Defendant-appellant Jeffrey W. Spinney challenges his convictions for aiding and abetting an armed bank robbery (count 2) *233and aiding and abetting the use of a firearm during and in relation to a crime of violence (count 3).1 In our view, the two crimes, despite superficial similarities, require the application of dissimilar legal standards. Because the evidence amassed by the government falls between these stools, we affirm the first conviction but reverse the second.
I. BACKGROUND
We limn the pertinent facts in the light most favorable to the government, see United States v. Ortiz, 966 F.2d 707, 710-11 (1st Cir.1992), cert. denied, — U.S. -, 113 S.Ct. 1005, 122 L.Ed.2d 154 (1993), paying particular heed to those details that arguably reflect what appellant knew and when he knew it.
On August 20, 1991, at around 3:00 p.m., Gerald Mohan, a conscientious FBI agent, noticed appellant (a person previously known to him) sitting on the steps of the Federal Building in Lowell, Massachusetts. Mohan decided to mount an impromptu surveillance. As he was positioning his vehicle, a blue Oldsmobile arrived at the scene. After appellant entered the Oldsmobile, it made several quick turns and then pulled to the curb. The driver (subsequently identified as Paul Kirvan) moved into the front passenger seat and appellant took the wheel. Kirvan and appellant proceeded to criss-cross the streets around the Lowell Institution for Savings (the Bank). Mohan testified that the pair’s driving pattern appeared to be part of a process of careful scrutiny.
Roughly fifteen minutes after beginning surveillance, Mohan followed the Oldsmobile to Academy Drive. There he observed a classic getaway “switch site” on a dead end approximately 0.3 miles from the Bank. Appellant and Kirvan next returned to the vicinity of the Federal Building. On Fayette Street, Kirvan alighted from the Oldsmobile (which he owned), entered a parked Chevrolet Monte Carlo (later ascertained to have been stolen some distance away), and began driving toward the Bank. Appellant followed him in the Oldsmobile, and Mohan followed both of them, caravan-style, in his own vehicle. When the two drivers veered in separate directions, Mohan lost sight of both cars. He circled in the general vicinity and, some four minutes later, glimpsed the Chevrolet at a standstill in the Bank’s parking lot.
Mohan made a U-turn and headed back to the Bank. His efforts were unavailing; at that precise moment, the Chevrolet accelerated rapidly out of the parking lot and passed him (going in the opposite direction). Kirvan was alone in the car. Mohan made yet another U-tum and unsuccessfully gave chase.
At approximately 3:25 p.m., ostensibly during the brief interval in which Mohan lost track of his quarry, a masked Kirvan entered the Bank, instructed those present not to move, jumped over the tellers’ counter, stuffed the contents of several cash drawers into a garbage bag, leapt back over the counter, and fled. Although Kirvan brandished a handgun at the height of the robbery, a teller testified that the weapon was not visible when he entered the Bank.
*234Other percipient witnesses reported that, mid-afternoon of the same day, they saw a Chevrolet Monte Carlo speed down Academy Drive. Two men, one holding a bag, emerged from the vehicle, crossed through two gates, and drove away in a waiting blue car. The witnesses were unable to identify either suspect positively, although one of the men “looked like” appellant. The Lowell police recovered the Chevrolet that afternoon. They found Kirvan’s blue Oldsmobile the next morning, abandoned not far from the switch site.
The government established that appellant and Kirvan had been close Mends for many years. Telephone toll records reflected seventy-three calls between the men’s residences in the nineteen days preceding the robbery, including eleven calls on August 19.
A federal grand jury indicted appellant on counts of conspiracy to commit bank robbery, see 18 U.S.C. § 371 (which count was later dismissed), aiding and abetting an armed bank robbery, see 18 U.S.C. § 2113(d), and aiding and abetting the use of a firearm during and in relation to a crime of violence, see 18 U.S.C. § 924(c). A jury trial eventuated. After appellant unsuccessfully moved for judgment of acquittal, the jurors returned a guilty verdict on each remaining count.2 The court sentenced appellant to serve 262 months in prison for armed robbery and imposed a mandatory sixty-month consecutive sentence in respect to the firearms charge. This appeal followed.
II. ARMED BANK ROBBERY
Appellant challenges the sufficiency of the evidence supporting his conviction on count 2. Our task is straightforward. We must ascertain whether, “after assaying all the evidence in the light most amiable to the government, and taking all reasonable inferences in its favor, a rational factfinder could find, beyond a reasonable doubt, that the prosecution successfully proved the essential elements of the crime.” United States v. O’Brien, 14 F.3d 703, 706 (1st Cir.1994). In performing this task, we do not pass upon the credibility of witnesses, see id., nor do we demand that the government disprove every hypothesis consistent with the defendant’s innocence, see United States v. Echeverri, 982 F.2d 675, 677 (1st Cir.1993).
The jury reached its verdict in this case on the basis of circumstantial evidence. Reliance on indirect, as opposed to direct, evidence in a criminal case is both permissible and commonplace. See O’Brien, 14 F.3d at 706 (observing that “the criminal law does not place a special premium on direct evidence”). In making such judgments, “juries are not required to examine the evidence in isolation, for ‘individual pieces of evidence, insufficient in themselves to prove a point, may in cumulation prove it.’” Ortiz, 966 F.2d at 711 (quoting Bourjaily v. United States, 483 U.S. 171, 179-80, 107 S.Ct. 2775, 2780-81, 97 L.Ed.2d 144 (1987)). Thus, when a jury draws inferences from circumstantial evidence, a reviewing court should refrain from second-guessing the ensuing conclusions as long as (1) the inferences derive support from a plausible rendition of the record, and (2) the conclusions flow rationally from those inferences. See id.
We add a cautionary note. Despite the deference that characterizes appellate review of jury verdicts, juries do not have carte blanche. The appellate function, properly understood, requires the reviewing court to take a hard look at the record and to reject those evidentiary interpretations and illations that are unreasonable, insupportable, or overly speculative. See, e.g., United States v. Valerio, 48 F.3d 58, 64 (1st Cir.1995); United States v. Loder, 23 F.3d 586, 589-92 (1st Cir.1994). This function is especially important in criminal eases, given the prosecution’s obligation to prove every element of an offense beyond a reasonable doubt.
In this instance, the jury convicted the appellant on count 2 as an aider and abettor. See 18 U.S.C. § 2(a). Under this theory of accomplice liability, Spinney would be guilty only if the government proved (1) *235that Kirvan (the principal) committed the substantive offense (armed bank robbery), and (2) that Spinney (the accomplice) became associated with the endeavor and took part in it, intending to ensure its success. See Nye & Nissen v. United States, 336 U.S. 613, 619, 69 S.Ct. 766, 769-70, 93 L.Ed. 919 (1949); Ortiz, 966 F.2d at 711 n. 1. Because the jury heard plethoric evidence from which it rationally could conclude that Kirvan committed armed bank robbery, we direct our analysis to the second of these two elements.
The central requirement of the second element is “a showing that the defendant consciously shared the principal’s knowledge of the underlying criminal act, and intended to help the principal.” United States v. Taylor, 54 F.3d 967, 975 (1st Cir.1995). In a prosecution for armed bank robbery, this shared knowledge requirement is binary; it extends both to awareness of the robbery and to comprehension that a weapon would likely be used. See United States v. Jones, 678 F.2d 102, 106 (9th Cir.1982) (explaining that, to convict under § 2113(d), the prosecution must “show that the defendant aided and abetted the principal both in the act of bank robbery and in the principal’s use of ‘a dangerous weapon or device’ during the act”) (citing other cases). Appellant maintains that neither finding is justified here. In the first place, he claims that the evidence does not adequately show that he knew Kirvan aspired to rob the Bank and nonetheless endeavored to help him. In the second place, he claims that the record is devoid of any proof that he knew about the actual or intended use of a gun.3
1. Shared Knowledge of the Bobbery. The first challenge need not occupy us for long. Appellant does not assert that he was “merely present” at the scene. See Ortiz, 966 F.2d at 711 (“Mere association between the principal and those accused of aiding and abetting is not sufficient to establish guilt; ... nor is mere presence at the scene and knowledge that a crime was to be committed sufficient to establish aiding and abetting.”) (internal quotation marks omitted). Rather, he focuses on the lack of direct evidence placing him at the switch site, in or near the Bank, or in Kirvan’s company at any time except immediately prior to the commission of the crime, thereby attempting to raise doubts about whether he had any specific intent to assist in the enterprise.
This argument is flawed in its presumption that a dearth of direct evidence somehow precludes jurors from drawing logical inferences based on available circumstantial evidence. Contrary to the burden of appellant’s thesis, it is precisely those situations that involve an absence of direct evidence in which circumstantial evidence must be most closely analyzed. See O’Brien, 14 F.3d at 706 (explaining that a lack of direct evidence spurs examination of indirect evidence). In the last analysis, the persuasive power of circumstantial evidence is attributable more to its relevance and probative force than to the presence of complementary direct evidence.
Having in mind Mohan’s observations, the events that transpired on Academy Drive, the vehicles abandoned in close proximity to the Bank, and the telephone logs, a completely rational juror need make only modest inferential leaps to arrive at a founded conclusion that the two long-time friends planned the crime, the flight, and the car switch. See United States v. Olbres, 61 F.3d 967, 971-72 (1st Cir.1995) (finding a “sturdy infrastructure,” provided by “circumstantial and suggestive” evidence, for making sound rational inferences); Taylor, 54 F.3d at 975 (similar); see also Ortiz, 966 F.2d at 711 (remarking that “[t]he sum of an evidentiary presentation may be greater than its constituent parts”) (quoting Bourjaily, 483 U.S. at 180, 107 S.Ct. at 2781). Since every necessary inference is adequately rooted in the record, we reject as meritless appellant’s assignment of error based on a supposed lack *236of proof that he knew of, and helped to further, Kirvan’s desire to rob the Bank.4
2. Shared Knowledge of the Weapon. Appellant’s stronger challenge is directed at the jury’s finding of shared knowledge, prior to the commission of the crime, that Kirvan would use a firearm. See United States v. Dinkane, 17 F.3d 1192, 1197 (9th Cir.1994) (holding that, for purposes of § 2113(d), aid-iiig and abetting requires prior knowledge of weapon); see generally United States v. de la Cruz-Paulino, 61 F.3d 986, 999-1000 (1st Cir.1995) (stating, in aiding and abetting case, that shared knowledge must be prior knowledge).
A participant in the holdup of a bank will be found to be an aider and abettor of an armed robbery only if the government can provide an additional piece of the puzzle: proof that the accomplice “knew a dangerous' weapon would be used [in the robbery] or at least ... was on notice of the likelihood of its use,” United States v. Sanborn, 563 F.2d 488, 491 (1st Cir.1977); accord United States v. Ferreira, 625 F.2d 1030, 1082 (1st Cir.1980). Refined to bare essence, appellant’s asseveration on this point is that the evidence, taken as a whole, is so sparse that it does not satisfy the Sanborn standard; there are simply no facts, he tells us, from which a reasonable juror could extrapolate to a finding of shared knowledge.5 We do not agree.
In terms, the Sanborn rubric requires only proof of “notice of ... likelihood” to satisfy this prong of the shared knowledge element in connection with a charge of aiding and abetting an armed bank robbery. This phrase is not self-defining. Hence, it is important to search out its meaning.
We start from the premise that the San-born court’s formulation of the shared knowledge requirement is not merely an awkward locution. Other courts have adopted it, see, e.g., United States v. McCaskill, 676 F.2d 995, 998 (4th Cir.), cert. denied, 459 U.S. 1018, 103 S.Ct. 381, 74 L.Ed.2d 513 (1982); Ingram v. United States, 592 A.2d 992, 1003 (D.C.App.), cert. denied, 502 U.S. 1017, 112 S.Ct. 667, 116 L.Ed.2d 757 (1991), and it stands in marked contrast — almost as point and counterpoint — to the “practical certainty” formulation that courts have developed for assessing the shared knowledge requirement applicable to aiding and abetting firearms charges brought under 18 U.S.C. § 924(c). See infra Part III. The glaring linguistic difference between the two formulations guides our inquiry.
Knowledge is a concept, not an absolute. In the law, as in life, “knowledge” means different things in different contexts. Accordingly, we believe it is useful to view the concept as a continuum.
At one end of the continuum is what the law commonly calls “constructive knowledge.” Constructive knowledge is the law’s way of recognizing that, given an awareness of certain subsidiary facts, a person is quite likely to know, can be expected to know, or at least should know that a further fact exists. See Black’s Law Dictionary 314 (6th ed. 1990) (“If one by exercise of reasonable care would have known a fact, he is deemed to have had constructive knowledge of said fact_”). By way of illustration, if an easily visible foreign object is on a staircase for an appreciable length of time, the law accepts the reasonableness of a conclusion that the occupier of the premises “knew” of its presence (even though there is no evidence that the occupier actually knew, by observation or report, of the object’s whereabouts).
At the other end of the continuum is what the law commonly calls “actual knowledge.” Actual knowledge, as the term implies, reduces the need for inference; it suggests the presence of particular evidence which, if credited, establishes conclusively that the person in question knew of the existence of the fact in question. See id. at 873 (defining *237actual knowledge as “positive, in contrast to imputed or inferred, knowledge of a fact”). To carry our example forward, if witnesses testify that the occupier himself placed the foreign object on the stair, or remarked its location, that testimony, if believed, establishes that the occupier actually knew of its presence.
The concepts of constructive and actual knowledge do not occupy the entire span of the continuum. Knowledge varies in origin, degree, and an array of other respects. These gradations are best visualized as way stations that dot the length of the hypothetical knowledge continuum. Notice of likelihood fits into the poorly charted area that stretches between the poles of the continuum. While we believe that, in a criminal case, the reasonable doubt standard requires that notice of likelihood comprise more than constructive knowledge simpliciter, its articulation evokes echoes of constructive knowledge and places the proof requirement closer to that end of the continuum. Actual knowledge, after all, is certain knowledge, see id., and likelihood is not the stuff of certainty.
Logically, then, the Sanborn rubric implies, in a section 2113(d) case, that the defendant’s shared knowledge need not amount to actual knowledge that his cohort intended to use a gun or other dangerous weapon in robbing the bank; indeed, San-born’s disjunctive phraseology, 563 F.2d at 491 (“knew ... or at least ... was on notice of the likelihood”), leaves no doubt that a conviction can be grounded on something less than actual knowledge. We conclude that an enhanced showing of constructive knowledge will suffice. See United States v. Grubczak, 793 F.2d 458, 463 (2d Cir.1986).
While this is a very close case, we think that the evidence clears the notice of likelihood hurdle. Our analysis builds on the human condition. Jurors are “not expected to ignore what is perfectly obvious,” Echeverri, 982 F.2d at 679, but, rather, “to take full advantage of their collective experience and common sense.” O’Brien, 14 F.3d at 708. In this ease, the scheme called for a lone robber to enter a bank during business hours with the intent of looting it. One would expect tellers, guards, customers, and other persons unsympathetic to an unauthorized withdrawal of funds to be on the premises. Under those circumstances, not even the most sanguine criminal would expect clear sailing without some menace in the wind. In short, the circumstances gave rise to constructive knowledge beforehand that the intruder would need a gun or some other dangerous device to accomplish the felons’ agreed goal. See United States v. Powell, 929 F.2d 724, 727 (D.C.Cir.1991) (stating, in dictum, that “possession of a gun ... is virtually essential in [perpetrating a bank robbery]”).
Here, moreover, Spinney was not merely a bit player (say, a lookout or a getaway driver), but a leading man. A jury could reasonably infer from the totality of the attendant circumstances, particularly from the host of telephone calls between Spinney and Kirvan and from Spinney’s participation in the elaborate reconnaissance mission, that he had a major role in planning the heist. Even assuming that there was no specific discussion of the use of a gun, evidence of a defendant’s substantial involvement over the course of several days in planning and orchestrating a robbery, when coupled with actual participation in carrying it out, permits a compelling inference that the defendant knew the salient details of the plot (e.g., the timing of the robbery, the bank’s identity and location, the planned entry by a lone robber). These circumstances seem to us to sustain a finding that Spinney was on notice that Kirvan likely would tote a gun in the course of the upcoming robbery. See Grubczak, 793 F.2d at 464 (relying on evidence of a defendant’s substantial involvement as a planner of and a “principal player[] in the robbery” to help ground “the inference that he had to have been aware of the likely use of a gun”); see also United States v. DeMasi, 40 F.3d 1306, 1316 (1st Cir.1994) (inferring knowledge that weapons would be employed from accomplice’s awareness of the conspirators’ overall plan), cert. denied, — U.S.-, 115 S.Ct. 947, 130 L.Ed.2d 890 (1995).
Appellant decries this approach, claiming that it necessitates the stacking of inference upon inference. In one sense, at least, this may be so — but “[t]he rule is not that an *238inference, no matter how reasonable, is to be rejected if it, in turn, depends upon another reasonable inference; rather, the question is merely whether the total evidence, including reasonable inferences, when put together is sufficient to warrant a jury to conclude that defendant is guilty beyond a reasonable doubt.” Dirring v. United States, 328 F.2d 512, 515 (1st Cir.), cert. denied, 377 U.S. 1003, 84 S.Ct. 1939, 12 L.Ed.2d 1052 (1964). Chains of inference are a familiar, widely accepted ingredient of any process of ratiocination. This method of reasoning, commonly called logic, is regularly relied upon in the realm of human endeavor, and should not be forbidden to a criminal jury.
Of course, the inferential chain must be strong6 — but here, the hypothesis upon which Spinney’s section 2113(d) conviction rests is not at all dubious. On this record, despite the lack of direct evidence and the uncertainties associated with that lack, the jury rationally could find Spinney to have been an architect of, and an active participant in, the robbery. Given these available findings, and the persuasive force of the other permissible inferences supported by the overall circumstances, we cannot say that the jury exceeded its proper province in concluding that Spinney was on notice of the likelihood that Kirvan would use a gun.7 See Sanborn, 563 F.2d at 490.
III. THE FIREARMS CHARGE
Appellant also challenges the sufficiency of the evidence in regard to his conviction under 18 U.S.C. § 924(c). Although here, too, appellant is charged as an aider and abettor, his assignment of error raises a somewhat different question. To prove that a defendant aided and abetted a violation of section 924(c), the government must establish that the defendant knew “to a practical certainty that the principal would be [using] a gun.” United States v. Torres-Maldonado, 14 F.3d 95,103 (1st Cir.) (quoting Powell, 929 F.2d at 728), cert. denied, — U.S.-, 115 S.Ct. 193, 130 L.Ed.2d 125 (1994); accord DeMasi, 40 F.3d at 1316.
The government strives to collapse the linguistically different standards for aiding and abetting liability under 18 U.S.C. §§ 2113(d) and 924(c), respectively, at the less taxing end of the knowledge continuum. We are uncomfortable with this esemplastic approach. Particularly when juxtaposed with “notice of ... likelihood,” we believe that “practical certainty” is a rubric that calls for proof verging on actual knowledge, see Model Penal Code § 2.02 at 236 n. 13 (1985) (“With respect to result elements, one cannot of course ‘know’ infallibly that a certain result will follow from engaging in conduct, and thus to some extent ‘knowledge,’ when applied to result elements, includes a contingency factor as well. This is expressed defi-nitionally in terms of whether the actor is ‘practically certain’ that the result will follow.”), and, thus, presents a considerably higher hurdle for the prosecution to overcome. Nor do we think that we are free to cut this hurdle down to size either by reading significantly dissimilar articulations to denote a single meaning or by treating one of them as a linguistic accident. Courts invite error *239when they try to weigh meaning only after placing a thumb on the scale, or when they too freely write off as malapropos words carefully chosen in the past. If principle is to prevail, we must give effect to the obvious difference in standards of knowledge.8
In this ease, the difference is dispositive. The government’s argument boils down to an assertion that the jury could infer that Spinney was practically certain of Kirvan’s anticipated use of a gun based on the evidence we have just reviewed, principally the confederates’ joint design of the robbery. Yet, the government adduced no evidence suggesting that firearms were actually contemplated in the planning stages, or that Spinney had any actual knowledge that Kirvan would be armed. Under the circumstances, we conclude that the government’s best evidence (that Spinney helped to mastermind the robbery), taken in the light most favorable to the verdict, even when coupled with the jury’s ability to make intuitive judgments, is insufficient to support the requisite inference of practical certainty.9 See, e.g., Powell, 929 F.2d at 729; United States v. Hamblin, 911 F.2d 551, 558-59 (11th Cir.1990), cert. denied, 500 U.S. 943, 111 S.Ct. 2241, 114 L.Ed.2d 482 (1991).
In a last-ditch effort to save the day, the government directs our attention to a series of drug cases in which knowledge of a vessel’s cargo was imputed to crew members. See, e.g., United States v. Guerrero-Guerrero, 776 F.2d 1071 (1st Cir.1985), cert. denied, 475 U.S. 1029, 106 S.Ct. 1233, 89 L.Ed.2d 342 (1986); United States v. Quejada-Zurique, 708 F.2d 857 (1st Cir.), cert. denied, 464 U.S. 855, 104 S.Ct. 173, 78 L.Ed.2d 156 (1983). But these scenarios are readily distinguishable from the case at bar. In those cases, guilty knowledge was inferred from the crew members’ close proximity to detectable quantities of drugs over an extended period of time.10 See Guerrero-Guerrero, 776 F.2d at 1074-75; Quejada-Zurique, 708 F.2d at 859-60. Here, Spinney did not even enter the bank, and there was no evidence that Kirvan had the weapon in his possession while he and Spinney were together before the robbery, let alone that it was visible or otherwise detectable at that juncture. Put bluntly, even though Spinney may have spent much time with Kirvan devising the plan, and was on notice of the likelihood that a gun would be used in the course of the robbery, there is simply no evidence to support a reasoned conclusion that Spinney was practically certain that Kirvan would be armed.
In sum, “likelihood” and “practical certainty” are not equivalent terms. Applying the practical certainty rubric, we hold that the jury reached its verdict on the section 924(c) count without an adequate evidentiary foundation.
IV. CONCLUSION
This is the rare case in which the evidence, viewed most congenially to the government, *240passes muster under the notice of likelihood test and, therefore, justifies the appellant’s conviction for aiding and abetting an armed bank robbery, 18 U.S.C. § 2113(d), but, nevertheless, fails the practical certainty test and, therefore, does not justify the appellant’s conviction for aiding and abetting the principal’s use of a firearm during and in relation to a crime of violence, 18 U.S.C. § 924(c). This result, though it is unusual, does not strike us as strange. See, e.g., Model Penal Code § 2.02 at 236-37 n. 13 (suggesting that, where knowledge is in issue, it is sometimes “meaningful to draw a line between practical certainty and awareness of substantial risk”). So it is here: though notice of likelihood must be proven beyond a reasonable doubt, the government can meet this burden by an evidentiary showing less than is required to prove that the defendant knew to a practical certainty that a gun would be used.
We need go no further. The law is full of complexities, and language is, at best, an imperfect device for capturing the energy of ideas. Here, interpreting the words used by the Sanborn court (“notice of ... likelihood”), on the one hand, and by the Powell court (“practical certainty”), on the other hand, “in the light of the tacit assumptions upon which it is reasonable to suppose that the language was used,” Ohio v. Agler, 280 U.S. 379, 383, 50 S.Ct. 154, 155, 74 L.Ed. 489 (1930) (Holmes, J.), it is unsurprising to discover a fork in the road.
The appellant’s conviction on count 2 is affirmed and the appellant’s conviction on count 3 is reversed.