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

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.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-04
DOI
https://doi.org/10.5281/zenodo.22295156
Primary Topic
Advanced Algebra and Geometry
Type
preprint
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preprint

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

David Gallouin
Zenodo (CERN European Organization for Nuclear Research)
Advanced Algebra and Geometry
preprint

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

David Gallouin
preprint en

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.

Zenodo (CERN European Organization for Nuclear Research)
Reduced inequalities
Advanced Algebra and Geometry
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