Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.Ordinary.DimSevenPlane

In the context of the 3-generation Standard Model with right-handed neutrinos, $B_0$ is a specific charge assignment (a vector in the charge space $\mathbb{Q}^{18}$) that serves as a basis element for a 7-dimensional plane. Under the identification of the charge space with a species-generation grid (where species index 0 corresponds to the left-handed quark doublet $Q$), $B_0$ assigns a charge of $1$ to the first generation of quark doublets ($Q_1$), a charge of $-1$ to the second generation of quark doublets ($Q_2$), and a charge of $0$ to all other fermion species across all generations.

In the context of the 3-generation Standard Model with right-handed neutrinos, let $V = \mathbb{Q}^{18}$ be the space of rational charges. Let $B_0 \in V$ be the basis charge assignment defined by assigning a charge of $1$ to the first-generation quark doublet ($Q_1$) and a charge of $-1$ to the second-generation quark doublet ($Q_2$), with all other fermion charges set to zero. Let $f$ denote the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). For any two charge assignments $S, T \in V$, the following identity holds: $$f(B_0, S, T) = 6 (S_0 T_0 - S_1 T_1)$$ where $S_0, T_0$ and $S_1, T_1$ are the charges of the first and second generation quark doublets in the configurations $S$ and $T$, respectively.

The charge assignment $B_1$ is a vector in the charge space of the 3-generation Standard Model with right-handed neutrinos, denoted as $(\text{SM } 3).\text{Charges}$. It serves as a basis element for a specific 7-dimensional plane where points satisfy the anomaly cancellation conditions. The vector is defined by assigning a charge of $1$ to the right-handed up-type quark ($u$) of the first generation and a charge of $-1$ to the right-handed up-type quark of the second generation, with all other fermion charges set to $0$. Using the species index $s \in \{0, \dots, 5\}$ and generation index $i \in \{0, 1, 2\}$, the assignment is: $$(B_1)_{s,i} = \begin{cases} 1 & \text{if } s=1, i=0 \\ -1 & \text{if } s=1, i=1 \\ 0 & \text{otherwise} \end{cases}$$ where $s=1$ corresponds to the right-handed up-type quark species.

In the context of a 3-generation Standard Model with right-handed neutrinos, let $V \cong \mathbb{Q}^{18}$ be the space of rational charges assigned to the fermions. Let $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. For the basis charge vector $B_1 \in V$ (which assigns a charge of $1$ to the first-generation right-handed up-type quark and $-1$ to the second-generation right-handed up-type quark, with all other charges being zero) and any two charge configurations $S, T \in V$, the trilinear form evaluates to: $$f(B_1, S, T) = 3(S_3 T_3 - S_4 T_4)$$ where $S_i$ and $T_i$ denote the $i$-th components of the charge vectors $S$ and $T$ respectively. The indices $3$ and $4$ correspond to the right-handed up-type quark species in the first and second generations.

The charge assignment $B_2$ is an element of the charge space $(\text{SM } 3).\text{Charges} \cong \mathbb{Q}^{18}$ for the 3-generation Standard Model with right-handed neutrinos. It is defined as one of the basis elements for a 7-dimensional plane of charges. Using the identification of the 18 total charges with 6 fermion species $s \in \{0, \dots, 5\}$ and 3 generations $i \in \{0, 1, 2\}$, $B_2$ is defined by: $$(s, i) \mapsto \begin{cases} 1 & \text{if } s = 2, i = 0 \\ -1 & \text{if } s = 2, i = 1 \\ 0 & \text{otherwise} \end{cases}$$ where the species index $s=2$ corresponds to the right-handed down-type quark $d$. In other words, $B_2$ assigns a charge of $1$ to the first generation of $d$ quarks and $-1$ to the second generation, with all other charges in the system being zero.

In the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. Let $B_2$ be the specific charge configuration (basis vector) that assigns a charge of $1$ to the first-generation right-handed down-type quark and $-1$ to the second-generation right-handed down-type quark, with all other charges being zero. For any two charge configurations $S$ and $T$ in the charge space $\mathbb{Q}^{18}$, the trilinear form satisfies: $$f(B_2, S, T) = 3 (S_6 T_6 - S_7 T_7)$$ where $S_i$ and $T_i$ denote the $i$-th components of the charge vectors $S$ and $T$, and the indices $6$ and $7$ correspond to the first and second generation right-handed down-type quarks, respectively.

In the 3-generation Standard Model with right-handed neutrinos, $B_3$ is a charge assignment (vector) in the space of charges $\mathbb{Q}^{18}$. It is defined such that the left-handed lepton doublet $L$ (species index 3) of the first generation (index 0) has a charge of $1$, and the left-handed lepton doublet $L$ of the second generation (index 1) has a charge of $-1$. All other fermion charges for all species and generations are zero.

Let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC) for the 3-generation Standard Model with right-handed neutrinos. For any two charge configurations $S, T \in \mathbb{Q}^{18}$, the value of the trilinear form evaluated at the basis vector $B_3$ and the vectors $S$ and $T$ is given by: $$f(B_3, S, T) = 2(S_9 T_9 - S_{10} T_{10})$$ where $S_i$ and $T_i$ denote the components of the charge vectors $S$ and $T$ at index $i \in \{0, \dots, 17\}$. Here, the indices 9 and 10 correspond to the charges of the left-handed lepton doublet ($L$) species for the first and second generations, respectively.

The charge vector $B_4$ is a basis element for a 7-dimensional plane in the charge space of the 3-generation Standard Model with right-handed neutrinos. This vector is defined by assigning a charge of $1$ to the right-handed charged lepton ($e$) of the first generation, a charge of $-1$ to the right-handed charged lepton of the second generation, and a charge of $0$ to all other fermion charges (including all generations of quarks, left-handed leptons, neutrinos, and the third generation of the right-handed charged lepton).

In the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). Let $B_4$ be the basis charge vector where the first-generation right-handed charged lepton has charge $1$, the second-generation right-handed charged lepton has charge $-1$, and all other fermion charges are zero. For any two charge configurations $S$ and $T$, the trilinear form evaluated at $(B_4, S, T)$ is: $$f(B_4, S, T) = S_{12} T_{12} - S_{13} T_{13}$$ where $S_{12}$ and $T_{12}$ are the charges of the first-generation right-handed charged leptons, and $S_{13}$ and $T_{13}$ are the charges of the second-generation right-handed charged leptons in the respective configurations.

The definition $B_5$ represents a specific charge assignment within the space of charges for the 3-generation Standard Model with right-handed neutrinos. It is defined such that the first-generation right-handed neutrino has charge $1$ and the second-generation right-handed neutrino has charge $-1$, while all other fermion charges (including all quarks, charged leptons, left-handed leptons, and the third-generation neutrino) are set to $0$. This assignment serves as one of the basis elements for a 7-dimensional plane of charges that satisfy the anomaly cancellation conditions.

For any two charge assignments $S, T$ in the charge space of the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. If $B_5$ is a basis charge assignment where the first-generation right-handed neutrino has charge $1$ and the second-generation right-handed neutrino has charge $-1$ (with all other fermion charges set to $0$), then the trilinear form evaluated at $B_5, S,$ and $T$ satisfies: $$f(B_5, S, T) = S_{15} T_{15} - S_{16} T_{16}$$ where $S_i$ and $T_i$ denote the rational charges of the $i$-th fermion representation in assignments $S$ and $T$, with indices 15 and 16 corresponding specifically to the first and second generation right-handed neutrinos.

The charge assignment $B_6$ is one of the basis elements for a seven-dimensional plane in the charge space of the three-generation Standard Model with right-handed neutrinos. It is defined as a vector in $(\text{SM } 3).\text{Charges} \cong \mathbb{Q}^{18}$ such that, under the identification of charges with species-generation pairs $(s, i) \in \{0, \dots, 5\} \times \{0, 1, 2\}$, we have: $$B_6(s, i) = \begin{cases} 1 & \text{if } s = 1, i = 2 \\ -1 & \text{if } s = 2, i = 2 \\ 0 & \text{otherwise} \end{cases}$$ Here, the species index $s=1$ corresponds to the right-handed up-type quark $u$, the index $s=2$ corresponds to the right-handed down-type quark $d$, and the generation index $i=2$ corresponds to the third generation.

In the context of the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). For the basis charge vector $B_6$ and any two charge configurations $S, T \in \mathbb{Q}^{18}$, the trilinear form satisfies the identity: $$f(B_6, S, T) = 3(S_5 T_5 - S_8 T_8)$$ where $S_i$ and $T_i$ denote the $i$-th components of the respective charge vectors. The indices 5 and 8 correspond to the charges of the third-generation right-handed up-type quark and the third-generation right-handed down-type quark, respectively.

The function $B$ maps an index $i \in \{0, 1, \dots, 6\}$ to a corresponding basis charge assignment $B_i$ in the charge space $(\text{SM } 3).\text{Charges} \cong \mathbb{Q}^{18}$ for the 3-generation Standard Model with right-handed neutrinos. Specifically, the mapping is defined as $B(i) = B_i$ for $i \in \{0, \dots, 6\}$, where the vectors $\{B_0, B_1, B_2, B_3, B_4, B_5, B_6\}$ form a basis for a 7-dimensional plane of charges where each point satisfies the anomaly cancellation conditions.

Consider the 3-generation Standard Model with right-handed neutrinos and let $V \cong \mathbb{Q}^{18}$ be its charge space. Let $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). Let $B_0, B_1, \dots, B_6$ be the basis vectors defined for the 7-dimensional plane in this charge space. For any index $i \in \{1, 2, \dots, 6\}$ and any charge configuration $S \in V$, the trilinear form satisfies the identity: $$f(B_0, B_i, S) = 0$$

In the 3-generation Standard Model with right-handed neutrinos, let $V$ be the space of rational fermion charges and $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors for the 7-dimensional plane of charges defined in this system. For any index $i \in \{0, 1, \dots, 6\}$ such that $1 \neq i$, and for any charge configuration $S \in V$, the trilinear form satisfies $f(B_1, B_i, S) = 0$.

In the 3-generation Standard Model with right-handed neutrinos, let $V$ be the charge space and $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors for a specific 7-dimensional plane in $V$. For any index $i \in \{0, 1, \dots, 6\}$ such that $i \neq 2$ and for any charge configuration $S \in V$, the trilinear form satisfies $f(B_2, B_i, S) = 0$.

Let $V \cong \mathbb{Q}^{18}$ be the charge space of the 3-generation Standard Model with right-handed neutrinos, and let $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors of a specific 7-dimensional plane in $V$. For any index $i \in \{0, \dots, 6\}$ such that $i \neq 3$, and for any charge vector $S \in V$, the trilinear form evaluated at $B_3$, $B_i$, and $S$ is zero: $$f(B_3, B_i, S) = 0$$

In the context of the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors for the specific 7-dimensional plane of charges defined in this system. For any index $i \in \{0, 1, \dots, 6\}$ such that $i \neq 4$, and for any arbitrary charge configuration $S$, the trilinear form evaluated at $B_4, B_i$, and $S$ is zero, i.e., $f(B_4, B_i, S) = 0$.

In the context of the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors of a specific 7-dimensional plane in the charge space. For any index $i \in \{0, \dots, 6\}$ such that $i \neq 5$, and for any charge assignment $S$, the trilinear form satisfies: $$f(B_5, B_i, S) = 0$$

In the context of the 3-generation Standard Model with right-handed neutrinos, let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC), and let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors for the 7-dimensional plane in the charge space. For any index $i \in \{0, 1, \dots, 6\}$ such that $i \neq 6$ and for any charge configuration $S$, it holds that $f(B_6, B_i, S) = 0$.

In the 3-generation Standard Model with right-handed neutrinos, let $V$ be the space of rational fermion charges and $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors for the 7-dimensional plane of charges defined in this system. For any distinct indices $i, j \in \{0, 1, \dots, 6\}$ (where $i \neq j$) and for any charge configuration $S \in V$, the trilinear form satisfies: $$f(B_i, B_j, S) = 0$$

Let $f$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC) for the 3-generation Standard Model with right-handed neutrinos. Let $\{B_i\}_{i=0}^6$ be the basis vectors defining a 7-dimensional plane within the rational charge space $\mathbb{Q}^{18}$. For any two indices $i, j \in \{0, 1, \dots, 6\}$, the trilinear form evaluated at the elements $B_i, B_i$, and $B_j$ vanishes: $$f(B_i, B_i, B_j) = 0$$

In the 3-generation Standard Model with right-handed neutrinos, let $V$ be the space of rational fermion charges and $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC). Let $\{B_0, B_1, \dots, B_6\}$ be the basis vectors for the 7-dimensional plane defined in this system. For any indices $i, j, k \in \{0, 1, \dots, 6\}$, the trilinear form evaluated at these basis vectors is zero: $$f(B_i, B_j, B_k) = 0$$

In the 3-generation Standard Model with right-handed neutrinos, let $\mathcal{A}_{\text{cube}}$ be the cubic anomaly cancellation condition (ACC) map and $\{B_i\}_{i=0}^6$ be the basis vectors for the 7-dimensional plane of charges. For any set of rational coefficients $f_i \in \mathbb{Q}$, the cubic ACC vanishes for the linear combination of these basis vectors: $$\mathcal{A}_{\text{cube}} \left( \sum_{i=0}^6 f_i B_i \right) = 0$$

In the 3-generation Standard Model with right-handed neutrinos, let $\{B_i\}_{i=0}^6$ be the basis vectors defined for the 7-dimensional plane of charges. For any set of rational coefficients $f_i \in \mathbb{Q}$, the resulting linear combination of charges $S = \sum_{i=0}^6 f_i B_i$ is a solution to the full set of anomaly cancellation conditions (including the gravitational, $SU(2)$, $SU(3)$, and cubic anomalies).

In the 3-generation Standard Model with right-handed neutrinos, the set of seven basis vectors $B = \{B_0, B_1, B_2, B_3, B_4, B_5, B_6\}$ in the charge space $(\text{SM } 3).\text{Charges} \cong \mathbb{Q}^{18}$ is linearly independent over the field of rational numbers $\mathbb{Q}$.

In the 3-generation Standard Model with right-handed neutrinos, there exists a set of seven charge assignment vectors $\{B_0, B_1, \dots, B_6\} \in (\text{SM } 3).\text{Charges}$ that are linearly independent over $\mathbb{Q}$, such that every linear combination $S = \sum_{i=0}^6 f_i B_i$ (where $f_i \in \mathbb{Q}$) is a solution to the full set of anomaly cancellation conditions (ACCs).