Physlib.StringTheory.FTheory.SU5.Quanta.FiveQuanta

The function extracts the underlying multiset of $SU(5)$ flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ from a `FiveQuanta` object $x$. It maps each quantum in the collection $x$ to its flux component, where $M$ represents the chirality flux and $N$ represents the hypercharge flux.

The function extracts the underlying multiset of charges $z \in \mathcal{Z}$ from a `FiveQuanta` object $x$. It maps each quantum in the collection $x$ to its charge component, representing the $U(1)$ charge of the 5-bar representation.

For a given `FiveQuanta` object $x$, which represents a collection of quanta consisting of $U(1)$ charges $z \in \mathcal{Z}$ and $SU(5)$ fluxes $\phi \in \text{Fluxes}$, this function defines a map from the set of charges $\mathcal{Z}$ to the additive monoid of fluxes. For any charge $z$, the function returns the sum of all fluxes $\phi$ associated with that specific charge within $x$. If a charge $z$ is not present in $x$, the map returns the identity flux $(0,0)$.

For any `FiveQuanta` object $x$ and any $U(1)$ charge $z \in \mathcal{Z}$, if $z$ is not an element of the multiset of charges associated with $x$ (denoted as $x.\text{toCharges}$), then the total flux associated with that charge in the collection (given by $x.\text{toChargeMap}(z)$) is equal to the zero flux $(0, 0)$.

Given a collection of quanta $x$ representing 5-bar representations of $SU(5) \times U(1)$, where each element is a pair consisting of a $U(1)$ charge $z \in \mathcal{Z}$ and a flux $f \in \text{Fluxes}$, the function `reduce` returns a new collection of quanta. This resulting collection contains only unique charges $z$, where the flux associated with each $z$ is the sum $\sum f_i$ of all fluxes that were associated with that specific charge in the original collection $x$.

Let $x$ be a collection of quanta representing 5-bar representations of $SU(5) \times U(1)$. The reduction of $x$, denoted as $x.\text{reduce}$, is the collection obtained by summing all fluxes associated with each distinct $U(1)$ charge $z \in \mathcal{Z}$. The theorem states that the resulting collection $x.\text{reduce}$ contains no duplicate elements.

For any collection of quanta $x$ representing 5-bar representations of $SU(5) \times U(1)$, let $x.\text{reduce}$ be the reduced collection obtained by summing all fluxes associated with each unique $U(1)$ charge. The deduplication of this reduced collection is equal to the reduced collection itself, i.e., $\text{dedup}(x.\text{reduce}) = x.\text{reduce}$.

For any collection of quanta $x$ representing 5-bar representations of $SU(5) \times U(1)$ with charges $z \in \mathcal{Z}$, the multiset of charges obtained from the reduced collection $x.\text{reduce}$ is equal to the deduplicated multiset of charges from the original collection $x$.

For any collection of quanta $x$ representing 5-bar representations of $SU(5) \times U(1)$ and any pair $p = (z, f)$ consisting of a $U(1)$ charge $z \in \mathcal{Z}$ and a flux $f \in \text{Fluxes}$, the pair $p$ is an element of the reduced collection $x.\text{reduce}$ if and only if the charge $z$ is present in the multiset of charges $x.\text{toCharges}$ and the flux $f$ is equal to the sum of all fluxes $f_i$ in $x$ that are associated with the charge $z$.

Let $x$ be a collection of quanta (a `FiveQuanta` object) representing 5-bar representations of $SU(5) \times U(1)$, where each element is a pair $(z, f)$ consisting of a $U(1)$ charge $z \in \mathcal{Z}$ and a flux $f \in \text{Fluxes}$. Let $x.\text{reduce}$ be the reduction of $x$ obtained by grouping elements by charge and summing their corresponding fluxes. For any charge $q \in \mathcal{Z}$ that is present in the underlying multiset of charges of $x$, filtering the collection $x.\text{reduce}$ for the charge $q$ results in a singleton set containing the pair $(q, \Phi)$, where $\Phi = \sum_{(z, f) \in x, z=q} f$ is the sum of all fluxes associated with the charge $q$ in the original collection $x$.

Let $x$ be a collection of quanta (a `FiveQuanta` object) representing 5-bar representations of $SU(5) \times U(1)$, where each element is a pair $(z, f)$ consisting of a $U(1)$ charge $z \in \mathcal{Z}$ and a flux $f \in \text{Fluxes}$. Let $x.\text{reduce}$ be the reduction of $x$ obtained by grouping elements by their $U(1)$ charges and summing the corresponding fluxes. The theorem states that the reduction operation is idempotent: $x.\text{reduce}.\text{reduce} = x.\text{reduce}$.

Let $\mathcal{Z}$ be the set of $U(1)$ charges and $x$ be a collection of quanta (a multiset of pairs $(z, \phi)$ where $z \in \mathcal{Z}$ and $\phi \in \text{Fluxes}$). Let $M$ be an additive commutative monoid, and let $f$ be a mapping that assigns to each charge $z \in \mathcal{Z}$ an additive homomorphism $f_z: \text{Fluxes} \to M$. The sum of $f_z(\phi)$ evaluated over all pairs in the collection $x$ is equal to the sum of $f_z(\phi')$ evaluated over the reduced collection $x_{\text{reduce}}$, where $x_{\text{reduce}}$ is obtained by consolidating $x$ such that all fluxes associated with the same charge are summed together.

For any collection of quanta $x$ representing 5-bar representations of $SU(5) \times U(1)$, if the multiset of its underlying $U(1)$ charges $x.\text{toCharges}$ contains no duplicate elements, then the reduction of $x$ (the operation that sums fluxes for identical charges) is equal to $x$ itself.

Let $x$ be a collection of quanta (a `FiveQuanta` object) representing 5-bar representations of $SU(5) \times U(1)$, where each quantum is a pair consisting of a $U(1)$ charge $z \in \mathcal{Z}$ and an $SU(5)$ flux $f \in \text{Fluxes}$. The function $x.\text{toChargeMap}: \mathcal{Z} \to \text{Fluxes}$, which assigns to each charge $z$ the sum of all fluxes associated with it in $x$, is invariant under the reduction operation $x.\text{reduce}$ (the operation that consolidates the collection by summing fluxes for identical charges). That is, $x.\text{reduce}.\text{toChargeMap} = x.\text{toChargeMap}$.

For a collection of quanta $F$ representing 5-bar representations of $SU(5) \times U(1)$, let $F.\text{toFluxesFive}$ be the multiset of its associated $SU(5)$ fluxes. If a flux $f$ is an element of the multiset of fluxes belonging to the reduced collection $F.\text{reduce}$ (the collection obtained by summing all fluxes associated with the same $U(1)$ charge), then $f$ is equal to the sum of some sub-multiset of the original fluxes $F.\text{toFluxesFive}$.

Let $F$ be a collection of quanta for the $\mathbf{\bar{5}}$ representations of $SU(5) \times U(1)$, and let $F.\text{toFluxesFive}$ be its underlying multiset of fluxes. If $f$ is a flux in the reduced collection $F.\text{reduce}$ (the collection where all fluxes sharing the same $U(1)$ charge are summed together), then $f$ is equal to the sum of some non-empty sub-multiset of $F.\text{toFluxesFive}$.

Let $F$ be a collection of quanta representing the $\mathbf{\bar{5}}$ representations of $SU(5) \times U(1)$. If the underlying multiset of $SU(5)$ fluxes, $F.\text{toFluxesFive}$, is an element of the set of flux configurations that satisfy the conditions for no chiral exotics and no zero fluxes ($\texttt{FluxesFive.elemsNoExotics}$), then the total number of chiral $L = (1, 2)_{-1/2}$ representations in the reduced collection $F.\text{reduce}$ is 3. The reduced collection $F.\text{reduce}$ is obtained by summing all $SU(5)$ fluxes associated with the same $U(1)$ charge.

Let $F$ be a collection of quanta for the $\mathbf{\bar{5}}$ (5-bar) representations of $SU(5) \times U(1)$, and let $F.\text{toFluxesFive}$ be the multiset of its associated $SU(5)$ flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$. If $F.\text{toFluxesFive}$ belongs to the set of flux configurations that have no chiral exotics and no zero fluxes (`FluxesFive.elemsNoExotics`), then the total number of anti-chiral $L = (1,2)_{-1/2}$ representations in the reduced collection $F.\text{reduce}$ is $0$.

Let $F$ be a collection of quanta representing $\mathbf{\bar{5}}$ (5-bar) matter curves in an $SU(5) \times U(1)$ F-theory model. Suppose the underlying multiset of $SU(5)$ fluxes, $F.\text{toFluxesFive}$, belongs to the set of configurations that satisfy the "no exotics" condition (`elemsNoExotics`), meaning the original configuration corresponds to exactly 3 generations of chiral matter. If we reduce the collection $F$ by summing the fluxes associated with each unique $U(1)$ charge to obtain $F.\text{reduce}$, then the total number of chiral $D = (\bar{3}, 1)_{1/3}$ representations in the resulting reduced flux multiset is equal to 3.

Let $F$ be a collection of quanta representing $\mathbf{\bar{5}}$ representations of $SU(5) \times U(1)$. Suppose the multiset of its underlying $SU(5)$ fluxes, $F.\text{toFluxesFive}$, belongs to the set of configurations with no chiral exotics, $\text{FluxesFive.elemsNoExotics}$. Then, for the reduced collection $F.\text{reduce}$ (obtained by summing fluxes for each unique $U(1)$ charge), the total number of anti-chiral $D = (\bar{3}, 1)_{1/3}$ representations is zero.

Let $F$ be a collection of quanta representing the $\mathbf{\bar{5}}$ (5-bar) representations of $SU(5) \times U(1)$. Suppose that the multiset of its underlying $SU(5)$ fluxes, $F.\text{toFluxesFive}$, is an element of the set of flux configurations that have no chiral exotics and no zero fluxes ($\text{FluxesFive.elemsNoExotics}$). Then, the reduced configuration $F.\text{reduce}$—formed by summing all $SU(5)$ fluxes associated with each unique $U(1)$ charge—satisfies the `NoExotics` condition. Specifically, the resulting spectrum contains exactly 3 generations of chiral $L = (1, 2)_{-1/2}$ and $D = (\bar{3}, 1)_{1/3}$ representations, and zero anti-chiral representations.

Let $F$ be a collection of quanta representing the $\mathbf{\bar{5}}$ (5-bar) representations of $SU(5) \times U(1)$. If the multiset of its underlying $SU(5)$ fluxes, $F.\text{toFluxesFive}$, belongs to the set of flux configurations that have no chiral exotics and no zero fluxes ($\text{FluxesFive.elemsNoExotics}$), then the underlying flux multiset of the reduced configuration $F.\text{reduce}$—obtained by summing all $SU(5)$ fluxes associated with each unique $U(1)$ charge—is also an element of $\text{FluxesFive.elemsNoExotics}$.

Given a flux $f = \langle M, N \rangle$, this function returns a multiset that decomposes $f$ into elementary units. The multiset contains $|M|$ copies of the flux vector $\langle 1, -1 \rangle$ and $|M + N|$ copies of the flux vector $\langle 0, 1 \rangle$.

Let $f = \langle M, N \rangle$ be a flux pair, where $M$ is the chirality flux and $N$ is the hypercharge flux. If $f$ is an element of a flux configuration $F$ belonging to the set of $\mathbf{\bar{5}}$ flux configurations with no chiral exotics ($\text{FluxesFive.elemsNoExotics}$), then the sum of the multiset of elementary fluxes obtained by decomposing $f$ (consisting of $|M|$ copies of $\langle 1, -1 \rangle$ and $|M+N|$ copies of $\langle 0, 1 \rangle$) is equal to $f$.

Given a `FiveQuanta` $x$, which is a multiset of pairs $(q, f)$ where $q \in \mathcal{Z}$ represents a charge and $f = \langle M, N \rangle$ represents a flux, this function decomposes $x$ into a new `FiveQuanta`. Each original pair $(q, f)$ is replaced by a collection of pairs $(q, f_i)$, where the $f_i$ are elementary fluxes. Specifically, for each $(q, f)$, the function produces $|M|$ copies of $(q, \langle 1, -1 \rangle)$ and $|M+N|$ copies of $(q, \langle 0, 1 \rangle)$. The resulting `FiveQuanta` maintains the same reduction (total flux per charge) as the original.

For any two `FiveQuanta` $x$ and $y$ over a charge space $\mathcal{Z}$, the decomposition of their sum is equal to the sum of their individual decompositions: $$(x + y).\text{decompose} = x.\text{decompose} + y.\text{decompose}$$

For a `FiveQuanta` $x$ (defined as a multiset of pairs $(q, f)$ where $q \in \mathcal{Z}$ is a charge and $f$ is a flux) and a specific charge $q \in \mathcal{Z}$, the operation of decomposing fluxes into elementary units commutes with filtering the multiset by charge. Specifically, filtering the decomposed multiset to retain only elements with charge $q$ yields the same result as decomposing the subset of $x$ that already consists only of elements with charge $q$.

Let $x$ be a `FiveQuanta` over a charge type $\mathcal{Z}$ with decidable equality. If the underlying multiset of $SU(5)$ fluxes of $x$ (denoted $x.\text{toFluxesFive}$) belongs to the set of configurations with no chiral exotics and no zero fluxes ($\text{FluxesFive.elemsNoExotics}$), then the decomposition of $x$ into elementary fluxes preserves the mapping from charges to their total flux. That is, $$(x.\text{decompose}).\text{toChargeMap} = x.\text{toChargeMap}$$ where the $\text{toChargeMap}$ function assigns to each charge $q \in \mathcal{Z}$ the sum of all fluxes associated with that specific charge within the multiset.

Let $x$ be a `FiveQuanta` over a charge type $\mathcal{Z}$ with decidable equality. If the underlying multiset of $SU(5)$ fluxes of $x$ (denoted $x.\text{toFluxesFive}$) belongs to the set of configurations with no chiral exotics and no zero fluxes ($\text{FluxesFive.elemsNoExotics}$), then the set of distinct charges in the decomposition of $x$ is equal to the set of distinct charges in the original $x$. That is, $$(x.\text{decompose}).\text{toCharges}.\text{dedup} = x.\text{toCharges}.\text{dedup}$$ where $\text{toCharges}$ extracts the multiset of charges and $\text{dedup}$ reduces the multiset to its underlying set of unique elements.

Let $x$ be a `FiveQuanta` object over a charge type $\mathcal{Z}$ with decidable equality, representing a multiset of pairs $(q, f)$ where $q \in \mathcal{Z}$ is a $U(1)$ charge and $f = \langle M, N \rangle$ is an $SU(5)$ flux pair. If the underlying multiset of fluxes of $x$ (denoted $x.\text{toFluxesFive}$) belongs to the set of configurations with no chiral exotics and no zero fluxes ($\text{FluxesFive.elemsNoExotics}$), then the reduction of the decomposition of $x$ is equal to the reduction of $x$ itself. That is, $$(x.\text{decompose}).\text{reduce} = x.\text{reduce}$$ where the `reduce` operation produces a collection of unique charges by summing all fluxes associated with each specific charge, and the `decompose` operation replaces each original pair $(q, \langle M, N \rangle)$ with $|M|$ copies of $(q, \langle 1, -1 \rangle)$ and $|M+N|$ copies of $(q, \langle 0, 1 \rangle)$.

Let $x$ be a `FiveQuanta` object, which represents a collection of $SU(5)$ $\mathbf{\bar{5}}$ (5-bar) matter curves, each associated with a $U(1)$ charge $q$ and a flux pair $f = \langle M, N \rangle$, where $M$ is the chirality flux and $N$ is the hypercharge flux. If the underlying multiset of fluxes of $x$, denoted by $x.\text{toFluxesFive}$, belongs to the set of 3-generation configurations with no chiral exotics (`FluxesFive.elemsNoExotics`), then the multiset of fluxes for the decomposition of $x$ is given by \[ x.\text{decompose.toFluxesFive} = \{ \langle 1, -1 \rangle, \langle 1, -1 \rangle, \langle 1, -1 \rangle, \langle 0, 1 \rangle, \langle 0, 1 \rangle, \langle 0, 1 \rangle \}. \] The decomposition process replaces each original flux $\langle M, N \rangle$ with $|M|$ copies of the elementary flux $\langle 1, -1 \rangle$ and $|M + N|$ copies of the elementary flux $\langle 0, 1 \rangle$.

Given a finite set of charges $c \subset \mathcal{Z}$, the function `liftCharge c` constructs a multiset of all possible `FiveQuanta` configurations that have no exotics, no duplicate charges, and no zero fluxes, such that their underlying set of charges is exactly $c$. The construction proceeds by: 1. Identifying all multisets $s$ of cardinality 3 where every element belongs to $c$. 2. Forming pairs of such multisets $(s_1, s_2)$ such that their multiset sum $s_1 + s_2$ contains every charge in $c$. 3. For each pair, creating a collection of quanta by assigning the flux $\langle 1, -1 \rangle$ to each charge in $s_1$ and the flux $\langle 0, 1 \rangle$ to each charge in $s_2$. 4. Applying the `reduce` operation to each collection, which sums the fluxes for identical charges to ensure each charge in the resulting configuration is unique.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. If $x$ is a collection of quanta belonging to the multiset $\text{liftCharge}(c)$, then the set of distinct charges associated with $x$ (obtained by converting its multiset of charges to a finite set) is equal to $c$.

For any finite set of charges $c \subset \mathcal{Z}$, if a configuration of 5-bar quanta $x$ belongs to the multiset $\text{liftCharge}(c)$, then the multiset of $U(1)$ charges associated with $x$, denoted as $x.\text{toCharges}$, contains no duplicate elements.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. If $x$ is a configuration of 5-bar quanta belonging to the multiset $\text{liftCharge}(c)$, then there exists a configuration $a$ such that: 1. $x$ is the reduction of $a$ (i.e., $a.\text{reduce} = x$); 2. The set of distinct charges associated with $a$ is exactly $c$; 3. The multiset of fluxes in $a$ consists of three $\langle 1, -1 \rangle$ pairs and three $\langle 0, 1 \rangle$ pairs.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. A configuration $x$ of 5-bar quanta belongs to the multiset $\text{liftCharge}(c)$ if there exists a configuration $a$ such that: 1. The reduction of $a$ (summing fluxes for each unique charge) is $x$, i.e., $a.\text{reduce} = x$. 2. The set of distinct charges in $a$ is exactly $c$. 3. The multiset of $SU(5)$ fluxes associated with $a$ is $\{\langle 1, -1 \rangle, \langle 1, -1 \rangle, \langle 1, -1 \rangle, \langle 0, 1 \rangle, \langle 0, 1 \rangle, \langle 0, 1 \rangle\}$.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. A configuration of 5-bar quanta $x$ belongs to the multiset $\text{liftCharge}(c)$ if and only if there exists a configuration $a$ such that: 1. $x$ is the reduction of $a$ ($a.\text{reduce} = x$), where reduction sums the fluxes for each unique charge; 2. The set of distinct charges associated with $a$ is exactly $c$; 3. The multiset of $SU(5)$ fluxes associated with $a$ consists of three $\langle 1, -1 \rangle$ pairs and three $\langle 0, 1 \rangle$ pairs.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. If a configuration $x$ of 5-bar quanta belongs to the multiset $\text{liftCharge}(c)$, then the multiset of $SU(5)$ fluxes associated with $x$ does not contain the zero flux $(0, 0)$.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. If a configuration of $\mathbf{\bar{5}}$ (5-bar) quanta $F$ is an element of the multiset $\text{liftCharge}(c)$, then its underlying multiset of $SU(5)$ fluxes, $F.\text{toFluxesFive}$, satisfies the `NoExotics` condition. This implies that the resulting physical spectrum contains exactly 3 generations of chiral $L$ and $D$ representations and zero anti-chiral representations.

Let $x$ be a `FiveQuanta` configuration over a charge type $\mathcal{Z}$, representing a multiset of pairs $(q, f)$ where $q \in \mathcal{Z}$ is a $U(1)$ charge and $f = \langle M, N \rangle$ is an $SU(5)$ flux pair (with $M$ being the chirality flux and $N$ being the hypercharge flux). Let $c \subset \mathcal{Z}$ be a finite set of charges. If $x$ satisfies the following four conditions: 1. **No chiral exotics**: The underlying multiset of fluxes satisfies the `NoExotics` condition, meaning the total spectrum contains exactly three generations of lepton doublets $L$ and down-type quarks $D$ with no anti-chiral counterparts. 2. **No zero fluxes**: The multiset of fluxes does not contain the zero flux $\langle 0, 0 \rangle$ (`HasNoZero`). 3. **Charge set equality**: The set of unique charges present in $x$ is exactly $c$. 4. **No duplicate charges**: Each charge in the configuration $x$ appears exactly once (the multiset of charges is `Nodup`). Then $x$ is an element of the multiset $\text{liftCharge}(c)$, which identifies valid 5-bar quanta configurations for the given set of charges.

Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges and $x$ be a configuration of $\mathbf{\bar{5}}$ (5-bar) quanta. Then $x$ is an element of the multiset $\text{liftCharge}(c)$ if and only if the following three conditions are satisfied: 1. The underlying multiset of $SU(5)$ fluxes of $x$, denoted $x.\text{toFluxesFive}$, belongs to the collection $\text{FluxesFive.elemsNoExotics}$. This implies that the configuration has no chiral exotics (resulting in exactly 3 generations of lepton doublets $L$ and down-type quarks $D$ with no anti-chiral counterparts) and contains no zero fluxes. 2. The set of distinct $U(1)$ charges associated with $x$ is exactly $c$. 3. The $U(1)$ charges in $x$ are unique, meaning the multiset of charges $x.\text{toCharges}$ contains no duplicates.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be commutative rings, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be a ring homomorphism. Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. Suppose $F$ is a configuration of $\mathbf{\bar{5}}$ (5-bar) quanta that is an element of $\text{liftCharge}(c)$, meaning it has no chiral exotics, no zero fluxes, and its unique set of charges is $c$. If we transform $F$ by mapping its charges under $f$ and then apply the reduction operation (which sums the $SU(5)$ fluxes for any charges that became identical under $f$), the resulting configuration is an element of $\text{liftCharge}(f(c))$, where $f(c)$ is the image of $c$ under $f$.

For a given collection of quanta for the 5-bar representation of $SU(5) \times U(1)$, denoted as $F$, let each element $i$ in the collection be associated with a $U(1)$ charge $q_i$ and a flux $N_i$. The anomaly coefficient of $F$ is defined as the pair of integers: \[ \left( \sum_i q_i N_i, \sum_i q_i^2 N_i \right) \] The first component corresponds to the mixed $U(1)$-MSSM gauge anomaly coefficient, and the second component corresponds to the mixed $U(1)_Y\text{-}U(1)\text{-}U(1)$ gauge anomaly coefficient.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be commutative rings, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be a ring homomorphism. Consider a collection $F$ of 5-bar representation quanta for $SU(5) \times U(1)$, where each element $i$ is associated with a $U(1)$ charge $q_i \in \mathcal{Z}$ and a flux $N_i$. The anomaly coefficient of $F$ is defined as the pair $A(F) = (\sum_i q_i N_i, \sum_i q_i^2 N_i)$. If the charges of $F$ are mapped under $f$ to produce a new collection $F'$ (where the charges are $f(q_i)$ and the fluxes $N_i$ remain unchanged), then the anomaly coefficient of $F'$ is the image of $A(F)$ under the product map $(f, f)$. That is, \[ A(F') = (f(\sum_i q_i N_i), f(\sum_i q_i^2 N_i)) \]

Let $F$ be a collection of quanta for the 5-bar representation of $SU(5) \times U(1)$, where each quantum $i$ is associated with a $U(1)$ charge $q_i$ and a flux $N_i$. The anomaly coefficient of $F$ is defined as the pair of sums $\left( \sum_i q_i N_i, \sum_i q_i^2 N_i \right)$. If $F$ is reduced by summing all fluxes that correspond to the same charge, the resulting collection $F_{\text{reduce}}$ has the same anomaly coefficient as the original collection. That is, \[ \text{anomalyCoefficient}(F_{\text{reduce}}) = \text{anomalyCoefficient}(F) \]