Physlib.StringTheory.FTheory.SU5.Quanta.IsViable

A configuration of quanta $x$ in an $SU(5) \times U(1)$ F-theory model is **viable** if and only if it satisfies the following conditions: 1. **Completeness**: The charge spectrum is complete, meaning it contains $U(1)$ charges for the Higgs fields $H_d$ and $H_u$, and the sets of charges for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations are non-empty. 2. **Phenomenological Safety**: The charge spectrum does not allow gauge-invariant potential terms that are phenomenologically constrained (such as $\mu, \beta, \Lambda, W_i, K_i$ which lead to proton decay or R-parity violation) and no such terms are regenerated by a single insertion of Yukawa-related singlets. 3. **Geometric Consistency**: There exists a codimension-one fiber configuration $I \in \{\text{same, nearestNeighbor, nextToNearestNeighbor}\}$ such that the charges in $x$ are elements of the allowed charge sets $S_{\bar{\mathbf{5}}}(I)$ and $S_{\mathbf{10}}(I)$. 4. **Uniqueness of Matter Charges**: The multisets of $U(1)$ charges associated with the $\bar{\mathbf{5}}$ and $\mathbf{10}$ matter curves do not contain duplicate elements. 5. **Top Yukawa Coupling**: The charge spectrum allows for a gauge-invariant top-quark Yukawa coupling. 6. **Matter Content and Fluxes**: The fluxes associated with the $\bar{\mathbf{5}}$ and $\mathbf{10}$ sectors contain no zero entries and result in exactly three generations of Minimal Supersymmetric Standard Model (MSSM) matter fields ($Q, U, E, L, D$) with no chiral exotics. 7. **Linear Anomaly Cancellation**: The configuration satisfies the linear anomaly cancellation condition: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q$ are the $U(1)$ charges and $N$ are the multiplicities (fluxes) for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations.

A configuration of quanta $x$ in an $SU(5) \times U(1)$ F-theory model is **viable** ($IsViable \, x$) if and only if the following conditions are satisfied: 1. **Charge Viability**: There exists a codimension-one configuration $I$ such that the charge spectrum of $x$, denoted $x.\text{toCharges}$, is an element of the multiset $\text{viableCharges}(I)$. 2. **Charge Uniqueness**: The multisets of $U(1)$ charges associated with the $\bar{\mathbf{5}}$ (F) and $\mathbf{10}$ (T) representation matter curves do not contain duplicate entries (`Nodup`). 3. **Flux Consistency**: The fluxes associated with the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations contain no zero entries (`HasNoZero`) and result in exactly three generations of MSSM matter fields with no chiral exotics (`NoExotics`). 4. **Anomaly Cancellation**: The linear anomaly cancellation condition is satisfied: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q$ are the $U(1)$ charges and $N$ are the multiplicities (fluxes) for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations.

A configuration of quanta $x$ in an $SU(5) \times U(1)$ F-theory model is viable ($IsViable \, x$) if and only if the following three conditions are satisfied: 1. There exists a codimension-one configuration $I$ such that the charge spectrum of $x$, denoted $x.\text{toCharges}$, is an element of the multiset of viable charge spectra $\text{viableCharges}(I)$. This implies the spectrum permits a top Yukawa coupling and is phenomenologically safe. 2. $x$ belongs to the multiset of configurations obtained by lifting its charge spectrum, $\text{liftCharge}(x.\text{toCharges})$. This condition ensures that the matter curves associated with $x$ have no chiral exotics, no zero fluxes, and unique $U(1)$ charges. 3. The linear anomaly cancellation condition is satisfied: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q$ are the $U(1)$ charges and $N$ are the multiplicities (fluxes) for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations.

A configuration of quanta $x$ in an $SU(5) \times U(1)$ F-theory model is viable ($\text{IsViable } x$) if and only if the following three conditions are satisfied: 1. There exists a codimension-one configuration $I$ such that the $U(1)$ charge spectrum of $x$, denoted $x.\text{toCharges}$, is an element of the multiset of viable charge spectra $\text{viableCharges}(I)$ that satisfies the $\text{IsAnomalyFree}$ property. 2. $x$ belongs to the multiset of configurations $\text{liftCharge}(x.\text{toCharges})$ obtained by lifting its charge spectrum, ensuring no chiral exotics and no zero fluxes. 3. $x$ satisfies the linear anomaly cancellation condition: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q_{H_d}$ and $q_{H_u}$ are the $U(1)$ charges of the Higgs particles, $q_i, q_a$ are the charges of the $\overline{\mathbf{5}}$ and $\mathbf{10}$ representations, and $N_i, N_a$ are their respective multiplicities (fluxes).

The multiset of viable quanta, `viableElems`, is the collection of 36 specific configurations of the type `Quanta` that satisfy the physical viability predicate `IsViable` in the context of $SU(5)$ $F$-theory. A configuration is considered viable if it contains the Higgs fields $H_d$ and $H_u$, matter in the $\mathbf{5}$ and $\mathbf{10}$ representations, satisfies linear anomaly cancellation, and supports a top Yukawa coupling while avoiding exotic chiral particles and phenomenologically dangerous couplings. The multiset explicitly enumerates these valid configurations, including several duplicate entries as specified in the formal definition.

For any configuration of quanta $x$ in the $SU(5)$ F-theory model, if $x$ is an element of the multiset of viable quanta $\text{viableElems}$, then $x$ satisfies the physical viability predicate $\text{IsViable } x$.

In the $SU(5)$ F-theory model, a configuration of quanta $x$ satisfies the physical viability predicate $\text{IsViable } x$ if and only if $x$ is an element of the explicitly enumerated multiset of viable quanta $\text{viableElems}$.

In the context of an $SU(5)$ F-theory model, for any configuration of quanta $x$ that satisfies the physical viability predicate $\text{IsViable}(x)$, the associated charge spectrum of $x$ (denoted $\text{toCharges}(x)$) necessarily satisfies the condition $\text{YukawaGeneratesDangerousAtLevel}(x, 2)$. This means that the insertions of at most two Yukawa-related singlets are capable of regenerating a phenomenologically dangerous superpotential term. Mathematically, the intersection of the multiset of charges formed by summing up to two Yukawa-associated charges, $Y_2(x)$, and the multiset of phenomenologically constraining superpotential charges, $P(x)$, is non-empty: \[ Y_2(x) \cap P(x) \neq \emptyset \]

In the $SU(5)$ F-theory model, let $x$ be a configuration of quanta that is physically viable (i.e., satisfies the predicate $\text{IsViable } x$). Then $x$ satisfies the quartic anomaly cancellation condition $$q_{H_d}^2 - q_{H_u}^2 + \sum_{i} q_i^2 N_i + 3 \sum_{a} q_a^2 N_a = 0$$ if and only if $x$ is one of the following six configurations (defined by the Higgs charges $q_{H_d}, q_{H_u}$ and the sets of charge-flux pairs for the $\overline{\mathbf{5}}$ and $\mathbf{10}$ representations): 1. $q_{H_d} = 2, q_{H_u} = -2, \mathcal{F} = \{(-1, \langle 1, 2 \rangle), (1, \langle 2, -2 \rangle)\}, \mathcal{T} = \{(-1, \langle 3, 0 \rangle)\}$ 2. $q_{H_d} = 2, q_{H_u} = -2, \mathcal{F} = \{(-1, \langle 0, 2 \rangle), (1, \langle 3, -2 \rangle)\}, \mathcal{T} = \{(-1, \langle 3, 0 \rangle)\}$ 3. $q_{H_d} = -2, q_{H_u} = 2, \mathcal{F} = \{(-1, \langle 2, -2 \rangle), (1, \langle 1, 2 \rangle)\}, \mathcal{T} = \{(1, \langle 3, 0 \rangle)\}$ 4. $q_{H_d} = -2, q_{H_u} = 2, \mathcal{F} = \{(-1, \langle 3, -2 \rangle), (1, \langle 0, 2 \rangle)\}, \mathcal{T} = \{(1, \langle 3, 0 \rangle)\}$ 5. $q_{H_d} = 6, q_{H_u} = -14, \mathcal{F} = \{(-9, \langle 1, 2 \rangle), (1, \langle 2, -2 \rangle)\}, \mathcal{T} = \{(-7, \langle 3, 0 \rangle)\}$ 6. $q_{H_d} = 6, q_{H_u} = -14, \mathcal{F} = \{(-9, \langle 0, 2 \rangle), (1, \langle 3, -2 \rangle)\}, \mathcal{T} = \{(-7, \langle 3, 0 \rangle)\}$

For any viable configuration of quanta $x$ in an $SU(5)$ F-theory model with integer $U(1)$ charges, its charge spectrum reduced modulo 2 is one of the following three configurations in $\text{ChargeSpectrum}(\mathbb{Z}_2)$: 1. $\langle q_{H_d} = 0, q_{H_u} = 0, Q_{\bar{\mathbf{5}}} = \{1\}, Q_{\mathbf{10}} = \{1\} \rangle$ 2. $\langle q_{H_d} = 0, q_{H_u} = 0, Q_{\bar{\mathbf{5}}} = \{0, 1\}, Q_{\mathbf{10}} = \{1\} \rangle$ 3. $\langle q_{H_d} = 0, q_{H_u} = 0, Q_{\bar{\mathbf{5}}} = \{0, 1\}, Q_{\mathbf{10}} = \{0\} \rangle$ Here, $q_{H_d}$ and $q_{H_u}$ denote the $U(1)$ charges of the Higgs fields $H_d$ and $H_u$ (mapped to $\mathbb{Z}_2$), while $Q_{\bar{\mathbf{5}}}$ and $Q_{\mathbf{10}}$ denote the sets of $U(1)$ charges for the matter particles in the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations, respectively.