Universal Quartic Scaling Law for Kerr-Type Interactions: Projection-Law Factorization Across Nonlinear Quantum Platforms

We present a rigorous derivation and numerical validation of a universal projection-law factorization for quartic nonlinear coupling rates across physically distinct platforms. The central result is that observable Kerr-type interactions - self-Kerr, cross-Kerr, and cross-phase modulation - factorize into a dimensionless projection coefficient and an intrinsic quartic energy scale. This structure follows from canonical quantization of a quartic interaction projected onto a finite normal-mode basis. We state this result as a formal proposition with clearly enumerated assumptions, provide a proof sketch based on second quantization of the anharmonic potential, and enumerate the domain of validity. We then implement the scaling law as a lightweight computational toolkit (UEFT-Designer) with platform-specific kernels for superconducting circuits, photonic microcavities, and epsilon-near-zero (ENZ) structures. A complete, step-by-step worked example for a superconducting quarton device - with full uncertainty propagation - predicts a cross-Kerr rate χ/2π= 361pm 13 MHz, agreeing with the independently measured value of 366pm 0.5 MHz \cite{Ye2025} to within 1.4%. A formal error-propagation analysis establishes that the dominant uncertainty is the Josephson-energy extraction uncertainty $sim2\%$, not the geometric projection factor. Cross-platform validation against five independent experiments spanning eight orders of magnitude in coupling strength confirms the universality of the factorization to within reported experimental uncertainties. The ghost-sector spectral correction introduced in earlier versions is reframed as an optional phenomenological self-energy model with no mandatory role in the validated scaling law.