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 K₂ₑ₈ₓ 1. 73 3, interpreted not as a fitted cosmological parameter, but as a geometric eigenvalue of structural space itself. The central structural quantity K = RddR 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 K₂ₑ₈ₓ = 3 acts as the first instability mode of cosmic structure. This implies: Big Bang = Cosmological Division at K₂ₑ₈ₓ, 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.