Sign-free kernel bound $I_1+I_2\le\max(d(A,C),d(B,D))$ for the matrix integral inequality

Let $n\ge 1$ and let $A,B,C,D$ be real symmetric positive definite $n\times n$ matrices. Let $S=S^{n-1}\subset\mathbb R^n$ be the unit sphere carrying the surface measure $\sigma$ of the mission (the polar-decomposition measure toSphere of Lebesgue measure, without normalisation), and for a symmetric matrix $M$ write $\beta_M(u,v)=u^{\mathsf T}Mv$. Define the two kernel integrals

The statement asserts

where $d(X,Y)=\iint_{S\times S}|\beta_{X-Y}(u,v)|/(\beta_X(u,u)\beta_Y(v,v))\,d\sigma\,d\sigma$ is the distance of CUHK-Shenzhen AI Math Problem 1.

Relation to the mission target. Since $\beta_{(A+B)-(C+D)}=\beta_{A-C}+\beta_{B-D}$ and both kernels share the denominator $\beta_{A+B}(u,u)\,\beta_{C+D}(v,v)$, the triangle inequality gives $d(A+B,C+D)\le I_1+I_2$ under the integral sign. Hence this statement implies the mission target RybinAI2026.P01.matrix_integral_inequality. It is a strengthening: it discards the possible cancellation between the two signed kernels and keeps only the two nonnegative ones.

Structure of the difficulty. Put $a(u)=\beta_A(u,u)/\beta_{A+B}(u,u)$, $c(v)=\beta_C(v,v)/\beta_{C+D}(v,v)$, $b=1-a$, $d=1-c$. Then $I_1=\int a(u)c(v)\,dP_1$ and $I_2=\int b(u)d(v)\,dP_2$, where $P_1=|\beta_{A-C}|/(\beta_A\beta_C)\,d\sigma\,d\sigma$ has total mass $d(A,C)$ and $P_2$ is the analogous measure for $(B,D)$. The pointwise bound $a(u)c(v)+b(u)d(v)\le 1$ settles the one-dimensional case and the case of proportional pairs, but $P_1$ and $P_2$ differ in general; the claim is that $\mathbb E_{P_1}[ac]\,d(A,C)+\mathbb E_{P_2}[bd]\,d(B,D)\le\max(d(A,C),d(B,D))$ nevertheless.

Evidence. Numerical quadrature in dimension $2$ (about $9{,}000$ random quadruples plus adversarial hill-climbing) and dimension $3$ ($300$ quadruples) found no violation; the ratio $(I_1+I_2)/\max(d(A,C),d(B,D))$ approaches $1$ only in the degenerate regime where one pair dominates the other. Two natural sufficient conditions are false and should not be attempted: the arithmetic-mean variant $\mathbb E_{\tilde P_1}[ac]+\mathbb E_{\tilde P_2}[bd]\le 1$ with $\tilde P_i\propto|\beta|/(\beta_{A+B}\beta_{C+D})$ (violations up to $1.27$), and stochastic dominance of the ratio pair $(\beta_B/\beta_A,\beta_D/\beta_C)$ under $P_1$ versus $P_2$.

Formalization note. The integrals are iterated Bochner integrals against surfaceMeasure n, exactly as in the mission's definition distance; bilinear M u v is $u^{\mathsf T}Mv$, and Matrix.PosDef includes symmetry. The hypothesis 0 < n mirrors the mission target.