Spectral Chaos Precedes Densification of Higher-Order Eigenstate Correlations in the Integrability-to-Chaos Crossover

We study how two distinct signatures of quantum chaos, the level spacing ratio ⟨r⟩and a four- eigenstate correlator (FESC), behave as integrability is gradually broken in the disordered spin-1/2 Heisenberg chain. Working within the Stot z = 0 sector at system sizes N = 12, 14, and 16 via exact diagonalization, we observe that these two diagnostics do not track each other across the crossover. The level spacing ratio settles into Wigner-Dyson statistics already at rather weak disorder, but the FESC stays sparse, well below its random-matrix expectation, over a wide range of disorder strengths, and never clearly approaches a dense regime before many-body localization sets in. Taken together, these observations point to a finite-size window in which the energy spectrum looks chaotic while the eigenstates still carry residual structure from integrability. The FESC thus serves as a sharper probe than ⟨r⟩ alone, picking up correlations that standard spectral measures miss.