Finite-Width Generation of a Transverse-Mode Multi-Weyl Hierarchy in Curved Quantum Waveguides

A scalar curved quantum waveguide is used to construct a programmable hierarchy of geometric-control multi-Weyl nodes in a genuinely transverse microscopic two-state sector. The protected doublet is the first excited degenerate subspace of an isotropic transverse harmonic confinement, dominated by the orthogonal oscillator modes ∣1,0⟩⊥ and ∣0,1⟩⊥. For a D2q-symmetric centreline, the two transverse modes carry opposite characters under the improper dihedral shift. Representation covariance then forbids every geometric off-diagonal coupling below total degree q. Finite transverse width supplies the complementary Pauli structure absent from the strict torsional gauge theory. Aharonov–Bohm flux tunes the resulting transverse mass through zero while preserving the dihedral selection rule. Full finite-width calculations at ℓ=0.02 realise a contiguous hierarchy for q=2,…,8: the participating eigenstates retain approximately 98.8%–99.3% weight in the intended first-excited transverse oscillator pair, the complex off-diagonal element has winding magnitude q, and independent Berry–Pancharatnam calculations using the full generalized eigenvectors give lower-band Chern-number magnitude ∣C∣=q. An explicit O(ℓ2) transverse reduction identifies how torsion and finite-width curvature anisotropy complete the microscopic Pauli algebra. Across q=2,…,8, the reduced theory reproduces the full-tube flux-mass velocity to better than 0.22%. A D4 benchmark reproduces both strong and weak quadratic harmonic amplitudes at sub-percent accuracy. Width scans show that the transverse mass and complementary geometric Pauli direction vanish as O(ℓ2), whereas the torsional component approaches a nonzero thin-width limit. The isolated transverse multi-Weyl node is therefore intrinsically finite-width and has a singular strict-width limit.