Finite-size spin fluctuations and fixed-index sweeps in a ferromagnetic spinor condensate

We find a dip in the optimized rotation quantum Fisher information of a zero-magnetization ferromagnetic spin-1 Bose-Einstein condensate near the excited-state quantum phase transition. Its displacement from ${\epsilon }_c=-q$ decreases on an inverse-size scale. At $q=0.6,0.8,1.0,1.2$, the dip value of $F_{\max }$ equals $2⟨S^2⟩$ for $N$ up to $3840$, linking it to total-spin fluctuations. A semiclassical orbit average normalized by the classical density of states describes the dip over $N=120$–$3840$ with maximum relative residual below $0.028$; a small power law fits comparably. Inserting the measured energy offsets directly into the semiclassical formula leaves residuals of $17$–$30\mathrm{\%}$, so $\kappa$ acts as an effective cutoff. Among retained states, spin-nematic squeezing occurs at the spectral edges. Separately, fixed-index excited-state sweeps have half-excitation times between $11.2$ and $13.1$ times their ground-state counterparts, with local size exponents increasing across the four sizes.