Marked correlation functions, in which galaxy properties such as luminosity or stellar mass are treated as marks, are widely used to test models of galaxy formation. In astronomy, however, these statistics are typically implemented as summary measures that do not preserve the joint structure of mark pairs conditioned on separation. In this work, we formulate galaxies as points $(x,m)$ on the product space $\mathbb{R}^3\times\mathcal{M}$, where $x$ denotes position and $m$ a mark, and introduce the joint pair correlation function $g(r;m_1,m_2)$ as the fundamental quantity describing mark-dependent clustering. We further define a diagnostic quantity $Δ_{\mathrm{ind}}(r;m_1,m_2)$ that locally quantifies deviations from the independence hypothesis relative to spatial clustering alone, thereby providing a projection-free description of which mark pairs are over- or underrepresented at a given separation scale. Within this framework, commonly used diagnostics such as the inhomogeneous cross-$J$ function are naturally interpreted as summary statistics obtained through averaging over mark sets and geometric-event-based reductions of the joint structure. This perspective clarifies that previously discussed marked effects, including assembly bias, correspond to projections of an underlying joint dependence, and that observationally accessible information is the existence of non-factorizable joint structure itself. The present formulation provides both a fundamental quantity and practical diagnostics for its characterization.

翻译：标记相关函数（将星系属性如光度或恒星质量视为标记）广泛应用于检验星系形成模型。然而在天文学中，这些统计量通常作为汇总度量实现，未保留标记对条件于间距的联合结构。本研究将星系形式化为乘积空间$\mathbb{R}^3\times\mathcal{M}$上的点$(x,m)$（其中$x$表示位置，$m$表示标记），引入联合对相关函数$g(r;m_1,m_2)$作为描述标记依赖性成团的基本量。我们进一步定义诊断量$\Delta_{\mathrm{ind}}(r;m_1,m_2)$，该量在局域尺度上量化了相对于空间成团性的独立性假设的偏离，从而提供了无投影的描述，揭示特定间距尺度上哪些标记对过量或不足。在此框架下，常用诊断如非齐次交叉$J$函数可自然解释为通过对标记集求平均及对联合结构进行基于几何事件的约化而获得的汇总统计量。该视角阐明，先前讨论的标记效应（包括装配偏差）对应于底层联合依赖性的投影，而观测上可获取的信息正是不可分解联合结构本身的存在性。本研究既提供了基本量，也为其刻画提供了实用诊断工具。