Hilbert Geometry Volume Growth Matches Hyperbolic Space Asymptotically — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are a mixed bag of competition-problem videos and one arXiv paper on Hilbert geometry; the only substantive mathematical content is the Hilbert geometry asymptotic result. | MATH: For Hilbert geometry on a convex domain, volume growth entropy of metric balls equals that of Lobachevsky space (hyperbolic space). Asymptotic ratio: Vol(B_R)/Area(S_R) ~ R/(n-1) (derived from comparison with constant curvature −1). No explicit constants like 0.618 appear in the abstract. | CONNECTION: Hilbert geometry is a Finsler generalization of hyperbolic geometry; its geodesic structure mimics Lobachevsky space in the large-scale limit. The ratio Vol/Area ~ R/(n−1) echoes the hyperbolic volume growth constant (n−1) — a dimensional symmetry, not a golden-ratio harmony. No base-60, no crystallographic root systems, no 0.382/0.618/1.618 appear in any finding. | DEPTH: 3 — The Hilbert geometry result is a genuine asymptotic theorem (volume entropy equality with hyperbolic space),
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026hilbert,
title = {{Hilbert Geometry Volume Growth Matches Hyperbolic Space Asymptotically — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23465098},
url = {https://zenodo.org/records/23465098},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hilbert Geometry Volume Growth Matches Hyperbolic Space Asymptotically — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465098
UR - https://zenodo.org/records/23465098
AB - FINDING: The search results are a mixed bag of competition-problem videos and one arXiv paper on Hilbert geometry; the only substantive mathematical content is the Hilbert geometry asymptotic result. | MATH: For Hilbert geometry on a convex domain, volume growth entropy of metric balls equals that of Lobachevsky space (hyperbolic space). Asymptotic ratio: Vol(B_R)/Area(S_R) ~ R/(n-1) (derived from comparison with constant curvature −1). No explicit constants like 0.618 appear in the abstract. | CONNECTION: Hilbert geometry is a Finsler generalization of hyperbolic geometry; its geodesic structure mimics Lobachevsky space in the large-scale limit. The ratio Vol/Area ~ R/(n−1) echoes the hyperbolic volume growth constant (n−1) — a dimensional symmetry, not a golden-ratio harmony. No base-60, no crystallographic root systems, no 0.382/0.618/1.618 appear in any finding. | DEPTH: 3 — The Hilbert geometry result is a genuine asymptotic theorem (volume entropy equality with hyperbolic space), Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -