Abstract This paper introduces the "Aureo Vortex, " a novel geometric model that maps numerical radicals into a 3D space defined by the golden ratio (φ ≈ 1. 618). Based on a comprehensive statistical analysis of the 241 highest-quality known ABC triples, we present empirical evidence that φ acts as a fundamental geometric threshold separating stable from anomalous arithmetic structures. Key Contributions & Version 2. 1 Updates: The NZ Signature: We define a geometric degeneracy metric (NZ) which serves as a robust predictor of ABC quality. Analysis reveals a strong negative correlation (r ≈ -0. 72, p 1) residing at an extremely low NZ signature (NZ ≈ 0. 0023). The Golden Threshold: We observe that of the top known triples, only 1. 24% exceed φ, forming a distinct statistical population. We propose the "Golden Threshold Conjecture, " stating that for triples with stable geometry, quality is strictly bounded by φ. Predictive Law: Non-linear regression confirms an exponential relationship (R² = 0. 9987), predicting an absolute geometric upper bound for ABC quality at qₘax ≈ 1. 631 as geometric stability approaches zero (NZ → 0). These findings suggest that violations of classical ABC bounds are not random accidents but predictable geometric phenomena confined exclusively to the low-NZ regime. Keywords ABC Conjecture, Number Theory, Golden Ratio, Aureo Vortex, NZ Signature, Diophantine Equations, Titanium Prime, Computational Discovery, Geometric Degeneracy, Arithmetic Geometry
Pirolo Andres Sebastian (Mon,) studied this question.