QWIM Quantum Wave Integration Model A Variational Theory of Phase Synchronization with Local Adaptive Noise: Mathematical Foundations and Neural Interpretation

AbstractWe present a revised formulation of QWIM (Quantum Wave Integration Model), addressing mathematical inconsistenciesidentified in peer review. The core contribution is a local adaptive noise modification of the Kuramoto model, in whicheach oscillator’s diffusion coefficient depends on the instantaneous global order parameter R. We work exclusively inthe Stratonovich convention and derive the corresponding Fokker–Planck operator with its spurious-drift correctionexplicitly. A careful linear stability analysis shows that the critical coupling in the thermodynamic limit isKc = 2(∆ + D0),where ∆ is the half-width of the frequency distribution and D0 is the baseline noise; the adaptivity parameter α entersonly at nonlinear order and governs the shape of the bifurcation (sharpness, hysteresis width), not the linear threshold.The variational (free-energy) picture is made precise: it is exact for identical oscillators in the co-rotating frame andprovides a useful approximation in the heterogeneous case. Phase switching between Wave (low-R) and Impulse (high-R)modes is generated endogenously by a slow fatigue variable K(t), eliminating the need for external resets. A speculativequantum biophysical substrate (NV centers in nanodiamonds) is retained but clearly labeled as a hypothesis, with orderof-magnitude feasibility estimates and genuinely discriminative experimental predictions provided.