A Galois–Siegel Grand Theory for Ramanujan's Prime Dissections: From the Common Cyclic Cover to Explicit Cubic Houses at Conductors 19, 37, 73, 97, and 109
Year: 2026
Type: preprint
Abstract: We build on a previously proved Galois–Siegel framework for Ramanujan's prime dissections. The normalized Euler, phi and psi radicands generate one cyclic modular-function extension with Galois group (Z/pZ)^*/{±1}, and prime-index subgroups yield the corresponding auxiliary equations. In Version 3.0 we make the cubic part explicit at conductors 19, 37, 73, 97 and 109. At p=37 we derive a full q-dependent cubic over a genus-two model of X_0(37) with globally square discriminant. Blind exact cyclotomic tests at 73, 97 and 109 recover the classical conductor-p cyclic cubic fields; the p=73 packet coordinate is a published fundamental unit, p=97 is Shanks S_8, and p=109 supplies a non-Shanks falsifier with an explicit bidirectional field transformation. Exact symbolic verification scripts accompany the release. Historical identity of the preferred coordinates and external bibliographic priority remain open.
Author: Gallouin, David
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208717046