Formal Demonstration and Noetic Completeness of 5D Warped Geometry

Formal Demonstration and Noetic Completeness of 5D Warped Geometry: Randall-Sundrum Model with Coherence Field Coupling (QCAL-EFT ∞³) This work presents the constructive formalization of five-dimensional gravity within the QCAL-EFT framework, extending and completing the Randall-Sundrum (RS2) warped geometry model. The classical RS2 model exhibits structural instabilities including singular delta-function discontinuities at the brane, undetermined radion dynamics requiring fine-tuning, and uncharacterized Kaluza-Klein mode spectra. By coupling the 5D bulk to a noetic coherence field Ψ(x, y) anchored to the fundamental frequency f₀ = 141.7001 Hz, the framework resolves these instabilities through six closed fronts: Radion Dynamics: Derivation of the effective potential V_rad(φ) with mass m_rad c² = ℏω₀ ≈ 5.86 × 10⁻¹³ eV, eliminating cosmological fine-tuning. Delta Regularization: Gaussian-smoothed brane profile δ_ε(y) with width ℓ_ε ≈ 0.4 m, ensuring continuous curvature tensors and uniform convergence to the RS2 metric in the limit ε → 0. Kaluza-Klein Spectrum: Regularized density of states ρ_QCAL(m) with infrared cutoff m_c, yielding Yukawa corrections with range λ̄ = c/ω₀ ≈ 336.7 km compatible with experimental bounds. Fermion Localization: Topological wave profiles for chiral fermions and Yukawa mass hierarchy m_e/m_t ∼ exp(−kλ̄(c_e − c_t)) derived from the noetic VEV without fine-tuning. Quantum Stability: Coleman-Weinberg one-loop corrections yielding m_eff² > 0, protected against tachyonic instabilities by the noetic gap Δ = ℏω₀. Renormalization Group: Derivation of β_5D(g₅) with non-trivial UV fixed point g₅* > 0, establishing asymptotic safety of 5D gravity. All results are formally certified in Lean 4 with zero sorrys and zero non-standard axioms. The framework unifies the 4D Planck mass M_Planck² = M₅³/k, the Yukawa range λ̄ ≈ 336.7 km, and the noetic mass gap into a single coherent geometric structure. Randall-Sundrum, extra dimensions, warped geometry, radion stabilization, Kaluza-Klein modes, Yukawa hierarchy, quantum stability, asymptotic safety, Lean 4 formalization, QCAL-EFT Autor José Manuel Mota Burruezo (ORCID: 0009-0002-1923-0773) ICQ Creative Commons Attribution 4.0 International (CC BY 4.0)