Sharp Norms of Rectangular Majorana Matrices and a Two-Matrix Obstruction to Strong Convergence

Let \(C_{m,n}\) be an \(m\times n\) array of \(mn\) distinct self-adjoint Clifford generators, each squaring to one. Its squared operator norm is \(2(m+n-2)\) for \(m,n\ge2\), and \(\max(m,n)\) when \(\min(m,n)=1\). A vector–spin Casimir identity and an attaining highest-weight summand give the proof. The square case answers the explicit CAR boundedness subquestion in the AIM Free Analysis list with the optimal uniform bound \(2\).

For \(d\) normalized square arrays using disjoint sets of jointly anticommuting generators, the joint star moments converge to a standard free circular family. For \(N\ge2\), the norm of the sum \(\sum_{r=1}^d X_r X_r^*\) is exactly \(2(d+1)-4/N\), while the free-limit norm is at most \((1+\sqrt d)^2\). Strong convergence fails already for two arrays.

The broader q-Gaussian approximation question is not resolved. The imported square calculation and Shlyakhtenko’s prior work on a different Hermitian fermionic ensemble are credited. This is a self-audited, AI-assisted, unrefereed preprint; no independent review, formal verification or absolute priority is claimed.