We present an analytical organizing principle underlying the dynamics of the periodically forced Duffing oscillator. In the massless limit the system reduces to a time-dependent cubic equation whose discriminant and Jacobian classify the canonical Duffing regimes and determine the number and stability of instantaneous equilibria. For the double-well oscillator, the same construction gives an exact forcing threshold $F_{\mathrm{bif}}$ separating intrawell and interwell dynamics. We show that this algebraic quantity governs the finite-mass dynamics far beyond the singular limit from which it originates: The boundary between intrawell and interwell motion converges to $F_{\mathrm{bif}}$ as $m\to 0$, and the families of pitchfork bifurcations responsible for asymmetric interwell states accumulate at a common point. Numerical simulations further show how the discontinuous jumps of the degenerate model are regularized at finite mass into short oscillatory transients that shrink as $m\to 0$. These results reveal the massless Duffing equation as the organizing center of the full finite-mass dynamics and establish a direct link between the algebraic structure of the degenerate problem and the global bifurcation structure of the oscillator.

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