A Theory of Geometric Structure: The Classification Law (R₂ ∧ C ≠ C₀)

This paper presents the Classification Law, a deterministic biconditional governing the existence of measurable geometric structure in directed relational systems: GEOMETRY EXISTS ⇔ (R₂ = 1) ∧ (C ≠ C₀) A system possesses stable, non-degenerate structural invariants (depth, width, spectral gap, connectivity) if and only if (1) its ordering contains no directed cycles (R₂, verified by acyclicity testing) and (2) its coupling exceeds the detectability threshold (average degree k̄ ≥ 1). Neither condition alone is sufficient. The law is validated across 17 domains: causal spacetime, sound, light, mathematics, gravity, quantum entanglement (via the Topological Constraint Index), color perception, black holes, iterative dynamics (Mandelbrot set), turbulence, large language model outputs, the golden angle as constrained growth optimizer, the observable universe, recursive self-application, the geometry of structural failure (Dark Geometry), and a philosophical extension to human life trajectories. Four computational validation suites (83 tests total: 61 proved, 8 disproved, 14 inconclusive) are included as reproducible Python scripts with fixed RNG seeds. All disproved claims are documented in Appendix B and removed from the main text. An ENSDF-scale analysis confirms that the complete nuclear decay graph (3,386 nuclides, IAEA data) is a DAG with R₂ = 1.0 , no cycle exists anywhere in the nuclear transition landscape. The failure taxonomy (DARK_LOOP, DARK_FRAGMENT, DARK_NULL) characterizes how systems lose geometric structure, with the key empirical finding that cycles are infinitely more destructive than disconnection at the single-edit level. Full reproducibility bundle included: paper, four validation suites, and README with environment specifications.