Ginzburg-Landau Theory with Elastic Incompatibility Describes the Liquid-G-phase-Crystal State Diagram

We derive a mechanically consistent Ginzburg-Landau theory for the metastable heterogeneous G state reported in deeply undercooled metallic liquids. The central result is that type-II-like mixed-state physics can emerge structurally with the random internal stress field playing the role of the magnetic field. Frozen eigenstrain due to elastic incompatibility between ordered cores and the amorphous matrix generates incompatible stress, while the non-affine internal strain in the liquid-like matrix in between the ordered cores relaxes that stress over a finite screening length $λ_σ$ (analogous to the London length in the superconducting problem). Structural order heals over a distinct mechanically-dressed length $ξ_G$; their ratio $κ_G=λ_σ/ξ_G$ provides the structural counterpart of the Ginzburg-Landau parameter. Because the matrix is transiently rigid yet locally relaxational only over an intermediate time window, the same mechanism selects crystal for slow protocols, a heterogeneous G state at intermediate rates, and a homogeneous L-like glass under rapid arrest. Residual incompatibility frustrates coarsening, allowing finite ordered cores and screened, orientation-dependent core interactions. The resulting texture is an inverted Abrikosov analogue: structural order resides in the cores, whereas incompatible-stress screening is carried by the surrounding disordered matrix. The theory yields direct simulation and experimental tests through stress-decay, structural-healing, non-affine-susceptibility, and kinetic measurements.