Calibrating the Brody exponent as a quantitative measure of short-range exclusion in 2D spatial point processes

Abstract

The Brody distribution, originally a phenomenological interpolation between Poisson and Wigner level-spacing statistics in quantum chaos, is calibrated here as a quantitative measure of short-range exclusion in 2D spatial point processes. Two results form the core. First, the 2D complete-spatial-randomness baseline is recalibrated to $β=0.96\pm0.15$, correcting the inappropriate 1D Poisson reference. Second, an empirical $β$--$r_{\text{excl}}$ calibration is validated against the effective hard-core radius with Spearman $ρ=0.988$. The framework is demonstrated on 58 manufactured surfaces (10 materials, 10 processes), phase-extracted interferometric profilometry of a certified roundness standard, and 2D binary embeddings of prime numbers. A sparse-integer control proves the prime $β=2.15$ signal is genuinely arithmetic ($Δβ=+0.68$ over random-integer control), while a Cantor-embedding null result ($β=1.40$, TOST $p<0.01$) demonstrates that 2D exclusion is embedding-created rather than intrinsic. Density-thinning experiments establish that $β$ captures exclusion strength rather than point density, while absolute values are density-dependent. A distinct CSR baseline for binary fields at low fill fraction is identified, with a decision table provided. The $β$--$r_{\text{excl}}$ calibration, the CSR baseline correction, and the control protocols together constitute a calibrated measurement framework for reproducible characterisation of short-range exclusion in 2D spatial point processes.