无边之动的几何:形态的公理化构造 / Geometry of Boundless Motion: An Axiomatic Construction of Form

无边之动的几何:形态的公理化构造(构造数学系列第四篇·几何层公理层4a)。观察公理:几何形态的存在方式=无边之动——必有能量输入、能量值不预设、真实边界消失、抽象=在动的基础上对具体性的最大化约束。两条公理(G-Ⅰ能量的存在与自由/G-Ⅱ边界与具体性的消失);三步几何内导:坐标阶梯(定理3.1——四参数内在微分几何不变量获得公理地位)、零点结构(定理4.1——四个零点=形态空间分区原点)、对数对称(定理4.2——tan 15°·tan 75°=1,四条恒等式Lean形式化零sorry);定理4.3给出'哪些分区边界几何必然/哪些经验'的分层答案。特例化:具体形态分类学与工程体系=公理+实现条件三件套的特例(与δ对齐/流对齐同族)。学术对照:形状文法与形状空间(工具层)/Thompson形态学(物理层——正面分道=新颖性核心)/耗散结构/旋涡尾迹/流变哲学——三层先例齐备而构造层空白。中文主版。 Geometry of boundless motion: an axiomatic construction of form (fourth paper of the constructive-mathematics series, geometry layer 4a). Observational axiom: the mode of being of geometric form is boundless motion — energy input necessary, the energy value not presupposed, real boundaries vanishing, abstraction as the maximal constraining of concreteness on the basis of motion. Two axioms (G-Ⅰ the existence and freedom of energy / G-Ⅱ the vanishing of boundaries and concreteness); three steps of geometric introjection: the coordinate ladder (Theorem 3.1 — four intrinsic differential-geometric invariants acquire axiomatic standing), the zero structure (Theorem 4.1 — four zeros as the partition origins of form space), logarithmic symmetry (Theorem 4.2 — tan 15°·tan 75°=1, four identities formalized in Lean with zero sorry); Theorem 4.3 gives the layered answer to 'which partition boundaries are geometrically necessary / which empirical'. Specialization: concrete morphological taxonomies and engineering systems = the axioms plus a triple of realization conditions (the same family as δ-alignment / flow alignment). Scholarly comparison: shape grammars and shape spaces (tool layer) / Thompson's morphometrics (physics layer — the frontal demarcation is the core of novelty) / dissipative structures / vortex wakes / process philosophy — three layers of precedent with the construction layer blank. English edition.