Dyadic matrix systems force noncommuting corners

Throughout, $M$ is a unital C*-algebra with a partial order making it star-ordered, $X$ is a complex Banach space, and $e:X^*\to M$ is an isometric conjugate-linear isomorphism, so $M$ is a von Neumann algebra with predual $X$ (the Lean hypothesis e). A family $(a_i)$ converges in the predual sense to $b$ (PredualTendsto) when $e^{-1}(a_i)(z)\to e^{-1}(b)(z)$ for every $z\in X$. Here $\tau$ is a positive linear functional on $M$ with $\tau(1)=1$ that lies in the predual (TraceInPredual: $\tau(x)=\overline{e^{-1}(x)(z)}$ for some $z\in X$), is tracial, and is faithful ($\tau(a^*a)=0$ implies $a=0$), and $\|x\|_2=\tau(x^*x)^{1/2}$ is the norm of τ.toPreGNS x. The map $E$ (average) sends $a$ to a central element $E(a)$, chosen by Classical.choose from exists_central_average, with $\|E(a)\|\le\|a\|$ and whose $2$-norm vector lies in the $2$-norm closed convex hull of the unitary conjugates of $a$. A projection $p$ is homogeneous (Homogeneous) when $E(p)=\operatorname{Re}\tau(p)\cdot 1$. A matrix system of size $n$ under a projection $p$ (System n p) is a family $(e_{ij})_{i,j<n}$ in $M$ with $e_{ij}^*=e_{ji}$, $e_{ij}e_{kl}=\delta_{jk}e_{il}$ and $\sum_ie_{ii}=p$. Suppose that for every $k$ there is a matrix system of size $2^{k+1}$ under $1$. Then NoAbelianCorners M holds: for every nonzero projection $p$, the corner algebra $\{a:pa=a=ap\}$ contains two noncommuting elements.

Formalization Note The statement is verbatim from OpenAI's Lean development, with the listed Definitions imported in place of the upstream modules; the definitions are supporting code for this statement. Only the keyword lemma, where used upstream, becomes theorem, and a leading include ... in or omit ... in moves to the preamble. It is a step of the Lean proof of the vanishing of higher bounded Hochschild cohomology.