SOUTER, Associate Justice.

This appeal is from the district court's denial of a preliminary injunction against enforcing two city ordinances prohibiting coercive or risky behavior by panhandlers, other solicitors, and demonstrators seeking the attention of motor vehicle drivers. We affirm.

I.

A.

For a decade, the public policy of the City of Worcester has been pushed and pulled by concerns about panhandling on its streets. In 2005, the City adopted a plan to reduce its prevalence that included public education about charitable organizations and increased efforts by social service agencies. The City posted signs reading "Panhandling is not the Solution!" and encouraged residents to redirect their contributions to charities serving the poor. Criticism of the signs led the City to take them down by August of 2006.

The issue became prominent again in the summer of 2012, when the City Manager sent a memorandum to the City Council describing a number of "[c]ommon concerns" about panhandling, including the perception that the City gave too little help to the needy, as well as the "fear/intimidation" of residents and "public safety" hazards arising from roadside solicitation. The memo reported that in the course of one year, Worcester police had been dispatched to 181 incidents of aggressive behavior by individuals suspected of panhandling, resulting in five arrests. The Manager observed that there was no "current mechanism for tracking or compiling statistics on panhandling or its impact on the community," and proposed a "multi-faceted, community-wide response that incorporates direct service providers, non-profit agencies, area businesses, policymakers, and public services."

The following October, the City Manager reported again, this time with data collected by a team of case workers and an outreach worker who had spent months educating 38 panhandlers about the resources and services available to them from the City. The report concluded that the "outcomes of the outreach worker's engagement efforts [were] encouraging," with a majority of the consulted panhandlers affirming "a desire to work with the outreach worker to obtain assistance." At the same time, the Manager noted that outreach efforts failed to address "another side of the issue": the "issue of public safety — when individuals are walking in and out of traffic to collect money in intersections, traffic islands, and roadways."

In light of that problem and the earlier police reports, the Manager advised the City Council to adopt two ordinances addressing the safety risks. The first was "An Ordinance Prohibiting Aggressive Begging, Soliciting and Panhandling in Public Places" (Aggressive Panhandling Ordinance), which would make it "unlawful for any person to beg, panhandle or solicit any other person in an aggressive manner." It would apply to "soliciting" in the form of "using the spoken, written, or printed word, bodily gestures, signs, or other means of communication with the purpose of obtaining an immediate donation of money or other thing of value," and it defined "aggressive" conduct at two levels. The definition included obviously threatening behavior, as by soliciting someone "in a manner ... likely to cause a reasonable person to fear immediate bodily harm," using "violent or threatening language," or blocking a person's right of way. It further covered a range of potentially coercive though not conventionally aggressive behaviors, including soliciting from someone waiting in line to buy tickets or enter a building; soliciting after dark, calculated as "the time from one-half hour before sunset to one-half hour after sunrise"; continuing to solicit from a person after the receipt of a negative response; and soliciting anyone within 20 feet of an entrance or parking area of a bank, automated teller machine, public transportation stop, pay phone, theater, or any outdoor commercial seating area like a sidewalk café. The text of the ordinance was preceded by a proposed "Declaration of Findings and Policy," which detailed the City's concerns about how the behaviors to be banned threatened the safety of Worcester residents. In particular, the declaration stated that "[p]ersons approached by individuals asking for money, objects or other things of any value are particularly vulnerable to real, apparent or perceived coercion when such request is accompanied by ... [certain forms of] aggressive behavior."

The second proposal, "An Ordinance Relative to Pedestrian Safety" (Pedestrian Safety Ordinance), targeted distractions on public roads:

The ensuing City Council debates were a mix of reactions. Some councilmembers objected that existing laws already regulated intimidating behaviors and several protested that the primary purpose of the ordinances was less to enhance public safety than to eliminate unsightly panhandling, despite the mayor's espousal of the proposals as aimed at resolving "purely a public safety issue." The most prominent reservations were about the effect the ordinance would have on Worcester's traditional "tag days": fundraisers and publicity campaigns for local charities, civic organizations, and political groups, whose participants commonly used traffic islands and medians. While several councilmembers denounced the tradition as an "accident waiting to happen," especially when children participated, others worried that prohibiting tag days would unduly harm local civic groups. Some of these qualms were addressed at a meeting of the Worcester Joint Public Health & Human Services and Municipal Operations Committee, where the City Solicitor said that the text of the Pedestrian Safety Ordinance allowed the police "an element of discretion" in identifying which roadside activity posed a threat to public safety and had to be stopped. The vote approving the proposals nonetheless included an express repeal of the City's existing provision for tag day permits.

That vote came in January of 2013, when the City Council adopted the Aggressive Panhandling Ordinance and the Pedestrian Safety Ordinance, codifying them at ch. 9, § 16(d) and ch. 13, § 77(a) of the Worcester Revised Ordinances, respectively. After a "grace period" during which the police distributed flyers telling panhandlers and other Worcester residents about the new ordinances, but made no arrests, the police began enforcement. Between March 1 and March 20, 2013, they arrested four individuals for violating the Aggressive Panhandling Ordinance, including one man arrested twice; all four were given multiple warnings about the new rules prior to arrest. The record shows no arrests for violation of the Pedestrian Safety Ordinance. When protestors staged a small demonstration against the ordinances in February of 2013, featuring individuals soliciting donations from traffic islands, the police did not disturb the protest.1

B.

Appellants Robert Thayer and Sharon Brownson are homeless people who regularly solicit donations on the sidewalks of Worcester, commonly stepping into the roads to receive contributions. Both have been warned by police that they faced arrest unless they stopped panhandling this way. Appellant Tracy Novick is an elected member of the Worcester School Committee who has customarily displayed political signs on median strips and traffic circles during the campaign season.

In May of 2013, Thayer, Brownson, and Novick brought this suit challenging the new ordinances as violating their rights under the First Amendment and the Due Process and Equal Protection Clauses of the Fourteenth. They sought declaratory and injunctive relief and monetary damages. The First Amendment claim has been presented as a facial challenge based on substantial overbreadth, and we continue to regard it as such here. The vagueness claim is necessarily of the as-applied variety. See Holder v. Humanitarian Law Project, 561 U.S. 1, 18-19, 130 S.Ct. 2705, 177 L.Ed.2d 355 (2010) ("We consider whether a statute is vague as applied to the particular facts at issue, for `[a] plaintiff who engages in some conduct that is clearly proscribed cannot complain of the vagueness of the law as applied to the conduct of others.'" (alteration in original) (quoting Hoffman Estates v. Flipside, Hoffman Estates, Inc., 455 U.S. 489, 495, 102 S.Ct. 1186, 71 L.Ed.2d 362 (1982))). The equal protection challenge seems to include both varieties, as we will mention again later.

The following October, the District Court for the District of Massachusetts denied a preliminary injunction. Although none of the appellants had been arrested under the new laws, the court found that all three had standing to sue, because all engage in activities that reasonably lead them to expect the ordinances to be enforced against them. But the court found that they had failed to meet their burden of demonstrating a likelihood of success on the merits of any of their constitutional claims. As to the First Amendment challenge, the court did not describe the claim as a facial overbreadth challenge. After noting that the ordinances prohibited all aggressive solicitations and roadside demonstrations, regardless of the speaker's message or ideology, it concluded that the regulations were content-neutral time, place, or manner restrictions subject only to intermediate scrutiny. Because the ordinances furthered a substantial interest in public safety and freedom from coercion, were narrowly tailored to prohibit only aggressive or distracting activity, and left open ample alternate spaces for solicitation, the court held that the appellants had shown no probability of demonstrating a violation. With regard to the appellants' due process claim that the ordinances were unconstitutionally vague, the district court dismissed the argument as "strained" and "disingenuous," finding that both ordinances provided sufficient detail to constrain the police's discretion in enforcing the new rules. Finally, the district court saw no probability of success in the appellants' equal protection challenge that the two ordinances discriminated against the poor and homeless. Having already noted the lack of evidence of uneven enforcement, the court found that the provisions applied facially to all groups and that the appellants had failed to rebut the record evidence suggesting that the City Council was motivated by legitimate concerns about coercion and safety.

II.

In assessing whether to grant or to deny a preliminary injunction, a district court must address four considerations: "(1) the plaintiff's likelihood of success on the merits; (2) the potential for irreparable harm in the absence of an injunction; (3) whether issuing an injunction will burden the defendants less than denying an injunction would burden the plaintiffs; and (4) the effect, if any, on the public interest." Gonzalez-Droz v. Gonzalez-Colon, 573 F.3d 75, 79 (1st Cir.2009) (internal quotation marks omitted). We review the district court's decision for abuse of discretion, id., and will not reverse unless the district court made a mistake of law, clearly erred in its factual assessments, or otherwise abused its discretion. Id.; see United States v. Lewis, 517 F.3d 20, 24 & n. 4 (1st Cir.2008) ("In practice, [the abuse of discretion] standard contemplates de novo review of abstract questions of law" because "a mistake of law is always an abuse of discretion.").

A.

i.

There is no dispute here that the combined speech and physical activity performed to deliver the messages occur in public forums. See United States v. Grace, 461 U.S. 171, 177, 103 S.Ct. 1702, 75 L.Ed.2d 736 (1983) ("[P]ublic places historically associated with the free exercise of expressive activities, such as streets, sidewalks, and parks, are considered, without more, to be public forums." (quotation marks omitted)). Thus, the first issue we address is whether the regulations are based on the content of the speech. If yes, the standard of scrutiny is strict: the regulation "must be narrowly tailored to promote a compelling Government interest," such that no "less restrictive alternative would serve the Government's purpose." United States v. Playboy Entm't Grp., 529 U.S. 803, 813, 120 S.Ct. 1878, 146 L.Ed.2d 865 (2000). If no, the standard is less demanding: the government "may impose reasonable restrictions on the time, place, or manner of protected speech," so long as "they are narrowly tailored to serve a significant governmental interest" and "leave open ample alternative channels for communication of the information." Ward v. Rock Against Racism, 491 U.S. 781, 791, 109 S.Ct. 2746, 105 L.Ed.2d 661 (1989) (quoting Clark v. Community for Creative Non-Violence, 468 U.S. 288, 293, 104 S.Ct. 3065, 82 L.Ed.2d 221 (1984)). Under this more lenient "intermediate" scrutiny, a law need not be the "least restrictive" means of achieving the government's interest, so long as the interest "would be achieved less effectively absent the regulation" and the law does not "burden substantially more speech than is necessary to further the government's legitimate interests." Id. at 798-99, 109 S.Ct. 2746.

In determining whether a particular regulation is content-neutral, the principal enquiry is "whether the government has adopted a regulation of speech because of disagreement with the message it conveys." Ward, 491 U.S. at 791, 109 S.Ct. 2746. A regulation that has "an incidental effect on some speakers or messages but not others" may still qualify as content-neutral so long as the regulation "is justified without reference to the content of the regulated speech." Id. (emphasis omitted). After examining the texts and independent evidence of intent behind the ordinances, we think there is no serious question that the district court was correct in finding that the restrictions were not based on the content of the speech within the terms of First Amendment doctrine.

To begin with, the text of the ordinances does not identify or affect speech except by reference to the behavior, time or location of its delivery, identifying circumstances that raise a risk to safety or that compromise the volition of a person addressed to avoid solicitation: it is aggressive, particularly obtrusive or alarming or risky solicitation that is forbidden, along with distracting activity on traveled roadways and traffic islands.

This is not to deny that certain subjects of speech and even certain messages are associated with the targeted behavior. Panhandling and solicitation of immediate donations convey messages of need, and waving placards at traffic islands may often be political expression. But if the mere association of certain behavior with certain subjects were to amount, in itself, to a content basis for First Amendment scrutiny, the point behind content discrimination would be lost. That point is to bar the government from suppressing speech because it disapproves the message, see Playboy Entm't Grp., 529 U.S. at 812, 120 S.Ct. 1878 ("Laws designed or intended to suppress or restrict the expression of specific speakers contradict basic First Amendment principles."), not to give every message maximum protection no matter how or where or when it is delivered, cf. Ward, 491 U.S. at 802, 109 S.Ct. 2746 ("That [a regulation] may reduce to some degree the potential audience for respondent's speech is of no consequence, for there has been no showing that the remaining avenues of communication are inadequate."). Even a statute that restricts only some expressive messages and not others may be considered content-neutral when the distinctions it draws are justified by a legitimate, non-censorial motive. See Hill v. Colorado, 530 U.S. 703, 724, 120 S.Ct. 2480, 147 L.Ed.2d 597 (2000) (finding statute that distinguishes "speech activities likely to have [certain undesirable] consequences from speech activities... that are most unlikely to have those consequences" to be content-neutral); Clatterbuck v. City of Charlottesville, 708 F.3d 549, 556 (4th Cir.2013) ("[N]ot every content distinction merits strict scrutiny; instead, a distinction is only content-based if it distinguishes content with a censorial intent to value some forms of speech over others....") (internal quotation mark omitted).

To be sure, there is evidence in the record that over the course of several years some public officials have been of a mind to suppress panhandling, though not other forms of solicitation, regardless of deportment, location, or circumstances, owing to the impression it gives about the social responsibility of the City government. But when there is further evidence to look to, the motives of discrete officials are not necessarily to be taken as the predominant intent of the local government. See Rhode Island v. Narragansett Indian Tribe, 19 F.3d 685, 699 (1st Cir. 1994) ("[T]he overarching rule is that `statements by individual legislators should not be given controlling effect'; rather, such statements are to be respected only to the extent that they `are consistent with the statutory language.'" (quoting Brock v. Pierce Cnty., 476 U.S. 253, 263, 106 S.Ct. 1834, 90 L.Ed.2d 248 (1986))).

The ordinances adopted here come with a preamble and accompanying evidence that provide good reason to accept the ostensible objects of the ordinances as the true ones, that is, not suppressing certain kinds of messages but regulating their delivery. The first of these reasons is the fairness of the City's working premise that there are particular, commonly acknowledged circumstances, unrelated to the expression of particular views and messages, in which solicitation can cause serious apprehensiveness, real or apparent coercion, physical offense, or even danger to the person addressed or to all parties. We are not dealing here, in other words, with a mere attempt to suppress a message that some people find distasteful for its content. Cf. Forsyth Cnty., Ga. v. Nationalist Movement, 505 U.S. 123, 134-35, 112 S.Ct. 2395, 120 L.Ed.2d 101 (1992) ("Speech cannot be ... punished or banned ... simply because it might offend a hostile mob."). A person can reasonably feel intimidated or coerced by persistent solicitation after a refusal, and can reasonably feel trapped when sitting in a sidewalk café or standing in line waiting for some service or admittance. And even the stout-hearted can reasonably fear assault when requests for money are made near an ATM where cash may have been obtained and so provide temptation to snatch a wallet or purse. These are not imaginary concerns that smell of pretext. As for the restrictions on using traveled roadways or traffic islands for solicitation or demonstration, it would be hard to gainsay the City Manager's conclusion that the previously unrestricted practice was "an accident waiting to happen" even though it had not happened yet. The whole point of soliciting or demonstrating at such places, after all, is to distract the attention of drivers to some degree. The City Council debates featured recurrent concerns, voiced both by drivers and by former participants in roadside demonstrations, that tag days and other expressive assemblies on medians were dangerous for participants and drivers alike. In sum, common experience confirms that the City has identified behavior and circumstances that it may fairly be concerned about, however much the behavior is associated with certain sorts of messages.

Not only are there thus affirmative indications of a behavioral objective behind the ordinances, but a dearth of the classic indicators of content basis. The most obvious manifestation of content basis, discrimination turning on a speaker's viewpoint, is of course absent here. While there are no restrictions on messages discouraging solicitation, as opposed to encouraging it, that is insignificant simply because there is no evidence or common experience of any such speakers operating on the sidewalks to dissuade potential donors.

Nor does its limitation to solicitations for "immediate" donations of money render the Aggressive Panhandling Ordinance content-based as First Amendment doctrine employs the term. Even assuming that the ban on immediate donations is a content distinction, an assumption which finds scant support in the case law, see Int'l Soc'y for Krishna Consciousness, Inc. v. Lee, 505 U.S. 672, 704-05, 112 S.Ct. 2701, 120 L.Ed.2d 541 (1992) (Kennedy, J., concurring) (finding that a ban on direct donations simply "limit[ed] the manner of that expression to forms other than the immediate receipt of money"); ISKCON of Potomac, Inc. v. Kennedy, 61 F.3d 949, 954-55 (D.C.Cir.1995) (holding that a ban on immediate donations "does not ... totally prohibit a type of expression or a specific message; rather, it merely regulates the manner in which the message may be conveyed"), that distinction alone does not render the ordinance content-based so long as it reflects a legitimate, non-censorial government interest. See Hill, 530 U.S. at 724, 120 S.Ct. 2480 (statute content-neutral, despite its express restrictions on "oral protest, education, or counseling," where content distinction furthers non-censorial government interest). In this case, the limitation of the restrictions in the Aggressive Panhandling Ordinance to solicitations for immediate donations of money reflects the relationship between aggressive street behaviors and certain categories of messages. "In-person solicitation of funds, when combined with immediate receipt of that money, creates a risk of fraud and duress that is well recognized, and that is different in kind from other forms of expression or conduct." Lee, 505 U.S. at 705, 112 S.Ct. 2701 (Kennedy, J., concurring in the judgments).

As for the more general category of subject matter discrimination (based on the subject of a solicitation, say, or of a political placard), this form of selectivity is not only missing here; its very absence is one of the grounds for the appellants' claim of overbreadth. Girl Scout cookie sellers and Salvation Army bell-ringers are as much subject to the Aggressive Panhandling Ordinance as the homeless panhandler. Nor do we discount the inclusion of the charitable solicitors within the scope of the regulation as merely cosmetic overlay. While it is apparently true that those who sold cookies or held out the tambourine were free to solicit ad lib before panhandling became common, that fact shows nothing more than the need for a public practice to reach some critical dimension before it is worth the effort to regulate, coupled with the City's sense that any regulating it does do must be evenhanded.

The same conclusion of no apparent intent to suppress a particular viewpoint or subject is true as to the Pedestrian Safety Ordinance. While it will unquestionably limit political campaigning, it draws no line by party or position or cause, and it covers solicitation for money as well as for votes.2 Cf. Acorn v. City of New Orleans, 606 F.Supp. 16, 22-23 (E.D.La.1984) (applying content-neutral intermediate scrutiny to ordinance prohibiting solely "solicitation for funds").

These considerations support acceptance of the preface to the Aggressive Panhandling Ordinance, with its statements of findings and policy pointing to behavioral, not censorial, objectives. That prefatory text recognizes the pertinence of the First Amendment to all of the regulated behavior, panhandling included, and it details the circumstances likely to give rise to actual or perceived coercion and fear. The preface and the operative provisions thus jibe in supporting a finding of the City's good faith, for which there is further confirmation in the City Manager's report proposing adoption of the ordinances. Cf. Clatterbuck, 708 F.3d at 559 (4th Cir.2013) (recognizing that an ordinance regulating monetary solicitations may be content-neutral where justified by non-censorial interests like public safety and where "the government's justification for the regulation [is] established in the record").

The appellants seek to turn this report into evidence of the City's discriminatory intent, emphasizing solely a statement at the beginning of the document. That sentence refers to the Manager's prior presentation to the City Council of "strategies aimed at reducing the incidence of panhandling in our community," a recollection that could be consistent with animus against the communicative content of panhandling (albeit not unequivocal evidence of it). But the report goes on in some detail to describe the City's investigations into the social conditions that led to the evident homelessness and begging and the City's efforts to connect the destitute with providers of food, shelter, and work, as well as its ongoing concerns about safety hazards resulting from panhandling. It gives an account of a municipal government trying to relieve the dangerous effects of poverty, not muzzle the poor. Taken as a whole, there is no basis for discounting the report's conclusion that public safety was the driving force of the proposal to draw the lines set out in the challenged ordinances.

In fine, the district court had a sufficient basis in text, common experience, and evidence of the City's intent to conclude that the ordinances were not designed to suppress messages expressed by panhandlers, Girl Scouts, the Salvation Army, campaigning politicians, or anyone else subject to restriction. The ordinances are therefore subject to scrutiny as content-neutral time, place, and manner regulations.

ii.

The First Amendment scrutiny applicable to content-neutral time, place, or manner regulations like the City's ordinances is the intermediate standard: they must be narrowly tailored to serve a significant governmental purpose while leaving open adequate alternative channels of communication. Ward, 491 U.S. at 791, 109 S.Ct. 2746. The standard of narrow tailoring, in turn, requires that a regulation promote the governmental objective more effectively than the law would do in its absence, without burdening substantially more speech than necessary in serving the chosen interest. Id. at 799, 109 S.Ct. 2746.

The appellants here have assumed that, regardless of whether they prevail on their claim that the ordinances are content-based restrictions, the burden rests on the City from the start to demonstrate that the applicable standard of scrutiny is satisfied. But that is not the law. The appellants have chosen to challenge these ordinances for facial overbreadth, a standard under which "a law may be invalidated as overbroad" only if "a substantial number of its applications are unconstitutional...." United States v. Stevens, 559 U.S. 460, 473, 130 S.Ct. 1577, 176 L.Ed.2d 435 (2010) (quoting Washington State Grange v. Washington State Republican Party, 552 U.S. 442, 449 n. 6, 128 S.Ct. 1184, 170 L.Ed.2d 151 (2008)).3 In a facial overbreadth challenge, the claimant has the initial burden to make at least a prima facie showing of such "substantial" overbreadth before any burden of justification, be it strict or intermediate, passes to the government.4 Virginia v. Hicks, 539 U.S. 113, 122, 123 S.Ct. 2191, 156 L.Ed.2d 148 (2003) ("The overbreadth claimant bears the burden of demonstrating, `from the text of [the law] and from actual fact,' that substantial overbreadth exists." (quoting New York State Club Ass'n v. City of New York, 487 U.S. 1, 14, 108 S.Ct. 2225, 101 L.Ed.2d 1 (1988))). And because the burden of persuasion when a preliminary injunction is sought follows the burden at the final merits stage, see Gonzales v. O Centro Espirita Beneficente Uniao do Vegetal, 546 U.S. 418, 429-30, 126 S.Ct. 1211, 163 L.Ed.2d 1017 (2006), the appellants must show a probability of their ultimate success in demonstrating substantial overbreadth at least to the prima facie degree, see Gonzalez-Droz, 573 F.3d at 79 (plaintiff seeking a preliminary injunction bears the burden of showing a "likelihood of success on the merits"). We do not think the appellants have made such a showing.

The point of weakness in the appellants' case for a preliminary injunction is their failure seriously to address their burden of persuasion that the ordinances' overbreadth is substantial. When dealing with a content-neutral speech restriction, we recognize a regulation as substantially overbroad if, but only if, it is susceptible to a substantial number of applications that are not necessary to further the government's legitimate interest. See Stevens, 559 U.S. at 473, 130 S.Ct. 1577. In this way, the substantial overbreadth standard anticipates the narrow tailoring component of the intermediate standard of scrutiny, if the challenge proceeds to a final merits determination. See Ward, 491 U.S. at 799, 109 S.Ct. 2746. The number of impermissible applications, in turn, is considered both in isolation and as compared against instances of plainly permissible restriction. Compare, e.g., Stevens, 559 U.S. at 474-76, 130 S.Ct. 1577 (finding ban on "depiction[s] of animal cruelty" substantially overbroad, absent any comparison of other likely applications, where the statute by its terms extended to footage of hunting or humane slaughter)5; City of Houston, Tex. v. Hill, 482 U.S. 451, 462, 466, 107 S.Ct. 2502, 96 L.Ed.2d 398 (1987) (finding ban on speech that "in any manner ... interrupt[s]" a police officer substantially overbroad, absent comparisons, where the statute was "not limited to fighting words" or "obscene or opprobrious language," but additionally "criminalize[d] a substantial amount of constitutionally protected speech"), with Hicks, 539 U.S. at 123-24, 123 S.Ct. 2191 (finding no substantial overbreadth where trespass policy applied to "strollers, loiterers, drug dealers, roller skaters, bird watchers, soccer players, and others not engaged in constitutionally protected conduct — a group that would seemingly far outnumber First Amendment speakers"); New York v. Ferber, 458 U.S. 747, 773, 102 S.Ct. 3348, 73 L.Ed.2d 1113 (1982) (finding no substantial overbreadth in an anti-child pornography statute because "we seriously doubt ... that [educational, medical, or artistic works featuring nude children] amount to more than a tiny fraction of the materials within the statute's reach"). After all, there is no good reason to allow facial challenges when the likelihood of unjustifiable applications is limited to a trivial number or dwarfed by perfectly constitutional impositions. The consequence is that when the infirmity raised by a facial challenge does not by its nature infect every possible application (as in a patent attempt to suppress unwanted political speech, say) the test looks both to absolute and comparative volume, and each informs a court's judgment about the seriousness of any burden.

And it is judgment that is necessarily involved. Despite the case law's vocabulary of size and "relative" likely applications, neatly mathematical estimates are not to be expected, if only because the required quantification is a predictive exercise based on common experience and such evidence as there may be about a regulation's likely applications. What is substantial and what is trivial, what is substantial in relation to another number, these are enquiries too various to be captured by simple arithmetic.

In this case, the district court did not address the appellants' burden of demonstrating prima facie substantial overbreadth. The court proceeded directly to hold the ordinances up to intermediate scrutiny, and denied a preliminary injunction on the grounds that the appellants failed to show a likelihood of ultimate success on the merits.6 We do not need to follow the district court's reasoning that far, however, because we find that the appellants have failed to make the prima facie showing necessary to trigger the government's burden of proving that the ordinances survive intermediate scrutiny. Based on their presentation of evidence, we find that (subject to one qualification) the appellants have not made a prima facie demonstration that the ordinances are susceptible to a substantial number of illegitimate applications, either in isolation or judged relative to their legitimate sweep.7

Appellants would have us see overbreadth, for instance, in the Aggressive Panhandling Ordinance's bans on soliciting within 20 feet of a bus stop even by a hand-held sign, or less than 20 feet away from people waiting in line to get into a restroom or theater, or even by a polite request for reconsideration after rejection. We agree that some of these prohibitions are at the far side of the reasonable reach of the City's objectives. But it is also true that people can feel intimidated or unduly coerced when they do not want to give to the solicitor standing close to a line they must wait in for a bus or a movie. Similarly, the meek repeat solicitor may justify no concern at all, but the one who shouts or even raises her voice is quite different.

Even the most intuitively appealing of all the appellants' claims, going to the ban on requests by using a sign within 20 feet of the listed lines and locations, does not prompt adequate support to meet their burden. Twenty feet is not very far, being within the range of audible conversation, and a sign request that close would reasonably give rise to discomfort to someone stuck at a bus stop, and could definitely produce apprehensiveness in someone obviously possessing fresh cash. The degree of each would probably vary depending on whether the signholder was just moving along or standing still, intent on one person. As to the moving solicitor, the 20 foot restriction at the bus stop is probably too broad, but the contrary is probably true in the case of a stationary sign-holder staring at a lone individual waiting for a bus. And as to the ATM patron, there is no apparent overbreadth, however we consider variables.

We could go on, but these examples point out the fairly debatable character of even the restrictions that are easiest to challenge, and the appellants make no attempt to show the relative likely frequencies of the ordinances' controversial versus obviously acceptable applications in the circumstances specified. The best we can conclude is that there is probably some overbreadth, but not apparently to a substantial degree. The upshot is that the appellants have shown neither absolutely nor comparatively that the provisions cited are susceptible to substantially overbroad application.

The same point is true with respect to the challenge to the Pedestrian Safety Ordinance, limiting traffic islands and roadways to intended travel as commonly understood. The appellants introduced a photo of a traffic island with hardly any nearby traffic and a sole pedestrian lingering there with his hands in his pockets, along with another photo of people holding three signs in a traffic circle. If the first were representative of all city traffic islands during all parts of the day when there are both traffic and people desiring to use the islands for some purpose other than crossing the street, there would probably be few if any applications of the ordinance that would serve to reduce highway hazard. Nor for that matter would there appear to be any communicative activity to be protected. But everyone knows that traffic islands do not look this way all day, and this fact is enough to suggest the probable insignificance of this piece of evidence. People who hold signs to get attention, for example, do not stand on a traffic island at times when cars are sparse; in the real world it is thus unlikely that there will be any occasion for the ordinance even to be applied under the highway conditions shown. See Hicks, 539 U.S. at 122, 123 S.Ct. 2191 (requiring a challenger to demonstrate substantial overbreadth "from the text of [the law] and from actual fact" (quoting New York State Club, 487 U.S. at 14, 108 S.Ct. 2225)). The same thing is true of the second photo, which shows sign carriers but no cars. Incidentally, however, the second photo does suffice to show how distracting the behavior of those sign carriers would be, and how dangerous, if they were displaying their signs during busy hours with many drivers who could be distracted. Again, the appellants' evidence indicates no substantial overbreadth in either positive or comparative terms, and the appellants have not directed us to record evidence indicating otherwise, or to evidence that the police are failing to differentiate between hazardous and benign conditions when ordering demonstrators to leave or be charged with a violation.8

The same frailty infects the appellants' concerns about overbroad applications of the ban against in-street solicitation in quiet, residential neighborhoods with few cars going back and forth; it is sensible to assume that in these streets there will be little or no solicitation. The application of the First Amendment is not to turn on implausible speculation. Cf. Hicks, 539 U.S. at 122, 123 S.Ct. 2191.

Based on this record as the appellants have directed our attention to it, they have failed to carry their burden of demonstrating likelihood of success in proving that the City's ordinances are substantially overbroad. While they certainly point to some instances in which applying the ordinances may raise constitutional concerns, they have provided no grounds to conclude even at the level of prima facie showing that the scope of any unjustifiable applications is or will be "substantial" in relation to the ordinances' plainly legitimate sweep.

B.

Beyond the overbreadth speech challenge under the First Amendment, the appellants have stated independent equal protection and due process claims under the Fourteenth. With regard to equal protection, the appellants gesture at both a facial and an as-applied challenge. They claim that the ordinances are facially invalid because they were motivated by the City's distaste for its poor and homeless. See Crawford v. Bd. of Educ., 458 U.S. 527, 544, 102 S.Ct. 3211, 73 L.Ed.2d 948 (1982) ("[A] law neutral on its face still may be unconstitutional [under the Equal Protection Clause of the Fourteenth Amendment] if motivated by a discriminatory purpose."). Their as-applied equal protection challenge rests on their contention that the City is selectively enforcing the ordinances against that group. See Martin v. Walton, 368 U.S. 25, 28, 82 S.Ct. 1, 7 L.Ed.2d 5 (1961) ("A law, fair on its face, may be applied in a way that violates the Equal Protection Clause of the Fourteenth Amendment.").

We have already spoken to the facial element in the course of the First Amendment discussion, which explains that the record indicates an effort aimed at certain dangerous behaviors, not a municipal intent to target a particular message or the class expressing it. With respect to the as-applied challenge, the appellants have provided no evidence suggesting a discriminatory pattern in the City's enforcement of either ordinance. The appellants focus on the fact that the City made no attempts to disperse a political protest in February of 2013 under the Pedestrian Safety Ordinance, while arresting four homeless people in March under the Aggressive Panhandling Ordinance. Putting aside whether the February protest took place during the "grace period" immediately after the ordinances' enactment, an issue unresolved by the record in its current state, different rates of arrest under two distinct ordinances do not provide evidence of selective enforcement of either one. The fact that the Aggressive Panhandling Ordinance may thus far have been enforced only against the poor, with no further details about the circumstances of those arrests or the police's greater leniency toward other groups, is not in itself probative of discrimination. If the full facts of the City's enforcement patterns since the filing of this suit point more strongly to intentional discrimination on any basis, the appellants may come forward with the evidence, and injured persons may file as-applied complaints on the basis of evidence. We only note that, because neither wealth nor homelessness is a suspect class under the Equal Protection Clause, see United States v. Myers, 294 F.3d 203, 209 (1st Cir.2002), any effort to show discriminatory enforcement will be subject to the deferential standard of rational basis scrutiny, Kadrmas v. Dickinson Pub. Sch., 487 U.S. 450, 458, 461-62, 108 S.Ct. 2481, 101 L.Ed.2d 399 (1988).

As for due process, the appellants zero in on the imprecision of certain language, implicating (in addition to potential overbreadth) constitutional concerns about fair notice. Under the Due Process Clause, a regulation may be void for vagueness if it "fails to provide a person of ordinary intelligence fair notice of what is prohibited, or is so standardless that it authorizes or encourages seriously discriminatory enforcement." United States v. Williams, 553 U.S. 285, 304, 128 S.Ct. 1830, 170 L.Ed.2d 650 (2008). The appellants correctly note that, when a statute "interferes with the right of free speech or of association, a more stringent vagueness test should apply." Hoffman Estates, 455 U.S. at 499, 102 S.Ct. 1186. But "perfect clarity and precise guidance have never been required even of regulations that restrict expressive activity." Williams, 553 U.S. at 304, 128 S.Ct. 1830. That a statute's language may "pos[e] difficult questions of exactly" what kinds of conduct is proscribed is not sufficient to sustain a vagueness challenge unless the "plaintiffs... provid[e] [a] specific articulation of the degree to which they seek to" engage in the conduct at issue. Humanitarian Law Project, 561 U.S. at 24-25, 130 S.Ct. 2705.

In this case, the fair notice requirement is said to be flouted, for example, by the provision of the Pedestrian Safety Ordinance that defines traffic islands as areas set aside by paint or construction and "not constructed or intended for use" by vehicles or pedestrians. Appellants ask how they are supposed to know the intent. But taking the definition whole, it is hard to imagine difficulty in understanding that a space identified by paint or structure and set in the midst of traffic lanes was itself not intended for vehicular or pedestrian use, save as a crosswalk. Appellants also contest the ordinance's exception for individuals using traffic islands and roadways for "some ... lawful purpose," other than getting out of motor vehicles or crossing the street. They posit that it could be reasonable to claim engaging in political speech and solicitation as "lawful purposes." But considering that the ordinance expressly exemplifies a "lawful purpose" as getting out of a car or crossing on a crosswalk, the appellants' suggestions cannot be taken as serious possibilities. This "lawful purpose" language is thus distinguishable from, say, a prohibition against any conduct that would unlawfully "interrupt any policeman in the execution of his duty" in an ordinance that could not have been meant to impose the etiquette of the drawing room on all police-pedestrian street encounters. See Hill, 482 U.S. 451, 107 S.Ct. 2502 (striking down as void for vagueness a statute making it "unlawful for any person to assault, strike or in any manner oppose, molest, abuse or interrupt any policeman in the execution of his duty" (emphasis omitted)).

Neither can the district court be faulted for failing to face reality in seeing no serious vagueness problem in the Aggressive Panhandling Ordinance's prohibition on "continuing to solicit" after receiving a negative response, even as applied to one holding a sign. In the expectable course, the person specifically solicited by means of a sign will move on and the solicitor will be in no danger of transgressing the ordinance; if the solicitor follows and gestures with the sign, he or she will have committed a violation. Thus, save for odd situations that can arise under almost any statute or regulation, the risk of perplexity is not serious.

Nor do we find any error in the district court's failing to see a probability of a due process violation in the discretion created by the Pedestrian Safety Ordinance's provision that a police officer "may" order a violator to stop and "may" arrest for non-compliance with the order. "As always, enforcement requires the exercise of some degree of police judgment," and the conferral of enforcement discretion does not render a law impermissibly vague as long as that judgment is appropriately "confined." Grayned v. City of Rockford, 408 U.S. 104, 114, 92 S.Ct. 2294, 33 L.Ed.2d 222 (1972) (upholding anti-noise ordinance applying outside schools against void-for-vagueness due process challenge because the ordinance required "demonstrated interference with school activities"). The ordinances in question here simply state explicitly what the law provides in any case: discretion in enforcing and prosecuting under a regulation intended to prevent hazardous action in the streets. If it should turn out in practice that the police exercise their discretion to enforce the ordinance arbitrarily, or in a discernibly discriminatory way, a further challenge may raise a due process or equal protection claim based on evidence to that effect.

Finally, the improbability of success on the appellants' vagueness claim is apparent from their failure to identify any specific instance of uncertainty on their part about the status of a traffic island, or the lawfulness of continuing to hold a solicitation sign, or the likelihood of police action, or any other detriment that might be attributable to a failure of textual concreteness. A vagueness claim must be specific and as applied to a given plaintiff, not facial. "A plaintiff who engages in some conduct that is clearly proscribed cannot complain of the vagueness of the law as applied to the conduct of others." Hoffman Estates, 455 U.S. at 495, 102 S.Ct. 1186. The appellants' due process claim comes down primarily to arguing that the "outer bounds of [the ordinances] are entirely unclear." Appellants' Reply Br. 21. Yet "even if the outermost boundaries of [a law] may be imprecise, any such uncertainty has little relevance ... where appellants' conduct falls squarely within the `hard core' of the statute's proscriptions." Broadrick v. Oklahoma, 413 U.S. 601, 608, 93 S.Ct. 2908, 37 L.Ed.2d 830 (1973).

III.

This case has proven to be time-consuming, owing to the plenary challenge to the ordinances, to the unsettled fit of various elements of governing law with each other, and to the apparent velocity with which the request for preliminary relief was prepared for hearing. The trial judge treated the prayer for an initial injunction with great care, and we have spent more time on it than we expected to do.

But at the end of this particular day, nothing has been reached except the issue of entitlement to a preliminary injunction. There is a lesson in this, both for counsel with the obligation to present the case and for the courts that must manage the litigation. Except for instances of facial challenge where a right to at least some preliminary relief is not reasonably debatable, there is great merit in combining litigation for both preliminary and permanent orders and expediting the evidentiary hearing as much as a fair opportunity for trial preparation will allow. Such procedural self-discipline will tend to soften the siren call of the repeatedly discouraged resort to the opportunity for facial challenges.

IV.

For the foregoing reasons, we AFFIRM the district court's denial of a preliminary injunction as to all provisions of the challenged ordinances save for the Aggressive Panhandling Ordinance's proscription on nighttime solicitation, see Worcester Revised Ordinances, ch. 9, § 16(e)(11), and REMAND to the district court for proceedings consistent with this opinion. The mandate will issue immediately, but without prejudice to any petition for rehearing.