Exact Robustness Laws for a Topologically Protected Class D Zero Mode: Universal Susceptibility, Noise-Structure Boundaries, and Closed-Form Boundary-Phase Corrections

We derive four exact results for a Class D (Bogoliubov–de Gennes) topologically protected zero mode, modeled as a real antisymmetric matrix of odd dimension. (1) The protected zero eigenvalue survives 99% random structural damage and is lifted only by breaking particle–hole symmetry, linearly in the breaking strength. (2) For generic symmetry-breaking noise, the linear splitting coefficient is an exactly Gaussian random variable with variance 1/2, independent of system size and of the specific zero-mode wavefunction — proved algebraically and confirmed symbolically, by exact-rational arithmetic, and by Monte Carlo with independently reproduced seeds. (3) This universality holds only for non-local noise: local (on-site) noise gives a variance equal to the wavefunction's inverse participation ratio, which shrinks with system size, while spatially correlated noise interpolates between this local-noise floor and a full, unprotected shift once the correlation length exceeds the mode's extent. (4) For a physically motivated oscillating zero-mode profile, we derive closed-form susceptibility corrections in oscillation frequency, envelope decay length, and boundary phase, including a pre-registered peak confirmed on independent lattice data to 0.03% and an exact closed form for the cost of not knowing the boundary phase. The results specialize the general zero-mode broadening theorem of Kieburg, Mielke and Splittorff (Phys. Rev. E, 2019 and 2020) to the single-mode Class D → A case most directly relevant to Majorana-based topological qubits, and are, to the author's knowledge, new in this closed form. Every numerical claim is verified by at least two independent methods (symbolic algebra, exact-rational arithmetic, fresh-seed Monte Carlo, complex-domain diagonalization); retracted or boundary-limited claims from the investigation are documented explicitly rather than omitted.