Finite-Size Evidence Against a Single Integrability-Breaking Scale for Spectral Chaos and ETH in an SU(2)-Invariant Spin Chain

We test whether a next-nearest-neighbor coupling supplies one finite-size control scale for both the spectral crossover from integrability and the onset of eigenstate thermalization in an open spin-$1/2$ XXZ chain at $\Delta=1$. Dense exact diagonalization was executed for $L=8,10,12$ in total-$S^z=0$ sectors resolved by reflection parity, with $\lambda\in\{0,0.05,0.1,0.2,0.3,0.5,0.7,1.0\}$. The computed diagnostics do not support a clean Poisson-to-Gaussian-orthogonal-ensemble rise in the mean spacing ratio: the reflection-even high-$\lambda$ values remain near $0.35$--$0.40$ rather than approaching $0.5307$. Off-diagonal matrix elements of a parity-preserving central-spin observable show partial $1/D$ consistency, but the adjacent-size ETH exponents drift from near one at small $\lambda$ to about $0.78$ and $0.72$ at $\lambda=1$. Half-chain entanglement increases toward a Page-like comparator, reaching about $0.86$--$0.92$ of the full-space Page value at $\lambda=1$. Independently extracted crossover scales do not coincide: spectral midpoint crossings are absent for $L=10,12$, whereas ETH midpoint estimates occur near $\lambda\simeq0.31$--$0.36$. At the executed sizes, the evidence is a finite-size failure atlas for the proposed single-scale collapse rather than confirmation of a universal crossover law.