Recognition Physics Institute

Independent research · Austin

A proposal to study inverse Galois over the rationals through rational points on four-branch Hurwitz curves. The argument separates the covering construction from the rational-point question.

Inverse Galois over the rationals as a rational point on a four-branch curve

Jonathan Washburn · 17 August 2026

A finite group with trivial center is the Galois group of a regular extension of ℚ(t) if and only if, for some number of branch points, the inner Hurwitz space has a rational point. That is Fried and Voelklein. Hilbert irreducibility then specializes to infinitely many realizations over ℚ. The abelian architecture has no such descent.

The moduli space of r branch points has dimension r − 3. Three branch points is rigidity. Five or more is expected to be empty of a general method. Four branch points is a curve. For groups with no rigid triple, inverse Galois over ℚ is a rational point on a rationally defined four-branch inner component. That sentence is derived. It is not a theorem that every such curve has a point, and it is not a construction of one.

The moduli space of r branch points has dimension r minus 3. At three branch points it is a point, rational as soon as the class data is, and that is where the Monster was realized. At four it is a curve, so a rational point is a Diophantine question controlled by genus. At five or more the components are of general type for large r. Below, a regular realization over the rational function field descends to the rationals by Hilbert irreducibility, and a realization over the maximal abelian extension has no such engine.

Dimension is what changes; the question stays the same one, whether a rationally defined component has a rational point.

Two architectures

Realizing a group over the maximal abelian extension of ℚ, even completely, does not reach ℚ. Only a regular realization over ℚ(t) has a descent engine.

The dimension ladder

Three branch points is where essentially every unconditional realization of a simple group over ℚ lives, the Monster included.

Falsifier

A finite group with trivial center, no rigid triple, and a rationally defined four-branch inner component proved to have no rational point; or a proof that some finite group is not a Galois group over ℚ; or a descent from the maximal abelian extension to ℚ that preserves the group.

The compiler on this site aims a quadratic step over a number field. It is a different object from a rational point on a Hurwitz space.