I think you're mistaken. When talking about political unity in the Middle Ages, the waters are very murky (after all, every lord was sovereign and voluntarily linked to a liege lord... in theory). But the theory, which you refer to, was far from practice.
Saying Italy was united is more like saying, in 2019, that Syria or Iraq are united countries, stable and whole. Sure, Bashar al-Assad is President of Syria, but there are vast swathes of the country which do not obey his laws and do not send him their taxes (the very essence of a State: collects taxes and enforces laws, in the present or in the past).
The HRE had theorical authority over it, but this authority was rarely enforced even before the 10th Century, and seldom considered after the 13th, even by the Emperor himself. The Pope sure wanted to fill that chair, but he couldn't. Which already tells us that what you said is wrong. Italy was not united. The "neglect" you cite from both the Papacy and the Empire were already symptoms of the Italian states considerable political disunion and autonomy. Rome was unconfortable for the Pope because of how politically unstable it was, because of how much the Pope became less and less relevant in local Italian politics, despite his enormous influence over kings and princes. The inner workings of Italian politics after the 1250's were so contrary to the Pope's interests that he simply found himself a more stable base of power under the wings of the King of France, who had newly gained sovereignity over Southern France and could redistribute power as he wished (itself another example of how much de jure ownership of land in the early Middle Ages meant very little; the King of France had to retake Southern France essentially from the lords of the Midi, who saw themselves as within France, but not under it, and certainly not vassals of the King of France, but I digress).
At the year 1200, not even Innocent III, the most powerful man in Europe (probably the most powerful Pope in history) had the capacity to consider Italy "under his tumb", and by 1250, after Barbarossa's defeat and Frederick II's abdication of German power and retreat to Naples and Sicily, Italy was effectively a land without external entanglements. All ready for Aragon and France to claw their way into it by the 1400's.
Everything is in theory, nothing ever works in human politics as do clocks, checking at specific times. Eccentricity always exists and political legitimacy is always mired by dissent, decentralism and a level of inadequate ruling. We speak of many states of the past as unified entities, lords over great realms, as single units, undivided. Yet, for the Holy Roman Empire, due to the western mindset as of late, following the Enlightenment, a bigotry and flawed understanding of the Holy Roman empire has arose. Yet in the same breath, these speakers and thinkers, refer to the Abbasid caliphate and the Sassanid empire as if it is singular entities. Each of these are arguably less centralized than the Holy Roman Empire in Italy and less able to claim to be a singular entity. The Sasanid empire might best be called 12 different independent kingdoms who simply placed puppet rulers upon the throne. The Abbasid was a regime that could not even issue taxes upon the majority of its subjects and was dependent upon vassals to a degree exceeding even French monarchs in its height of noble power.
The empire by comparison, as is argued by scholars who are learned in fields not pertaining to recent European history or revisionist dogma, we find in a world-wide basis, the Holy Roman empire and the French kingdom was actually symbols of centralism, especially in terms of theoretical ideology. Namely, feudalism implies a level of centralism in terms of ruling legitimacy that in some areas would be uncomfortable. Firstly, in feudalism we presuppose a liege-lord, who by either divine ordination, Papal confirmation, by noble lineage or tribal custom, is the master over a realm. As master of said realm, he distributes the lands to his vassals in the form of demenses and fiefs. He holds all lands by title and commits his allies to rule lands on his behalf. This supposes a singular authority and a conception that ever beyond one's post, there is another above himself. In other lands, such a conception was not always existing, yet they are termed by western scholars and rightfully so, as singular entities.
So regarding your notion that if a king does not rule exactly to theory or that the theory is not a Louis XIV model or some other modernist conception of firm handed rule, if this is your position, I would like you to admit or agree that under this view, we shoudl revise history to say the vast majority of states who have existed, were not truly so. Rather, they were just conglomerations of states and really did not rule them. It is hard to imagine for instance, claiming Assyria ruled Syria when they had to reconquer the land some dozen times (meanwhile such rebellions of this calibre never erupted against the Emperor's legitimacy to rule), while the Empire is some sort of mythic entity nonexistent outside of Heidelburg or Regensburg. My view is opposite of yours, it is that the Empire, while very decentralized, even for feudalism, was ultimately a singular entity, both in Lotharingia, Germany and in Italy. Its failures to hold onto Italy firmly, have less to do with its reigning ideology, but more to do with the various events that occurred in the Empire and Europe at large that made such universal empires less permissible in Europe, this is covered in detail by Giorgio Falco in his 'Holy Roman Republic.'
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Syria is a country in civil war, wherein many different parties question fundamental legitimacy of a particular ruler. Italian states at times rejected a particular emperor, but did not question the notion of the Kingdom of Italy or the Roman Empire. They also questioned the notion of how one was to attain an imperial throne. We call upon Dante and his 'De Monarchia,' wherein he made the claim that the Holy Roman Emperor was the divinely appointed and majestic master of Italy and in his opinion, of earth entirely. While simultaneously, Dante made an even stronger case for Imperial absolutism, claiming that the Papacy had no power over Imperial succession and that the electors of the Empire should only be advisory positions.
" If this is so, and there is none higher than He, only God elects and only God confirms. Whence we may further conclude that neither those who are now, nor those who in any way whatsoever have been, called Electors, have the right to be called;rather shoudl they be entitled heralds of divine providence. Whence it is that those in whom is vested the dignity of proclamation suffer dissension among themselves at times, when all or part of them being shadowed by the passions, they discern not the face of God's dispensation. It is established clearly, that the authority of the Emperor (he refers to the HRE) descends without mediation from the fountain of divine and universal authority.This fountain, pure and true, flows multifarious channels from the divine abundance of excellence." -Dante Alighieri espousing absolute imperial authority in Europe
" Wherefore a twofold directive agent was necessary to man, in accordance with the twofold end ; the Supreme Pontiff to lead the human race to life eternal by means of revelation, and the Emperor to guide it to temporal felicity by means of philosophic instruction." - Dante Alighieri espousing the two stars model
"Methinks I have now approached close enough to the goal I had set myself, for I have taken the kernels of truth from the husks of falsehood, in that question which asked whether the office of Monarchy was essential to the welfare of the world, and in the next which made inquiry whether the Roman people rightfully appropriated the Empire, and in the last which sought whether the authority of the Monarch derived from God immediately, or from some other. But the truth of this final question must not be restricted to mean that the Roman Prince shall not be subject in some degree to the Roman Pontiff", for felicity that is mortal is ordered in a fore measure let Caesar after felicity honor that Peter is immortal. Wherefore let Caesar honor Peter as a first born son should honor his father, so that, refulgent with the light of paternal grace, he may illumine with greater radiance the earthly sphere over which he has been set by Him who alone is Ruler of all things spiritual and temporal." -Dante Alighieri once more espouses the two star model of Charles I (the origin of this model).
The preeminent scholar of Italy for the time in 1300, understood the role of the Emperor and held with stern gaze, that the Emperor was the liege lord over Italy. We are dealing with a set of Italian states that rejected the Empire's right to rule, but rather an Italy that was assured of its certain liberties and were also of two-minds as to the custom of rule. Was the kingdom of Italy inherited as:
1. Inherited by the empire by right and as such he was divinely appointed by God. This is the opinion of the Hoehnstaufen clan, Frederick II and Dante.
-or-
2. Upon the demise of an emperor, the kingdom goes into flux as does Rome, with the Papacy inheriting the title and then redistributing the title back to the elected king of Germany or whom they choose, making the Pope the feudal lord of the Empire of Rome. This was the opinion of Innocent III, Boniface VIII and other advocates of true Papal Absolutism. It is also the view of the Papacy in the 8th century, John of Damascus outlined it well when he said:
‘….if any anyone attempts to teach you anything contrary to what the Catholic Church has received from the Scripture, Holy Apostles, the fathers and the synods, and that has kept to this day, pay no heed to him. Even if an angel, even if the Emperor tells you something other than what you have received from the Church, let them be anathematized.’
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Regarding the Papacy it would depend. At least many of Italy accepted the Papal feudal overlordship prior to Frederick I, when they accepted an Emperor over them that was confirmed by the Papacy. All of the Italian states also accepted the Papal powers of interdiction, which presupposes Papal feudal authority over said realm. Innocent III regarding Italy too was able to enforce councils upon them which affirmed the existing custom, that Italy was inherited principally by the Papacy and distributed by him afterward back to the Kingdom of Germany, this is the 'translatio imperrii.' Legally speaking, the Papacy was the land lord of Italy before Innocent III and after Innocent III certainly.
What you refer to as disunity in the Papal regime, can be broken into different types and for different times:
-The battle between specifically Roman clans.
-The protection of localism among the varied members of the Duchy of Spoleto and the former Byzantine province.
The second did not exist prior to the Papal flight from Rome to Avignon. The first, was an issue faced by any state..... Byzantium had constant sources of intrigue, none of which came from foreign entourage, but most came directly from its capitol. Yet we do not claim, 'Byzantium did not rule Constantinople.' Yet far more emperors of Byzantium found their end at the hand of their court and a bureaucrat, than did a Pope faced against the swords of the Roman barons. The Papacy fled from Rome though for two reasons:
-The danger certainly imposed by Roman barons who had become strong on account of treachery and the French state.
-An attempt by the Papacy to appease the French monarchy.
If both did not exist at once, they would not have fled. Byzantium never fled Constantinople due to no benefit gained from leaving it (this is a bit wrong, in that some emperors simply ruled from the frontlines or in Anatolia, rather than Constantinople; we could say this is what the Papacy did,, they never relinquished their rule over said Duchies). The Papacy however, by this flight, left Italy to divide itself and reach the state of the fabled 'shadow kingdom,' permitting the rise of local lords and the preeminence of Venice and other usurpers to Papal and Imperial legitimacy.
My point: it is too hasty to judge Papal monarchy and the Empire based upon the situation exhibited upon their fall and decline. It is as if we are to judge the Assyrian empire based upon its situation at its decline when it ruled only a few towns and villages. Such is not how we should conduct history.
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Naples and Sicily are not dejure Italian lands. The inclusion of them into our discussion is anachronistic, those lands were never part of the Kingdom of Italy as it was conceived.