First-Order Topological FFLO Transition and Superconducting Diode Sign Reversal in Altermagnetic Nanowires

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state conventionally emerges via a second-order phase transition driven by finite magnetization. Here we show that a spin-orbit-coupled nanowire proximitized to $d$-wave altermagnets -- with zero net magnetization -- can realize topological FFLO states through a first-order transition, marked by a sharp sign-reversing superconducting diode effect. The altermagnetic field generates band-resolved competing pairing channels, giving rise to a double-valley free energy landscape whose global minimum switches discontinuously. It consequently leads to a first-order topological FFLO transition with simultaneous jumps in the Cooper pairing amplitude and finite center-of-mass momentum. Remarkably, this discontinuous topological reconfiguration substantially enhances the diode efficiency and drives a characteristic sharp sign reversal across the transition. The mechanism of such exotic phenomena is captured by Ginzburg--Landau theory. Our results provide a field-free altermagnetic route to topological FFLO states and identify their direct transport fingerprint.