Reduction from PARTITION to MAX-CUT
Since the graph is complete, any set $S$ will induce a cut of weight

$$\sum_{i\in S, j\notin S} c_ic_j ~~=~~ \sum_{i\in S} c_i \cdot \sum_{j\notin S} c_j.$$

Denote the two sums in the latter formula by $A$ and $B$. The problem of finding a maximal cut then means maximizing $A\cdot B$, while $A+B$ is fixed. It is well-known that the maximum of this product is achieved for $A=B$. In this case the product becomes $\left(\frac 12 \sum_{i=1}^n c_i\right)\cdot \left(\frac 12 \sum_{i=1}^n c_i\right) =  \frac 14 \left(\sum_{i=1}^n c_i\right)^2$. So $W$ should not be a quarter of the sum of squares, but a quarter of the square of the sum.

So the reason you could not prove the correctness of the reduction is, that the reduction as given is not correct. (And I'm somewhat embarrassed that I only noticed this after your followup question in the comment.)