The Oculomotor Analemma: Torsional Oscillation of the Pupil–Macula Axis, Möbius Topology, and Structural Convergence with the Solar Analemma

The figure-eight geometry of the solar analemma emerges from two well-established astronomical mechanisms: the Earth's axial tilt (~23.4°) and orbital eccentricity. The result is a closed trajectory with a specific topological property — a Möbius crossing point at the geometric centre — not shared by arbitrary superpositions of two oscillators. We propose that the same Möbius topology governs the oculomotor system through the G→X→Q→N cycle of the Kinematic Energy PHramework (KEPH), where the involutive operator X² = −Id generates precisely this topological structure. In the visual system: the pupil occupies the G pole (luminous potential entry), angle kappa — the structural misalignment between the pupillary axis and the foveal axis — is the X operator (torsional transition), and the macula/fovea occupies the Q pole (crystallized image). The superposition of a short pupillary cycle (~28 days) and a long macular cycle (~365 days), coupled through the directional asymmetry of angle kappa, generates a temporal trajectory with the formal structure of an analemma — including the Möbius crossing point as the structural inversion of the visual field at the transition X. The convergence is not between two arbitrary double oscillators: it is between two systems governed by the same X² = −Id topology, which generates a figure-eight with a structurally determined crossing point rather than a generic lemniscate. Four verifiable predictions are derived, including the novel prediction that solar eclipses — the precise alignment of the pupil-pole (Moon) and the macula-pole (Sun) along the structural Z→N axis — represent the astronomical analogue of the X transition in the oculomotor cycle.