The Geometric Origin of the Intrinsic Relaxation Time in Quantum Transport: A Fine-Structure Topological Projection

In condensed matter physics and quantum transport theory, the relaxation time (τ) is traditionally treated as an empirical phenomenological parameter, originating from the classical Drude model to describe electron scattering. While modern quantum geometric frameworks correctly identify the symmetric quantum metric as a source of dissipation, the evaluation of the relaxation time fundamentally lacks a closed-form quantum-geometric derivation at the zero-temperature, disorder-free limit. This letter proposes a novel geometric-kinematic derivation. By treating the fundamental particle as a dynamic spatial topology where the fine-structure constant (α) acts as the geometric projection residual between macroscopic divergence and microscopic rotation, we show that the intrinsic decoherence time scale can be perfectly derived from the electron's Compton period (T_c). We demonstrate that a fourth-order topological projection yields τ_geom = T_c · α^(−4) ≈ 2.85 × 10^(−12) s. This result closely aligns with universally observed relaxation timescales in clean samples, suggesting that intrinsic dissipation is not merely a statistical collision process, but a deterministic geometric breakdown of spatial phase-locking.