A Galois-Siegel Framework for Three Prime Dissections of Ramanujan

Ramanujan recorded that, for prime order p, the prime dissections of the Euler function and of his theta functions phi and psi involve (p-1)/2 p-th roots of functions, and he further indicated that quadratic, cubic, and quintic equations arise in determining those functions. Berndt later emphasized the interpretive gap, while Berndt-Rebak described the cubic, quintic, and septic theories as apparent pieces of a broader “grand theory.” This preprint gives a uniform structural answer. From the three exact prime dissections it isolates normalized p-th radicands and proves that a distinguished radicand from each family is a primitive generator of the same cyclic modular-function extension K_p/K_p^0 with Galois group (Z/pZ)^*/{+-1}, of degree (p-1)/2. For every subgroup H, the coefficients of the H-orbit polynomial generate the exact fixed field K_p^H, so every prime-index step yields a genuine cyclic auxiliary equation of that prime degree. The fixed-field tower is identified as the connected 2-adic pullback of the standard diamond-operator tower above X_0(p). A single generalized-eta row further unifies the three branch systems; its total product is eta(tau)/eta(p tau), recovering the classical eta Hauptmodul coordinates at the prime genus-zero levels p=5,7,13. Explicit theta/distribution formulas are given for every subgroup-resolvent coefficient. The general theories of Siegel units, modular-unit distributions, diamond actions, genus-zero modular equations, and cyclic resolvents are treated as prior art. The candidate contribution is their simultaneous realization by the three exact Ramanujan prime-dissection radicand systems. External specialist correctness and priority review have not yet occurred.