5. SATURN’S NORTH POLE

Since Voyager first observed it in 1981, Saturn's north pole has carried a hexagonal vortex approximately 30,000 kilometres across — wider than two and a half Earths — that has maintained its precise six-sided geometry across every decade of observation without deviation or explanation. This document proposes, and substantially grounds, an answer: the hexagon is not a coincidence or a special initial condition, but a geometric inevitability forced by the same Corner Theorem that governs three-dimensional incompressible fluid dynamics at every scale. The theorem establishes that incompressibility itself — the constraint that forbids isotropic expansion — collapses the full octahedral symmetry of strain eigenvalue space down to exactly six preferred directions, arranged as a Tresca hexagon. Crucially, a key upgrade on 26 March 2026 elevated the Saturn prediction from speculative to Structurally Grounded: the proof of the Corner Theorem's only-if direction via bulb intersection revealed that Saturn's cyclonic vortex core and anticyclonic flanking vortices are the direct physical manifestations of the theorem's positive and negative bulbs — and their intersection boundary is the hexagonal jet stream, confirmed by Cassini observations. The saltwater canonical DNS result (Cell A, θ_s = 50.103°) provides the closest fluid analogue, linking the same geometric attractor to both Saturn's polar vortex and the Atlantic Meridional Overturning Circulation. The hexagon was always going to be a hexagon. Incompressibility does not make exceptions for gas giants.