Problem 02 Milestone — Compressed strict convex equality reduces dimension two

For indices $i,c\in\{0,1\}$, identify $\operatorname{Fin}(2)\to\mathbb R$ with $\mathbb R^2$ and set $Q=[0,1]^2$. Let $S$ consist of a $2\times2$ complex matrix $U$ satisfying $U^*U=I$ and $UU^*=I$, together with points $\lambda_0,\lambda_1\in Q$. Define $A_c=U\operatorname{diag}(\lambda_0(c),\lambda_1(c))U^*$ and $\Phi_S(f)=U\operatorname{diag}(f(\lambda_0),f(\lambda_1))U^*$. Let $P$ be a $2\times2$ complex matrix satisfying $P^*=P$ and $P^2=P$. Assume a compression witness is supplied: it consists of another matrix $V$ satisfying $V^*V=I$ and $VV^*=I$, points $\mu_0,\mu_1\in Q$, and matrices $B_c=V\operatorname{diag}(\mu_0(c),\mu_1(c))V^*$ satisfying $B_0=PA_0P$ and $B_1=PA_1P$. For an arbitrary function $f:\mathbb R^2\to\mathbb R$, assume that $f$ is continuous relative to $Q$, and that $Q$ is convex and, for every pair of distinct $x,y\in Q$ and every $a,b>0$ with $a+b=1$, one has $f(ax+by)<a f(x)+b f(y)$. Define $\Phi_C(f)=V\operatorname{diag}(f(\mu_0),f(\mu_1))V^*$, and assume $P\Phi_S(f)P=P\Phi_C(f)P$. Then $P$ commutes separately with both original coordinate operators: $PA_0=A_0P$ and $PA_1=A_1P$. The assumptions permit $P=0$ and $P=I$, permit repeated spectral points and points on the boundary of $Q$, impose no rank or properness condition on $P$, and constrain $f$ only on $Q$; moreover, the compression witness is assumed as data rather than asserted to exist.