Is this a valid use of big-O notation?
If these computations appear in a paper for me to review, I would like to ask the author to rewrite.


  $$\exp(-x^2/2)\leq \exp\left(-K_1\log n^{c+1/2\,}\right), \text{ where } K_1&gt;0$$


This is wrong. The direction of inequality should be switched. For example, let $n\to\infty$ and $x$ be constant 1, the left hand side remains a constant while the right hand side goes to 0. This error means "Computation 2" is wrong from the start.


  The last quantity $n^{-O(1)}$ represents the set of polynomially-bounded functions.


Instead of $n^{-O(1)}\,$, $n^{O(1)}$ represents asymptotically the set of polynomially-bounded functions.