Phase-Wave Differential Mathematical Analysis

A neural signal can differ from its recent context in several ways at once: relative phase, amplitude, duration, and spatial propagation. Existing analyses often measure these quantities separately or summarize interaction with a scalar synchrony statistic. This paper defines a phase-wave differential (PWD) as a receiver-relative, edge-conditioned vector of departures from a strictly past baseline. The construction starts with a causal complex filter bank, expresses phase at a receiver relative to a putative sender, estimates circular-phase and log-amplitude references using only earlier samples, and combines wrapped phase departure, log-amplitude departure, event duration, and a coordinate-normalized lagged phase gradient. The operator is invariant to a common phase rotation and, for its amplitude-departure component, to common positive amplitude scaling. Its interpretation is conditional on the selected filter, reference, receiver, edge direction, delay, phase-quality gate, and spatial coordinates. Delay and expected phase offset are not generally identifiable from one narrow band; low-amplitude phase, common input, volume conduction, waveform asymmetry, evoked latency fields, and reference choice can all mislead. An existing implementation passed eight deterministic causal and shape checks on generated signals, while a separate equal-input synthetic comparator did not show a PWD-feature advantage. A frozen Allen Visual Coding Neuropixels development procedure then compared an identical 16-feature shared raw branch plus eight PWD random features against the same shared branch plus eight additional raw random features. Fourteen development sessions, 921 eligible VISp units, three fixed projection seeds, and 908,295 causal samples were analyzed. Every session-level primary increment was negative: the cross-session median raw-minus-PWD held-out Poisson-deviance difference was -0.001099 (descriptive session-bootstrap 95% interval -0.001301 to -0.000872; two-sided exact sign test, 0 of 14 positive, *p*=0.000122). Thus, the full tested PWD branch underperformed the capacity-matched raw-only branch. The controls narrow that failure. Intact PWD outperformed its within-block circular-shift null and the source-receiver role swap in all 14 sessions. Removing amplitude and duration also worsened the PWD branch in all 14 sessions. By contrast, removing the phase components improved performance in all 14 sessions. These controls do not rescue the failed primary benchmark; they indicate that the tested phase representation was the most immediate redesign target, while aligned, directional, amplitude-duration structure retained limited predictive information relative to its own nulls. The five-session final test remains unopened. PWD is therefore retained as a defined candidate observable, while the present 25-feature implementation fails its grouped development criterion and requires independently reviewed redesign before any new confirmatory protocol.