Wave-Propagation Geometry Emerges from Scalar Travel-Time Relations

Abstract: Scalar travel-time relations can identify propagation geometry without velocity inputs, ray labels or equation supervision during learning. Physical interpretation is applied after learning through the high-frequency isotropic eikonal relation (Aki and Richards, 2002). Here we show that local value differences constrain derivatives through sampling geometry and smoothness. Controlled neural fields recover propagation directions with $5.34\pm0.07<sup>{\circ}$ median angular error, and local P- and S-wave velocities in a prospectively locked, test-sealed realization. Classical representations also recover geometry. Scalar-matched perturbations separate value accuracy from derivative reliability; regularized inversion can recover structure when direct gradients fail. A field trained on 2.90 million Chinese arrivals retains continental crust--mantle boundary structure: correlation with a withheld reference is $ρ=0.910$ raw and 0.592 after quadratic detrending (spatial-shift $p=0.020$). Direct catalogue values yield $ρ=0.909$, and thickness contrasts are compressed. Manual-only California arrivals support broad velocity organization with limited lateral fidelity. Synthetic quantum fields containing phase information support probability-flow and trajectory readouts without supervising those quantities. These results establish conditional identification across representations and physical settings: informative scalar relations can constrain geometry, while known physical relations supply its interpretation. They do not imply that accurate scalar fitting alone guarantees reliable physical derivatives.