There is no maximal K-degree

@misc{liu2026there,
  title = {{There is no maximal $K$-degree}},
  author = {Lu Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.11054},
  eprint = {2609.11054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11054},
}

TY  - UNPB
TI  - There is no maximal K-degree
AU  - Lu Liu
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.11054
UR  - https://arxiv.org/abs/2609.11054
AB  - The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real x is more complex to describe than every prefix (of the same length) of real y, then it is seen as x is more complex to describe than y. It is wondered if there is a real x so that no other reals are strictly more complex (to describe) than x. The behavior of Kolmogorov complexity functions generated by reals (namely n↦ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal K-degree. In this paper, it is shown that there is no real with maximal K-degree. Actually, for almost all real x, we can uniformly computably find another real whose K-degree is strictly above x. The same construction also derives the following: (a) There is no maximal $C $-degree. (b) Every n-random real x admits an n-random real y such that y>_Kx and y>_C x; moreover, the difference of K-complexity and C-complexity of y,x (namely K(yupharpoonright n)−K(xupharpoonright n) and C(yupharpoonright n)−C(xupharpoonright n)) do not tend to infinity.
ER  -