Spectral Gap Protection of Qubit Coherence via a Fourth-Order Scalar Bath

We study a qubit coupled to a bosonic bath described by a fourth-order scalar field Lagrangian, whose dispersion relation omega^2 = c^2(k^4 + alpha*k^2 + m^2) supports a tunable spectral gap. Below the gap, the local density of states vanishes exactly, suppressing both energy relaxation (T1) and pure dephasing (T2) at leading order. We derive the canonical quantization of the theory from first principles, establish an exact closed-form expression for the local density of states, and map out a phase diagram in (alpha, m^2) parameter space with a sharp gap-closing transition at m^2 = alpha^2/4. At the critical line, the momentum coupling (partial_t phi) uniquely yields an Ohmic bath, placing the system at the spin-boson quantum phase transition boundary, while direct field and gradient couplings produce infrared-divergent spectral functions due to the universal 1/sqrt(omega_k) mode normalization of soft critical modes. The framework provides a concrete design principle for spectral-gap-based qubit protection and connects fourth-order field theory to the spin-boson literature.