When Does Reconstruction Identify BM Tensor Representations?

The Bhattacharya–Mesner (BM) decomposition represents a third-order tensor through factors indexed by complementary pairs of data indices. We study identifiability of these factors from a fully observed tensor. On the entrywise-nonzero stratum, we characterize all term-preserving transformations and construct a canonical normalization. A full-column-rank Jacobian in the normalized coordinates certifies local identifiability, determines reconstruction-loss curvature, and yields local perturbation bounds. For cubic tensors, parameter counting gives a necessary term-count ceiling of $N/3+O(1)$ for generic local identifiability. Below this ceiling, separable template ratios generate continuous families of inequivalent decompositions with identical reconstructions. Under explicit full-column-rank assumptions, the slice-wise mixing families have dimension $pr(r-1)$ in one mode, with corresponding formulas in the other modes; these ambiguities also occur for positive factors. At rank one, we establish global identifiability, closed-form recovery, and a membership test expressed entirely in tensor entries. Exact finite-field witnesses and synthetic experiments examine gauge freedom, structural ambiguity, numerical conditioning, and optimization behavior.