Instead of damping amplitude, an image writes a coarse field of local phase delays. A uniform complex signal passes through two operations in opposite orders. Their difference exposes changes in the phase map.
Inspect actual trained maps — seed 41 ↗ · Seed 97 ↗ · Full inspection metrics JSON ↗
32×32 RGB pixels; 3,072 bytes raw.
8×8 quantized phase map; 64 bytes.
Hue is phase, not amplitude.
Magnitude of [Sₓ,Uτ] applied to a uniform query.
Magnitude of [Sᵧ,Uτ] applied to the same query.
Pure phase has unit intensity everywhere. A deliberately blind control.
The commutator reveals spatial variation in a phase-delay map. A uniformly delayed field produces exactly zero wake under periodic boundary conditions; a localized delay changes neighboring phases and gives a nonzero response. Pure phase multiplication and periodic shifts preserve energy, whereas attenuating transport loses it.
Those are mathematical controls, not trained vision-model wins. Directly reading the phase map or its finite differences may solve a task just as well. The Python v2 benchmark makes those comparisons on held-out synthetic scenes, including an equally sized amplitude-coded baseline.
A true eikonal arrival-time gradient is not necessarily a persistent physical wake after a pulse disappears. We simulate a phase mask under prescribed discrete operations, not refracted rays, membrane electrophysiology or lipid biochemistry.