Oscillatory Credit Assignment: When Phase-Relative Neural Updates Approximate Gradient Learning

This preprint reconstructs a September 2024 Self Aware Networks conjecture about phase-wave differentials and asks when a receiver-relative neural phase observation can participate in gradient-like learning. It separates a phase departure from an objective-bearing teaching factor, synapse-specific eligibility, and the resulting parameter gradient; derives factor-error, alignment, descent, and nonidentifiability results; reports a bounded Lean verification layer; and supplies a frozen deterministic phase-event reference application with matched comparators, ablations, adverse results, robustness tests, 1,068 evidence rows, and byte-identical reruns. The results support only a conditional computational model. They do not establish that phase alone assigns credit, that a particular phase calibration is biologically unique, or that the application is a cortical simulation. ## Evidence boundary The application is deterministic, synthetic, phase reduced, and deliberately small. Lean checks only the declared scalar algebraic and logical leaves. Neither layer establishes a cortical implementation, physiological noise threshold, phase necessity, unique calibration superiority, or empirical confirmation of the complete biological conjecture. The manuscript is a preprint and has not been peer reviewed.