摘要本文系统研究 EML-Membrane 映射的周期轨道结构。该映射定义为 T (M )c (x, y) = (sin(log(x2+c)) − y, sin(log(y2 + c)) − x),其中 c = 0.5 为最优工作点。我们建立了以下核心结果:零点序列 Bn = √enπ − c 的完整理论;四族周期 2 轨道的分类定理(Type I/II/III/IV-a);统一的奇偶稳定性选择定则;Type I 质量谱的闭式公式 mn = − ln(1 − 4e−nπ + 2e−2nπ );Type III 质量谱的指数衰减律 mij ≈ 4e−sπ/2;弦-膜模桥定理 c = 0.5 = (1 − 2c∗)2 = 4c∗(1 − c∗);动力学降维定理(EML-String 是 EML-Membrane 基频模式的投影);以及 Type V 三弦耦合的初步结构。所有核心结论均附解析推导和独立数值验证,证明和验证代码随文公开。关键词:EML 映射;周期轨道;稳定性选择定则;质量谱;弦-膜模桥;动力学降维AbstractWe systematically investigate the periodic orbit structure of the EML-Membrane mapping, de-fined as T (M )c (x, y) = (sin(log(x2 + c)) − y, sin(log(y2 + c)) − x), with optimal operating pointc = 0.5. We establish: the complete theory of the zero sequence Bn = √enπ − c; classification the-orem for four families of period-2 orbits (Type I/II/III/IV-a); unified parity-based stability selectionrule; closed-form mass spectrum formula for Type I, mn = − ln(1 − 4e−nπ + 2e−2nπ ); exponentialdecay law for Type III, mij ≈ 4e−sπ/2; the string-membrane modular bridge c = 0.5 = (1−2c∗)2 =4c∗(1 − c∗); the dynamical dimensional reduction theorem (EML-String as the projection of EML-Membrane’s fundamental mode); and preliminary structure of Type V three-string coupling. All coreconclusions are accompanied by analytic derivations and independent numerical verification. Proofand verification code is provided.Keywords: EML mapping; periodic orbits; stability selection rule; mass spectrum; string-membrane modular bridge; dynamical dimensional reduction