Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.Ordinary.FamilyMaps

For a natural number $n$, this defines a $\mathbb{Q}$-linear map from the space of linear solutions to the anomaly cancellation conditions for the 1-generation Standard Model (with right-handed neutrinos) to the space of linear solutions for the $n$-generation model. It acts by taking a solution $S \in (SM \ 1).\text{LinSols}$ and replicating its charges across all $n$ generations via the universal embedding $\text{familyUniversal}$. The resulting charge vector is shown to satisfy the three linear anomaly cancellation conditions (gravitational, $SU(2)$, and $SU(3)$) required to be an element of $(SM \ n).\text{LinSols}$.

For a natural number $n$ representing the number of fermion generations, the map `SMRHN.SM.familyUniversalQuad` embeds the space of 1-generation anomaly solutions into the space of $n$-generation solutions. Given a solution $S \in (SM \ 1).\text{QuadSols}$ (a set of charges for the six fermion species in one generation that satisfy the gravitational, $SU(2)$, and $SU(3)$ linear anomaly cancellation conditions), the map applies the universal embedding $\text{familyUniversal}_n$ to replicate these charges across all $n$ generations. The resulting charge vector is then proven to satisfy the corresponding linear anomaly cancellation conditions for the $n$-generation system, thus defining a valid element of $(SM \ n).\text{QuadSols}$. Since the quadratic sector of this system contains no equations, any charge vector satisfying the linear conditions is a solution to the quadratic sector.

For a natural number $n$, let $(SM \ 1).\text{Sols}$ and $(SM \ n).\text{Sols}$ be the spaces of anomaly-free charge assignments for the 1-generation and $n$-generation Standard Model with right-handed neutrinos (without hypercharge), respectively. This map takes a 1-generation anomaly-free solution $S$ and constructs an $n$-generation solution by applying the universal embedding $\text{familyUniversal}_n$ to its charge vector. This embedding replicates the charges of the six fermion species $(q_Q, q_u, q_d, q_L, q_e, q_\nu) \in \mathbb{Q}^6$ across all $n$ generations. The resulting $6n$-dimensional charge vector is proven to satisfy the $n$-generation linear (gravitational, $SU(2)$, and $SU(3)$) and cubic anomaly cancellation conditions, thus forming a valid element of $(SM \ n).\text{Sols}$.