From Giant Envelopes to Rapidly Contracting Cores in Rotating Population III Stars from $500$ to $1000M_{\odot }$

Population III (Pop III) stars link the chemical enrichment of the early Universe to the formation of the first massive black holes, yet their evolution remains poorly understood at extreme masses. Using GENEC, we evolve rotating zero metallicity models from $500$ to $1000M_{\odot }$ at $v\mathrm{/}{v}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=0.1$ through core Si burning, and a $1000M_{\odot }$, $v\mathrm{/}{v}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=0.4$ model evolved through He exhaustion. In the fiducial $1000M_{\odot }$ model, convection and rotational transport carry freshly synthesized C and O into H burning layers, producing $62.1M_{\odot }$ of primary N by He exhaustion. Surface mass loss removes $20.2M_{\odot }$, including $1.8\times {10}^{-3}{M}_{\odot }$ of N, with $\log (\mathrm{N}\mathrm{/}\mathrm{O})=0.68$, $\log (\mathrm{N}\mathrm{/}\mathrm{C})=1.48$, and $\log (\mathrm{C}\mathrm{/}\mathrm{O})=-0.80$. By the final hydrostatic model, the inner $100M_{\odot }$ is confined within $0.088R_{\odot }$ while the photosphere remains several thousand solar radii away. As C, Ne, O, and Si burning accelerate, chemical mixing freezes out and advanced products remain buried beneath the envelope. At core Si exhaustion, the central angular velocity reaches $4\times {10}^{-3}{\mathrm{s}}^{-1}$ while the surface rotates at $5.25\times {10}^{-8}{\mathrm{s}}^{-1}$, a contrast of $7.6\times {10}^4$. Contraction drives this spin up, while inner specific angular momentum changes by less than $1\mathrm{\%}$. The maximum $j_{\mathrm{e}\mathrm{q}}\mathrm{/}{j}_{\mathrm{I}\mathrm{S}\mathrm{C}\mathrm{O}}$ is $0.313$, limiting disk formation in our inside out estimate. These models show that Pop~III stars can produce strong primary N signatures and rapidly rotating black hole progenitors while their envelopes determine which products enrich the early Universe.