Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.OrthogY3B3.ToSols

For a charge assignment $R$ in the space of anomaly-free assignments perpendicular to $Y_3$ and $B_3$ (denoted as `AnomalyFreePerp`), the property holds if the conditions $\alpha_1(R) = 0$, $\alpha_2(R) = 0$, and $\alpha_3(R) = 0$ are all satisfied. Geometrically, these conditions ensure that the "quad line" in the plane spanned by $R, Y_3$, and $B_3$ lies within the cubic anomaly hypersurface, and the "cube line" lies within the quadratic anomaly hypersurface.

For a charge assignment $R$ in the space of anomaly-free assignments perpendicular to $Y_3$ and $B_3$ (denoted as `AnomalyFreePerp`), the property $\text{LineEqProp}(R)$ is decidable. This property holds if the conditions $\alpha_1(R) = 0$, $\alpha_2(R) = 0$, and $\alpha_3(R) = 0$ are all satisfied.

For an anomaly-free solution $R \in \mathbb{Q}^{20}$ of the Minimal Supersymmetric Standard Model (MSSM) anomaly cancellation system, the property `LineEqPropSol` is the condition that the following equality holds: $$f(R, R, Y_3) B(B_3, R) - f(R, R, B_3) B(Y_3, R) = 0$$ where $f$ is the symmetric trilinear form `cubeTriLin` associated with the cubic anomaly condition, $B$ is the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly condition, and $Y_3, B_3$ are the specific anomaly-free charge vectors defined for the MSSM system. This condition is equivalent to the projection of $R$ onto the subspace orthogonal to $Y_3$ and $B_3$ satisfying the `LineEqProp` property.

Given an anomaly-free solution $T$ to the Minimal Supersymmetric Standard Model (MSSM) anomaly cancellation conditions, this function computes a rational number defined by the formula: $$(Y_3 \cdot B_3) \alpha_3(\text{proj}(T))$$ where $\cdot$ denotes the dot product on the space of MSSM charges, $Y_3$ and $B_3$ are specific anomaly-free solutions, $\text{proj}(T)$ is the projection of $T$ onto the subspace orthogonal to $Y_3$ and $B_3$, and $\alpha_3$ is a specific function defined on this orthogonal subspace. This rational value appears in the definition of the map `toSolNS` acting on solutions, and it being equal to $0$ is equivalent to the solution $T$ satisfying the property `LineEqPropSol`.

Let $T$ be an anomaly-free solution to the MSSM anomaly cancellation conditions. Then $T$ satisfies the condition $f(T, T, Y_3) B(B_3, T) = f(T, T, B_3) B(Y_3, T)$ if and only if the line equation coefficient of $T$, defined as $(Y_3 \cdot B_3) \alpha_3(\text{proj}(T))$, is equal to zero. Here, $f$ is the symmetric trilinear form corresponding to the cubic anomaly, $B$ is the symmetric bilinear form corresponding to the quadratic anomaly, and $\text{proj}(T)$ is the projection of $T$ onto the subspace orthogonal to the anomaly-free solutions $Y_3$ and $B_3$.

Let $R$ be an anomaly-free solution of the MSSM anomaly cancellation system. Then $R$ satisfies the property $\text{LineEqPropSol } R$ (which is defined by the equality $f(R, R, Y_3) B(B_3, R) = f(R, R, B_3) B(Y_3, R)$) if and only if its projection onto the subspace orthogonal to $Y_3$ and $B_3$, denoted by $\text{proj}(R)$, satisfies the property $\text{LineEqProp}$. The property $\text{LineEqProp}$ holds for the projection if the conditions $\alpha_1(\text{proj}(R)) = \alpha_2(\text{proj}(R)) = \alpha_3(\text{proj}(R)) = 0$ are all satisfied.

For a charge assignment $R$ perpendicular to $Y_3$ and $B_3$ (an element of `MSSMACC.AnomalyFreePerp`), this proposition represents the condition that the plane spanned by $R$, $Y_3$, and $B_3$ lies entirely in the quadratic surface. Mathematically, it is defined as the conjunction of three equations: \[ B(R, R) = 0 \land B(Y_3, R) = 0 \land B(B_3, R) = 0 \] where $B$ is the symmetric bilinear form associated with the quadratic anomaly cancellation condition (`quadBiLin`), and $Y_3$ and $B_3$ are the specific anomaly-free solutions for the MSSM.

For a charge assignment $R$ perpendicular to the solutions $Y_3$ and $B_3$ (an element of `MSSMACC.AnomalyFreePerp`), the proposition $\text{InQuadProp}(R)$ is decidable. This proposition requires that $R$ satisfies the conditions $B(R, R) = 0$, $B(Y_3, R) = 0$, and $B(B_3, R) = 0$, where $B$ is the symmetric bilinear form associated with the quadratic anomaly cancellation condition (`quadBiLin`).

For an anomaly-free solution $R \in \text{MSSMACC.Sols}$, this defines a proposition that evaluates to true if the plane spanned by the solutions $R$, $Y_3$, and $B_3$ lies entirely in the quadratic surface. Mathematically, this condition requires that $R$ is orthogonal to both $Y_3$ and $B_3$ with respect to the symmetric bilinear form $B$ (represented by `quadBiLin`) associated with the MSSM quadratic anomaly cancellation condition. This is explicitly expressed as the logical conjunction $B(Y_3, R) = 0 \land B(B_3, R) = 0$.

For an anomaly-free solution $T \in \text{MSSMACC.Sols}$, the quadratic coefficient is defined as the rational number: \[ 2 (Y_3 \cdot B_3)^2 (B(Y_3, T)^2 + B(B_3, T)^2) \] where $Y_3$ and $B_3$ are specific anomaly-free solutions, $\cdot$ denotes the dot product on the charge space $\mathbb{Q}^{20}$, and $B$ is the symmetric bilinear form associated with the quadratic anomaly cancellation condition. This value is zero if and only if $T$ is orthogonal to both $Y_3$ and $B_3$ with respect to $B$ (i.e., $B(Y_3, T) = 0$ and $B(B_3, T) = 0$).

For an anomaly-free solution $T$ in the MSSM anomaly cancellation system, the condition `InQuadSolProp T` (which states that $T$ is orthogonal to the solutions $Y_3$ and $B_3$ with respect to the quadratic symmetric bilinear form $B$, i.e., $B(Y_3, T) = 0$ and $B(B_3, T) = 0$) holds if and only if the quadratic coefficient `quadCoeff T` is zero, where $$\text{quadCoeff}(T) = 2 (Y_3 \cdot B_3)^2 (B(Y_3, T)^2 + B(B_3, T)^2)$$ and $\cdot$ denotes the dot product on the rational charge space $\mathbb{Q}^{20}$.

Let $R$ be an anomaly-free solution to the MSSM anomaly cancellation conditions. Let $B$ be the symmetric bilinear form associated with the quadratic anomaly condition (`quadBiLin`), and let $\text{proj}(R)$ be the projection of the linear component of $R$ onto the subspace orthogonal to the fixed charge vectors $Y_3$ and $B_3$ with respect to the dot product. The condition `InQuadSolProp R`, which states that $B(Y_3, R) = 0$ and $B(B_3, R) = 0$, holds if and only if the condition `InQuadProp (proj R)` holds, which states that $B(\text{proj}(R), \text{proj}(R)) = 0$, $B(Y_3, \text{proj}(R)) = 0$, and $B(B_3, \text{proj}(R)) = 0$.

Let $f: V \times V \times V \to \mathbb{Q}$ be the symmetric trilinear form `cubeTriLin` representing the cubic anomaly cancellation condition for the MSSM. For a charge assignment $R$ that is perpendicular to the specific solutions $Y_3$ and $B_3$, the property `InCubeProp` is satisfied if the following three conditions hold: $$f(R, R, R) = 0, \quad f(R, R, B_3) = 0, \quad \text{and} \quad f(R, R, Y_3) = 0$$ This property indicates that the entire plane spanned by the vectors $R, Y_3,$ and $B_3$ is contained within the cubic surface defined by the equation $f(X, X, X) = 0$.

For a charge assignment $R$ belonging to the subspace of anomaly-free charges perpendicular to the reference solutions $Y_3$ and $B_3$ (denoted as `MSSMACC.AnomalyFreePerp`), the property $\text{InCubeProp}(R)$ is decidable. This property holds if the symmetric trilinear form $f$ associated with the cubic anomaly cancellation condition satisfies the following three equations in $\mathbb{Q}$: $$f(R, R, R) = 0, \quad f(R, R, B_3) = 0, \quad \text{and} \quad f(R, R, Y_3) = 0$$ Decidability ensures that there is an algorithmic procedure to determine whether a given $R$ satisfies these conditions.

For an anomaly-free solution $R$ in the MSSM charge space $V \cong \mathbb{Q}^{20}$, the property `InCubeSolProp R` is satisfied if the symmetric trilinear form $f$ (associated with the cubic anomaly cancellation condition) satisfies the following two equations: $$f(R, R, B_3) = 0 \quad \text{and} \quad f(R, R, Y_3) = 0$$ where $B_3$ and $Y_3$ are specific reference solutions. This condition ensures that the entire plane spanned by the charge vectors $R, B_3,$ and $Y_3$ lies within the cubic surface defined by $f(X, X, X) = 0$.

For an anomaly-free solution $T$ in the MSSM charge space $\mathbb{Q}^{20}$, the rational number $\text{cubicCoeff}(T)$ is defined as: $$3 (Y_3 \cdot B_3)^3 \left( c(T, T, Y_3)^2 + c(T, T, B_3)^2 \right)$$ where $Y_3$ and $B_3$ are specific fixed anomaly-free charge assignments, $\cdot$ denotes the dot product on the vector space of charges, and $c(S_1, S_2, S_3)$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition. This coefficient is zero if and only if the solution $T$ satisfies the property `InCubeSolProp`.

For any anomaly-free solution $T$ in the MSSM charge space ($T \in \text{MSSMACC.Sols}$), the property `InCubeSolProp T` holds if and only if the cubic coefficient $\text{cubicCoeff}(T)$ is equal to zero. Here, `InCubeSolProp T` is the property that the symmetric trilinear form $f$ (associated with the cubic anomaly cancellation condition) satisfies $f(T, T, B_3) = 0$ and $f(T, T, Y_3) = 0$, where $B_3$ and $Y_3$ are specific reference anomaly-free solutions. The cubic coefficient is defined as: $$\text{cubicCoeff}(T) = 3 (Y_3 \cdot B_3)^3 \left( f(T, T, Y_3)^2 + f(T, T, B_3)^2 \right)$$ where $\cdot$ denotes the dot product on the vector space of charges.

Let $R$ be an anomaly-free solution for the MSSM anomaly cancellation system ($R \in \text{MSSMACC.Sols}$). Let $\text{proj}(R)$ be the projection of the linear components of $R$ onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$. The property `InCubeSolProp R`, which states that the symmetric trilinear form $f$ associated with the cubic anomaly condition satisfies $f(R, R, B_3) = 0$ and $f(R, R, Y_3) = 0$, holds if and only if the property `InCubeProp` holds for $\text{proj}(R)$. Specifically, `InCubeProp (proj R)` means that $f$ evaluated on the projection satisfies $f(\text{proj}(R), \text{proj}(R), \text{proj}(R)) = 0$, $f(\text{proj}(R), \text{proj}(R), B_3) = 0$, and $f(\text{proj}(R), \text{proj}(R), Y_3) = 0$.

The type `InLineEq` represents the set of charge assignments $R$ in the space `AnomalyFreePerp` (charge assignments perpendicular to the hypercharge $Y_3$ and baryon number $B_3$) that satisfy the property `LineEqProp R`. This property is defined by the condition $\alpha_1(R) = \alpha_2(R) = \alpha_3(R) = 0$. Geometrically, these conditions ensure that the "quad line" in the plane spanned by $R, Y_3$, and $B_3$ lies within the cubic anomaly hypersurface, and the "cube line" lies within the quadratic anomaly hypersurface.

The type `InQuad` represents the set of charge assignments $R$ in the space `AnomalyFreePerp` (those perpendicular to the hypercharge $Y_3$ and baryon number $B_3$) that satisfy both the linear conditions $\alpha_1(R) = \alpha_2(R) = \alpha_3(R) = 0$ and the quadratic conditions $B(R, R) = 0$, $B(Y_3, R) = 0$, and $B(B_3, R) = 0$, where $B$ is the symmetric bilinear form associated with the quadratic anomaly cancellation condition. Geometrically, these conditions ensure that the plane spanned by $R$, $Y_3$, and $B_3$ lies entirely within the quadratic anomaly surface and satisfies specific constraints relative to the cubic anomaly hypersurface.

The type `InQuadCube` is the subtype of charge assignments $R$ in the space `AnomalyFreePerp` (charge assignments perpendicular to $Y_3$ and $B_3$) that satisfy the linear conditions $\alpha_i(R) = 0$ for $i \in \{1, 2, 3\}$, the quadratic conditions $B(R, R) = B(Y_3, R) = B(B_3, R) = 0$, and the cubic conditions $f(R, R, R) = f(R, R, B_3) = f(R, R, Y_3) = 0$. Geometrically, this represents the set of vectors $R$ such that the entire plane spanned by $\{R, Y_3, B_3\}$ is contained within both the quadratic anomaly surface (defined by the bilinear form $B$) and the cubic anomaly hypersurface (defined by the trilinear form $f$).

The type `NotInLineEqSol` represents the set of anomaly-free solutions $R$ for the MSSM anomaly cancellation system that do not satisfy the property: $$f(R, R, Y_3) B(B_3, R) = f(R, R, B_3) B(Y_3, R)$$ where $f$ is the symmetric trilinear form associated with the cubic anomaly condition, $B$ is the symmetric bilinear form associated with the quadratic anomaly condition, and $Y_3, B_3$ are specific anomaly-free charge vectors.

The type `MSSMACC.AnomalyFreePerp.InLineEqSol` represents the set of anomaly-free solutions $R \in \text{MSSMACC.Sols}$ that satisfy the condition $$f(R, R, Y_3) B(B_3, R) = f(R, R, B_3) B(Y_3, R)$$ but do not satisfy the condition that the plane spanned by $R$, $Y_3$, and $B_3$ lies in the quadratic surface (i.e., it is not the case that $B(Y_3, R) = 0$ and $B(B_3, R) = 0$ simultaneously). Here, $f$ denotes the symmetric trilinear form associated with the cubic anomaly condition, $B$ denotes the symmetric bilinear form associated with the quadratic anomaly condition, and $Y_3, B_3$ are specific basis charge vectors.

The type `MSSMACC.AnomalyFreePerp.InQuadSol` represents the set of anomaly-free solutions $R \in \text{MSSMACC.Sols}$ for the MSSM system that satisfy the properties `LineEqPropSol R` and `InQuadSolProp R`, but do not satisfy `InCubeSolProp R`. A solution $R$ belongs to this type if: 1. It satisfies the quadratic plane condition: $B(Y_3, R) = 0$ and $B(B_3, R) = 0$, where $B$ is the symmetric bilinear form associated with the quadratic anomaly condition, and $Y_3, B_3$ are the reference charge vectors. 2. It satisfies the linear equation property: $f(R, R, Y_3) B(B_3, R) = f(R, R, B_3) B(Y_3, R)$, where $f$ is the symmetric trilinear form associated with the cubic anomaly condition. 3. It does not satisfy the cubic plane condition, meaning it is not the case that $f(R, R, B_3) = 0$ and $f(R, R, Y_3) = 0$ both hold.

The type `MSSMACC.AnomalyFreePerp.InQuadCubeSol` is the subtype of anomaly-free solutions $R \in \text{MSSMACC.Sols}$ that simultaneously satisfy three conditions: 1. **Line equality property**: The condition `LineEqPropSol R` holds, meaning $f(R, R, Y_3) B(B_3, R) = f(R, R, B_3) B(Y_3, R)$. 2. **Quadratic plane property**: The condition `InQuadSolProp R` holds, meaning the plane spanned by $\{R, Y_3, B_3\}$ lies within the quadratic anomaly surface ($B(Y_3, R) = 0$ and $B(B_3, R) = 0$). 3. **Cubic plane property**: The condition `InCubeSolProp R` holds, meaning the plane spanned by $\{R, Y_3, B_3\}$ lies within the cubic anomaly surface ($f(R, R, Y_3) = 0$ and $f(R, R, B_3) = 0$). Here, $B$ is the symmetric bilinear form associated with the quadratic anomaly cancellation condition, $f$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition, and $Y_3, B_3$ are specific reference charge vectors.

Given an anomaly-free charge assignment $R$ perpendicular to $Y_3$ and $B_3$ (represented by the type `MSSMACC.AnomalyFreePerp`), this definition produces a quadratic solution to the anomaly cancellation conditions for the MSSM (represented by the type `MSSMACC.QuadSols`).

For any anomaly-free charge assignment $R$ that is perpendicular to the specific solutions $Y_3$ and $B_3$, let $S = \text{toSolNSQuad}(R)$ be the resulting charge vector constructed to satisfy the linear and quadratic anomaly cancellation conditions. This theorem states that $S$ also satisfies the cubic anomaly cancellation condition, i.e., $\text{accCube}(S) = 0$.

For any anomaly-free charge assignment $R$ that is perpendicular to the solutions $Y_3$ and $B_3$ (represented by $R \in \text{MSSMACC.AnomalyFreePerp}$), the charge vector of the quadratic solution derived from $R$, denoted as $(\text{toSolNSQuad}(R))_1$, is equal to the charge vector constructed by the function $\text{planeY}_3\text{B}_3$ using $R$ and the rational coefficients $\alpha_1(R)$, $\alpha_2(R)$, and $\alpha_3(R)$.

The function `toSolNS` maps a quadruple $(R, a, b, c) \in \text{AnomalyFreePerp} \times \mathbb{Q} \times \mathbb{Q} \times \mathbb{Q}$ to a solution in the MSSM anomaly cancellation system $\text{MSSMACC.Sols}$. This solution is constructed by taking the quadratic solution $S = \text{toSolNSQuad}(R)$ (which is proven to satisfy the cubic anomaly cancellation condition by `toSolNSQuad_cube`) and scaling it by the rational factor $a$ via the scalar action of $\mathbb{Q}$ on the space of solutions. Note that the components $b$ and $c$ are ignored in this specific mapping.

Given an anomaly-free solution $T$ in the MSSM anomaly cancellation system, this function maps $T$ to a quadruple $(R, c_1, c_2, c_3)$ in the product space of $\text{AnomalyFreePerp}$ and $\mathbb{Q}^3$. The mapping is defined as: $$( \text{proj}(T), (\text{lineEqCoeff}(T))^{-1}, 0, 0 )$$ where: - $\text{proj}(T)$ is the projection of the solution's charge vector onto the subspace of charges orthogonal to the hypercharge $Y_3$ and baryon number $B_3$. - $\text{lineEqCoeff}(T)$ is a rational coefficient computed from $T$ and its projection. - The multiplicative inverse $(\cdot)^{-1}$ in $\mathbb{Q}$ follows the convention that $0^{-1} = 0$. This map acts as a right inverse to the construction `toSolNS` for solutions where $\text{lineEqCoeff}(T) \neq 0$.

Let $T$ be an anomaly-free solution of the MSSM anomaly cancellation system that does not satisfy the line equation property ($T \in \text{NotInLineEqSol}$). Let $\text{toSolNSProj}$ be the projection mapping $T \mapsto (\text{proj}(T), \text{lineEqCoeff}(T)^{-1}, 0, 0)$. Then applying the construction map $\text{toSolNS}$ to this projection recovers the original solution $T$.

Given an element $R$ of the type `InLineEq` (which represents charge assignments perpendicular to the hypercharge $Y_3$ and baryon number $B_3$ satisfying $\alpha_1(R) = \alpha_2(R) = \alpha_3(R) = 0$) and three rational coefficients $c_1, c_2, c_3 \in \mathbb{Q}$, this function constructs a full solution to the anomaly cancellation conditions (ACC). The function specifically maps the input $(R, c_1, c_2, c_3)$ to the charge assignment defined by the quadratic construction $\text{lineQuad}(R, c_1, c_2, c_3)$. Because $R$ satisfies the `InLineEq` property, the resulting point is mathematically guaranteed to satisfy the cubic anomaly condition $\text{accCube} = 0$, thereby forming a complete solution in `MSSMACC.Sols`.

Given an anomaly-free solution $T \in \text{InLineEqSol}$ for the MSSM, this function maps $T$ to a quadruple $(R, c_1, c_2, c_3)$ consisting of a charge assignment $R \in \text{InLineEq}$ and three rational coefficients $c_1, c_2, c_3 \in \mathbb{Q}$. The components are defined as follows: 1. The assignment $R = \text{proj}(T)$ is the projection of the charge vector $T$ onto the subspace orthogonal to $Y_3$ and $B_3$ with respect to the dot product. 2. The first coefficient is $c_1 = Q(T)^{-1} \cdot B(B_3, T)$. 3. The second coefficient is $c_2 = Q(T)^{-1} \cdot (-B(Y_3, T))$. 4. The third coefficient is $c_3 = Q(T)^{-1} \cdot [B(B_3, T)(B_3 \cdot T - Y_3 \cdot T) - B(Y_3, T)(Y_3 \cdot T - 2(B_3 \cdot T))]$. where $Q(T)$ is the quadratic coefficient `quadCoeff T`, $B$ is the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly condition, and $\cdot$ denotes the dot product `dot` on the MSSM charge space. This map serves as a right-inverse to the solution construction map `inLineEqToSol`.

For any charge assignment $R$ in the space $\text{InLineEq}$ (the subtype of charge assignments perpendicular to the hypercharge $Y_3$ and baryon number $B_3$ satisfying the conditions $\alpha_1(R) = \alpha_2(R) = \alpha_3(R) = 0$) and for any rational numbers $c_1, c_2, c_3, d \in \mathbb{Q}$, the map $\text{inLineEqToSol}$ that constructs a solution to the MSSM anomaly cancellation conditions satisfies the following homogeneity property: $$\text{inLineEqToSol}(R, d \cdot c_1, d \cdot c_2, d \cdot c_3) = d \cdot \text{inLineEqToSol}(R, c_1, c_2, c_3)$$ where the dot on the right-hand side represents the scalar action of $\mathbb{Q}$ on the space of solutions $\text{MSSMACC.Sols}$.

For any anomaly-free solution $T$ in the set $\text{InLineEqSol}$ for the MSSM, the composition of the projection map $\text{inLineEqProj}$ and the solution construction map $\text{inLineEqToSol}$ recovers the original charge assignment of $T$. That is, $\text{inLineEqToSol}(\text{inLineEqProj}(T)) = T.\text{val}$.

Let `InQuad` be the subtype of charge assignments $R$ perpendicular to the hypercharge $Y_3$ and baryon number $B_3$ that satisfy the specific quadratic conditions $B(R, R) = 0$, $B(Y_3, R) = 0$, and $B(B_3, R) = 0$, as well as the linear conditions $\alpha_i(R) = 0$. The function `inQuadToSol` maps a 4-tuple consisting of such a charge assignment $R \in \text{InQuad}$ and three rational coefficients $(a_1, a_2, a_3) \in \mathbb{Q}^3$ to a full solution of the MSSM anomaly cancellation conditions (ACCs). This solution is constructed by taking the linear combination $a_1 R + a_2 Y_3 + a_3 B_3$ (referred to as `lineCube`) and verifying that it satisfies both the quadratic and cubic anomaly conditions.

Let $R \in \text{InQuad}$ be a charge assignment perpendicular to $Y_3$ and $B_3$ satisfying the quadratic conditions $B(R, R) = B(Y_3, R) = B(B_3, R) = 0$ and the linear conditions $\alpha_i(R) = 0$. For any rational coefficients $c_1, c_2, c_3 \in \mathbb{Q}$ and any rational scalar $d \in \mathbb{Q}$, the map $\text{inQuadToSol}$ to the MSSM anomaly cancellation solutions satisfies the homogeneity property: $$\text{inQuadToSol}(R, d \cdot c_1, d \cdot c_2, d \cdot c_3) = d \cdot \text{inQuadToSol}(R, c_1, c_2, c_3)$$ where the scalar multiplication on the right-hand side is the natural action of $\mathbb{Q}$ on the space of solutions $\text{MSSMACC.Sols}$.

For an anomaly-free solution $T$ in the subtype `InQuadSol` (meaning $T$ satisfies the quadratic plane condition and the specific linear relationship between the cubic and quadratic forms, but does not lie entirely in the cubic surface), the function `inQuadProj` maps $T$ to a quadruple $(R, c_1, c_2, c_3) \in \text{InQuad} \times \mathbb{Q} \times \mathbb{Q} \times \mathbb{Q}$. The components are defined as follows: 1. $R = \text{proj}(T)$ is the projection of the charge vector $T$ onto the subspace orthogonal to $Y_3$ and $B_3$. 2. The rational coefficients $(c_1, c_2, c_3)$ are given by: $$c_1 = \frac{f(T, T, B_3)}{\text{cubicCoeff}(T)}$$ $$c_2 = \frac{-f(T, T, Y_3)}{\text{cubicCoeff}(T)}$$ $$c_3 = \frac{f(T, T, B_3) \cdot (B_3 \cdot T - Y_3 \cdot T) - f(T, T, Y_3) \cdot (Y_3 \cdot T - 2 B_3 \cdot T)}{\text{cubicCoeff}(T)}$$ where $f$ is the symmetric trilinear form `cubeTriLin`, $\cdot$ is the symmetric bilinear dot product on the MSSM charge space, and $\text{cubicCoeff}(T)$ is the normalizing rational factor. This map acts as a right-inverse to the solution construction map `inQuadToSol`.

For any anomaly-free solution $T$ in the subtype `InQuadSol` (those satisfying the quadratic plane condition and a specific linear relationship between cubic and quadratic forms, but not lying entirely in the cubic surface), applying the solution construction map `inQuadToSol` to the result of the projection map `inQuadProj(T)` recovers the original solution $T$. Specifically, if $T$ is projected to a quadruple $(R, c_1, c_2, c_3)$ where $R = \text{proj}(T)$ and $c_1, c_2, c_3$ are rational coefficients derived from the trilinear form $f$, then the linear combination $c_1 R + c_2 Y_3 + c_3 B_3$ is equal to $T$.

Let $R$ be a charge assignment in the subtype `InQuadCube`, meaning that the plane spanned by $\{R, Y_3, B_3\}$ lies entirely within both the quadratic and cubic anomaly surfaces of the MSSM anomaly cancellation system. Given $R$ and rational coefficients $b_1, b_2, b_3 \in \mathbb{Q}$, the function `inQuadCubeToSol` produces a full anomaly-free solution $S \in \text{MSSMACC.Sols}$. The solution is constructed by taking a linear combination of the vectors in the plane: $$S = b_1 R + b_2 Y_3 + b_3 B_3$$ where $Y_3$ is the hypercharge solution and $B_3$ is the baryon number minus lepton number solution. Since $R \in \text{InQuadCube}$, any such linear combination is guaranteed to satisfy the quadratic condition $\text{accQuad}(S) = 0$ and the cubic condition $\text{accCube}(S) = 0$ in addition to the four linear anomaly conditions.

Let $R \in \text{InQuadCube}$ be a charge assignment such that the plane spanned by $\{R, Y_3, B_3\}$ lies in both the quadratic and cubic anomaly surfaces. For any rational coefficients $c_1, c_2, c_3 \in \mathbb{Q}$ and any scalar $d \in \mathbb{Q}$, the map `inQuadCubeToSol` satisfies: $$\text{inQuadCubeToSol}(R, d c_1, d c_2, d c_3) = d \cdot \text{inQuadCubeToSol}(R, c_1, c_2, c_3)$$ where the map `inQuadCubeToSol` constructs a solution to the anomaly cancellation conditions via the linear combination $c_1 R + c_2 Y_3 + c_3 B_3$.

Let $T$ be an anomaly-free solution in the subspace `InQuadCubeSol`, meaning $T$ is a solution to the MSSM anomaly cancellation conditions such that the plane spanned by $\{T, Y_3, B_3\}$ lies within both the quadratic and cubic anomaly surfaces. The map `inQuadCubeProj` sends $T$ to a quadruple $(R, c_1, c_2, c_3) \in \text{InQuadCube} \times \mathbb{Q} \times \mathbb{Q} \times \mathbb{Q}$. The first component $R$ is the projection of $T$ onto the subspace of charges orthogonal to $Y_3$ and $B_3$, defined as: $$R = (B_3 \cdot T - Y_3 \cdot T) Y_3 + (Y_3 \cdot T - 2 (B_3 \cdot T)) B_3 + (Y_3 \cdot B_3) T$$ The scalar components are given by: $$c_1 = (Y_3 \cdot B_3)^{-1} (Y_3 \cdot T - B_3 \cdot T)$$ $$c_2 = (Y_3 \cdot B_3)^{-1} (2(B_3 \cdot T) - Y_3 \cdot T)$$ $$c_3 = (Y_3 \cdot B_3)^{-1}$$ where $\cdot$ denotes the symmetric bilinear form `MSSMACC.dot`. This map provides a right-inverse to the map `inQuadCubeToSol` on the specified subtype.

Let $T$ be an anomaly-free solution in the subspace `InQuadCubeSol`, meaning that the plane spanned by $\{T, Y_3, B_3\}$ lies within both the quadratic and cubic anomaly surfaces of the MSSM. Then, applying the map `inQuadCubeToSol` to the projection of $T$ given by `inQuadCubeProj` recovers the original charge assignment $T$, i.e., $\text{inQuadCubeToSol}(\text{inQuadCubeProj}(T)) = T$.

The function `toSol` maps a quadruple $(R, a, b, c)$ consisting of a charge assignment $R \in \text{MSSMACC.AnomalyFreePerp}$ (charges perpendicular to $Y_3$ and $B_3$) and three rational coefficients $a, b, c \in \mathbb{Q}$ to a complete solution in the space of MSSM anomaly cancellation solutions $\text{MSSMACC.Sols}$. The mapping is defined by a case-wise construction based on the geometric properties of the plane $\text{span}\{R, Y_3, B_3\}$: 1. If $R$ satisfies `LineEqProp`, `InQuadProp`, and `InCubeProp` (meaning the plane $\text{span}\{R, Y_3, B_3\}$ lies entirely within both the quadratic and cubic anomaly surfaces), the solution is given by `inQuadCubeToSol`, taking the form $a R + b Y_3 + c B_3$. 2. Otherwise, if $R$ satisfies `LineEqProp` and `InQuadProp` (meaning the plane lies in the quadratic surface and satisfies specific cubic constraints), the solution is given by `inQuadToSol`. 3. Otherwise, if $R$ satisfies only `LineEqProp`, the solution is given by `inLineEqToSol`. 4. In all other cases (where the standard linear constraints for $R$ in the plane do not hold), the solution is constructed using the non-standard map `toSolNS`. This map is designed to be a surjection from $\text{MSSMACC.AnomalyFreePerp} \times \mathbb{Q}^3$ onto the space of solutions $\text{MSSMACC.Sols}$.