Small-time controllability without polynomial reachable neighborhoods

We construct a polynomial control-affine system in five dimensions, with two controls in $[-1,1]$, that is small-time locally controllable at the origin but admits no polynomial lower bound for its reachable neighborhoods. Every point on the negative fourth axis reachable in time at most $t$ has distance less than $t^{11}e^{-1/(6t)}/4$ from the origin. Nevertheless, exact positive and negative fourth-coordinate loops, followed by linear steering in the other coordinates, reach a full neighborhood for each $t>0$. The construction combines a critical quartic identity with a logarithmic estimate and concentration near its equality profile. It gives a negative answer to the polynomial-growth question for analytic systems with bounded measurable controls.