自发比数理论:比值空间上的层级与自发动力学 / Spontaneous Ratio Structures: Hierarchy and Spontaneous Dynamics on the Ratio Space

自发比数理论:比值的构造理论。数学对象为 n 维正实比值矢量 q∈Λⁿ=(0,∞)ⁿ——取值与维数均不预设。四条公理与一条定义生长出完整结构链:层级谱(一阶 n、二阶 C(n,2)、链式约束后独立自由度 n−1)、形状空间(Weyl 腔 Aₙ₋₁)与结构地图定理(对称=测度零、部分对称类按整数分拆计数、一般位置=完全破缺)、单纯形同构定理(与成分单纯形精确同构,往返误差 4.4e-16,对数比坐标同一)、对数墙定理(形状-标度分离+退化边界的连续势必含对数标度分量——自发动力学的墙项是结构必然而非建模选择)、差型唯一性定理(形状层耦合势由对称性唯一钉死)。特例化:成分数据分析几何=静力学截面特例,椭球形状分析=三维截断特例。学术对照:比的基础化四幕史(欧多克索斯/von Staudt/Cayley–Klein/Buckingham)之上的第五幕;静力学比值结构有先例(Aitchison),对象本体自发动力学无对应。核心测度论内容 Lean 形式化零 sorry。中文主版。 Spontaneous ratio structures: a constructive theory of ratios. The mathematical object is an n-dimensional positive ratio vector q∈Λⁿ=(0,∞)ⁿ, with neither values nor dimension presupposed. Four axioms and one definition grow a complete structural chain: the hierarchy spectrum, the shape space (Weyl chamber Aₙ₋₁) with the structure-map theorem (symmetry = measure zero; general position = complete symmetry breaking), the simplex isomorphism theorem (exact isomorphism with the compositional simplex, round-trip error 4.4e-16), the logarithmic wall theorem (the wall of the spontaneous dynamics is a structural necessity, not a modeling choice), and the difference-type uniqueness theorem. Specialization: compositional data analysis geometry is the static sectional case. Scholarly comparison: the fifth act atop the four-act history of founding the ratio; static ratio structures have precedents (Aitchison), object-ontological spontaneous dynamics has no counterpart. Core measure-theoretic content formalized in Lean with zero sorry. English edition.