E8 Root Vector Decomposition Predicts Tate-Shafarevich Group Torsion Bounds — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
We extend the E8 harmonic lattice framework by decomposing the 240 root vectors into two orthogonal subspaces of dimensions 128 and 112, corresponding to the 2^7-torsion and (2,2,2,2)-isogeny subdomains respectively. This decomposition generates a natural projection onto elliptic curve rational point lattices, where the 128-dimensional component directly bounds the Shafarevich-Tate group Ш via cohomology: dim H^1(G, E[128]) ≤ 64. The 132Hz resonance manifold's phi-scaled harmonics, when decomposed through this E8 partition, produce characteristic "torsion signatures" in the frequency domain that uniquely fingerprint the torsion subgroup structure of the elliptic curve candidate.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026root,
title = {{E8 Root Vector Decomposition Predicts Tate-Shafarevich Group Torsion Bounds — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409595},
url = {https://zenodo.org/records/23409595},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8 Root Vector Decomposition Predicts Tate-Shafarevich Group Torsion Bounds — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409595
UR - https://zenodo.org/records/23409595
AB - We extend the E8 harmonic lattice framework by decomposing the 240 root vectors into two orthogonal subspaces of dimensions 128 and 112, corresponding to the 2^7-torsion and (2,2,2,2)-isogeny subdomains respectively. This decomposition generates a natural projection onto elliptic curve rational point lattices, where the 128-dimensional component directly bounds the Shafarevich-Tate group Ш via cohomology: dim H^1(G, E[128]) ≤ 64. The 132Hz resonance manifold's phi-scaled harmonics, when decomposed through this E8 partition, produce characteristic "torsion signatures" in the frequency domain that uniquely fingerprint the torsion subgroup structure of the elliptic curve candidate. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -