The sequence of Plato's three analogies (Sun, Line, Cave) in the Republic implies a unified system, yet its structure remains an unsolved aporia. This paper postulates a generative numerical "Symmetry Matrix" that functions as the "upper octave" of the Divided Line. It is the metaphysical reality—the complete Platonic doctrine of Principles and Ideas (One > Indefinite Dyad > Genus > Eidos)—which, as a unified system, is analogous to the Sun in the Sun Analogy. This four-part matrix forms the arithmetic core beneath the Gnomon, a complex coordinate system that contains the entire ancient Quadrivium. From this system, the exact proportions of the Divided Line, (1:2 // 2:4), are derived. The "lower octave" is the exact analogue: it begins where Plato's doctrine ends (at the world of Ideas) and casts its shadow over the graded system of the Line (Knowledge/Opinion), pointing towards the shadows in the subsequent Cave Allegory. The symbolic meaning of the Gnomon as a sundial's shadow-caster (the "Square" and "Diagonal" in Republic 510E) thus links all three analogies. This hypothesis resolves the aporias, provides the arithmetic core for the Gnomon, and proves that the entire ancient Quadrivium (Arithmetic, Geometry, Harmonics, Cosmology) is a single, unified system derived from the matrix's inherent properties.