Layered Bose-Einstein Cosmology (LBEC): A Unified Model for the Dark Sector The Layered Bose-Einstein Cosmology, or LBEC, is a unified theoretical model for the dark sector of the universe. It is based on a relativistic two-component scalar field that functions as a Bose-Einstein Condensate. In this model, dark matter is explained as stable, localized structures known as topological hedgehog solitons. These solitons have a mass of approximately 10^-22 electron volts and a halo radius of about one kiloparsec, with their stability ensured by quantum pressure that resolves the small-scale structure problem. Dark energy is an emergent property of the field's self-interactions, with its behavior governed by a Renormalization Group flow. This dynamic nature offers a compelling solution to the Hubble tension, as the model predicts a Hubble constant of approximately 69.1 kilometers per second per megaparsec. A key and unique feature of the LBEC model is its two-layer structure. This layered structure arises from the minimization of a Ginzburg-Landau energy functional and allows the model to reconcile the universe's large-scale isotropy with enhanced lensing shear power. The enhanced lensing shear is a distinct, testable signature that sets the LBEC model apart from the standard cosmological model. The theory also provides a direct path for experimental verification through analogue gravity. It proposes that cosmic gravitational waves are analogous to phonons in a laboratory BEC. The model explicitly derives a scaling factor that maps laboratory frequencies to cosmic frequencies, with a predicted strain amplitude of about 10^-22, which is detectable by the LISA mission. All cosmological predictions, including the matter power spectrum and the Hubble constant, are supported by numerical simulations. The model is a coherent, internally consistent, and testable alternative to the standard Lambda Cold Dark Matter paradigm.