We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions
d≥3, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in
[m,M], we show that the least uniform reconstruction-derivative error is
max{1−M(M−m)/2,0}, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale
0.05, the mean theoretical bound is
0.155, about
84% of the mean normalized training error
0.185 across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below
6×10−6.