The dark side of early galaxies: $\texttt{geko}$ uncovers dark-matter fractions at $z\sim4-6$
JWST/NIRCam slitless spectroscopy enables dynamical mass measurements for typical star-forming galaxies only a billion years after the Big Bang. We model the H$α$ morpho-kinematics of 163 galaxies at redshift $z\approx4$-6 from FRESCO and CONGRESS (with JADES imaging), using the $\texttt{geko}$ code, and infer rotational velocities and dispersions within $r_{\rm e}$. Our sample spans $\log M_{\star}\approx7$-10 and $\log M_{\rm dyn}\approx9$-11. Gas masses are estimated via scaling relations, yielding baryonic masses and dark-matter (DM) fractions $f_{\rm DM}(r<r_{\rm e})$ within the H$α$ half-light radius. We find high median fractions of $\langle f_{\rm gas}\rangle=0.77$ and $\langle f_{\rm DM}\rangle=0.73$, where $f_{\rm gas}$ is measured with respect to the baryonic mass and $f_{\rm DM}$ with respect to the DM+baryonic mass. About two-thirds of systems are DM-dominated within $r_{\rm e}\sim0.5-1$ kpc. Both $f_{\rm gas}$ and $f_{\rm DM}$ decrease with stellar mass, consistent with simulations. The stellar Tully-Fisher relation shows a tentative offset to higher $v_{\rm circ}$ at fixed $M_{\star}$ and substantial intrinsic scatter, suggesting that the relation is only beginning to emerge at $z\sim5$. We measure a negative correlation between $f_{\rm DM}$ and baryonic surface density $Σ_{\rm bar}$, weaker but broadly consistent with trends at cosmic noon and at $z\sim0$. Qualitatively comparing with modified NFW profiles coupled to an empirical stellar-to-halo mass relation suggests that the lowest $f_{\rm DM}$ ($\lesssim0.4$) require cored inner DM profiles, while the highest fractions favour cuspier profiles, potentially reflecting adiabatic contraction. Overall, the elevated $f_{\rm gas}$ and $f_{\rm DM}$ at $z\gtrsim4$ are compatible with progenitors of baryon-dominated systems at $z\sim2$ and naturally anticipate overmassive black holes at fixed $M_{\star}$.