Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.MinimalSuperSet

Given finite sets of possible charges $S_5, S_{10} \subset \mathcal{Z}$ and a charge spectrum $x = (q_{H_d}, q_{H_u}, Q_5, Q_{10})$, the **minimal super set** of $x$ is the collection of all charge spectra $y$ obtained by adding exactly one charge from the allowed sets to a component of $x$ that does not already contain it. Specifically, it is the union of the following sets, excluding $x$ itself: 1. If $x.q_{H_d}$ is empty, the set of spectra where $q_{H_d}$ is set to some $c \in S_5$ and all other components remain as in $x$. 2. If $x.q_{H_u}$ is empty, the set of spectra where $q_{H_u}$ is set to some $c \in S_5$ and all other components remain as in $x$. 3. The set of spectra where $Q_5$ is replaced by $Q_5 \cup \{c\}$ for some $c \in S_5 \setminus x.Q_5$. 4. The set of spectra where $Q_{10}$ is replaced by $Q_{10} \cup \{c\}$ for some $c \in S_{10} \setminus x.Q_{10}$.

For any finite sets of charges $S_5, S_{10} \subset \mathcal{Z}$ and any $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$, if $y$ is an element of the minimal super set of $x$ relative to $S_5$ and $S_{10}$, then $x$ is a subset of $y$ ($x \subseteq y$).

Let $S_5$ and $S_{10}$ be finite sets of charges in $\mathcal{Z}$. For any $SU(5)$ charge spectrum $x$, the spectrum $x$ is not a member of its own minimal super set relative to $S_5$ and $S_{10}$, denoted as $x \notin \text{minimalSuperSet}(S_5, S_{10}, x)$.

Let $S_5$ and $S_{10}$ be finite sets of charges in $\mathcal{Z}$. For any charge spectra $x$ and $y$, if $y$ is a member of the minimal super set of $x$ relative to $S_5$ and $S_{10}$, then $x \neq y$.

Let $\mathcal{Z}$ be a type of charges, and let $S_5, S_{10} \subset \mathcal{Z}$ be finite sets of allowed charges. For any $SU(5)$ charge spectrum $x$, if a spectrum $y$ is a member of the minimal super set of $x$ relative to $S_5$ and $S_{10}$, then the cardinality of $y$ is exactly one greater than the cardinality of $x$: $$\text{card}(y) = \text{card}(x) + 1$$ where the cardinality $\text{card}(x)$ is defined as the total number of charges present across all components ($q_{H_u}, q_{H_d}, Q_5, Q_{10}$) of the spectrum.

Let $S_5$ and $S_{10}$ be finite sets of charges in $\mathcal{Z}$ and let $x = (q_{H_d}, q_{H_u}, Q_5, Q_{10})$ be an $SU(5)$ charge spectrum. For any charge $z \in S_5$, if $z \notin Q_5$, then the charge spectrum $(q_{H_d}, q_{H_u}, Q_5 \cup \{z\}, Q_{10})$ is a member of the minimal super set of $x$ relative to $S_5$ and $S_{10}$.

Let $S_5$ and $S_{10}$ be finite sets of allowed charges in a type $\mathcal{Z}$. For any $SU(5)$ charge spectrum $x = (q_{H_d}, q_{H_u}, Q_5, Q_{10})$, if $z$ is a charge such that $z \in S_{10}$ and $z \notin Q_{10}$, then the charge spectrum obtained by adding $z$ to the set $Q_{10}$—specifically the spectrum $(q_{H_d}, q_{H_u}, Q_5, Q_{10} \cup \{z\})$—is a member of the minimal super set of $x$ relative to $S_5$ and $S_{10}$.

Let $S_5, S_{10} \subset \mathcal{Z}$ be finite sets of allowed charges. Suppose $x$ is an $SU(5)$ charge spectrum where the down-type Higgs charge component $q_{H_d}$ is empty (represented as `none`), such that $x = (\text{none}, q_{H_u}, Q_5, Q_{10})$. For any charge $z \in S_5$, the charge spectrum $y = (z, q_{H_u}, Q_5, Q_{10})$, which is identical to $x$ except that its $q_{H_d}$ component is set to $z$, is a member of the minimal super set of $x$ relative to $S_5$ and $S_{10}$.

Let $S_5, S_{10} \subset \mathcal{Z}$ be finite sets of allowed charges. Suppose $x$ is an $SU(5)$ charge spectrum where the up-type Higgs charge component $q_{H_u}$ is empty (represented as `none`), such that $x = (q_{H_d}, \text{none}, Q_5, Q_{10})$. For any charge $z \in S_5$, the charge spectrum $y = (q_{H_d}, z, Q_5, Q_{10})$, which is identical to $x$ except that its $q_{H_u}$ component is set to $z$, is a member of the minimal super set of $x$ relative to $S_5$ and $S_{10}$.

Let $S_5, S_{10} \subset \mathcal{Z}$ be finite sets of allowed charges. Let $x$ and $y$ be $SU(5)$ charge spectra such that $y$ belongs to the set of spectra whose charges are contained in $S_5$ and $S_{10}$ (i.e., $y \in \text{ofFinset}(S_5, S_{10})$). If $x$ is a proper subset of $y$ ($x \subseteq y$ and $x \neq y$), then there exists a charge spectrum $z$ in the minimal super set of $x$ relative to $S_5$ and $S_{10}$ such that $z \subseteq y$.

Let $S_5$ and $S_{10}$ be finite sets of charges in $\mathcal{Z}$, and let $p$ be a property defined on $SU(5)$ charge spectra. Suppose that for any spectrum $x$, if $p(x)$ holds, then $p(z)$ holds for every $z$ in the minimal super set of $x$ relative to $S_5$ and $S_{10}$ (i.e., $z \in \text{minimalSuperSet}(S_5, S_{10}, x)$). If $x$ and $y$ are charge spectra such that $x \subseteq y$, $p(x)$ is true, and $y$ is a spectrum whose charges are all contained in $S_5$ and $S_{10}$, then $p(y)$ is true. This result is proved by induction on the cardinality difference $n = \text{card}(y) - \text{card}(x)$, where the cardinality $\text{card}(x)$ is the total number of charges across all components of the spectrum.

Let $T$ be a multiset of $SU(5)$ charge spectra, $S_5$ and $S_{10}$ be finite sets of charges, and $p$ be a predicate on charge spectra. Suppose the following conditions hold: 1. Every charge spectrum $x \in T$ is complete (i.e., it possesses $q_{H_d}$, $q_{H_u}$, and non-empty sets of charges $Q_{\bar{5}}$ and $Q_{10}$). 2. For every $q_{10} \in S_{10}$, any spectrum $y$ formed by adding $q_{10}$ to the $Q_{10}$ component of some $x \in T$ satisfies the property that if $y \notin T$, then $p(y)$ is false. 3. For every $q_5 \in S_5$, any spectrum $y$ formed by adding $q_5$ to the $Q_{\bar{5}}$ component of some $x \in T$ satisfies the property that if $y \notin T$, then $p(y)$ is false. Then, for any $x \in T$ and any $y$ in the minimal super set of $x$ relative to $S_5$ and $S_{10}$, if $y \notin T$, then $p(y)$ is false.

Let $T$ be a multiset of $SU(5)$ charge spectra and let $S_5, S_{10}$ be finite sets of charges. Let $p$ be a predicate on charge spectra such that for any two spectra $x$ and $y$, $x \subseteq y$ and $\neg p(x)$ implies $\neg p(y)$ (equivalently, the property $p$ is downward-closed under the subset relation). Suppose the following conditions hold: 1. Every charge spectrum $x \in T$ is complete. 2. For every $q_{10} \in S_{10}$, any spectrum $y'$ formed by adding $q_{10}$ to the $Q_{10}$ component of some $x \in T$ satisfies $y' \notin T \implies \neg p(y')$. 3. For every $q_5 \in S_5$, any spectrum $y'$ formed by adding $q_5$ to the $Q_{\bar{5}}$ component of some $x \in T$ satisfies $y' \notin T \implies \neg p(y')$. Then, for any $x \in T$ and any $y$ with charges in $S_5$ and $S_{10}$ such that $x \subseteq y$, if $n = \text{card}(y) - \text{card}(x)$, then $y \notin T$ implies $\neg p(y)$.

Let $T$ be a multiset of $SU(5)$ charge spectra and let $S_5, S_{10}$ be finite sets of charges. Let $p$ be a predicate on charge spectra that is downward-closed under the subset relation, such that for any two spectra $x$ and $y$, $x \subseteq y$ and $\neg p(x)$ implies $\neg p(y)$. Suppose the following conditions hold: 1. Every spectrum $x \in T$ is complete (i.e., it contains $q_{H_d}$, $q_{H_u}$, and non-empty sets $Q_{\bar{5}}$ and $Q_{10}$). 2. For every charge $q_{10} \in S_{10}$ and every $x \in T$, the spectrum $y'$ formed by inserting $q_{10}$ into the $Q_{10}$ component of $x$ satisfies $y' \notin T \implies \neg p(y')$. 3. For every charge $q_5 \in S_5$ and every $x \in T$, the spectrum $y'$ formed by inserting $q_5$ into the $Q_{\bar{5}}$ component of $x$ satisfies $y' \notin T \implies \neg p(y')$. Then, for any $x \in T$ and any spectrum $y$ whose charges are contained in $S_5$ and $S_{10}$, if $x \subseteq y$ and $y \notin T$, then $\neg p(y)$.