This monograph presents a rigorous, single-parameter toy model based on the stochastically perturbed parabolic map x_{n+1} = x_n + x_n^3 + η ξ_n (ξ_n ~ N(0,1)). Using a 200-million-bin matrix-free transfer operator (PETSc/SLEPc), we resolve spectral gaps to Δ₁ ≈ 1.58×10⁻⁴ at η = 8×10⁻⁷ and extend the asymptotic regime to η = 5×10⁻¹⁰. Exact Fokker–Planck continuous limit and Freidlin–Wentzell quasipotential V(x) = |x|⁴/2 • Δ_k ∼ η² scaling with logarithmic corrections below η ≈ 3×10⁻⁹ • Localisation of the slowest mode at the parabolic cusp x = 0 • Spatial inhomogeneity of effective potential curvature U''(x) producing compressed spectral ratios 1 : 3.1 : 5.6 • Quantum equivalence via Hopf–Cole transform to a quartic oscillator Hamiltonian • Probability cavitation: exponential concentration onto a shrinking quartic spine (width ∼ η^{1/2}) while tails become super-Gaussian voids A speculative extension embeds the mechanism in the GRST toy model: a hexagonal seed + quadratic iteration + 12 pentagonal defects yields SM-like lepton mass ratios ≈ 1 : 205 : 3460 purely from inverse spectral gaps, with emergent gravity and cosmology from residual vacuum energy after cavitation (ρ_Λ ∼ 10⁻¹²⁰ M_Pl⁴). All derivations, numerical code (PETSc operator + quantum-graph diagonalisation), and raw data are included. The work is a proof-of-concept toy system demonstrating how weak noise at a parabolic fixed point can generate enormous hierarchies, quantum localisation, and cosmological features from one scale. It is not claimed to describe the real world.