The Gauge Freedom of Algorithmic Complexity — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Kolmogorov complexity's invariance theorem reveals that algorithmic information content is defined only up to an additive constant dependent on the choice of universal Turing machine — a gauge-like freedom in the definition of "absolute" complexity. | MATH: Let KU(x) = minp {|p| : U(p)=x}. Invariance theorem: for any two universal machines U, V, KU(x) ≤ KV(x) + cU,V and vice versa, so |KU(x) − KV(x)| ≤ cU,V where cU,V is a constant independent of x. This is formally analogous to gauge freedom in physics: physical laws are invariant under transformations that add a constant to a potential (e.g., Aμ → Aμ + ∂μ χ adds a total derivative, changing action by a boundary term — similarly, changing the reference machine adds a constant to all complexities). The additive constant cU,V is not computable (no effective procedure finds it), mirroring the non-observability of absolute gauge potentials. | CONNECTI
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026gauge,
title = {{The Gauge Freedom of Algorithmic Complexity — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409848},
url = {https://zenodo.org/records/23409848},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Gauge Freedom of Algorithmic Complexity — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409848
UR - https://zenodo.org/records/23409848
AB - FINDING: Kolmogorov complexity's invariance theorem reveals that algorithmic information content is defined only up to an additive constant dependent on the choice of universal Turing machine — a gauge-like freedom in the definition of "absolute" complexity. | MATH: Let K_U(x) = min_p {|p| : U(p)=x}. Invariance theorem: for any two universal machines U, V, K_U(x) ≤ K_V(x) + c_U,V and vice versa, so |K_U(x) − K_V(x)| ≤ c_U,V where c_U,V is a constant independent of x. This is formally analogous to gauge freedom in physics: physical laws are invariant under transformations that add a constant to a potential (e.g., A_μ → A_μ + ∂_μ χ adds a total derivative, changing action by a boundary term — similarly, changing the reference machine adds a constant to all complexities). The additive constant c_U,V is not computable (no effective procedure finds it), mirroring the non-observability of absolute gauge potentials. | CONNECTI Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -