The reason why we cannot find proof for something like the Collatz Conjecture is because its a function of reality that transcends the universe in which we have based our math. We cannot find harmony/logic in this sequence, as it eventually collapses to a loop, because our logic (with which we would define such things as harmony, for instance) is a feature of the universe in which we live. The "Collatz Sequence" is a function of reality that transcends the universe in which we live. To describe it mathematically, we would have to use logic based in higher levels (dimensions). This kind of logic would trickle down. For instance: all mathematical statements in what we call the universe (which needs to be understood as a dimension with dimensions in it) are true in every dimension below it, in our four spacetemporal ones, and the many more that exist beneath, seemingly below our perceptions. However, they are not necessarily true in the dimensions above.
Consider reality (as opposed to "just" the universe) an attempt by a flow of energy to prove mathematical laws. It will bend into lesser dimensions (universes) with different laws of nature/mathematics. Universe 1 would claim that 1=1. Then Universe 2 and 3, which would exist inside Universe 1, and be lesser, would postulate that: 1+1=2 and 1+1=3 in attempts to prove the initial point (and so on). It would then keep "bending" into lesser dimensions in ever more complex attempts to prove said statement. Not until proof is found will the energy "regress" back to the source and relay the proof. What we get is a hierarchy of universes/dimensions, most of which will calculate for what we call infinity in vain attempts at finding proof.
Our universe is just another dimension in this hierarchy, as is the four commonly known dimensions spacetime, and the lower dimensions currently being proposed by string theory and suchlike models. A simple thought, for instance, is a new dimension, bending deeper into reality to prove a point. If it proves it, it regresses back to the source, the consciousness that initiated the thought.
Likely, the mathematics of this universe are flawed, which is also the reason why this universe still exists, and why it is still calculating (expanding, as cosmology would say). This universe is just one more dimension trying to prove the viability of a set of "natural laws" (or mathematics). They are flawed in the sense that they are not harmonic, but that is what drives the existence of this universe. It forces reality to probe ever deeper by virtue of self-propagation, ever bending into smaller dimensions in pursuit of proof of the mathematics (what this pursuit is, and why, is a different discussion).
Reality, if we are to describe it with the abstract tool of language, as opposed to the stricter tool of mathematics, a data collection system. Proving a point. In the case of the Collatz problem, that seems destined to fail, but when?
The pattern we observe it in the Collatz conjecture breaks down to a loop in the end, but let's describe that as an optical illusion. There is no loop; it's a spiral - a spiral into a lesser dimension that we are simply not "attuned" to notice. If you watch a spiral from above (from a different perspective/dimension) it will be completely identical to a loop.
Just a few thoughts.

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