Theorem 13 — for $d_A=2$ the normalized states form the Bloch ball and $G_A\cong SO(3)$

NON-BLIND READ-BACK — NOT INDEPENDENT TESTIMONY. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, with full knowledge of the source paper and of the intended meaning. It is not a blind audit and must not be mistaken for independent testimony; an independent, blind read-back should replace it before the proposal is relied upon.

Fix $T$ satisfying all six principles, a system $A$, and a maximal perfectly distinguishable family of pure states $\varphi_0,\varphi_1$ of $A$ with exactly two members. The conclusion is that there exists a real-affine map $f:\mathbb R^{D_A}\to\mathbb R^3$ such that: (i) $f$ is injective on the set of deterministic states of $A$; (ii) the image under $f$ of the set of deterministic states is exactly the closed Euclidean unit ball $\{x:\ x_1^2+x_2^2+x_3^2\le1\}$; (iii) for every linear map $\mathcal U$ of $\mathbb R^{D_A}$ that is a reversible transformation of $A$ there is a real $3\times3$ matrix $R$ in the special orthogonal group ($R^{\mathsf T}R=I$, $\det R=1$) with $f(\mathcal U\rho)=Rf(\rho)$ for every deterministic state $\rho$; (iv) conversely, for every such $R$ there is a reversible transformation $\mathcal U$ of $A$ with $f(\mathcal U\rho)=Rf(\rho)$ for every deterministic state $\rho$.