Projection effects in star-forming regions: I. Nearest-neighbour statistics and observational biases

Stars form as molecular clouds fragment into networks of dense cores, filaments, and subclusters. The characteristic spacing of these cores is a key observable imprint of fragmentation physics and is commonly measured using nearest-neighbour (NN) statistics. However, NN separations are derived from projected two-dimensional (2D) positions, while fragmentation occurs in three dimensions (3D). Using spherical and fractal toy models, we show that the standard geometric deprojection factor of $4/π\simeq1.27$ is inadequate because projection not only foreshortens separations but also rewires the NN network, while finite angular resolution merges close neighbours and inflates apparent spacings. We quantify these competing biases with Monte Carlo experiments spanning a wide range of morphologies, sample sizes, and effective resolutions. From these we derive an empirical correction factor that depends on both sample size and resolution: for small ($N\lesssim10$) or poorly resolved samples ($\lesssim$10 resolution elements across the field), intrinsic NN spacings exceed projected values by only 20 to 40%, whereas for well-sampled ($N\gtrsim100$), well-resolved data ($\gtrsim$30-50 resolution elements), true 3D separations are typically larger by a factor of $\sim$2. This calibration enables observers to convert measured 2D NN spacings into corresponding 3D estimates, with typical morphology-driven uncertainties of order 30 to 40%, and we demonstrate how it alters inferred fragmentation scales in observed and simulated core populations. [abridged]