The Continuous-Emergent Universe Model: A Covariant Formulation of Relational Geometry, Constrained Dynamics, Two-Field Scalar Cosmology, and Minisuperspace Operator Regulation

The Continuous-Emergent Universe Model is a unified cosmological research framework built on the principle that space, time, probability, cosmic expansion, and regional habitability acquire operational meaning strictly through dynamical relations among fields. Space is defined by metric separation (ds² = gμνdx^μ dx^ν) vanishing into rank-deficient metric boundaries (rank(g) < 4). Global time is non-existent (H_⊥Ψ = 0), with duration (dtJ) emerging correlationally from configuration displacement (dσQ). State space operates on a non-atomic continuous measure (μ(₀\) = 0), guaranteeing uncountably infinite structural diversity ($P=0$ duplication). Anisotropic stress is resolved via a two-field scalar sector comprising an isotropic spatial wall triplet (ΦI = κ xI) and a quintessence field ($Q(t)$), naturally generating dark energy (ρDE ~ MPl² H₀² ~ 10⁻¹²⁰ MPl⁴) and regional Effective Field Theory (EFT) parameter drift. FLRW minisuperspace quantization with an operator regulator $F(v)$ and Dirichlet domain boundary conditions guarantees zero probability flux (Jᵛ(0,Q) = 0) across zero volume, proving quantum singularity exclusion.