Fundamental Dynamic Structures IV: The Three-Dimensional Vortex Sector of the Logarithmic Medium — Ring Kinematics, Outgoing Core Instabilities, Kelvin Waves, and Filament Topology

We extend the two-dimensional FDS vortex analysis to three dimensions for the conservative repulsive logarithmic Schrödinger medium \[i\,\partial_\lambda\psi=-\tfrac12\nabla^2\psi+\ln(|\psi|^2)\psi,\] in units \(\hbar=m_{\rm eff}=\rho_0=b=1\). Four groups of results are reported with separate claim status. **(1) Ring kinematics.** An analytic solid-angle phase initializer with a numerically converged two-dimensional core profile gives \(2Rv=0.96(4)\ln(8R)+0.15\) for \(R=3,4,5\); a fixed-slope fit gives \(C=-0.013\pm0.014\) in \(2Rv=\ln(8R)-C\), and a held-out \(R=6\) run agrees with that fit to \(0.85\%\). The profile-energy constant \(a_0=0.4182\) closes a separate energy–momentum estimate to approximately \(4\)–\(5\%\). A matched cubic Gross–Pitaevskii control confirms the predicted sign of the medium-dependent velocity shift. **(2) Finite-cylinder and outgoing spectra.** The straight-line Bogoliubov–de Gennes problem separates in \((\ell,k_z)\). The finite cylinder displays bands and window cascades, but outgoing Evans/Jost matching and complex-absorbing-potential continuation reorganize their interpretation: the \(q=2,\ell=2\) and \(q=3,\ell=2\) core modes continue as smooth outgoing unstable branches, while a finite Dirichlet cylinder may enhance, suppress, or miss them. Representative outgoing growth rates are \(0.00967\) for \(q=2,k_z=0\), and \(0.01206,0.01040,0.00130\) for \(q=3,k_z=0,0.2,0.6\). The weak \(q=3,\ell=3\) branch seen with an absorber is not independently confirmed by direct outgoing matching and remains unresolved. No complex mode is detected in the tested unit-winding channels. **(3) Nonlinear closure and finite geometry.** A numerically resolved \(q=3,\ell=2,k_z=0.20036\) finite-cylinder mode was evolved to \(\lambda=100\); a filament-based \(k_z=1\) amplitude fit gives \(0.0239\pm0.0074\), compared with the BdG rate \(0.02521\). Unconstrained co-moving solves nominally initialized as \(q=\pm3\) do not retain a multiplicity-three zero: at three resolutions they resolve into \(3+3\) unit-vortex bundles. Their collective unstable modes are reproducible, with maximum growth \(0.088\)–\(0.098\), and mode-seeded nonlinear fits agree over their early linear windows; these are cluster modes, not spectra of a co-moving multiply-wound core. A retained tracker audit of an earlier nominal \(q=\pm2\) co-moving solve likewise resolves a \(2+2\) unit-strand cluster; its collective mode \(\omega=0.98951\pm0.08265i\) closes nonlinearly to \(3.3\%\) at \(\lambda=100\), but the raw complex background was not preserved, so this result is supplementary rather than a primary reproducibility-grade claim. **(4) Filament dynamics.** A density-qualified dual-cell tracker represents plaquette winding as an integer 2-cochain and reconstructs periodic filament components, bundles, and axial spectra. It detects a controlled ring topology-change event near \(\lambda=6.54\), although the universal \(|\lambda-\lambda_0|^{1/2}\) separation law is not quantitatively closed at the present grid. The unit-line Kelvin branch is real and negative-Krein; its low-\(k\) coefficient in \(\omega=\omega_0-A_Rk^2\ln(1/k)-B_Rk^2\) rises from \(A_R=0.284\) at \(R_{\rm dom}=40\) to \(0.453\) at \(R_{\rm dom}=160\), approaching the hydrodynamic coefficient \(1/2\). All infinite-domain statements are numerical outgoing/CAP evidence rather than analytic spectral theorems. Together, the calculations support a channel-resolved three-dimensional selection pattern: the tested \(q=2\) and \(q=3\) sectors possess outgoing \(\ell=2\) instabilities and unconstrained travelling solves relax into unit-strand bundles, whereas no complex mode is detected for \(q=1\) in the tested channels. This is a numerical, channel-limited result rather than a proof of complete nonlinear stability or universal unit-winding selection.