A Finite Four-Ball Traversing a Three-Plane: A Geometric Framework for a Closed, Cyclic Universe with a Bounded Maximum Size

We propose that the history of the universe is the sequence of cross-sections traced as a finite four-dimensional ball moves, always forward, through a fixed flat three-dimensional hyperplane. The cross-section has radius a = sqrt(R^2 - xi_c^2), with R the ball's radius and xi_c its center's fourth-axis position; a is identified with the curvature radius of a closed, homogeneous, isotropic three-sphere universe: globally closed, locally flat. The one kinematic postulate: monotonically forward passage at finite nonzero speed. Expansion, maximum, and contraction are one continuous forward motion read from three dimensions; the "turnaround" is a property of the slice, not of the ball. The three-dimensional acceleration we observe, distinct from the passage's four-dimensional acceleration, must be transient: because a <= R and the slice flattens as a -> R, observed acceleration ends before maximum size and contraction follows, whatever the dark-energy content, unlike an eternally expanding cosmological-constant-dominated future. Matter, dark energy, and entropy enter as distinct components. An effective Friedmann equation reduces the passage to one parameter, the present fraction of maximum size, and reproduces the standard-contents expansion history (present deceleration, acceleration onset, age) with one representative closure and nothing else tuned. Requiring the slice's history to match measurement gives a finite-passage theorem: a complete passage exists if and only if the universe is strictly closed, the slice-inferred dark energy ultimately dilutes, and the poles are radiation dominated; the last is already observed. A constant four-dimensional dark energy meets it through the projection, so a cosmological constant remains compatible. A first-principles passage law, the parameter's value, and entropy's mechanism are left as a research agenda.