On the Universality of Probe Complexity in $\mathcal{N}=4$ SYM

We investigate Krylov complexity for single-trace operators dual to open strings attached to giant gravitons in planar $\mathcal{N}=4$ super Yang-Mills theory. We show that in protected and few-body sectors, Krylov dynamics is governed by orthogonal polynomial theory associated to the seed spectral measure, leading to bounded Lanczos coefficients determined solely by spectral support. In particular, for fixed magnon number $M$ and open-string length $L\rightarrow\infty$, we derive $a_n=2Mg$ and $b_n\rightarrow Mg$, demonstrating integrable, band-limited dynamics. This establishes that such sectors are insufficient to test recently proposed gravity-side universality of operator complexity growth. We therefore formulate a finite-density program in which magnons scale with system size, and propose a concrete universality test: whether the leading Krylov growth depends only on coarse thermodynamic data $(\rho,\varepsilon)$ and not on microscopic probe structure. This provides a precise boundary-field-theory framework for testing gravitational universality conjectures.