Predicting sub-cycle transient peak currents from inverters within the first few milliseconds after fault inception is essential for instantaneous overcurrent protection coordination, yet the most commonly used standard for short-circuit calculation, like IEC~60909-0 and ANSI C037.010 addresses only quasi-steady-state currents. This paper proposes a closed-form analytical framework that decouples inverter fault current into hardware natural response and control-commanded response. The hardware natural response, governed by inverter output filter and passive grid resonances, dominates the first transient peak and requires only nameplate and filter data - no proprietary controller information. The control-commanded response, shaped by the current loop, phase-locked loop, and fault ride-through strategy, governs subsequent waveform evolution. Building on unified hardware modeling, the framework extends separately to Grid-Following and Grid-Forming inverters. Validation against electromagnetic transient simulations and power hardware-in-the-loop tests on three commercial inverters confirms prediction accuracy.