bibkey: lovitzpetrov2023kruskal
authors: Benjamin Lovitz and Fedor Petrov
year: 2023
title: A generalization of Kruskal's theorem on tensor decomposition
doi: 10.1017/fms.2023.20
url: https://arxiv.org/abs/2103.15633v2
claim: A connected family of nonzero product tensors has connected families of factors on both tensor components.
strata_touched: []
license: citation-only
triage: anchor

Connected product families and their factor families

Source and locator

Literature status: literature-attested.

The primary mathematical locator is arXiv:2103.15633v2, Proposition 5, in section 4, printed pages 10–11. A vector family is connected when no nonempty proper partition splits its span as a direct sum. Proposition 5 proves that connectedness of a family of nonzero product tensors implies connectedness of each factor family. The proof applies a linear functional to a nonzero vector in the intersection of the two product spans.

The arXiv record identifies the journal version as Forum of Mathematics, Sigma 11 (2023), e27, with the DOI above. The proposition number used here refers specifically to the arXiv v2 text.

RT application and boundary

In RT IV §114, the product Gram has rank seven and a one-dimensional kernel whose every coordinate is nonzero. Its eight product records form a circuit, hence a connected family. The cited implication rules out a direct-sum split of the first factor records into two rank-two halves. The public proof includes that special case directly by applying the projection onto one record span along the other.

The new RT content is the same-common-stress curved-tangent argument, positive original stress-fibre blocks, the two-fibre stress-rank saturation, and their conditional minimum exclusions. The circuit inheritance step itself is known. This note attributes only that step; it does not claim that the cited paper proves an RT channel-norm bound, a complete rank-(7,4) exclusion, or the unrestricted quarter optimum.