a forged green card, quite likely knew that it had to be forged. A jury might think that such knowledge would be widespread among unlawful entrants — or at least those who, like Amaya, had been in this country for a substantial period and had previously sought work.
*45The state of knowledge in the community is not an adjudicative fact about Ama-ya’s own knowledge, which would be subject to strict standards of proof. Compare Fed.R.Evid. 201. Rather, the likely state of knowledge in the community is a “background” fact about how the world works, and such facts if plausible can be inferred without direct proof. This is an issue not much discussed in the cases and not subject to very clear standards, see United States v. Amado-Nunez, 357 F.3d 119, 121-22 (1st Cir.), cert. denied, — U.S. -, 124 S.Ct. 2864, 159 L.Ed.2d 284 (2004), and we need not resolve the problem here.
It is enough for relevance that unlawful entry increases the likelihood that Amaya did acquire the false green card on the black market and therefore knew it to be forged. True, one might ask why Amaya needed a forged green card if he had a temporary work card from the INS (but perhaps he got the forgery before the work card). Anyway, “considerations [that] arguably reduce the probative value of [particular acts] do not destroy the relevance of the acts altogether.” NationWide Check Carp. v. Forest Hills Distribs., Inc., 692 F.2d 214, 219 (1st Cir.1982).
In moving to exclude the evidence, Ama-ya argued not only lack of relevance but also undue prejudice; evidence, although relevant, may still be excluded because its prejudicial impact substantially outweighs its probative value. Fed.R.Evid. 403; United States v. Balthazard, 360 F.3d 309, 313-14 (1st Cir.2004). In this case the argument for such an exclusion under Rule 403 is far from frivolous. It was not expressly resolved by the district court only because the evidence was excluded on other grounds.
Proof of Amaya’s unlawful entry is prejudicial in the sense intended by Rule 403. This is not because it hurts Amaya— all relevant evidence by the government does that — but because it introduces a factor into the case that might encourage the jury to dislike or disapprove of the defendant independent of the merits. See United States v. Varoudakis, 233 F.3d 113, 122 (1st Cir.2000) (evidence “invit[ing] the jury to render a verdict on an improper emotional basis”).
The extent of likely prejudice should not be overstated: some jurors might not care whether Amaya was an unlawful entrant and others might well assume — even without the government’s direct proof — that Amaya had entered unlawfully, just because of proof that the green card was not genuine. Still, Amaya would be better off without proof of unlawful entry, especially in a case that some jurors might think was overcharged and should not have been brought; after all, Amaya had a temporary employment card.
Curiously, the government suggests that proof of unlawful entry is essential to its case, but it is hard to see why. If the government proved that Amaya used a forged green card and did not get it from the INS, a jury could impute knowledge of falsity to Amaya — at least absent proof from Amaya that the forged card was acquired in (surely unusual) circumstances leading him to believe that it was valid. Unlike a single forged $20 bill, illegal green cards with name, photo and alien number do not show up in one’s pocket by chance.
Thus, we have a debatable case under Rule 403. The proof of unlawful entry is not crucial to the government’s case, which is adequate without such proof and which is bolstered by the proof only to a limited degree; at least in theory, Amaya could have entered unlawfully and still somehow ended up with a false green card that he believed to be valid. And the unlawful *46entry evidence is prejudicial. Yet the rule provides for exclusion only if the prejudice “substantially” outweighs probative value.
Ordinarily the district judge is entitled to latitude in applying Rule 403 to particular facts, partly because the issue turns on a balancing of unquantifiable considerations, United States v. Pitrone, 115 F.3d 1, 8 (1st Cir.1997); Dente v. Riddell, Inc., 664 F.2d 1, 5 (1st Cir.1981), and in this case the call is close enough that a judgment either way is defensible. But we cannot affirm the district judge on a ground he did not adopt, unless the issue could be decided only one way. Cf. Vincent v. Louis Marx & Co., Inc., 874 F.2d 36, 41 (1st Cir.1989).
Amaya is entitled to present this alternative Rule 403 ground for exclusion to the district judge; and if the judge excludes the evidence on this ground (as appears likely), his resolution must stand whether or not the government then chooses to dismiss the case. If the case goes forward and Amaya chooses to testify, the district judge has explicitly reserved the question whether unlawful entry evidence might be admissible for impeachment.
One issue remains. Amaya has suggested that the government failed to explain adequately to the district court why the unlawful entry evidence was relevant. One objecting to evidence must state the ground, unless it is clear from context, Fed.R.Evid. 103(a)(1); United States v. Carrillo-Figueroa, 34 F.3d 33, 39 (1st Cir.1994), but the rules do not say how much a party arguing for admissibility must say to preserve a proffer beyond making clear the substance of the offer.
Here the proffer was preserved. The government told the district judge that its evidence of unlawful entry was relevant to an element of the offense and did not mislead the district judge; it was not asked to elaborate on the reasoning behind its relevance argument nor given much opportunity to do so. Further, the government’s concern about proving knowledge had already been aired at the earlier hearing that led to excluding the agent’s testimony on Amaya’s inculpatory statements.
Under these circumstances, the exclusion of the certificate of unlawful entry and the grant of the motion in limine excluding testimony must be reversed and the matter remanded to the district court without prejudice to its consideration of the Rule 403 objection. The government has said almost nothing about the other certificate so we need not address it.
It is so ordered.
. The district court did not dispute the prosecutor’s claim that his office was unaware of the agent’s encounter with Amaya until December 31, 2002, and that the evidence was then promptly disclosed; but in any event the government has not sought review of the suppression order.
. The judge excluded both the certificate and related testimony before hearing from the prosecutor and then, when requested, gave the prosecutor an opportunity to change the court's mind. Although the prosecutor may not have explicitly stated that he wanted the court to reconsider its ruling as to the testimony as well as the certificate, the relevance issue is common to both and the prosecutor did more than enough to make plain the government’s view that the unlawful entry evidence was relevant to an element of the crime. Thus, we treat the objection as fully preserved. Cf. Beech Aircraft Corp. v. Rainey, 488 U.S. 153, 174-75 & n. 22, 109 S.Ct. 439, 102 L.Ed.2d 445 (1988).
. On its face, the local rule applies only to materials listed in Fed.R.Crim.P. 16(a)(1)(A)-(D); but subsection (E)'s provisions were located in (C) at the time of the local rule's last amendment in 1998. The parties have ignored the relocation and, as it does not alter the outcome in this case, so do we.
. The possibility remains that Amaya might have entered unlawfully but adjusted his status to that of a lawful permanent resident thereafter. According to the government, the certificate excluded by the trial court also indicated that Amaya's status had not been upgraded.