A strict pointwise tensor-train approximation bound by finite singular-subspace tie handling

For prescribed tensor-train rank bounds, a deterministic algorithm in the exact
arithmetic/SVD model returns an approximation with squared error strictly
smaller than $(d - 1)$ times the optimal squared error whenever that optimum is
positive, and reconstructs exactly at zero optimum. It performs at most $n_1$
tensor-train SVD completions. The only modification to the standard
construction is to test a finite collection of equally optimal first-cut
singular subspaces when a positive singular value straddles the truncation
boundary. This addresses the strict pointwise inequality as posed. It
does not produce a uniform constant $c < d - 1$: an explicit fixed-format
family shows that its error ratios can approach $d - 1$ already when $d = 3$.