Periodic disorder invariant as a reliable topological index for finite disordered superconducting nanowires

Building on our recent proposal of the periodic disorder invariant (PDI) as a reliable topological indicator for finite disordered superconductors, we present a detailed theoretical analysis establishing its formal connection to existing topological invariants. We show that it is mathematically equivalent to the real-space Pfaffian topological invariant defining class-D topology and clarify its direct relation to the ground-state fermion parity under periodic and anti-periodic boundary conditions. We further compare the PDI with the scattering matrix topological invariant and topological visibility, demonstrating that the PDI is free from lead coupling and quasi-Majorana-induced biases that affect scattering-based indicators. These results establish the PDI as a robust and physically transparent topological indicator for diagnosing topological superconductivity in finite, disordered semiconductor-superconductor hybrid nanowires and emphasize its usefulness in analyzing and interpreting future Majorana experiments.