  We have carried out new neutron spectroscopic measurements on single crystals
of La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ from 0.12 to 0.26 using time-of-flight
techniques. These measurements allow us to follow the evolution of parallel
spin stripe fluctuations with energies less than 33 meV, from x=0.12 to 0.26.
Samples at these hole-doping levels are known to display static (on the neutron
scattering time scale) parallel spin stripes at low temperature, with onset
temperatures and intensities which decrease rapidly with increasing x.
Nonetheless, we report remarkably similar dynamic spectral weight for the
corresponding dynamic parallel spin stripes, between 5 meV to 33 meV, from the
1/8 anomaly near x=0.12, to optimal doping near x=0.19 to the quantum critical
point for the pseudogap phase near x=0.24, and finally to the approximate end
of superconductivity near x=0.26. This observed dynamic magnetic spectral
weight is structured in energy with a peak near 17 meV at all dopings studied.
Earlier neutron and resonant x-ray scattering measurements on related cuprate
superconductors have reported both a disappearance with increasing doping of
magnetic fluctuations at ($\pi$, $\pi$) wavevectors characterizing parallel
spin stripe structures, and persistant paramagnon scattering away from this
wavevector, respectively. Our new results on
La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ from 0.12 < x <0.26 clearly show persistent
parallel spin stripe fluctuations at and around at ($\pi$, $\pi$), and across
the full range of doping studied. These results are also compared to recent
theory. Together with a rapidly declining x-dependence to the static parallel
spin stripe order, the persistent parallel spin stripe fluctuations show a
remarkable similarity to the expectations of a quantum spin glass, random t-J
model, recently introduced to describe strong local correlations in cuprates.
