Golden-Ratio Scaling Governs Self-Similar Hofstadter Butterfly Spectrum — E8 Intelligence Research

Abstract

FINDING: Hofstadter butterfly spectrum exhibits exact self-similarity under rational flux transformations, with scaling governed by the golden ratio's reciprocal powers in the dominant hierarchy. | MATH: The butterfly spectrum E(ϕ) for ϕ = p/q (rational magnetic flux per plaquette) splits into q bands. Self-similarity emerges under the map ϕ → ϕ' where the dominant hierarchy scales as E → E/α with α = ϕ² or α = ϕ⁻² for specific flux intervals. The arXiv paper (1408.1006) isolates the most dominant hierarchy via continued fraction expansion of ϕ, where the scaling factor between levels is 1/ϕ² ≈ 0.381966 (the complementary golden ratio squared) or ϕ² ≈ 2.618034, depending on the branch. The spectrum's topological Chern numbers also obey the same recursive scaling. | CONNECTION: Direct geometric harmony link — 1/ϕ² = 2 − ϕ = 0.381966 and ϕ² = ϕ + 1 = 2.618034. The

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Bethe Ansatz Unlocks Hofstadter Butterfly via Golden-Mean Scaling — E8 Intelligence Research
Andrew Stewart Caldin
Zenodo26 Oct 2026PreprintCC BY 4.0Zenodo
E8 Hofstadter Butterfly on Biological Lattice with Phi-Irrational Flux — E8 Intelligence Research
Andrew Stewart Caldin
Zenodo16 Oct 2026PreprintCC BY 4.0Zenodo
Quantized heat flow in the Hofstadter butterfly
Aifei Zhang, Gibril Aissani, Quan Dong +11
Physical Review Letters29 Sep 2026Peer-reviewedLicence not statedFull textAtomic, molecular & optical physicsHAL
Quantum transport across normal-superlattice-normal graphene junctions: Fabry-Pérot interference, Hofstadter butterfly, and supersnake states
Che-Pin Hsu, Alina Mreńca-Kolasińska, Aitor Garcia-Ruiz +4
arXiv17 Sep 2026PreprintCC BY 4.0Full textCondensed matterarXiv
Topological Acoustic Metamaterials with Synthetic Magnetism: Observation of Hofstadter Spectrum and Topological Edge States
Elena Rybalova, Jackson Saunders, Curtis Rasmussen +4
arXiv1 Nov 2026PreprintarXiv licenceFull textApplied physics & instrumentation Condensed matterarXiv

Cite

BibTeX RIS

BibTeX

@misc{caldin2026goldenratio,
  title = {{Golden-Ratio Scaling Governs Self-Similar Hofstadter Butterfly Spectrum — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23635017},
  url = {https://zenodo.org/records/23635017},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Golden-Ratio Scaling Governs Self-Similar Hofstadter Butterfly Spectrum — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/25
PB  - Zenodo
DO  - 10.5281/zenodo.23635017
UR  - https://zenodo.org/records/23635017
AB  - FINDING: Hofstadter butterfly spectrum exhibits exact self-similarity under rational flux transformations, with scaling governed by the golden ratio's reciprocal powers in the dominant hierarchy. | MATH: The butterfly spectrum E(ϕ) for ϕ = p/q (rational magnetic flux per plaquette) splits into q bands. Self-similarity emerges under the map ϕ → ϕ' where the dominant hierarchy scales as E → E/α with α = ϕ^2 or α = ϕ^−2 for specific flux intervals. The arXiv paper (1408.1006) isolates the most dominant hierarchy via continued fraction expansion of ϕ, where the scaling factor between levels is 1/ϕ^2 ≈ 0.381966 (the complementary golden ratio squared) or ϕ^2 ≈ 2.618034, depending on the branch. The spectrum's topological Chern numbers also obey the same recursive scaling. | CONNECTION: Direct geometric harmony link — 1/ϕ^2 = 2 − ϕ = 0.381966 and ϕ^2 = ϕ + 1 = 2.618034. The  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -

Abstract

Related

Cite