Dihedral Programming of Topological Charge in Curved Quantum Waveguides

We construct a scalar curved quantum-waveguide hierarchy in which dihedral symmetry programs both the leading geometric release order of a protected two-level degeneracy and its topological charge. For a family with D2q symmetry, representation theory forbids off-diagonal geometric coupling below order q. The protected microscopic doublet is formed by longitudinal symmetry sectors: for even q the degeneracy occurs at zero flux, while for odd q it occurs at half flux. The resulting local effective Hamiltonian contains two independent q-th-order geometric Pauli components together with a flux-controlled mass term. Strict thin-waveguide and finite-width calculations are performed for q=2,…,8. In every case both symmetry-allowed q-th-order channels are nonzero, the complex off-diagonal matrix element winds q times around the degeneracy, and an independent closed-surface calculation gives lower-band Chern number C=+q in the stated orientation convention. The sequence therefore realises charges 2,3,4,5,6,7,8 within one scalar microscopic construction. The results establish a direct hierarchy D2q⟶geometric release order q⟶winding q⟶∣C∣=q. The construction does not invoke electron spin or an appended phenomenological two-state degree of freedom; the protected pair consists of longitudinal quantum-waveguide symmetry sectors. A separate transverse-mode microscopic realisation remains an open problem.