Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.BoundPlaneDim

For a given natural number $n$, this proposition states that there exists an $n$-dimensional linear subspace (a "plane") within the space of charges for the 3-generation Standard Model with right-handed neutrinos such that every point in this subspace is a solution to the anomaly cancellation conditions (ACCs). Formally, there exists a set of $n$ linearly independent charge vectors $B_i \in \mathbb{Q}^{18}$ such that every rational linear combination $\sum_{i=1}^n f_i B_i$ satisfies the linear, quadratic, and cubic ACCs of the system `PlusU1 3`.

For any natural number $n$, if there exists an $n$-dimensional linear subspace (a "plane") within the 18-dimensional space of charges for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ symmetry such that every point in the subspace is a solution to the anomaly cancellation conditions, then there exists a set of $11 + n$ vectors in the space of charges $\mathbb{Q}^{18}$ that is linearly independent over $\mathbb{Q}$.

For the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry (SMRHN $U(1)$ system), if there exists an $n$-dimensional linear subspace (a "plane") within the 18-dimensional space of rational charges $\mathbb{Q}^{18}$ such that every vector in this subspace satisfies the linear, quadratic, and cubic anomaly cancellation conditions (ACCs), then the dimension of this subspace must satisfy $n \le 7$.