Coupled-Channel Quantum Tunneling with an Additional Confined Coordinate

This exploratory preprint studies a controlled quantum-scattering toy model with an additional harmonically confined coordinate w. It compares a fixed-width one-dimensional barrier, the exact shrinking observable slice V(x,0), and the full coupled (x,w) Hamiltonian. The identical-slice control shows that the non-monotonic transmission turn is not unique to an additional coordinate, while the coupled calculation produces a quantitatively distinct transmission curve and a converged transverse-broadening signal. The central hypothesis remains S_E = f(w), with A = S_E / ℏ the dimensionless tunnelling exponent. The matched-state stress test independently changes longitudinal energy or transverse confinement and retunes the barrier to reproduce center broadening Δw_rms = 0.08, 0.18, and 0.35. The corresponding WKB actions spread by 50.2%, 31.1%, and 22.7% relative to the baseline; the exact-scattering exponent agrees. Thus a universal one-variable proxy A = F(Δw_rms) fails without refitting. Version 3.3 left the numerical data and conclusions unchanged and added the companion tutorial Born-Oppenheimer to Tunnelling Action. That tutorial distinguishes the coordinate w from W_rms(x), ΔW(x), and the center scalar Δw_rms; explains the local and branch-dependent role of W_rms'; treats derivative-coupling gauge dependence; and identifies the gauge-invariant quantum metric g_xx as a candidate for a future matched-state test. Version 3.4 leaves the numerical data and conclusions of Version 3.3 unchanged. It adds the companion tutorial From Kaluza-Klein Compactification to a Tunnelling Decay-Rate Residual. The tutorial derives k_w = n/R and κ_eff² = κ_x² + n²/R² for a separable compact-circle toy model, clarifies that the zero mode reproduces ordinary tunnelling while excited modes further suppress it, and defines the model-independent residual Δκ² = κ_eff² - κ_x². It also states the mode-assignment and low-energy limitations and distinguishes this compact model from the harmonically confined, nonseparable Version 3.3 Hamiltonian. These additions are pedagogical and do not constitute evidence for a fundamental fourth spatial dimension. The work is non-peer-reviewed. The seven files contain the unchanged Version 3.3 main manuscript in PDF and DOCX, the Born-Oppenheimer companion tutorial in PDF and DOCX, the new Kaluza-Klein compactification companion tutorial in PDF and DOCX, and the reproducibility supplement with calculation scripts, CSV data, metadata, convergence checks, and figures. AI assistance is disclosed in all documents.