Magnetic spins and charges interact strongly in high-temperature superconductors. New physics emerges as layers of copper oxide are tuned towards the boundary of the superconducting phase. As the pseudogap increases the characteristic spin excitation energy decreases. We show that our well-annealed YBa2Cu3O6+x (YBCO6+x) single crystals are orthorhombic and superconducting but not antiferromagnetically ordered. Near the critical concentration for superconductivity for x = 0.35 the spins fluctuate on two energy scales, one a relaxational spin response at ~2 meV and the other a slow central mode that is resolution-limited in energy (<0.08 meV) but broad in momentum. The gradual formation on cooling of a central mode over a range of momenta suggests that the spin ground state from which coherent superconducting pairing emerges may be quantum disordered. We show that YBCO6.35 adopts a homogeneous state that consists of highly-organized frozen sub-critical three-dimensional spin correlations. The continuous spin evolution indicates that a single quantum state occurs in contrast to claims from site-based probes that lightly doped YBCO undergoes a transition to antiferromagnetic Bragg order followed by a sharp transition to a cluster glass phase. For x = 0.35, where Tc = 18 K is reduced to 1/5 of Tcmax, the spin ground state is reached without a sharp transition and consists of short correlations extending over only 8 Angstrom between cells and 42 Angstrom within the planes. Polarized neutrons show the angular spin distribution to be isotropic unlike the AF insulator. Since moment is conserved we interpret this as evidence for hole-induced spin rotations rather than decay.
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