The Big Bang is Not an Origin: A Geometric Phase Transition in the BECU Framework

We show that the Big Bang is not an initial singularity, but a structural phase transition governed by measurable geometric thresholds. Within the BECU (Baryonic Emergent Coupling Universe) framework, gravity is regulated by the structural parameter K = (R/ρ)(dρ/dR), which consistently explains galaxy rotation curves, the Baryonic Tully-Fisher Relation (BTFR), morphology dependence, gravitational lensing offsets, Bullet Cluster separation, and late-time cosmic acceleration. The same critical threshold, Kcrit ≈ 1.75, is independently recovered across galactic dynamics and cosmological H(z) fits, with Bayesian MCMC posterior analysis yielding H0 ≈ 74.65, Ωm ≈ 0.330, Kcrit ≈ 1.73, and χ² ≈ 7.38. We further identify a second threshold, Ktrigger ≈ 4.5, corresponding to collapse-induced instability: Collapse → Trigger → Expansion. This suggests that the Big Bang is not the origin of spacetime from nothing, but a geometric phase transition of gravity itself, providing a unified structural alternative to dark matter, dark energy, and an explicit cosmological constant.