The transition from effortful to automatic processing is a defining feature of skill acquisition, and converging evidence implicates theta-gamma phase-amplitude coupling as the neural signature of this transition. The Chrono-Resonant Field Model (CRFM) formalizes this observation into a computational theory: hierarchical automaticity should manifest as a measurable change in the phase organization of cortical field dynamics, and top-down predictions from higher levels should accelerate this transition at lower levels. Testing this prediction requires a physical substrate that generates phase-rich dynamics and a local learning rule that does not rely on backpropagation. We implement such a system using continuous Landau-Ginzburg wave fields trained by Equilibrium Propagation, constructing a three-layer architecture that processes speech at progressively longer timescales: phonemes (40 classes), words (501 classes), and sentence types (3 classes) from the TIMIT corpus. The architecture uses no neurons; field settling dynamics, spatial coarse-graining, holographic matched-filter readout, threshold-gated parameter updates, and top-down boundary conditions on lower-layer settling dynamics are the complete set of computational primitives. We report four principal findings. First, single-layer automaticity manifests as contraction of the Temporal Binding Index distribution rather than a rise in its mean, with Layer 1 converging to a stable phase-response attractor across samples. Second, top-down predictions from Layer 3 drive Layer 2's Temporal Binding Index from 0.349 to 0.510 over 20 training epochs, closing 92% of the gap to Layer 1's frozen value. Third, classification and prediction transmission dissociate: Layer 3 classifies equally well with or without coherent phase dynamics, but only coherent dynamics produce top-down signals capable of driving the cascade. Fourth, a multi-time-point readout exploits the underdamped regime to raise single-layer accuracy by nearly three percentage points over endpoint-only reading. These results provide a physically realizable, backpropagation-free proof-of-concept for CRFM-predicted hierarchical automaticity.