Complementary tensor relations and bipartite matching kernels
Year: 2026
Type: preprint
Abstract: Let D_i be a nonzero tensor occupying every leaf space except V_i, with a common additional factor space X. The paper studies relations sum_i v_i D_i = 0 with v_i in V_i. For n leaves, it proves the characteristic-free bound dim R <= n-1 and characterizes equality by a common factor in X and signed maximal minors of a matrix with separated column variables. This gives sharp bounds for completing any nonempty subset of the last row of a bipartite matching tensor. It then classifies maximal kernels for two fixed rows and three leaves, over every field and with zero edge blocks allowed, provided the three complementary tensors are nonzero. In characteristic different from two, a maximal kernel with six nonzero blocks forces common root lines and at least two aligned leaf pairs. In characteristic two, the combined space of contracted rows must have dimension two; this criterion extends to all square determinant tensors, and hence to matching tensors in characteristic two. A separate normal form treats zero blocks. Finally, the necessity of the root-line conclusion gives a sharp cut-rank obstruction on an eight-vertex matching network, with nonzero examples attaining the bound for every positive dimension of the vertex space at vertex 8, over every field of characteristic different from two. Files. paper.pdf is the manuscript, version 3.2 (18 pages). paper.tex is its LaTeX source. Disclosure. The author directed the project and used OpenAI Codex and Anthropic Claude to produce the mathematics, proofs and prose; the paper's own statement gives the details. The author has not verified the mathematics, and no independent human mathematical verification is claimed. No AI system is an author; the author publishes under a pseudonym. The checks made were AI checks by systems that also contributed to the paper; they are not peer review and not external verification. Novelty and significance are not established: literature searches found no prior statement of the main results, which is absence of evidence only. Corrections are welcome at [email protected].
Author: Ignoramus
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225731862
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