circuit analysis - How do we know loop currents exist?

I've been practising loop analysis questions, but don't really understand why the method is valid.

Say you have a circuit that only consists of resistors and some power source. Assume the circuit is valid (contains no contradictions)

How do we know that the currents in each node can be expressed as a sum of loop currents? Just because current exists in the circuit, doesn't necessarily imply that the current can be expressed as loop currents.

So could someone give me a proof that if a current exists through the circuit, its necessarily expressible as a sum of "loop currents"?

Example

For instance let C1, C2 be actual currents flowing through some nodes. They aren't abstractions - these are measurable currents that actually exist.

Now essentially loop analysis expresses these real, measurable currents as the sum of abstract loop currents. You represent the current in nodes that are on the "sides" of two loops as sums of the loop currents in those loops. Without loss of generality, for some loop currents L1, L2 say you express the physical currents as:

C1 = L1 + L2

C2 = L1 - L2

What essentially confuses is that loop analysis assumes that some values L1 and L2 exist that meet the above criteria. In other words, it assumes that the above system of equations has a unique solution.

So why is it okay to assume that the actual currents, the physical currents, flowing through the nodes can always be expressed as loop currents?

For an even more concrete example, consider this circuit:

Note how the real, physical currents are being expressed as sums of loop currents. I don't understand why this step is always valid. How do we know there exists some Ia, Ib, Ic and Id that fulfill the above system of equations?