Symmetry-Protected Degeneracy in Orbital-Tower Spectra: the L+1 Law, its Proof, and the Golden-Ratio Prediction

We define a family of hopping Hamiltonians on “orbital towers” — nested rings of size Nk = 2bkcoupled by an angle-preserving map between consecutive shells — and prove that their spectrumcarries symmetry-protected levels whose multiplicity is L + 1 at every depth L, independent of thecoupling strength t. Our central result is a complete hand proof, via a second-order perturbativemechanism, that the protected positions are exactly the critical points of the energy folding mapE → 2Tb(E/2) (the b-fold Chebyshev map), and that the selection rule is parity under centralinversion: odd states have no descending shadow and all survive (L of them), while even states shiftby a uniform amount t2 · 2/(3b) except for the bottom-orbit state (L + 1 in total). The frameworkpassed a pre-registered predictive test: migration of the protection, in base 5, to the golden ratio±φ, proved algebraically in Q( √5). We close with the physical reading (renormalization, Van Hovepoints, a measurable experimental signature in waveguide-array photonics) and the verification code.