Tensor Degenerations for Matrix Multiplication

@misc{sun2026tensor,
  title = {{Tensor Degenerations for Matrix Multiplication}},
  author = {Zhi-Wei Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13971},
  eprint = {2610.13971},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13971},
}

TY  - UNPB
TI  - Tensor Degenerations for Matrix Multiplication
AU  - Zhi-Wei Sun
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.13971
UR  - https://arxiv.org/abs/2610.13971
AB  - Let 𝕂 be an arbitrary field. We study tensor degenerations for square matrix multiplication over the field 𝕂, and mainly prove for any integer n≥5 the uniform dyadic estimate underlineR_𝕂(M_2^n) ≤ 2^2n+1 which implies that ω(𝕂)≤ 2+1/n, where the border rank underlineR_𝕂(T) is the least r for which the scalar extension of T to an algebraic closure of 𝕂 lies in the Zariski closure of the tensors of rank at most r, and ω(𝕂) is the matrix multiplication exponent over the field 𝕂. This improves the current best record ω(𝕂)<2.371177 greatly, and answers affirmatively the long-standing open question whether the value ω(𝕂) is exactly $2$. The implementation disclosure and appendices in this paper give fixed source maps, interpolation data, terminal incidence ledgers, and deterministic check counts for independent reconstruction.
ER  -