There is so much to respond to here that one doesn't know where to start.
In Section 28, Chapter 1 of Book 1, in On War, Clausewitz describes the "remarkable" or "paradoxical trinity". . .
War is more than a true chameleon that slightly adapts its characteristics to the given case. As a total phenomenon its dominant tendencies always make war a paradoxical trinity --composed of primordial violence, hatred and enmity, which are to be regarded as a blind natural force; of the play of chance and probability within which the creative spirit is free to roam; and of its element of subordination, as an instrument of policy, which makes it subject to reason alone.
The first of these aspects mainly concerns the people; the second the commander and his army; the third the government. The passions that are to be kindled in war must already be inherent in the people; the scope which the play of courage and talent will enjoy in the realm of probability and chance depends on the particular character of the commander and the army; but the political aims are the business of government alone.
These three tendencies are like three different codes of law, deep rooted in their subject and yet variable in their relationship to one another. A theory which ignores any one of them or seeks to fix an arbitrary relationship between them, would conflict with reality to such an extent that for this reason alone it would be totally useless.
Our task therefore is to develop a theory that maintains a balance between these three tendencies, like an object suspended between three magnets.
In On War, Clausewitz describes different types of theory: a theory of the art of Napoleonic war, a theory of politics and the general theory of war. This section is the capstone to the entire general theory of war, which is a theory of war, not warfare, and able to cover all wars since it deals with social and moral, not material elements. In fact the mention of "people", "army" and "government" are the first mention of any material elements in the entire chapter, this being the most important chapter in terms of the general theory. To spell it out once more, the Clausewitzian trinity is "primordial violence, hatred and enmity, which are to be regarded as a blind natural force; of the play of chance and probability within which the creative spirit is free to roam; and of its element of subordination, as an instrument of policy, which makes it subject to reason alone." To these can be matched material elements (which are not of course "the trinity"), but can be used to help explain it. Whereas the moral elements are common to all wars, the material elements make each war unique. One could replace people, army and government with tribe, warriors and chief and the general theory would still apply. So the trinity is about war as subordinated to politics/policy, not about the state. War is seen as part of political/social intercourse which is not limited to actions between states.
Thus "trinitarian warfare" is not a Clausewitzian concept, but a 4GW concept, so let them take it on as their own, FINALLY. That is except THEIR concept for what it is, essentially a strawman to which they compare their competing concept of non-trinitarian warfare.
Then perhaps we can actually start to discuss how 4GW links tactics with operational art and gains a military aim in support of a political purpose, but that of course is not possible if one rejects Clausewitz, since Clausewitz is and remains the basis for strategic theory.
Imo 4GW is simply a reified concept, which its promoters see as actually existing in reality, something which is an absurdity from a Clausewitzian perspective. This essentially puts the cart before the horse since strategic theory is seen as drawing concepts from military history, not "existing" before the history actually occurs.
Strategic theory is about a "system" of concepts used to analyze war, not conflict (such as the metaphorical wars on drugs, between the sexes, on poverty, against dandruff, among parents and teenagers, etc). War is the organized use of violence within or between political communities. Robbing a bank is not warfare and neither is a lone gunman shooting unarmed people. Both are violent crimes since neither criminal represents a political community in any legitimate (as judged by the political community involved) way.
This is one of the greatest inconsistencies of 4GW. Its proponents claim that 4GW is just about anything ("the tactics of non-trinitarian conflict"), but if it is just about anything, how is there to be any pattern that could form the basis of theory? It is precisely these patterns present in war as a social phenomenon that Clausewitz used to form the general theory.
The war in Afghanistan is not a "4GW war", but a war of distinct political communities confronting an occupying foreign coalition which wishes to impose its will on some of the locals, who resist. The Taliban, far from being a "4GW entity", represent the former government who are fighting to regain political power. A foreign occupier imposing his will has never succeeded in Afghanistan before, using buzzwords like "4GW" or "COIN" to describe military operations and confusing the issue, isn't going to make it happen this time.
Until we rediscover the basic truth, that wars occur within a political (broadly defined of course since "politics" is defined as the struggle for power and resources within a political community) context we will continue to misread and misunderstand the conflicts in which we have become engaged. Who benefits? Perhaps those for whom the reified concept conceals the actual political purposes. Notice that 4GW takes no account of the distinction between "Empire" and "State", focusing instead of the patently false assumption that all states are in some deterministic death spiral. This distinction, more than that between "dying state" and "4GW entity" is something important for all those interested in US foreign policy to consider today.
Changes in political conditions lead to changes in, an evolution of the nature of war. War being an interaction causes these changes to affect both sides but in different ways.
I have been working on a post on the development of operational art which will come up soon. For those interested in following up on the weaknesses of 4GW, one of the best articles is Matt Armstrong's The misleading theory of fourth generation warfare.
and of course LtCol Echevarria's 4th GW and other Myths .