Resonant suppression of the pi-type Josephson component in a Kitaev-ladder junction

We investigate Josephson transport through a finite normal segment connecting two Kitaev-ladder superconductors and show that discrete-level detuning controls the sign of the equilibrium Josephson coupling through a branch-selection mechanism. Away from resonance, detuning separates two competing low-energy branches, allowing inter-leg hybridization to clamp the 0-phase-favoring branch while leaving the π-favoring branch dispersive. The resulting negative first harmonic evolves from a branch-switching response at weak coupling into a smooth π-junction regime. At resonance the two clamping thresholds coincide, so the finite interval in which only one branch remains dispersive disappears. Once the inter-leg coupling exceeds the resonant clamping threshold, both low-energy branches become independent of the Josephson phase. The remaining finite-energy states then generate a 0-type Josephson coupling. At ϕ=π, the helicity decomposition yields an exact clamping identity for arbitrary finite normal-segment length. Within the controlled sweet-point finite-segment theory, serial propagation fixes the off-resonant fourth-order harmonic to be negative, whereas an isolated resonance leaves a positive residual harmonic within the resonant clamping regime. Numerical calculations demonstrate that the resonance-induced suppression persists away from the solvable phase choice. Thus normal-level resonance does not merely enhance Josephson transmission; it changes which low-energy branches contribute to the Josephson coupling.