I just wanna start by saying that I love this thread because it embodies everything that makes this forum great. By reading each post, I was able to learn a lot about both the relevant history, the different contextual understandings thereof, and the evolution of legal definitions over time. I'll throw my two cents, but as a lover of semantics, legalistic arguments, and discussions of historical nuances, I just wanna say that reading this thread has been awesome.
I'm inclined to agree with this:
Without an agreed upon working definition of what exactly was the "Roman Empire" the question isn't answerable.
The whole thread has been spent arguing void of well-defined context, and thus everyone has been talking in circles. Unfortunately, the anachronistic language used by mainstream historians is often insufficient to determining the validity of such claims. So first, I will attempt to provide a little context.
Tradition has it that the Romans were first rules by kings until abuses by the monarchs led to the partition of the powers of king (or rex) into several different offices. The rex's "imperium" (or ability to command the obedience of citizens from which the word "empire" eventually evolved) went to the consuls (originally called praetors), who acted as the heads of state and heads of government for the burgeoning Roman Republic. Other powers of the rex went to various elected offices including the pontifex maximus, censors, and the urban prefecture. Shenanigans ensued including the conflict of the orders, the creation of the tribunate as an office independent of the executive magistracies, the conquest of the Mediterranean, the co-option of the tribuneship into the senate, and eventually, the fall of the Republic. The important thing to remember here is that, for the entire period from 509 BCE to 14 CE, the senate (derived from "senex" or "elders") had
almost no actual legal authority. All legislative authority lay with the assemblies, which did not change until the reign of Tiberius, when the senate was finally incorporated into the official structure of the government. So what happened in the meantime? The fundamental contradiction between the formal power of the magistrates and the informal power (or "auctoritas") of individual private senators led to civil war and eventually, the rise of Augustus. Throughout his stewardship of the Republic, he gradually reconsolidated the powers of consul, tribune, pontifex maximus, and censor back into a single person - himself. However, this was done ad hoc and without a clear legal formula, and so applying words like "emperor" to Augustus and his successors betrays an anachronistic misunderstanding of the institutional framework within which Augustus lived. For the entirety of the "principate", the emperors were essentially private individuals wielding informal powers coupled with a few formal powers (tribunician authority, consular and proconsular imperium, and others). This framework gradually became more and more formal. By the time of at least Vespasian, the "imperial powers" were granted en bloc to any new emperor and the word "imperator", which had once signified a victorious commander in war, gradually morphed into the political title which we call the "emperor". However, legally throughout this period, until at least the third century, Rome was still legally a Republic. Sole legislative authority came from the senate and sole executive authority came from their grants of imperium to the magistrates and to the person of the emperor. The office of emperor became detached from Rome itself (the city) and eventually Rome itself ceased to be the capital. In that period, the empire became Christian (though this was not made official until the reign of Theodosius), the West fell, and the East suffered numerous territorial losses. All of this is to say that the evolution of Roman institutions was based strongly on precedent. The emperors powers were defined in the precedents of the republican magistracies; the succession of later emperors was informed by the precedents set by the succession of early emperors. Even the terminology in use remained the same, in spite of wildly divergent definitions over time.
So how can one argue that the Roman Empire was continuous throughout this whole period? What is generally cited is the continuity of laws throughout this period. In spite of the rise and fall of different dynasties, different constitutional arrangements, different state religions, and different capital cities, the general body of Roman law, which had begun with the authorship of the twelve tables and culminated in the Digest of Justinian remained in continuous use. However, one might argue that the ecclesiastical laws of the papacy represented a similar continuity. However, I will argue that the precedent set by Theodosius I, whereby the Eastern emperor had the right to appoint the Western emperor, which was implicitly recognized by Odoacer, thus his dispatch of the Western regalia to Constantinople. By this precedent - and the Romans valued their precedents dearly thus why they clung to the republic even as its institutions broke down - only the Eastern Emperor had the right to appoint a successor to the western throne. Even in cases wherein the succession in the East was uncertain, the West had not, since the reign of Julian the Apostate, been able to replace the Eastern emperor.
Of course, we are applying concepts like statehood, succession, and ecclesiastical primacy that would not have even been conceivable for the men who formulated the early principate. I think the continuity of institutions means that the Byzantine empire is unambiguously the Roman Empire. However, such discrete concepts as "successor states" are modern, post-Westphalian abstractions. Thus, without defining exactly what we mean by "Roman Empire", as the above post stated, we are talking in circles. In my opinion, the definition of "Roman Empire" is generally: the political and legal institutions which originated in Italy to eventually rule the Mediterranean before a long and gradual decline. In this context, the so-called "Byzantine empire" factors neatly into this decline, and the continuity of institutions in the broadest sense makes it the sole legitimate successor to the Roman empire. Not that other states did not similarly adopt Roman-like institutions, but merely that their widespread and continuous practice within a nominally sovereign and continuous political system, make the Byzantines the "most Roman-like" state to emerge out of the chaotic 5th century. But ultimately, these decisions are arbitrary and only make sense retroactively. By the strictest interpretation of the definition I gave, the last legitimate Roman emperor was Nero, after which every ruler from Galba to Constantine XI was a usurper. The absurd implications of this clearly illustrate why such strict definitions are not often used.
I'm not trying to convince anyone of anything. In some sense, almost every Western state is a "successor to the Roman empire". Why we have these discussions at all are really a reflection of what each of us prioritizes within the study of history. I, personally, understand history to be a gradual evolution, clash, and amalgamation of institutions primarily constrained by population and the environment. Anyone else might view history as primarily religious, or military, or economic, or ideological, or sociological, or ecological, or whatever. I think there's no incorrect answer to this question. I think the Byzantine empire is unambiguously the Roman empire for all the reasons stated above, but the idea that there is only one "legitimate" successor to such a vast and diverse political amalgamation that existed for so long and in so many different forms as the Roman Empire is patently ridiculous and displays a misunderstanding of the different historical conceptualizations of abstractions like "legitimacy" in the first place.