Exact Gravitational Collapse Time, Cloud Fragmentation, and Topological Invariants of Bonnor-Ebert Oscillations via the F lambda Class

Gravitational collapse of interstellar gas clouds is driven by self-amplifying density feedback: increasing density strengthens gravitational attraction, which compresses the gas further, which raises the density. We show that this process is exactly captured by rho_dot = rho exp(lambda rho), where rho is the normalized density (rho/rho_Jeans) and lambda > 0 encodes the pressure-to-gravity ratio. This is the F_lambda class of a previous work — the unique ODE of the form u_dot = g(u) Phi(lambda u) whose associated function space is stable under both natural derivation operators and which admits a transcendental non-elementary linearizer, the exponential integral E1. Exploiting this exact linearization, we derive: a closed-form collapse time T*_coll = E1(lambda rho0) in units of the free-fall time t_ff, the full A_n hierarchy rho_dot = rho^n exp(lambda rho) modeling isothermal (n=1), adiabatic (n=2), radiation-pressure (n=3), relativistic (n=4), and naked-singularity (n=5) collapse regimes, calibration on four astrophysical objects: Barnard 68, Class 0 protostar, Ophiuchus dense core, giant molecular cloud, a PDE model of spatial cloud fragmentation (P_lambda^+) with exact solution, producing multiple stellar formation sites simultaneously, a topological invariant Omega = pi/2 - Si(mu rho0) for Bonnor-Ebert oscillations, counting oscillation periods exactly, and a real-time collapse alert computable from radio observations. The classical Jeans criterion gives a free-fall time t_ff = sqrt(3pi/32G rho) independent of the feedback parameter lambda. The F lambda framework gives T* = E1(lambda rho0), which depends explicitly on lambda and classifies collapse regime analytically.