Time--Information--Complexity Unified Variational Principle\\ in Computational Universes:\\ Computational Worldlines on Control--Scattering Manifold\\ and Task Information Manifold

In previous works on the ``computational universe'' series, we abstracted the universe as discrete object U_{comp} = (X,T,C,I), constructing discrete complexity geometry (complexity distance, volume growth, and discrete Ricci curvature based on configuration graph) and discrete information geometry (based on task-aware relative entropy and Fisher structure) on it, and gave continuous limit of complexity geometry under unified time scale scattering mother scale: a control manifold M with Riemannian metric G. However, these geometric structures still separately characterize ``time/resource cost'' and ``information quality/task-relevant states'', lacking a framework to unify both under a single variational principle. This paper, building on control manifold (M,G) and task information manifold (S_Q,g_Q), introduces joint manifold $ E_Q = M \times S_Q, constructing on it a time--information--complexity joint action A_Q, thereby characterizing ``computational trajectories'' in computational universe as minimal curves on joint manifold (computational worldlines). Specifically, we first give action at discrete level A_Q^{disc}(\gamma) = \sum_k \big( \alpha\,C(x_k,x_{k+1}) + \beta\,d_{info,Q}(x_k,x_{k+1}) - \gamma\,\Delta I_Q(x_k,x_{k+1}) \big), proving that under appropriate scaling, this discrete action family \Gamma-converges as h\to 0 to continuous action A_Q[\theta(\cdot),\phi(\cdot)] = \int_0^T \Big( \tfrac12 \alpha^2 G_{ab}(\theta)\theta^a\theta^b + \tfrac12 \beta^2 g_{ij}(\phi)\phi^i\phi^j - \gamma\,U_Q(\phi) \Big)\,dt, where \theta(t)\inM is control trajectory, \phi(t)\inS_Q is task information state, U_Q is task-related information potential function (e.g., negative information quality). Then we derive Euler--Lagrange equations on joint manifold E_Q, proving that minimal trajectories satisfy coupled ``geodesic equations with potential'': control part evolves along geodesics of (M,G) but receives feedback from gradient of U_Q with respect to \phi; information part evolves along geodesics of (S_Q,g_Q) but is modulated by control trajectory \theta. Furthermore, using standard variational methods and \Gamma-convergence theory, we prove: under unified time scale and local Lipschitz assumptions, discrete optimal computational paths converge in the limit to minimal worldlines on joint manifold, achieving rigorous correspondence between ``optimal algorithms in discrete computational universe'' and ``continuous time--information--complexity worldlines.'' This paper concludes with discussion of minimization problems with resource constraints: maximizing task information quality under fixed time budget or complexity budget. We give equivalent Lagrange multiplier form, thereby characterizing ``optimal information acquisition strategy under given budget'' as a class of geodesic flows with effective potential. Results of this paper provide variational foundation at intrinsic dynamics level for subsequent construction of categorical equivalence between ``computational universe \leftrightarrow$ physical universe.''