Probability in QM: derivation or interpretation? It is known that coordinates $C_k\in\mathbb{C}$ of the QM-state vectors $|\psi\rangle$ has an interpretation as probability weights $p_k$ in the whole state through the formula like $|C_k|^2=p_k$. We thus have the two (independent) math operation on the coefficients: making a modulus $C_k\to |C_k|$ and their squaring $|C_k|\to|C_k|^2$. Does anybody know a derivation of that transition, i.e., $C_k\to |C_k|^2=p_k$, from plausible/forceful physical arguments? The key words in the question are 'derivation' and 'forceful'. In other words, can we throw out the word 'interpretation' in the context and derive the rule $|C_k|^2$ is a probability weight? And reversely, the probability weight is exactly $|C_k|^2$ and only this formula?
I've knew right now about the 2005 paper by Aaronson but observed there the following. He begins there with a norm of the space, not with a scalar product of two independent vectors of $H$. The problem here, in my view, is that I prefere the object $(\psi,\phi)$ to be more fundamental (in the QM context) than $(\psi,\psi)$. This is because the derivation of the former requires some extra/exotic arguments on properties of the norm (polarization procedure). Consideration in the reverse direction does not. So, one could rephrase the question about 'interpretation/derivation' of $|(\psi,\phi)|^2$, not about $|(\psi,\psi)|^2$.
 A: There is no derivation of the probabilistic nature of the wavefunction: it is an interpretation (postulate), the only one that makes the theory consistent.
From Sakurai, Modern Quantum Mechanics:


Schrödinger published his famous wave equation in February 1926 in the famous paper Quantisierung als Eigenwertproblem (Quantization as an Eigenvalue Problem), but at the beginning no one know how to interpret the function $\psi(\vec r,t)$ which appeared in it.
Quoting from Wikipedia:

At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with how elastic scattering of a wave packet representing a particle off a target appears; it spreads out in all directions. While a scattered particle may scatter in any direction, it does not break up and take off in all directions.

The same year, in June, Born published the paper Zur Quantenmechanik der Stroßvorgänge (On the quantum mechanics of collisions), in which he solved the Schrödinger equation for a scattering problem and concluded that the probabilistic interpretation of the solution was the only possible one.
The point is that if you want to build a consistent theory starting from the Schrödinger equation, then the object that appears in it must be interpreted in a way such that
$$\mid \psi(\vec r,t) \mid ^2 dV = \text{Probability that the particle is found in the volume}\ dV \ \text{after a measurement}$$
