
526 F.3d 991 (2008)
GENERAL AUTO SERVICE STATION, et al., Plaintiffs-Appellants,
v.
CITY OF CHICAGO, Defendant-Appellee.
Nos. 05-2515, 06-2672.
United States Court of Appeals, Seventh Circuit.
Argued June 4, 2007.
Decided May 16, 2008.
*993 Thomas E. Baltz (argued), Grey, Grey & Baltz, Chicago, IL, for Plaintiffs-Appellants.
Kerrie Maloney Laytin (argued), Mara S. Georges, Office of the Corporation, Counsel Appeals Division, Chicago, IL, Defendant-Appellee.
Before RIPPLE, ROVNER, and SYKES, Circuit Judges.
*992 ROVNER, Circuit Judge.
The plaintiffs-appellants are the owners and the agent of a building in Chicago that for more than four decades has displayed an advertising sign without a permit. Current zoning provisions prohibit the sign given its size and proximity to a residential district, but the plaintiffs contend that because the sign was lawful when first erected, they have a right to continue displaying the sign. After the City of Chicago declared the sign unlawful and ordered it removed, the plaintiffs-appellants filed suit under 42 U.S.C. § 1983 challenging the City's action on a variety of constitutional grounds. The district court resolved the suit in the City's favor. We affirm.

I.
Bernard A. Heerey purchased the building located at 1127-33 North State Street in Chicago in 1961. Pursuant to a 1984 trust agreement, Cosmopolitan Bank and Trust now holds legal title to the property. Heerey assigned his beneficial interest in the property to General Auto Service Station,[1] an Illinois corporation which in 1998 merged with and into General Auto Service Station LLC (collectively, "GASS"). F.A.Y. Properties, Inc. ("F.A.Y.") is an Illinois corporation that acts as an agent and lessor for GASS. Heerey was the sole *994 owner of both GASS and F.A.Y. Heerey died in 1999 at the age of 79 without immediate family to survive him. The terms of his will provide that once certain bequests are made, the assets in his estate, which include a number of properties on Chicago's north side, are to be placed in a charitable trust in order to fund distributions to religious and charitable organizations. As of the date this case was tried in December 2004, the probate of Heerey's estate was ongoing and the trust had not yet been funded.
The south exterior wall of the building at 1127-33 North State Street is a desirable location for advertising. The building is four stories tall, and soon after Heerey purchased the property, the buildings to its immediate south were torn down, eventually to be replaced by a one-story building. As a result, the exposed south wall of the building at 1127-33 North State is visible to motorists and pedestrians passing through the nearby intersection of North Rush and State Streets, in the heart of Chicago's Gold Coast.
In 1962, Heerey, without first seeking or obtaining a permit from the City, had an advertising sign painted on the south wall of the building. That wall has been continuously used for advertising since that time. In the ensuing years, Heerey and later GASS, through F.A.Y., leased the advertising space on the building to a variety of entities, including Whiteco Industries, Inc., whose successor, Lamar Advertising Company, holds the current lease. We shall refer to these lessees collectively as "Whiteco." For the one-year period ending July 14, 2008, GASS is being paid $99,074.48 for the lease of the sign.
When Heerey first began using the side of his building for advertising in 1962, section 86.1-4 of the City's electrical code provided in relevant part that "[i]t shall be unlawful to proceed with the erection, enlargement, alteration or rehang of any illuminated sign or illumination of signboard unless a permit therefore shall have first been obtained from the Commissioner of Buildings." The district court understood this language to require a permit only for an illuminated sign. Because the original sign on Heerey's building was not illuminated, the court concluded that no permit was required. R. 126 at 16. The City accepts that ruling for purposes of this appeal, as do we.
At all times relevant to this suit, the property at 1127-33 North State Street has been located within a district zoned by the City for business use. However, the sign on the south wall of the building has always been within 75 feet of a district zoned for residential use.
As of 1962, section 8.9(5) of the City's zoning ordinance prohibited advertising signs "within 75 feet of any property in a Residence District." The district court took the phrase "any property" in this provision to mean any improved property. Although the sign on Heerey's building is within 75 feet of a residential district, when first erected it was not within 75 feet of any improved property in such a district. Consequently, the court assumed that the sign was lawful when first painted on Heerey's building. R. 126 at 15. Although the City contested this point below, it has not quarreled with the district court's assumption for purposes of this appeal. Consequently, we like the district court shall assume that the sign was lawful when first erected.
In subsequent years, however, Heerey made two changes vis-à-vis the advertising on his building for which a permit was indubitably required under the then-governing terms of the City's electrical code. First, at some point between 1962 and 1969, Heerey illuminated the sign by installing lights near the roof line of his building. Section 86.1-4 of the code, *995 which we quoted from above, required that a permit be issued by the Commissioner of Buildings for, among other things, the "erection" of an illuminated sign and the "illumination of signboard." Second, between 1975 and 1979, Heerey allowed the lessee of the advertising space on his building to combine what had been two signs into a larger, single sign by covering the building windows that had separated the two signs. Then, in 1979, that sign was enlarged to its current dimensions of 26 feet by 59 feet. These changes again triggered the permit requirement of section 86.1-4, which applied to any "enlargement" or "alteration" made to "any illuminated sign." The district court thus concluded that Heerey was required to obtain a permit before illuminating and enlarging the signage on his building. R. 126 at 16-17. So far as the record reveals, Heerey did not obtain a permit before he made any of these modifications.[2]
In 1986, the City finally took notice that the sign lacked a permit. On September 3 of that year, it issued a violation notice to Heerey's property management company asserting that the illuminated sign had been installed without a permit in violation of section 86.1 of the electrical code. App. 193. A hearing before the City's electrical compliance board was scheduled and continued on several occasions, but the City took no further action on the 1986 notice.
In 1990, the provision of the City's zoning ordinance regarding the distance between advertising signs and residential districts was toughened. Whereas before, section 8.9(5) of the ordinance (as construed by the district court) permitted no advertising sign within 75 feet of improved property in a residential district, section 8.9(7) of the revised ordinance now provided that "[n]o advertising sign having a face which exceeds 100 square feet shall be permitted within 250 feet of a Residence District" and that "no advertising sign" of any size "shall be permitted within 75 feet of a Residence District." App. 237. A "grandfather" provision, section 6.7-1(a), was added to the ordinance providing that any sign not conforming to the new distance requirements (among others) which had been "lawfully erected pursuant to a permit lawfully issued prior to the effective date of this section, may remain in use as a legal non-conforming sign." App. 227.[3]
On September 29, 1994, the City issued a second violation notice to Heerey, this one charging that because the face of the sign was larger than 100 square feet and was within 250 feet of a residential district, the sign violated the distance requirement adopted in 1990. App. 113. Heerey's attorney forwarded the notice to Whiteco and asked that it handle the matter. Whiteco in turn submitted a permit application to the City.App. 113A. That application was approved initially by the City's zoning department on January 12, 1995. Zoning employee William Merritt determined that the sign was a "legal non-conforming" use, and he therefore stamped the permit application *996 to indicate that the sign "conforms to zoning ordinance." App. 107 ¶ 36, App. 113B. At that point, the approval of the City's building department was required, and Whiteco submitted its permit application to that department for review. However, for any sign that exceeded 100 square feet in size, the municipal code required City Council approval in addition to a permit. App. 285. On April 19, 1995, the building department noted on a "plan correction sheet" issued to Whiteco that a copy of the Council's order approving the sign "must be attached to the permit application." App. 208. Whiteco had written to the City alderman whose ward included the State Street property one week earlier requesting his support and assistance in obtaining such approval. But for reasons that the record does not reveal, Whiteco never obtained an order from the City Council approving the sign. For its part, the City took no further action on Whiteco's 1994 permit application.
On August 22, 1997, the City issued a third notice of violation to Heerey's property management company directing Heerey to "[o]btain permit for outdoor flat building sign installed without permit." App. 108 ¶ 40; App. 116. Heerey's attorney again forwarded the notice to Whiteco asking that it handle the matter. Whiteco appeared as the respondent before the City's Department of Administrative Hearings on September 30, 1997, and admitted that it was not in compliance with the permit requirement. The hearing officer entered the following findings and order:
Respondent is found to be in violation of the Chicago Building Code with regards to the afore stated charges; however, Respondent afforded opportunity of mitigation via compliance. Compliance must be evidenced by reinspection and/or other evidence corroborating conditions. This matter is continued for hearing on fines....
App. 117 (footnote omitted).
Whiteco subsequently attempted to bring the sign into compliance with the building code by applying for a permit, but its efforts met with failure. On April 21, 1998, the City's Zoning Administrator denied zoning certification for the sign on the grounds that the sign violated the provision of the zoning ordinance regarding the proximity of signs to residential districts and was not grandfathered because there was no prior permit for the sign on file. Given the lack of zoning certification, the City did not issue a permit for the sign. Whiteco appealed the denial of zoning certification to the City's Zoning Board of Appeals ("ZBA"), but the ZBA sustained the denial on September 14, 1998.
The following month, a City hearing officer entered a final order on the 1997 notice of violation imposing a fine of $250 and ordering Whiteco to remove the sign within 90 days in the event it did not obtain a permit. App. 161. By the terms of the City's zoning ordinance, GASS as the (effective) owner of the property is liable for the violation to the same extent as its lessee, Whiteco. The City's order has yet to be enforced; to this day, Whiteco's successor continues to lease the sign and display advertising on the south wall of GASS's building.
Whiteco filed suit in the Circuit Court of Cook County seeking judicial review of the ZBA's decision, arguing in part that the ZBA had overlooked the fact that no permit had been required when the sign was first erected in 1962 and that, consequently, the sign qualified as a legal, non-conforming use under the grandfather provision of the zoning ordinance. The Circuit Court denied review on March 2, 2000. The court found that Whiteco had failed to present any evidence to the ZBA in support of the notion that the sign constituted *997 a legal, non-conforming use. Whiteco appealed, but the Illinois Appellate Court affirmed in September 2001 and the Illinois Supreme Court subsequently denied Whiteco's petition for leave to appeal.
In the meantime, GASS, as the owner of the property, was seeking relief of its own. Late in 1999, GASS filed a declaratory judgment action in state court challenging the relevant provisions of the zoning ordinance and the City's failure to grant it a permit in 1994 on a variety of grounds. At the same time, GASS sought leave to intervene in Whiteco's action so that it too could challenge the ZBA's refusal to order zoning certification for the sign. The state court denied GASS's request to intervene in Whiteco's suit,[4] and GASS eventually dismissed its own state-court suit in favor of the instant federal suit.
GASS (along with F.A.Y. and Cosmopolitan Bank, but for the sake of simplicity, we shall refer only to GASS) filed this section 1983 action against the City in January 2000. GASS's second amended complaint[5] asserted a variety of constitutional claims, four of which are pursued in this appeal. First, GASS asserted that the advertising sign constituted commercial speech that was not misleading and which the City had no legitimate or substantial interest in restricting or prohibiting. The City's invocation of its zoning ordinance to prohibit the sign therefore deprived GASS of its rights under the First Amendment to the U.S. Constitution. R.122, Count I. Second, GASS contended that the City, in failing to process its 1994 application for a permit, violated its right to procedural due process under the U.S. Constitution. Id., Count II. Third, GASS asserted that in refusing to recognizing its sign as a lawful, nonconforming use under the grandfather provision found in section 6.7-1 of the zoning ordinance, the City had applied its laws in a way that did not promote the public health, safety, morals, or general welfare and therefore had deprived GASS of substantive due process under the U.S. Constitution. Id., Count III. Fourth, GASS claimed that the City's application of the 1990 zoning provisions, and its refusal to grandfather the sign, amounted to a retroactive application of laws enacted after the sign's erection that deprived GASS of its vested property and contractual rights in a harsh, oppressive, and arbitrary manner, thereby again depriving GASS of its right to substantive due process. Id., Count IV.[6] GASS sought a declaratory *998 judgment on each of its claims and an injunction barring the City from enforcing the zoning ordinance as to its sign and/or requiring the City to issue a permit for the sign.
Initially, the district court dismissed the suit without prejudice, abstaining pursuant to Younger v. Harris, 401 U.S. 37, 91 S.Ct. 746, 27 L.Ed.2d 669 (1971), in deference to Whiteco's then-ongoing suit for administrative review in state court. R. 37, 2001 WL 558148; R. 44, 2002 WL 221582. We reversed that dismissal in Gen. Auto Serv. Station LLC v. City of Chicago, 319 F.3d 902, 905 (7th Cir.2003). We reasoned in relevant part that because GASS's request to intervene in Whiteco's state court action had been denied, GASS had not had the opportunity to defend its interests as the owner of the property in that action. Id. at 905.
On remand, the district dismissed GASS's First Amendment and procedural due process claims. The court concluded that the real focus of the First Amendment claim was on the distinction that the grandfather provision drew between signs erected pursuant to a permit and signs, like GASS's, which were erected at a time when no permits were required. As such, the court reasoned, this claim was nothing more than a restatement of GASS's separate substantive due process claim.
Plaintiffs claim that the zoning ordinance violates the First Amendment because it allows advertising signs that pre-dated the 1990 amendments and which now violate those amendments to remain only if the signs were initially erected pursuant to a permit, regardless of whether such a permit was even required at the time the sign was erected. Plaintiffs' sign happens to be an advertising sign, but the provisions they really challenge here  the legal, non-conforming use and permit provisions, by which the City only "grandfathers in" those signs that were issued pursuant to a permit  apply to all signs, not just advertising signs. Plaintiffs nowhere argue that the size and distance restrictions on advertising signs found in Section 8.9(7) will not pass constitutional muster. Thus, the court agrees with the City that the Plaintiffs' challenge is not aimed at the ordinance's regulation of commercial speech.
Moreover, as the court noted in its earlier decision [dismissing the suit on abstention grounds], "Plaintiffs' First Amendment argument is nothing more than a restatement of its arguments that the City's refusal to `grandfather' Plaintiffs' sign deprives it of due process." General Auto Service Station, 2001 WL 558148, at *6. Plaintiffs cannot convert what is essentially a due process claim into one under the First Amendment merely by noting that theirs is an advertising sign. Plaintiffs have not effectively alleged a First Amendment Challenge to the zoning ordinance....
R. 93, 2004 WL 442636, at *4-*5. The court found the procedural due process claim to be untimely. That claim was based on the City's purported failure to process Whiteco's 1994 permit application. But GASS knew by August 1997, when the City issued yet another notice of violation, that the City had not processed the 1994 application. Consequently, in the court's view, the two-year statute of limitations on this claim began to run no later than August 1997 and had expired well before GASS filed this suit in January 2000. Id. at *8 The court rejected the City's contention that the substantive due process claims were also untimely. Id. at *9.
The district court conducted a one-day bench trial on the substantive due process claims, and ultimately found in favor of the City on those two claims. The court ruled in the first instance that the claims were *999 time barred, contrary to its earlier holding on the motion to dismiss. Both claims were focused on the City's preferential treatment of non-conforming signs that were erected with a permit. However, as with the procedural due process claim, the court concluded that the August 1997 notice of violation put the plaintiffs on notice that the City was not treating the non-permitted sign as grandfathered. "Though the Zoning Administrator did not officially deny Whiteco's 1998 permit application until April 21, 1998, Plaintiffs knew the circumstances that support the claim they are presenting here  i.e., the differential treatment of permitted and non-permitted legal, non-conforming signs  by September 1997" when Heerey's attorney forwarded the 1997 violation notice to Whiteco. R. 126 at 12. Therefore, GASS was obliged to file suit on these claims no later than September 1999, which it did not. Id. The court ruled alternatively that the claims failed on their merits. The premise of these claims was that because no permit was required when GASS erected its sign, the sign constituted a legal, non-conforming use just as if it had been erected with a permit. For that reason, in GASS's view, it was irrational to treat permitted signs differently from non-permitted signs for purposes of the grandfather provision. But GASS was wrong in characterizing its sign as a legal non-conforming use, the court found. Although a permit was not required when Heerey installed the first sign in 1962, a permit was required when he later illuminated and enlarged the sign, and a permit had never been obtained. Id. at 16-17. GASS rejoined that the permit requirement as it existed at those times amounted to an unconstitutional prior restraint on speech and should be treated a nullity (rendering its non-permitted sign legal) but the court was not convinced. "Plaintiffs have not explained how Chapter 86.1, which at all relevant times applied to all illuminated signs and not just to advertising signs, constitutes a regulation of commercial speech sufficient to implicate the First Amendment." Id. at 17.
GASS moved to amend the judgment, challenging the court's rationale in disposing of the substantive due process claims, but the court denied the motion. The court reiterated that GASS was made aware by the 1997 violation notice that the City would not deem its sign covered by the grandfather clause, thus obligating GASS to file suit no later than 1999. R. 144, 2006 WL 1460017, at *2-*3. As to the merits, the court again rejected GASS's contention that the permit requirement, as it existed in the 1960s and 1970s when GASS's sign was illuminated and then expanded, constituted an unlawful prior restraint on speech. The court pointed out that a key danger posed by licensing schemes that grant unfettered discretion to government officials is that a potential speaker will be intimidated into censoring his own speech. Id. at *4 (citing City of Lakewood v. Plain Dealer Publ'g Co., 486 U.S. 750, 759, 108 S.Ct. 2138, 2144, 100 L.Ed.2d 771 (1988)). That danger was not implicated here: the permit applied only to illuminated signs, leaving non-illuminated signs open as an alternative outlet for unregulated speech; and, at the same time, the permit requirement applied to all illuminated signs, drawing no distinctions between commercial and noncommercial speech. Id. at *5. Finally, the court rejected as untimely GASS's argument, made in a supplemental brief in support of the motion to amend, that it retained, at a minimum, a vested property interest in the sign in a non-illuminated, un-enlarged state given that no permit was required when Heerey first erected the sign in 1962. "This is the first time that Plaintiffs have claimed a property interest in the Wall Sign other than as currently configured, and Plaintiffs make no attempt to explain why they failed to raise this claim ... *1000 previously." Id. The court noted alternatively that GASS had never firmly established that its sign as originally erected truly complied with all applicable ordinances. Id. The court had assumed that the sign was consistent with the pre-1990 version of the zoning ordinance, but GASS had never actually proven to the satisfaction of the court that the sign was in fact legal in its non-illuminated, non-enlarged state. Id.

II.

A. Substantive Due Process
GASS has asserted two claims for deprivation of its Fourteenth Amendment right to substantive due process. See generally County of Sacramento v. Lewis, 523 U.S. 833, 840, 118 S.Ct. 1708, 1713-14, 140 L.Ed.2d 1043 (1998).
The claim that a person is entitled to "substantive due process" means, as we understand the concept, that state action which deprives him of life, liberty, or property must have a rational basis  that is to say, the reason for the deprivation may not be so inadequate that the judiciary will characterize it as "arbitrary."...
Jeffries v. Turkey Run Consol. Sch. Dist., 492 F.2d 1, 3-4 (7th Cir.1974) (Stevens, J.). See also Indiana Land Co. v. City of Greenwood, 378 F.3d 705, 711 (7th Cir. 2004).
Both of GASS's due process claims are premised on the notion that its sign was an established and lawful use of its property before the City's zoning ordinance was modified in 1990 to prohibit advertising signs within 250 feet of a residential district, and that, consequently, GASS has a vested property interest in continuing to display its sign of which it may not be deprived without substantive due process. For purposes of a due process claim, property interests are created and defined by an independent source, such as a contract or state law. E.g., Miyler v. Vill. of E. Galesburg, 512 F.3d 896, 898 (7th Cir.2008) (citing Bd. of Regents of State Colls. v. Roth, 408 U.S. 564, 92 S.Ct. 2701, 33 L.Ed.2d 548 (1972)). Illinois case law recognizes that a property owner can acquire a property interest in continuing a land use that was lawful when commenced and rendered unlawful (i.e., non-conforming) only by a subsequent legislative enactment. See Taylor v. Zoning Bd. of Appeals of City of Evanston, 375 Ill.App.3d 585, 314 Ill.Dec. 562, 874 N.E.2d 927, 933 (2007) (coll.cases).
Once a landowner obtains such a state-created property right, the due process clause of the Fourteenth Amendment circumscribes, but does not eliminate, the government's ability to deprive him of that interest. As we observed in Lee v. City of Chicago, "[S]ubstantive due process is not `a blanket protection against unjustifiable interferences with property.'" 330 F.3d 456, 461 (7th Cir.2003) (quoting Schroeder v. City of Chicago, 927 F.2d 957, 961 (7th Cir.1991)). And it does not confer on federal courts a license to act as zoning boards of appeals. Ctrs., Inc. v. Town of Brookfield, Wis., 148 F.3d 699, 704 (7th Cir.1998); Albiero v. City of Kankakee, 122 F.3d 417, 420 (7th Cir.1997). It is instead a modest limitation that prohibits government action only when it is random and irrational. "Unless a governmental practice encroaches on a fundamental right, substantive due process requires only that the practice be rationally related to a legitimate governmental interest, or alternatively phrased, that the practice be neither arbitrary nor irrational." Lee, 330 F.3d at 467 (citing Washington v. Glucksberg, 521 U.S. 702, 728, 117 S.Ct. 2258, 2271, 138 L.Ed.2d 772 (1997)). A property owner challenging a land-use regulation as a violation of due process is therefore obliged to show that the regulation is arbitrary *1001 and unreasonable, bearing no substantial relationship to public health, safety, or welfare. E.g., Greater Chicago Combine & Ctr., Inc. v. City of Chicago, 431 F.3d 1065, 1071 (7th Cir.2005). And because the right infringed upon is an interest in property rather than life or liberty, the property owner must first establish either an independent constitutional violation or the inadequacy of state remedies to redress the deprivation before a court will consider whether the interference with property is arbitrary or irrational. Lee, 330 F.3d at 467.
In this case, GASS contends that the City's conduct in prohibiting its sign was arbitrary and irrational in two ways. It contends first that the application of the zoning ordinance failed to promote the public health, safety, morals, and/or welfare because it arbitrarily grandfathered advertising signs that had been erected pursuant to a permit but not signs for which no permit had been required. (Count III.) Second, it contends that the City's application of the zoning ordinance was apparently retroactive, resulting in an infringement of GASS's vested property rights in a harsh and oppressive manner. Specifically, GASS notes that the City demanded in 1997 that it obtain a permit for a sign that had been erected some thirty years earlier, but then denied GASS's application for a permit based on current zoning restrictions that did not exist when the sign was installed. (Count IV.)
There is an initial question as to the timeliness of these claims. The district court concluded, and the City argues here, that the 1997 notice of violation was sufficient to put Heerey and GASS on notice that it intended to treat permitted and non-permitted signs differently for purposes of grandfathering, and thus supplied the plaintiffs with all of the information they needed in order to assert a due process claim. Under that view, GASS was obliged to file suit no later than 1999, within the two-year statute of limitations for section 1983 claims that we have borrowed from Illinois law. See, e.g., Jenkins v. Vill. of Maywood, 506 F.3d 622, 623 (7th Cir.2007). GASS replies that it was not yet clear in 1997 that the City would not belatedly grant the plaintiffs a permit for the sign; indeed, in 1995, the City zoning department had signaled an inclination to grant such a permit when its employee Merritt characterized the sign as a legal non-conforming use. GASS also makes a more fundamental contention that because it seeks only declaratory and injunctive relief (its sign still exists today, so it has yet to suffer any injury for which it might seek compensation), its substantive due process claims are not subject to a statute of limitations. See Lavey v. City of Two Rivers, 994 F.Supp. 1019, 1023 (E.D.Wis. 1998) (rejecting argument that section 1983 challenge to local sign ordinance was untimely, noting that plaintiff sought only declaratory and injunctive relief), judgment aff'd, 171 F.3d 1110 (7th Cir.1999); cf. entitled to judgment as a matter of law Nat'l Adver. Co. v. City of Raleigh, 947 F.2d 1158, 1168 (4th Cir.1991) (cited in Lavey) ("it is doubtful that an ordinance facially offensive to the First Amendment can be insulated from challenge by a statutory limitations period"); Maldonado v. Harris, 370 F.3d 945, 955 & n. 6 (9th Cir.2004) (seconding Fourth Circuit's doubts and collecting district court cases in like vein).
We need not resolve the timeliness question. The expiration of the statute of limitations is an affirmative defense that does not affect our subject matter jurisdiction to hear the case. E.g., Jogi v. Voges, 480 F.3d 822, 836 (7th Cir.2007). We may assume that even if GASS's claim is subject to a two-year statute of limitations, the limitations period did not begin to run when Heerey received the 1997 notice of *1002 limitation, as it was not yet clear at that time that the City would refuse to grant a permit for the sign. But even if we assume that the due process claims are timely, they nonetheless fail on their merits.
Intrusion upon a cognizable property interest is a threshold prerequisite to a substantive due process claim of the type that GASS asserts. See, e.g., Am. Mfrs. Mut. Ins. Co. v. Sullivan, 526 U.S. 40, 59, 119 S.Ct. 977, 989, 143 L.Ed.2d 130 (1999) ("The first inquiry in every due process challenge is whether the plaintiff has been deprived of a protected interest in `property' or `liberty.'"); Swartz v. Scruton, 964 F.2d 607, 609 (7th Cir.1992); Garza v. Miller, 688 F.2d 480, 485 (7th Cir.1982). "In other words, no abstract right to substantive due process exists under the Constitution." Zorzi v. County of Putnam, 30 F.3d 885, 895 (7th Cir.1994). Consequently, the lack of a protected property interest is fatal to a substantive due process claim. Id. at 894.
GASS's substantive due process claims, as we have said, are premised on the notion that because its advertising sign constituted a legal use of its property prior to the 1990 amendments to the Chicago zoning ordinance, GASS has a protected property interest in continuing this prior, legitimate, albeit now non-conforming, use of the property. This might be true of the original, non-illuminated sign that Heerey first had painted on his building in 1962. Although that sign was painted on the building without a permit, the district court determined or assumed that the relevant provisions of the zoning ordinance and electrical code did not require a permit for such a sign, and we (like the parties) have accepted her ruling in that regard as correct. Consequently, the sign in its original form arguably constituted a prior nonconforming use in which GASS has a protected property interest and of which it may not be deprived arbitrarily. And GASS, of course, contends that it is arbitrary and irrational for the City to treat as grandfathered signs that were erected with a permit but not signs, like its own, that were lawfully erected without a permit.
But even granting that GASS may have had a protected property interest in the original, non-illuminated sign, it could not have acquired such an interest in the later manifestations of the sign. When Heerey illuminated the sign in the late 1960s, and when he expanded the sign in the 1970s, he was obligated by the express terms of the City's electrical code to obtain a permit, which, so far as the evidence reveals, he did not do. Thus, the expanded and illuminated sign that existed in 1990, when the City amended the zoning ordinance to prohibit advertising signs within 250 feet of a residential district, was not an authorized use of the property that GASS had any protected interest in continuing as a legal non-conforming use. Taylor, 314 Ill. Dec. 562, 874 N.E.2d at 933 (citing Wright v. County of DuPage, 316 Ill.App.3d 28, 249 Ill.Dec. 456, 736 N.E.2d 650, 659 (2000)); City of Marengo v. Pollack, 335 Ill.App.3d 981, 270 Ill.Dec. 354, 782 N.E.2d 913, 917 (2002). And absent a protected property interest in the sign, GASS's substantive due process claims necessarily fail. E.g., Zorzi, 30 F.3d at 895.
GASS does not dispute that the terms of the electrical code required Heerey to obtain a permit when he illuminated and expanded the sign, but GASS argues that the permit requirement itself was constitutionally invalid. Specifically, GASS contends that the permit requirement amounted to an invalid prior restraint on speech (as advertising, the sign constituted commercial speech, see Metromedia, Inc. v. City of San Diego, 453 U.S. 490, 501, 101 S.Ct. 2882, 2889, 69 L.Ed.2d 800 (1981) (plurality)), in that the relevant terms of *1003 the permitting scheme placed no limits on the City's discretion to deny a permit nor any time constraints on the City's decision to grant or deny a permit. See generally FW/PBS, Inc. v. City of Dallas, 493 U.S. 215, 225-27, 110 S.Ct. 596, 604-05, 107 L.Ed.2d 603 (1990) (plurality); Weinberg v. City of Chicago, 310 F.3d 1029, 1045 (7th Cir.2002); see also, e.g., Solantic LLC v. City of Neptune Beach, 410 F.3d 1250, 1272 (11th Cir.2005) (permitting scheme for signs amounted to unconstitutional prior restraint given lack of time limits on city's obligation to act on permit applications); Desert Outdoor Adver., Inc. v. City of Moreno Valley, 103 F.3d 814, 818-19 (9th Cir.1996) (sign permit ordinance constituted unlawful prior restraint in view of city's unbridled discretion to deny sign permits). GASS adds that the permit application itself requested information about the content of the proposed sign (inquiring, for example, what the "sign will read"), suggesting that the City may have been making content-based decisions about which permit applications it would grant and which it would deny.[7] In any case, given the asserted defects in the permitting scheme as it existed when Heerey illuminated and expanded the sign, GASS contends that the permit requirement should be treated as a nullity with which Heerey was not obligated to comply. See Shuttlesworth v. City of Birmingham, 394 U.S. 147, 151, 89 S.Ct. 935, 939, 22 L.Ed.2d 162 (1969) ("a person faced with such an unconstitutional licensing law may ignore it and engage with impunity in the exercise of free expression for which the law purports to require a license").
We may accept without deciding that the features of the City's permitting scheme on which GASS has focused indeed rendered the permit requirement incompatible with the First Amendment, such that Heerey was entitled to ignore it. Indeed, Heerey did ignore it. So far as the record reveals, he not only never obtained a permit authorizing him to illuminate and then expand the sign on his building, but never bothered applying for one; nonetheless, he proceeded to illuminate and enlarge the sign. His speech, or more properly, that of his lessees, was never restrained. Of course, this does not preclude GASS from attacking the validity of the permit requirement. Shuttlesworth, 394 U.S. at 151, 89 S.Ct. at 939. But the question for us is whether, by virtue of the defects in the permitting scheme as it existed in that time period, Heerey and GASS also obtained a protected property interest in the sign as a lawful, non-conforming use, just as if Heerey had obtained a permit or as if the electrical code had not required a permit for the illumination or expansion of an existing sign.
Our decision in Petra Presbyterian Church v. Vill. of Northbrook, 489 F.3d 846 (7th Cir.2007), cert. denied, ___ U.S. ___, 128 S.Ct. 914, 169 L.Ed.2d 786 (2008), decided shortly after we heard argument in this case, reveals that the answer to this question is no. The plaintiff in Petra was a religious congregation that had purchased a warehouse in the hopes of converting the facility into a church. The property was located in an area of a suburban Chicago village that was zoned for industrial use. Although a 1988 zoning ordinance permitted other membership organizations, including community and youth centers, fraternal associations, and political clubs, to operate facilities in an industrial district, the ordinance did not allow a church to locate there absent rezoning of the property and issuance of a *1004 special use permit. Before purchasing its property, Petra applied for the necessary rezoning and permit. However, after neighboring landowners voiced opposition to the request, the local planning commission recommended to the village board of trustees that the application be denied, and the trustees signaled their inclination to acquiesce in the commission's advice. Before the denial could be formalized, Petra withdrew its application but it nonetheless proceeded to purchase the property and, without authorization, began using the warehouse as a church. In the meantime, Congress had enacted the Religious Land Use and Institutionalized Persons Act of 2000 ("RLUIPA"), which among other things prohibits a local government from imposing or implementing a land use regulation that "treats a religious assembly or institution on less than equal terms with a nonreligious assembly or institution." 42 U.S.C. § 2000cc(b)(1). Fearing that its 1988 ordinance might be invalid under RLUIPA because it treated religious organizations less favorably than other membership organizations in terms of their ability to operate in an industrial district, the village in 2003 adopted a revised ordinance that banned all membership organizations from industrial districts. Existing organizations were grandfathered, but Petra, which had never received permission to operate a church in the industrial district, was not. After the new ordinance was enacted, the village sought and obtained an injunction in state court effectively barring Petra from conducting worship services in its facility. The plaintiff filed suit in federal court under RLUIPA and lost in the district court.
Among other arguments, Petra contended on appeal that although it had never obtained the requisite rezoning and permit to operate a church within the industrial zone of the village, it nonetheless had obtained a vested interest in continuing to operate a church there. Petra claimed that when it bought the property, it was relying on the invalidity of the 1988 zoning ordinance, which by treating religious organizations less favorably than other membership organizations violated not only RLUIPA but the free-exercise clause of the First Amendment. By virtue of that reliance, Petra maintained that it had obtained an indefeasible right to use the warehouse for a church. So in Petra's view, even if the zoning ordinance enacted in 2003 barring all membership organizations from the industrial zone were valid, Petra would still have the right to operate its church there like any other grand-fathered organization  in other words, as if it had obtained the village's permission to do so in the first instance.
We categorically rejected the notion that Petra could have acquired any right to operate a church in reliance on the invalidity of the zoning ordinance that was in effect when it purchased the property and established its church:
We cannot find any basis, whether in cases or other conventional sources of law, or in good sense, for the proposition that the federal Constitution forbids a state that has prevented a use of property by means of an invalid (even an unconstitutional) enactment to continue to prevent that use by means of a valid one. From the proposition that the Village should not have discriminated in the industrial zone in favor of secular membership organizations it does not follow that when it eliminated the discrimination by banning all membership from the zone, this entitled the victim of the discrimination to claim, by way of remedy, discrimination in its favor. So strange a rule of estoppel could hardly be thought an imperative of due process, especially since, although property rights are protected (to a degree) by the due process clauses of the Constitution, *1005 the scope of those rights is determined by state law within broad limits that include the conditions under which a governmental act can render a property right indefeasible. Crown Media, L.L.C. v. Gwinnett County, 380 F.3d 1317, 1325 & n. 18 (11th Cir.2004); Coral Springs Street Sys., Inc. v. City of Sunrise, 371 F.3d 1320, 1333 (11th Cir. 2004); Lakeview Dev. Corp. v. City of South Lake Tahoe, 915 F.2d 1290, 1294-95 (9th Cir.1990).
And if there were such a rule of constitutional law ("vesting by estoppel"), it would be inapplicable to this case. If the 1988 ordinance violated RLUIPA, as Northbrook comes close to conceding, Petra didn't have to comply with it. But that doesn't mean that it acquired an immunity from all zoning regulation. It knew or should have known that Northbrook could redo its ordinance to comply with the "less than equal terms" provision of RLUIPA in one of two ways: by permitting religious organizations in the industrial zone, or by forbidding all membership organizations in the zone. Petra could not reasonably assume that the Village would choose the first option. And since it therefore did not reasonably rely on the illegality of the 1988 ordinance in going ahead and buying the property, but instead assumed the palpable risk that a new, valid ordinance would continue the ban on its desired use of the property, it has no ground for blocking the Village from enforcing the amended ordinance against it, on the theory that the Village pulled the rug out from under it by changing the ordinance.
489 F.3d at 849 (emphasis in original).
GASS finds itself in essentially the same position as the plaintiff in Petra. GASS wishes to continue using its property in a manner that the current version of the City's zoning ordinance prohibits, because its sign is larger than 100 square feet and is within 250 feet of a residential district.[8] GASS's only means of preserving its sign is to establish that the sign was a lawful use prior to the zoning change. The problem for GASS, as it was for Petra, is that neither GASS nor Heerey ever obtained the permits that were required for its use; Heerey illuminated and expanded the sign without even seeking a permit. So even before the current zoning took effect, GASS's sign, like Petra's church, was an un authorized use of its property. GASS cannot claim a protected interest in continuing to display advertising that, once illuminated and expanded, was illegal from the start. To circumvent that obstacle, GASS seeks to rely on constitutional defects in the permitting scheme in effect in the 1960s and 1970s, contending that these defects rendered the permit requirement a nullity and rendered its illuminated and expanded sign a lawful use of the property notwithstanding its lack of the requisite permits for such a sign. GASS, just like the plaintiff in Petra, is retrospectively relying on the supposed illegality of former incarnations of the zoning ordinance to transform its unlicensed use of property into an authorized use and to give it a property interest in continued use of that property despite subsequent changes to the zoning code. GASS's creative bootstrapping effort fails for precisely the same reasons that Petra's did.
We note that, consistent with our decision in Petra, Illinois cases also reject the notion that a property owner may rely on purported defects in prior versions of the law as the springboard to claim a vested interest in a property use for which it never obtained permission. As the court reasoned in City of Elgin v. All Nations *1006 Worship Ctr., 369 Ill.App.3d 664, 308 Ill. Dec. 9, 860 N.E.2d 853, 857 (2006):
[V]ested rights are acquired by attempting to comply with an ordinance as written.... [W]hen a party expends substantial time and effort attempting to comply with an ordinance as it then exists and then the legislative body amends the ordinance, the party may acquire a vested right to proceed under the old ordinance. [Citation omitted.] Here, however All Nations proceeded in violation of the zoning ordinance as written. It is difficult to see how All Nations can claim a vested right to ignore the existing ordinance.
(Emphasis in original.) See also Nat'l Adver. Co. v. Vill. of Downers Grove, 204 Ill.App.3d 499, 149 Ill.Dec. 604, 561 N.E.2d 1300, 1304-05 (1990). As it is Illinois law to which GASS must look for a property right, these holdings confirm that GASS lacks the requisite basis for a due process claim.
As a fallback, GASS contends that at the very least, it has a protected property interest in continuing to display a smaller, non-illuminated sign on its building. Because, as the district court assumed, no permit was required for the sign or signs Heerey originally had painted on his building in the early 1960s, that type of sign was an authorized use and GASS argues that it may assert a right to display at least that type of advertising, even if it has no claim to the larger, illuminated sign for which it never sought the requisite permits.
But this is an argument that GASS made too late in the day. GASS first advanced this argument in a meaningful way in its supplement to its post-trial motion under Rule 59 to amend the judgment. R. 131. Arguments that could have been made earlier but are instead raised for the first time in a Rule 59 motion are waived. E.g., Estremera v. United States, 442 F.3d 580, 587 (7th Cir.2006). The argument was hinted at parenthetically in the pretrial order's recitation of GASS's proposed legal conclusions (see R. 111 Ex. L at 8 ¶ 44), and again in the opening statement that GASS's lawyer made at trial, R. 166 at 16, but those references alone were insufficient to alert either the City or the court to the argument that GASS finally developed and supported in the supplement to its Rule 59 motion. As the district court noted, GASS offered no explanation for its tardiness in pursuing this argument. R. 144, 2006 WL 1460017, at *5. We do not reach the merits of the argument, concluding that GASS waived it.
Because GASS has not established that its sign was lawful before the 1990 zoning change that banned advertising signs within 250 feet of a residential district, it has no protected property interest in the sign that would support a substantive due process claim. On that basis, we affirm the entry of judgment against GASS and in favor of the City on the substantive due process claims set forth in Counts III and IV of GASS's second amended complaint.

B. Procedural Due Process
In a cursory argument spanning all of two paragraphs in its opening brief (GASS Br. at 43-44), GASS argues that the district court incorrectly dismissed as untimely its procedural due process claim. The argument is so brief that GASS has waived it. See, e.g., Robinson v. Alter Barge Line, Inc., 513 F.3d 668, 675 (7th Cir. 2008). GASS does not even set out the elements of a procedural due process claim. See Mema v. Gonzales, 474 F.3d 412, 421 (7th Cir.2007); Estate of Allen v. City of Rockford, 349 F.3d 1015, 1022 n. 9 (7th Cir.2003).

C. First Amendment
Finally, GASS contends that its complaint alleged a valid claim for the threatened and impending denial of its *1007 First Amendment right to free speech based on the commercial content of the sign. Recall that the district court dismissed this claim because the focus of GASS's challenge was not on a provision of the zoning ordinance having to do with the nature or content of the sign, but rather on the grandfather provision of the ordinance which allowed non-conforming signs erected pursuant to a permit to remain after the 1990 zoning changes but not signs erected without a permit. That provision, the court pointed out, applied to all signs, not just advertising signs; consequently, GASS's claim did not in fact have to do with the regulation of commercial speech. Moreover, as a challenge to the grandfather provision, the court viewed this claim as simply a restatement of GASS's substantive due process claims. "Plaintiffs cannot convert what is essentially a due process claim into one under the First Amendment merely by noting that theirs is an advertising sign." R. 93, 2004 WL 442636, at *5.
GASS contends that the district court overlooked the First Amendment implications of its challenge. Although the grandfather provision found in section 6.7-1(a) of the zoning ordinance applies to a variety of signs, not just those displaying advertising, as relevant here it operates as an exemption from the limits imposed by section 8.9(7) of the ordinance, which prohibits advertising signs in excess of 100 square feet from being within 250 of a residence district. "The regulation at issue, therefore, is a regulation of commercial speech." GASS Amended Br. 46. As such, GASS reasons, it must be scrutinized under the four-part test for regulation of commercial speech set forth in Cent. Hudson Gas & Elec. Corp. v. Pub. Serv. Com'n of New York, 447 U.S. 557, 566, 100 S.Ct. 2343, 2351, 65 L.Ed.2d 341 (1980). That test requires in part that a restriction on legitimate, non-misleading commercial speech must be designed to advance a substantial government interest. See id. at 564, 100 S.Ct. at 2350. GASS contends that the City has no legitimate interest in grandfathering non-conforming advertising signs that were erected pursuant to a permit but not those, like its own, that were erected without one; that distinction, in GASS's view, is irrational and arbitrary. GASS Amended Br. 47. It likens the distinction to the one that the City of Cincinnati drew between the use of newsracks on the public way to distribute commercial bills, which the city prohibited, and the use of such racks to distribute newspapers, which the city allowed. The Supreme Court deemed that distinction unconstitutional in City of Cincinnati v. Discovery Network, Inc., 507 U.S. 410, 113 S.Ct. 1505, 123 L.Ed.2d 99 (1993), finding there to be no reasonable fit between the city's interest in aesthetics and safety and its decision to ban only those newsracks distributing commercial bills.
Try as it might, GASS has not succeeded in stating a claim for the regulation of commercial speech. True enough, section 8.9(7) of the zoning ordinance is a regulation of commercial speech, in that it specifically bans advertising signs larger than 100 square feet from being located within 250 feet of a residence district. But GASS does not contend that the constitutionality of section 8.9(7) itself is suspect. It does not even analyze section 8.9(7) pursuant to the Central Hudson test. Its attack, as the district court recognized and as GASS's appellate brief makes plain, is on the grandfather provision of section 6.7-1(a), and the differing treatment that provision grants to previously permitted signs as distinct from signs that were enacted without a permit. Yet, the grandfather provision is not a provision that is triggered by the commercial content of the sign in question. As GASS concedes, section 6.7-1(a) applies to all manner of signs, not just advertising signs. As such, we agree with the City that the grandfather *1008 provision is a content-neutral time, place, and manner restriction on speech rather than a regulation of commercial speech. See Discovery Network, 507 U.S. at 428, 113 S.Ct. at 1516; see also, e.g., Solantic LLC v. City of Neptune Beach, supra, 410 F.3d at 1268 n. 15.
The City has gone on to analyze the grandfather provision as a time, place and manner restriction, but GASS itself has made no argument that the provision is unconstitutional under that framework, thereby waiving any such contention. For these reasons, we believe the district court correctly dismissed GASS's First Amendment claim.

III.
GASS has reaped hundreds of thousands of dollars in revenues from its sign despite the fact that the sign has been unauthorized for the past three or more decades. Those days are at an end. Because GASS's predecessor Heerey failed to obtain the requisite permits to illuminate and then expand the sign in the 1960s and 70s, the sign was not a legal land use that gave rise to a protected property interest. Absent such a vested property right, GASS cannot mount a substantive due process challenge to the 1990 amendments to the City's zoning ordinance, which prohibited signs as large as GASS's within 250 feet of a residential district and grandfathered only those existing, non-conforming signs which had been erected pursuant to a permit. GASS has not shown how the First Amendment is implicated by the content-neutral provisions of the 1990 grandfather provision. GASS's procedural due process claim has been waived. For all of these reasons, we AFFIRM the district court's judgment.
NOTES
[1]  The name of the corporation was a holdover from a time when Heerey and his parents owned an automobile parts business on Chicago's north side. Heerey closed that business in the 1960s.
[2]  Although GASS's counsel indicated at trial that the plaintiffs believed Heerey did obtain a permit for illumination of the sign, he conceded that GASS had been unable to locate any such permit. R. 166 at 15, 23.
[3]  In 2004, the City eliminated the distinction for grand-fathering purposes between permitted and non-permitted signs. Section 6.7-1(a) was repealed and replaced by two new provisions, found at sections 17-15-0502 and 17-15-0503 of Chicago's Municipal Ordinance, which collectively permit a non-conforming sign to remain in use so long as the sign complied with the zoning and sign regulations in effect at the time the sign was established. As it is section 6.7-1(a) that is relevant for purposes of GASS's appeal, we shall refer exclusively to this section and speak of it in the present tense, although it has since been superseded.
[4]  There is a dispute between the parties as to whether the City opposed GASS's petition for leave to intervene in Whiteco's suit. We were led to believe in Gen. Auto Serv. Station LLC v. City of Chicago, 319 F.3d 902, 905 (7th Cir. 2003), which we discuss below, that the City actively had opposed GASS's petition. The City contends that we were mistaken in that regard, although GASS and the other appellants do not share that view. We need not resolve this dispute for purposes of the instant appeal.
[5]  GASS filed the second amended complaint with the district court's permission after the bench trial had concluded, in order to conform the allegations to the evidence. For the sake of convenience, we shall refer to that version of the complaint, which includes the First Amendment and procedural due process claims that the court had dismissed prior to trial in addition to the substantive due process claims that the court disposed of after trial.
[6]  In addition to its federal claims, GASS asserted parallel state claims under the pertinent provisions of the Illinois Constitution, although the state claims have not been discussed on appeal. Illinois courts apply a limited version of the lockstep doctrine, adhering to U.S. Supreme Court precedent concerning the federal constitution when interpreting cognate provisions of the state constitution. See People v. Colon, 225 Ill.2d 125, 310 Ill.Dec. 396, 866 N.E.2d 207, 223 (2007) (citing People v. Caballes, 221 Ill.2d 282, 303 Ill.Dec. 128, 851 N.E.2d 26, 42 (2006)). In the absence of any argument to the contrary, we may assume that our analysis as to GASS's federal constitutional claims would also apply to its state constitutional claims.
[7]  We note, however, that GASS relies for this point on the permit applications Whiteco completed in 1994 and 1997, App. 113A, 118, rather than on any proof as to what information the permit applications required in the 1960s and 1970s.
[8]  No question is raised as to the validity of the current prohibition.
