Conditional independence testing (CIT) is fundamental to modern statistical inference in areas related to causal discovery and variable selection. While marginal independence is relatively well-understood, despite multiple advances, no existing non-parametric CIT provides a unified, efficient, and statistically guaranteed solution across heterogeneous data. We introduce a graph-based test statistic comparing kernel similarities of the response within composite neighborhoods that use exact matching on discrete components and $k_n$-nearest-neighbor matching on continuous ones. The raw statistic, related to prior constructions, suffices under fully discrete conditioning. However, when at least one conditioning variable is continuous, we instead use a local-polynomial debiased variant that cancels the local smoothing bias. We rigorously establish its asymptotic null distribution across all data-type combinations. We further prove a dimension-free $n^{-1/4}$ detection threshold under local alternatives, eliminating the phase transition that affects geometric estimators in high dimensions. Finally, we develop efficient algorithms with near-quadratic complexity and analytic graph-based calibration, bypassing the cubic bottlenecks of global kernel methods.

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