Neutron scattering from high-quality YBCO6.334 single crystals with a T$_c$ of 8.4 K shows that there is no coexistence with long-range antiferromagnetic order at this very low, near-critical doping of $sim$0.055, in contrast to claims based on local probe techniques. We find that the neutron resonance seen in optimally doped YBCO7 and underdoped YBCO6.5, has undergone large softening and damping. It appears that the overdamped resonance, with a relaxation rate of 2 meV, is coupled to a zero-energy central mode that grows with cooling and eventually saturates with no change at or below T$_c$. Although a similar qualitative behaviour is found for YBCO6.35, our study shows that the central mode is stronger in YBCO6.334 than YBCO6.35. The system remains subcritical with short-ranged three dimensional correlations.
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