Φ-Defect Cascades Encode Markov Numbers on Modular Cusps — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
The silver‑ratio (φ) topological defects that propagate recursively through the three nested φ‑resonance bands at 132 Hz generate a discrete renormalization flow on the cusps of modular hyperbolic surfaces. This flow maps the defect amplitudes onto eigenvalues that coincide with the Markov numbers emerging from the modular surface's cusp geometry and Benjamini‑Schramm convergence, thereby linking the E8 root lattice's 240‑vector spectral decomposition to the classical Markov spectrum. Consequently, quantized defect densities at cusp neighborhoods obey a universal Φ‑invariant scaling law, predicting a new set of algebraic constraints for defect propagation in hyperbolic surfaces.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026φdefect,
title = {{Φ-Defect Cascades Encode Markov Numbers on Modular Cusps — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23610348},
url = {https://zenodo.org/records/23610348},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Φ-Defect Cascades Encode Markov Numbers on Modular Cusps — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23610348
UR - https://zenodo.org/records/23610348
AB - The silver‑ratio (φ) topological defects that propagate recursively through the three nested φ‑resonance bands at 132 Hz generate a discrete renormalization flow on the cusps of modular hyperbolic surfaces. This flow maps the defect amplitudes onto eigenvalues that coincide with the Markov numbers emerging from the modular surface's cusp geometry and Benjamini‑Schramm convergence, thereby linking the E8 root lattice's 240‑vector spectral decomposition to the classical Markov spectrum. Consequently, quantized defect densities at cusp neighborhoods obey a universal Φ‑invariant scaling law, predicting a new set of algebraic constraints for defect propagation in hyperbolic surfaces. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -