Volume Growth Entropy in Hilbert Geometry Matches Hyperbolic Space — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The search results are a mix of competition-problem videos (windmill problem, "hardest easy geometry," USAMO problems) and one arXiv paper on Hilbert geometry asymptotics. The only substantive mathematical content is the Hilbert geometry paper; the rest are pedagogical artifacts without extractable equations. MATH: From the Hilbert geometry paper (arXiv:0711.0446v1): - Volume growth entropy of metric balls in Hilbert geometry equals that of Lobachevsky space (hyperbolic space). - Asymptotic ratio: \( \frac{\text{Vol}(B_R)}{\text{Area}(S_R)} \sim \frac{R}{n-1} \) (for dimension \(n\)), matching hyperbolic space behavior. - No explicit constants like 0.382, 0.618, 0.786, 1.618, 2.618 appear in the abstract. CONNECTION: Hilbert geometry is a projective generalization of hyperbolic geometry. Its metric balls share the same exponential growth rate as Lobachevsky space — implying a deep projective invariance of the entropy. No direct link to golden-ratio or base-60 struc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222628332
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