The Universal Structural Threshold

This work presents the strongest mathematical backbone of the BECU (Binding–Expansion Critical Universe) framework: the identification of a universal structural threshold [Kcrit ≈ 1.73 ≈ √3,] interpreted not as a fitted cosmological parameter, but as a geometric eigenvalue of structural space itself. The central structural quantity [K = R/ρ/dR] governs galaxy rotation curves, the BTFR slope, gravitational lensing response, late-time cosmic acceleration, Big Freeze evolution, black-hole saturation, and the final BOOM transition interpreted as cosmological division rather than singular creation. Using exact power-law derivation, cross-scale validation, and unified consistency tests, this work shows strong evidence that [Kcrit = √3] acts as the first instability mode of cosmic structure. This implies: [Big Bang = Cosmological Division at Kcrit,] replacing singular origin with structural recurrence. The framework unifies galaxies, lensing, cosmology, and cyclic evolution under a single geometry-driven principle, suggesting that dark matter and dark energy are observational projections of structural field geometry rather than separate hidden sectors.