Finite mean angular conditioning from a bounded semialgebraic graph

The manuscript proves that the angular condition number of the tensor rank
decomposition has finite mean for random real tensors of order $d \ge 3$, mode
sizes $n_j \ge 2$ and rank $r \ge 3$, in every format where complex tensors of
rank $r$ are generically identifiable. The input has density proportional to
$e^{-\|A\|^2/2}$ with respect to volume on the smooth identifiable rank-$r$
locus, as in Beltrán, Breiding and Vannieuwenhoven's conjecture; the angular
condition number is the norm of the derivative of the map to the individually
normalised rank-one summands. On the unit sphere of the locus, the graph of
normalised decompositions is a bounded semialgebraic set with finite fibres,
hence of finite volume, and by the area formula this volume bounds the integral
of the derivative norm. Homogeneity of degree $-1$ then reduces the expectation
to a convergent radial integral. No format-uniform bound is claimed.