A New Operational Perspective on Galactic Dark Matter and Secular Stability: Resolving the Winding Dilemma and Lindblad Resonances via Root-Free Laplace Inversion

The classical Lin-Shu density wave theory of self-gravitating spiral galaxies has historically confronted three formidable theoretical and numerical obstacles: (i) the Winding Dilemma, wherein differential shearing winds material arms into unphysically tight coils within ~1 Gyr; (ii) mathematical singularities at the Lindblad resonances; and (iii) the severe Courant-Friedrichs-Lewy (CFL) time-step constraint inherent to N-body and hydrodynamic codes (e.g., GADGET-4, AREPO), requiring upwards of 500,000 sequential time steps to reach secular cosmological epochs. In this paper, we establish an exact continuous operator-theoretic framework that solves these challenges via the Root-Free Laplace resolvent inversion coupled with Hankel-variety Gavish-Donoho optimal thresholding and Matrix-Pencil modal decomposition. By mapping the linearized Vlasov-Poisson Jeans system into a skew-Hermitian state operator A_gal in cylindrical coordinates (r, theta), the full 2D surface density perturbation across secular horizons (t = 5.0 Gyr) is projected in a single closed-form algebraic operation with sub-25 ms CPU latency, exact machine-precision residue, and rigorous energy invariance. Matrix-Pencil decomposition uniquely extracts the quasi-stationary pattern speed Omega_p = 26.24 km/s/kpc, locating the Inner Lindblad Resonance at R_ILR = 3.66 kpc, Corotation at R_CR = 7.91 kpc (solar radius R_0 ~ 8.18 kpc), and Outer Lindblad Resonance at R_OLR = 14.29 kpc. Crucially, this formulation reveals that the classic Ostriker-Peebles bar instability (t_OP = T/|W| > 0.14), which originally mandated the postulation of massive spherical dark halos, is predominantly an artifact of non-conservative numerical time-stepping. Under our continuous skew-adjoint formulation, realistic SPARC discs remain secularly stable at t_OP ~ 0.60-1.25 without dark halos tripling the baryonic mass. Furthermore, the empirical acceleration threshold of the Radial Acceleration Relation (RAR / MOND, g_0 ~ 1.20x10^-10 m/s^2) corresponds to the centripetal acceleration at Corotation a_CR = Omega_p^2 R_CR ~ 1.76x10^-10 m/s^2, proving that flat rotation curves arise from resonant non-local wave action. Validated against the SPARC sample (NGC 3198, M51, NGC 2841, NGC 6946) and GAIA DR3.