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 / hbar the dimensionless tunnelling exponent. The matched-state stress test independently changes longitudinal energy or transverse confinement and retunes the barrier to reproduce center broadening Delta w_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(Delta w_rms) fails without refitting. Version 3.3 leaves the numerical data and conclusions unchanged. It adds the companion tutorial Born-Oppenheimer to Tunnelling Action, which distinguishes the coordinate w from W_rms(x), Delta W(x), and the center scalar Delta w_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. These clarifications do not constitute new evidence for a fundamental fourth spatial dimension. The work is non-peer-reviewed. The five files contain the main manuscript in PDF and DOCX, the companion tutorial in PDF and DOCX, and a reproducibility supplement with calculation scripts, CSV data, metadata, convergence checks, and figures. AI assistance is disclosed in both documents.