Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.PlaneNonSols

The vector $B_0$ is a charge assignment in the 18-dimensional rational vector space $\mathbb{Q}^{18}$ corresponding to the 3-generation Standard Model with right-handed neutrinos. It is defined as the unit vector $(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$, which assigns a charge of $1$ to the first generation left-handed quark doublet $Q_0$ and a charge of $0$ to all other fermions. This vector serves as one of the basis elements for an 11-dimensional plane in the space of charges.

The vector $B_1$ is a basis element for the 11-dimensional plane of charges in the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. It is an element of the rational vector space $\mathbb{Q}^{18}$, specifically defined as the vector where the charge of the first-generation right-handed up-type quark $u_0$ is 1, and all other 17 charges are 0: $$B_1 = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$$

The charge assignment $B_2 \in \mathbb{Q}^{18}$ is defined as the vector $(0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$. In the context of the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, this vector represents a configuration where the right-handed down-type quark of the first generation ($d_0$) has charge $1$, and all other $17$ fermions have charge $0$. This vector serves as a basis element for an eleven-dimensional plane in the charge space.

The charge assignment $B_3$ is a vector in the space of charges $\mathbb{Q}^{18}$ for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. It serves as one of the basis elements for an 11-dimensional plane in the charge space where no non-trivial solutions to the anomaly cancellation conditions (ACCs) exist. Specifically, $B_3$ is defined as the unit vector where the charge of the second-generation left-handed quark doublet $Q_1$ is $1$, and all other charges are $0$: $$B_3 = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) \in \mathbb{Q}^{18}$$

The charge assignment $B_4$ is a specific vector in the 18-dimensional vector space of charges $\mathbb{Q}^{18}$ for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. It is defined as the vector $B_4 = (0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) \in \mathbb{Q}^{18}$, where the eighth component is $1$ and all other components are $0$. This vector serves as one of the basis elements for an 11-dimensional plane in the space of charges.

The definition $B_5$ represents a specific charge assignment vector in the space of charges $\mathbb{Q}^{18}$ for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. It is defined as the coordinate vector: $$B_5 = (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) \in \mathbb{Q}^{18}$$ This vector serves as one of the basis elements used to construct an 11-dimensional plane in the charge space on which there are no non-trivial solutions to the anomaly cancellation conditions.

The charge assignment $B_6 \in \mathbb{Q}^{18}$ is a basis vector used to construct an 11-dimensional plane in the charge space of the 3-generation Standard Model with right-handed neutrinos. It is defined as the vector where all charge components are zero except for the 13th component, which is 1: $$B_6 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)$$ In the context of the ordered fermion charges $(Q_0, u_0, d_0, L_0, e_0, \nu_0, \dots, Q_2, u_2, d_2, L_2, e_2, \nu_2)$, this vector corresponds to assigning a unit charge to the third-generation left-handed quark doublet $Q_2$ and zero to all other fermions.

This definition specifies a charge assignment $B_7$ within the 18-dimensional vector space of charges $\mathbb{Q}^{18}$ for the 3-generation Standard Model with right-handed neutrinos. It is defined as the vector: $$B_7 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0)$$ This vector serves as one of the basis elements for an 11-dimensional plane in the charge space on which no non-trivial solutions to the anomaly cancellation conditions exist.

The charge vector $B_8$ is an element of the 18-dimensional rational vector space $\mathbb{Q}^{18}$, representing the charge assignments for the 3-generation Standard Model with right-handed neutrinos and an extra $U(1)$ symmetry. This specific vector is one of the basis elements used to construct an 11-dimensional plane of charges. It is defined as the configuration where the charge of the right-handed down-type quark in the third generation ($d_2$) is 1, and all other charges are 0.

The charge assignment $B_9$ is a vector in the space of charges $\mathbb{Q}^{18}$ for the three-generation Standard Model with right-handed neutrinos. It is defined as the vector $(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0)$, where the charge of the third generation lepton doublet is $L_2 = 1$ and the charge of the third generation charged lepton is $e_2 = 2$, with all other fermion charges being zero. This vector serves as one of the basis elements for the 11-dimensional plane of charges on which no non-trivial solutions to the anomaly cancellation conditions exist.

For the anomaly cancellation system of the Standard Model with three generations ($n=3$) of fermions and right-handed neutrinos, the charge space is the vector space $\mathbb{Q}^{18}$. The charge assignment $B_{10}$ is defined as the basis vector $(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1) \in \mathbb{Q}^{18}$. This corresponds to the physical state where the right-handed neutrino of the third generation, $\nu_2$, is assigned a charge of $1$, while all other fermion charges are $0$. This vector is one of the eleven basis elements used to construct a specific 11-dimensional plane in the charge space.

For the anomaly cancellation system of the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, this function defines a sequence of eleven charge assignments $B_i \in \mathbb{Q}^{18}$ for $i \in \{0, 1, \dots, 10\}$. These vectors $\{B_0, B_1, \dots, B_{10}\}$ serve as a basis for an 11-dimensional plane in the rational vector space of charges. This specific plane is constructed to show that no non-trivial solutions to the anomaly cancellation conditions exist within it.

In the context of the 3-generation Standard Model with right-handed neutrinos, let $\{B_k\}_{k=0}^{10}$ be the basis vectors for the specific 11-dimensional plane of charges in the rational vector space $\mathbb{Q}^{18}$. For any two distinct indices $i, j \in \{0, 1, \dots, 10\}$ where $i \neq j$, the symmetric bilinear form $B$ associated with the quadratic anomaly cancellation conditions satisfies $B(B_i, B_j) = 0$.

In the context of the 3-generation Standard Model with right-handed neutrinos, let $\{B_k\}_{k=0}^{10}$ be the basis vectors for the 11-dimensional plane in the charge space $\mathbb{Q}^{18}$. For any index $i \in \{0, \dots, 10\}$ and any sequence of rational coefficients $f_k \in \mathbb{Q}$, the symmetric bilinear form $B$ associated with the quadratic anomaly cancellation conditions satisfies: $$B(B_i, \sum_{k=0}^{10} f_k B_k) = f_i B(B_i, B_i)$$ where $B(S, T)$ represents the symmetric bilinear map `quadBiLin`.

This definition specifies a sequence of 11 rational coefficients $q_i \in \mathbb{Q}$ (for $i \in \{0, \dots, 10\}$) used in a quadratic form defined over an 11-dimensional plane in the charge space. The coefficients are defined such that $q_i = 1$ for $0 \le i \le 8$ and $q_i = 0$ for $i = 9$ and $i = 10$. In the context of a basis $\{B_i\}$, these correspond to the values of the bilinear form $\text{quadBiLin}(B_i, B_i)$.

In the context of the 11-dimensional plane in the charge space for the 3-generation Standard Model with right-handed neutrinos, for any index $i \in \{0, 1, \dots, 10\}$, the predefined quadratic coefficient $q_i$ is equal to the value of the symmetric bilinear form $B$ evaluated on the basis vector $B_i$ with itself, that is, $q_i = B(B_i, B_i)$.

In the context of the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, the quadratic anomaly cancellation condition $accQuad$ evaluated on the linear combination of these basis vectors satisfies: $$accQuad\left(\sum_{i=0}^{10} f_i B_i\right) = \sum_{i=0}^{10} q_i f_i^2$$ where $q_i$ are the predefined quadratic coefficients associated with the basis vectors $B_i$, which correspond to the values of the symmetric bilinear form $B(B_i, B_i)$.

In the context of the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the charge configuration $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then for every index $k \in \{0, \dots, 10\}$, the product of the quadratic coefficient $q_k$ and the square of the coordinate $f_k$ is zero: $$q_k f_k^2 = 0$$ where $q_k$ are the predefined quadratic coefficients associated with the basis vectors $B_k$.

In the context of the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a linear combination $S = \sum_{i=0}^{10} f_i B_i$ of these basis vectors with rational coefficients $f_i \in \mathbb{Q}$ is a solution to the anomaly cancellation conditions, then the first coefficient $f_0$ must be zero.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of a specific 11-dimensional plane in the rational charge space. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the charge configuration $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the coefficient $f_1$ must be zero: $$f_1 = 0$$

In the anomaly cancellation system of the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the linear combination $\sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the coefficient $f_2$ must be equal to zero, i.e., $f_2 = 0$.

In the anomaly cancellation system of the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_0, B_1, \dots, B_{10}\}$ be the basis vectors for the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any set of rational coefficients $f_i \in \mathbb{Q}$, if the linear combination $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the coefficient $f_3$ must be zero.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the linear combination $\sum_{i=0}^{10} f_i B_i$ satisfies the anomaly cancellation conditions, then the coefficient $f_4$ must be equal to zero, i.e., $f_4 = 0$.

Consider the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. Let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a charge configuration $S = \sum_{i=0}^{10} f_i B_i$, where $f_i \in \mathbb{Q}$, is a solution to the anomaly cancellation conditions, then the coefficient $f_5$ must be zero.

Consider the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. Let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the charge assignment $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the coefficient $f_6$ must be zero.

Consider the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. Let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a charge assignment $S = \sum_{i=0}^{10} f_i B_i$, where $f_i \in \mathbb{Q}$, is a solution to the anomaly cancellation conditions, then the coefficient $f_7$ must be zero.

Consider the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry. Let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a charge assignment $S = \sum_{i=0}^{10} f_i B_i$, where $f_i \in \mathbb{Q}$, is a solution to the anomaly cancellation conditions, then the coefficient $f_8$ must be zero.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the charge assignment $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the sum reduces to the last two components: $$\sum_{i=0}^{10} f_i B_i = f_9 B_9 + f_{10} B_{10}$$

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any rational coefficients $f_i \in \mathbb{Q}$, if the charge assignment $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the coefficients of the last two basis vectors must satisfy the linear relation $f_{10} = -3 f_9$.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any sequence of rational coefficients $f_i \in \mathbb{Q}$, if the charge assignment $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then the sum reduces to a linear combination of the last two basis vectors involving only the coefficient $f_9$: $$\sum_{i=0}^{10} f_i B_i = f_9 B_9 + (-3 f_9) B_{10}$$

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a charge assignment $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions for some rational coefficients $f_i \in \mathbb{Q}$, then the coefficient $f_9$ must be zero.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a charge assignment $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions for some rational coefficients $f_i \in \mathbb{Q}$, then the coefficient $f_{10}$ must be zero.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. If a charge assignment $S = \sum_{i=0}^{10} f_i B_i$ (where $f_i \in \mathbb{Q}$) is a solution to the anomaly cancellation conditions, then for every $k \in \{0, 1, \dots, 10\}$, the coefficient $f_k$ must be zero.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, let $\{B_i\}_{i=0}^{10}$ be the basis vectors of the 11-dimensional plane in the rational charge space $\mathbb{Q}^{18}$. For any rational coefficients $f_i \in \mathbb{Q}$, if the linear combination $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then $S$ must be the zero vector.

The set of eleven basis vectors $\{B_i\}_{i=0}^{10}$ in the charge space of the 3-generation Standard Model with right-handed neutrinos (and an additional $U(1)$ gauge symmetry) is linearly independent over the rational numbers $\mathbb{Q}$.

In the anomaly cancellation system for the 3-generation Standard Model with right-handed neutrinos and an additional $U(1)$ gauge symmetry, there exists a set of eleven basis vectors $\{B_i\}_{i=0}^{10}$ in the rational charge space $\mathbb{Q}^{18}$ that are linearly independent over $\mathbb{Q}$, such that for any rational coefficients $f_i \in \mathbb{Q}$, if the linear combination $S = \sum_{i=0}^{10} f_i B_i$ is a solution to the anomaly cancellation conditions, then $S$ must be the zero vector.