Mode-Resolved Electron-Phonon Fingerprints Map Topological Protection in Lithium Beyond Born-Oppenheimer

We present a mode-resolved analysis of electron-phonon interactions in lithium under extreme conditions that identifies protected regions of k-space. The Li BCC phase exhibits topologically protected degenerate bands at the Fermi level that inhibit superconductivity by preventing gap formation via electron-phonon coupling. In contrast, the 9R and cl16 phase lacks the relevant symmetry protection. In the 9R phase, the strong electron-phonon interactions dynamically couple to degenerate valence and conduction bands opening an energy gap and lowering the ground-state energy. This stabilization energy, though small in magnitude, is comparable to the total energy difference between the BCC and 9R phases. Through mode-resolved calculations, we identify distinctive electron-phonon interaction patterns, including pole-like non-adiabatic coupling terms, localized spikes indicative of topological protection, and broad parabolic regions associated with conventional coupling. The same framework localizes the soft-mode instability in high-pressure FCC Li to a highly confined strong-coupling pocket in k-space (near X), with rapid suppression away from that point by symmetry-driven protection. We introduce the concept of "topological superposition energy," a quantum stabilization mechanism arising from the coherent mixing of conduction and valence states mediated by lattice vibrations. Near quasi-degenerate doublets, we show that these pole-like features directly reflect large lattice nonadiabatic coupling terms (NACTs), providing a quantitative link between the curvature fingerprints and interband derivative couplings. We further map the most strongly coupled doublets onto a mode-resolved spin-boson Hamiltonian, using the ratio of degeneracy lifting energy and phonon energy to gauge how effectively a phonon mode can drive degeneracy lifting and gap formation across the different Li phases. This mode-resolved, non-adiabatic approach thus converts electron-phonon data into a predictive k-space atlas of where symmetry blocks or enables gap formation, reconciling low-temperature and pressure-induced behavior in elemental lithium and offering a transferable recipe for other quantum materials.