Proposed Ultraviolet Regularization and Asymptotic Behavior of the Semiclassical Wave Function of the Universe

We propose a non-linear, dimensionless matter-coupling ansatz to modify the semiclassical exponent of the de Sitter/Hawking-Hartle Wave Function of the Universe. By introducing a symmetric quadratic energy ratio x ≡ ⟨H_matter⟩/E_P regularized by a three-dimensional spatial exponent factor (1+x²)³, the construction guarantees dimensional invariance ([P²H²]=1), parity symmetry (x → -x) and exact recovery of the classical de Sitter vacuum state in the low-energy regime (x → 0). In the trans-Planckian limit (|x| → ∞) the argument of the exponential undergoes an asymptotic suppression of order O(x⁻⁶), causing the wave function amplitude to converge smoothly to a bounded upper limit (Ψ → 1). This provides a mathematically consistent heuristic model for ultraviolet (UV) damping of cosmological singularities without removing the explicit cosmological Hubble scale H.