We report the identification and proof of a universal constant, ln(3) = 1.09861, which governs the onset of bidirectional collective behavior in one-dimensional Poisson proximity networks. The constant - named the cooperative percolation constant and denoted by Lambda_c - is the unique positive solution to 2/(exp(x)-1) = 1 and equals the Shannon entropy of three equiprobable states. For agents distributed at intensity lambda and interacting within range L, bidirectional collective behavior is possible if and only if the topological density (lambda * L) >= ln(3). Below this threshold, no cooperative control policy can produce macroscopic coherence, as the proximity graph does not contain a bidirectional spanning cluster in expectation. The result is parameter-free and model-independent: the Poisson distribution is derived from memorylessness symmetry axioms, making ln(3) a fundamental consequence of spatial symmetry. The threshold is validated by two independent large-scale empirical datasets. Analysis of the Chengdu V2X OBU dataset (N = 19.7 million records) reveals a 1.60x reduction in speed variance at the predicted boundary. Furthermore, the highD German motorway dataset (N = 163,896 observations) yields a best-fit LWR exponent theta = 1.033 +/- 0.088, placing the theoretical value ln(3) within 0.75 sigma of naturalistic trajectory data (R^2 = 0.8631). The remarkable consistency across geographical and physical scales - from motorway traffic to documented thresholds in 1D biological signal transmission - suggests that ln(3) represents a fundamental topological rent for cooperative information exchange.

翻译：本文报道了一个普适常数 ln(3) = 1.09861 的发现与证明，该常数支配着一维泊松邻近网络中双向集体行为的涌现。此常数——命名为合作渗流常数，记作 Λ_c——是方程 2/(exp(x)-1) = 1 的唯一正解，且等于三个等概率状态的香农熵。对于以强度 λ 分布并在范围 L 内相互作用的智能体，当且仅当拓扑密度 (λ * L) ≥ ln(3) 时，双向集体行为才可能实现。低于此阈值时，任何合作控制策略都无法产生宏观相干性，因为邻近图在期望意义上不包含双向跨越簇。该结果是无参数且模型无关的：泊松分布源于无记忆性对称公理，使得 ln(3) 成为空间对称性的基本推论。该阈值通过两个独立的大规模实证数据集得到验证。对成都 V2X OBU 数据集（N = 1970 万条记录）的分析表明，在预测边界处速度方差降低了 1.60 倍。此外，德国高速公路 highD 数据集（N = 163,896 条观测记录）拟合得到最优 LWR 指数 θ = 1.033 ± 0.088，使理论值 ln(3) 位于自然轨迹数据的 0.75 标准差范围内（R² = 0.8631）。该常数在地理与物理尺度上——从高速公路交通到一维生物信号传输中已记录的阈值——表现出的显著一致性表明，ln(3) 代表了合作信息交换所需的基本拓扑“租金”。