20年悬案：信息几何中曲率值域被证明为半整数，但仅限特定网络结构。
Quantization of Ricci Curvature in Information Geometry
- 通过贝塔函数消去机制，证明树状与完全图二元网络曲率必为半整数
- 发现环状结构反例，推翻普遍量化猜想，曲率可非半整数
- 揭示离散网络正曲率与高斯网络负曲率的符号二分规律
2004年研究二元贝叶斯网络（bitnets）的信息几何时，作者提出猜想：基于Fisher信息度量计算的体积平均里奇标量⟨R⟩恒为正半整数，即⟨R⟩ ∈ (1/2)Z。本文历经20年终予解决：对树状结构和完全图bitnets，通过普适的Beta函数抵消机制予以证明；但通过显式环状反例证明该猜想在一般情形不成立。研究进一步拓展至高斯DAG网络，发现符号二分现象——离散bitnets具正曲率，而高斯网络构成可解李群且具负曲率。
原文摘要 · Abstract (English)
In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.
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