BPS spectra of $\operatorname{tr}[\Psi^p]$ matrix models for odd $p$

We compute exact finite-rank BPS generating functions for the fermionic matrix model with single-trace supercharge $Q_p=\operatorname{tr}(\Psi^p)$ at $(p,N)=(5,3),(5,4),(5,5),(7,4)$, together with partial data at $(7,5)$. In all complete computed cases, the charge-resolved spectrum exhibits an overdetermined factorization -- a power of $p$, times an onset monomial $x^{q_{\min}}$, times $(1+x)^N$, times a palindromic reduced polynomial -- despite the loss of Casimir solvability at $p\ge 5$. We prove rank palindromicity $r_R=r_{N^2-p-R}$ from the exterior top-degree pairing; at $(5,5)$, the ten low-charge ranks and the minimal divisibility condition $(1+x)\mid\mathcal{R}_{5,5}$ determine the remaining middle rank, and direct computation confirms the full generating function. For fixed $p$, the mod-$p$ Witten indices give a closed-form index floor; together with the trivial Hilbert-space upper bound, this places any accumulation point of $N^{-2}\log Z_{\mathrm{BPS}}^{(p,N)}$ in the window $[\log(2\cos\frac{\pi}{2p}),\,\log 2]$. A rank-projection tower gives rigorous lower bounds on the projection-fortuitous cohomology. In matched $\mathcal{N}=2$ SYK examples at $N_f=16$, the BPS count saturates the index floor, whereas the single-trace matrix model has nonzero index excess and broader charge support.