Problem 01 Goal — Matrix integral inequality

For every natural number $n>0$ (including $n=1$), let $E_n=\mathbb{R}^n$ with its Euclidean norm, let $S_n=\{u\in E_n:\lVert u\rVert_2=1\}$, and let $\sigma_n$ be the measure on $S_n$ obtained by applying the polar-decomposition construction $\operatorname{toSphere}$ to Lebesgue volume on $E_n$, without probability normalization. For a real $n\times n$ matrix $M$, define $\beta_M(u,v)=u^{\mathsf T}Mv=\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}u_iM_{ij}v_j$, and, for arbitrary real $n\times n$ matrices $X,Y$, define the total-valued quantity

Then, for every four real $n\times n$ matrices $A,B,C,D$ such that, for each $M\in\{A,B,C,D\}$, every nonzero $x\in E_n$ satisfies $x^{\mathsf T}Mx>0$, one has

Matrix addition and subtraction here are entrywise; in particular, the numerator on the left uses $(A+B)-(C+D)$. No symmetry assumptions are stated separately. Since unit vectors are nonzero, the positivity hypotheses make every denominator occurring in these three quantities strictly positive, including those involving $A+B$ and $C+D$. The statement nevertheless contains no explicit measurability or integrability assumptions: the integrals are Lean’s totalized Lebesgue integrals, which take the value $0$ when the relevant function is not integrable; likewise, the underlying real division is total and assigns $a/0=0$, although zero denominators do not occur under the stated hypotheses.