We ask whether dependency topology correlates with functional load-bearing organization as recoverable geometry -- not as a metaphor, but as a measurable structural property detectable by multilayer network analysis. Across seven independent substrates, we show that hub persistence and rank divergence under the Functional Proximity Law recover operational organization that domain experts describe as logic: axiomatic load-bearing structure in formal mathematics, control and contract structure in legacy software, conserved hub grammar across approx. 600 million years of neural evolution, catalytic role organization in a published prebiotic autocatalytic network, carry-path dominance in a 4-bit digital circuit, betweenness persistence in the ISCAS85 c432 standard benchmark (n=196), and a directional formal-systems replication in the Coq Corelib (n=17). A key methodological finding: degree-based hub persistence is weak between physical wiring and simulation state-correlation layers (r=0.21 in c432), while betweenness-based persistence is stronger (r=0.77 in the 4-bit ALU post-hoc; r=0.34 in c432). The ISCAS85 pre-registered primary hypothesis was CONFIRMED (degree r=0.426, p=0.002, Spearman r=0.551). The formal-systems claim is supported by two proof-assistant corpora: Lean 4 mathlib4 (CONFIRMED, r=0.777, p=0.004) and Coq Corelib (PARTIAL, direction confirmed, r=0.288, p=0.287, n=17, underpowered). All seven experiments were pre-registered before analysis.

翻译：我们探究依赖拓扑是否可作为可恢复的几何结构——不是隐喻，而是可通过多层网络分析检测的可测量结构属性——与功能性负载组织相关联。在七个独立基质中，我们证明：在功能邻近律下，枢纽持久性与秩发散可恢复领域专家描述为逻辑的操作组织：形式数学中的公理负载结构、遗留软件中的控制与契约结构、跨越约6亿年神经演化的保守枢纽语法、已发表的生命起源自催化网络中的催化角色组织、4位数字电路中的进位路径主导性、ISCAS85 c432标准基准（n=196）中的介数持久性，以及Coq核心库（n=17）中的方向性形式系统复制。一个关键方法论发现：基于度的枢纽持久性在物理布线与仿真状态关联层之间较弱（c432中r=0.21），而基于介数的持久性更强（4位ALU事后分析中r=0.77；c432中r=0.34）。ISCAS85预注册主假设获得确认（度r=0.426，p=0.002，斯皮尔曼r=0.551）。形式系统的主张得到两个证明辅助语料库支持：Lean 4 mathlib4（确认，r=0.777，p=0.004）与Coq核心库（部分支持，方向确认，r=0.288，p=0.287，n=17，统计功效不足）。全部七项实验均在分析前预注册。