Euclidean Self-Return: Complex Metric Rotation, Vacuum Fluctuations, Quantum Diffusion, and Cosmogenesis
We develop Euclidean self-return as a geometric and deterministic mechanism for vacuum fluctuations, quantum diffusion, and rare cosmogenic formation. The construction begins with a complex signature angle through which a Lorentzian metric rotates around the degeneracy of the corresponding real metric path, reaches the exact Euclidean section, and returns to a Lorentzian section without losing invertibility. A positive Hermitian metric-speed norm defines Euclidean inertia, while conserved or retardedly stabilized phase momentum produces persistent oriented winding. Within the declared one-normal metric sector, a common Euclidean calibration and common constitutive law imply pointwise universality of the rotation energy, asymptotic winding rate, and local fluctuation spectrum, without requiring identical local phases or histories. Projection of the deterministic parent dynamics yields resolved drift, memory, and an orthogonal history force. Contact-Anosov mixing, a quadratic thermodynamic scaling, and a Ford–Kac–Mazur response model produce a finite effective noise kernel and the inverse-mass diffusion law \[ D_{\rm q}(m)=\frac{\mathcal A_*}{2m}. \] After the empirical identification \(\mathcal A_*=\hbar\), time-symmetric diffusion recovers the Bohm quantum potential and the uncertainty scale \(\Delta x\,\Delta p_{\rm fl}\geq\hbar/2\). The rotating projection has zero phase mean but nonzero quadratic mean; deterministic dephasing converts this persistent oscillatory activity into a strictly positive Green–Kubo coefficient. The same framework supports three compatible cosmogonic regimes: stationary rare nucleation, formation during nonstationary high-energy parent relaxation, and a conserved phase-domain branch. In the third branch, ordinary integrable mixing is first shown to erase macroscopic phase averages. A nonlinear conserved phase field then supplies metastable zero bias, two oppositely oriented locked states, exact global signed-charge conservation, and rare domain nucleation. Spatially heterogeneous retention barriers create hotspots in which multiple same-sign remnants have enhanced conditional probability. A positive pair-capture kernel and an open Skyrme locking basin give a conditional positive probability for universe-forming collisions, while opposite topological charge remains in distant domains or the diffuse background. Cosmological expansion is interpreted as dilution of matter on a fixed mother space rather than expansion of the mother metric. Collision recoil and rotation-generated topological stress yield a positive material virial, whereas dense matter screens the transmission of this stress into relative material motion. An explicit density-dressed propagator gives \(S(\rho)=(1+\alpha\rho)^{-2}\), so dilution progressively unmasks the washing channel. Finally, a linear response identity transfers the inhomogeneous formation-stress spectrum into a nonnegative late-time density spectrum, providing a mathematical route from early rotational stress to voids and large-scale material structure. Astronomical identification remains an empirical test of the completed model. ### Keywords **Euclidean self-return; complex metric rotation; Euclidean inertia; vacuum fluctuations; deterministic homogenization; quantum diffusion; topological stress; phase-domain nucleation; matter screening; cosmogenesis; cosmic voids; Skyrme topology**