Equality $X_+ = \int_0^\infty I_{\{X > x\}}\mathrm{d}x$ is obvious and is an equality between random variables. It cannot follow from equalities between expectations. Moreover, Mathoverflow is devoted to questions about research.

The argument given in my post still apply in this case.

What is the norm considered? Frobenius norm? Operator norm associated to canonical Euclidean norm on $\mathbb{R}^b$? It should be precised.

I do not understand what the problem is, since for any invertible $n \times n$ real matrix $X$, the matrix $XAX^\top$ is congruent to $A$, hence is still symmetric positive-definite (and invertible). Therefore any orthogonal matrix $O$ works.

@Drew Brady The calculation on MSE deals with $\max_{0 \le k \le n} |S_k|$, which is greater than $\max_{0 \le k \le n} S_k$. In the present situation, the right constant is $\sqrt{2/\pi}$ in the asymptotic regime $n \to +\infty$.

@მამუკაჯიბლაძე Is it easy to prove that the order of 2 modulo $p^\nu$ is $p^{\nu-1}$ times the order of 2 modulo $p$? I am a bit surprised by this statement.

My impression is that every diffuse (i.e. without atom) finite measure on a Polish space has the same property (providing every measure with same mass by suitable push-forward), see en.wikipedia.org/wiki/Standard_probability_space

@MrPie Did you see the addendum?

You still have to correct the inequality $F_k \le F \le F_{k+1}$, saying that it holds $k$ when is even or replacing $k$ by $2k$. A natural smooth interpolating function will probably have local extrema close to the integer points, but possibly not exactly at the integers. I will think about it to produce a formula.

@Mr Pie. It is absolutely not clear. What are the quantities $F_k(\alpha,\beta)$? Partial sums? The inequalities $F_k \le F \le F_{k+1}$ car not be true for all integer $k$. Do you want them when $k$ is even? Are you looking for a functions $k \mapsto F_k(\alpha,\beta)$ which interpolates of the partial sums with minima at odd $k$ and maxima at even $k$? Is there a good reason to desire those properties?

Ah, yes. The title should reflect more faithfully the purpose of the OP, instead of misleading the reader.