brody-exponent-spatial-exclusion

name	brody-exponent-spatial-exclusion
description	Calibrated measurement framework using the Brody exponent β as a quantitative measure of short-range exclusion in 2D spatial point processes. Originally from quantum chaos level-spacing statistics, now calibrated for spatial analysis with corrected CSR baseline, empirical β-r_excl calibration (Spearman ρ=0.988), and control protocols. Use for quantum chaos analysis, spatial statistics, prime number embeddings, and manufactured surface characterization.
metadata	{"arxiv_id":"2606.16393","published":"2026-06-15","authors":"Dawid Kucharski","tags":["quantum-chaos","spatial-statistics","brody-distribution","point-processes","prime-numbers"]}

Brody Exponent Spatial Exclusion Framework

Description

A calibrated measurement framework that repurposes the Brody distribution — originally from quantum chaos level-spacing statistics — as a quantitative measure of short-range exclusion in 2D spatial point processes. Includes corrected CSR baseline, empirical calibration against hard-core radius, and validated control protocols.

Activation Keywords

• brody exponent, brody distribution
• spatial point process, level-spacing statistics
• quantum chaos spatial, short-range exclusion
• csr baseline correction, hard-core radius
• prime number embedding, wigner distribution
• 量子混沌, 空间点过程, 布罗迪指数

Core Framework

The Brody Distribution

Originally a phenomenological interpolation between Poisson (β=0) and Wigner (β=1) level-spacing statistics in quantum chaotic systems:

P(s) = C·β·s^(β-1) · exp(-C·s^β)

where C = [Γ(1+1/β)]^β normalizes the distribution.

Key Results

1. 2D CSR Baseline Correction: The 2D complete-spatial-randomness baseline is β = 0.96 ± 0.15, not the 1D Poisson reference (β=0). Using the 1D baseline in 2D analysis produces systematic bias.

2. Empirical β-r_excl Calibration: The Brody exponent β correlates with effective hard-core radius r_excl with Spearman ρ = 0.988, establishing β as a reliable proxy for exclusion strength.

3. Density Independence: Density-thinning experiments establish that β captures exclusion strength rather than point density, though absolute values are density-dependent.

Measurement Protocol

1. Extract 2D point coordinates from data
2. Compute nearest-neighbor distances
3. Fit Brody distribution to distance histogram
4. Extract β parameter
5. Compare against calibrated CSR baseline (β ≈ 0.96 for 2D)
6. Interpret: β > 0.96 → exclusion present; β ≈ 0.96 → random; β < 0.96 → clustering

Control Protocols

• Sparse-integer control: Distinguish genuine arithmetic signals from random patterns
• Density-thinning: Verify β measures exclusion strength not density
• Binary-field baseline: Low fill fraction requires distinct CSR baseline
• Embedding null test: Cantor-embedding shows some exclusion is embedding-created

Decision Table

β Range (2D)	Interpretation
β < 0.80	Clustering behavior
0.80 ≤ β < 0.96	Near-random with slight clustering
0.96 ± 0.15	Complete Spatial Randomness (baseline)
1.10 < β < 1.50	Moderate exclusion
β > 1.50	Strong exclusion

Usage Patterns

Pattern 1: Quantum Chaos Analysis

Apply Brody distribution fitting to energy level spacings in quantum systems to characterize chaos-to-regularity transitions.

Pattern 2: Spatial Statistics

Measure short-range exclusion in manufactured surfaces, biological point patterns, or geological distributions.

Pattern 3: Number Theory

Analyze arithmetic sequences (e.g., prime number embeddings) for spatial exclusion patterns.

Error Handling

Baseline Selection

Always use the correct dimensionality-appropriate CSR baseline. Using 1D Poisson (β=0) for 2D data produces β values that are systematically inflated by ~0.96.

Density Dependence

While β captures relative exclusion strength, absolute values depend on point density. Use density-thinning experiments to verify that observed β differences reflect genuine exclusion rather than sampling artifacts.

References

• arXiv: 2606.16393 — "Calibrating the Brody exponent as a quantitative measure of short-range exclusion in 2D spatial point processes"