The Universal Structural Threshold in BECU–OLON: K crit = 3 , Fourier Embedding, and Predictive Galaxy Morphology

This work presents the foundational Paper C of the BECU–OLON framework: the mathematical proof and observational validation of the universal structural threshold Kcrit = sqrt(3) ≈ 1.732051. The structural parameter K = (R/ρ)(dρ/dR) is shown to govern galaxy rotation curves, the emergence of flat rotation curves, morphology transitions, lensing behavior, and the cosmological interpretation of the Big Bang as a Cosmological Division rather than an initial singular explosion. The threshold Kcrit = sqrt(3) is derived from exact structural analysis of power-law density profiles, variational stability, and the first instability eigenmode, establishing it as a theorem-level result rather than a phenomenological fit. A covariant Fourier-space embedding K(k,z) ~ k|δ(k,z)| / (1 + δ) is derived to provide the bridge between real-space galaxy structure and Boltzmann cosmology (CLASS/CAMB), showing that BECU–OLON remains perturbatively safe while preserving high-l Planck precision and BAO structure. Real SPARC galaxy analysis confirms universal threshold crossing across observed systems, with critical crossing radii concentrated near the disk turnover region (median ~2.8, mean ~3.6). Flat rotation curves emerge preferentially when the crossing occurs within a narrow structural band, indicating that galaxy morphology is controlled by the placement of the critical structural threshold rather than by hidden dark matter halos. The results support BECU–OLON as a geometry-driven infrared completion of ΛCDM, where dark sectors emerge from structural response rather than invisible particles. Dark matter asks where the missing mass is.BECU asks whether the structure was measured correctly.