Physlib.StringTheory.FTheory.SU5.Quanta.TenQuanta

For an object $x$ of type `TenQuanta` $\mathcal{Z}$, which represents a collection (multiset) of quanta where each quantum is a pair $(c, f)$ consisting of a $U(1)$ charge $c \in \mathcal{Z}$ and a flux $f$, this function returns the multiset of all fluxes $f$ associated with these quanta. This is defined by mapping the projection onto the second component across the multiset $x$, resulting in an object of type `FluxesTen`.

For an object $x$ of type `TenQuanta` $\mathcal{Z}$, which represents a collection of 10-dimensional representations in an $SU(5) \times U(1)$ model, the function returns the multiset of $U(1)$ charges in $\mathcal{Z}$ associated with those representations. This is defined by projecting each element in the collection $x$ (which consists of pairs of charges and fluxes) to its first component, the charge.

For an object $x$ of type `TenQuanta` $\mathcal{Z}$, which represents a multiset of pairs $(c, f)$ consisting of a $U(1)$ charge $c \in \mathcal{Z}$ and a flux $f \in \text{Fluxes}$, the function `toChargeMap` returns a map that associates each charge $z \in \mathcal{Z}$ with the sum of all fluxes $f$ in $x$ whose corresponding charge $c$ is equal to $z$. Mathematically, for a given $z$, the value is $\sum_{(c, f) \in x, c = z} f$.

For an object $x$ of type `TenQuanta` $\mathcal{Z}$ and a $U(1)$ charge $z \in \mathcal{Z}$, if $z$ is not contained in the multiset of charges of $x$ (denoted by $x.\text{toCharges}$), then the total flux associated with $z$ in $x$ (calculated by $x.\text{toChargeMap}(z)$) is equal to the zero flux $0$.

For a given `TenQuanta` $x$ consisting of a collection of pairs $(q, \phi)$, where $q \in \mathcal{Z}$ is a $U(1)$ charge and $\phi$ is a flux, the function `reduce` returns a new `TenQuanta` where each distinct charge $q$ present in $x$ is associated with the sum of all fluxes $\phi$ that were paired with $q$ in the original collection. Formally, for each unique charge $q$ in $x$, the reduced collection contains a single pair $(q, \Phi_q)$ with $\Phi_q = \sum_{(q_i, \phi_i) \in x, q_i=q} \phi_i$.

For any `TenQuanta` $x$, the reduced collection $x.\text{reduce}$ (obtained by summing the fluxes for each distinct charge) contains no duplicate elements.

For any collection of 10-dimensional representation quanta $x$ (a `TenQuanta`), the reduction $x.\text{reduce}$ is obtained by summing all fluxes associated with each unique charge. This theorem states that the deduplication of this reduced collection is equal to the reduced collection itself: $$x.\text{reduce}.\text{dedup} = x.\text{reduce}$$

For a `TenQuanta` $x$ over a set of $U(1)$ charges $\mathcal{Z}$ (a collection of charge-flux pairs), let $x.\text{toCharges}$ be the multiset of its charges and $x.\text{reduce}$ be the reduced collection where fluxes for each unique charge are summed together. The multiset of charges of the reduced collection is equal to the multiset of charges of the original collection with all duplicates removed, i.e., $x.\text{reduce}.\text{toCharges} = x.\text{toCharges}.\text{dedup}$.

Let $x$ be a collection of 10-dimensional representation quanta (a `TenQuanta`), and let $p = (q, \Phi)$ be a pair consisting of a $U(1)$ charge $q \in \mathcal{Z}$ and a flux $\Phi$. Then $p$ is an element of the reduced collection $x_{\text{reduce}}$ if and only if $q$ is a charge present in the original collection $x$, and the flux $\Phi$ is equal to the sum of all fluxes $\phi_i$ associated with the charge $q$ in $x$. Mathematically, this is expressed as: $$(q, \Phi) \in x_{\text{reduce}} \iff q \in \text{charges}(x) \land \Phi = \sum_{(q_i, \phi_i) \in x, q_i=q} \phi_i$$

For a `TenQuanta` $x$ (a collection of charge-flux pairs) and a $U(1)$ charge $q$ present in the multiset of charges of $x$, the result of filtering the reduced version of $x$ (where fluxes are summed for each distinct charge) for the charge $q$ is a singleton set containing the pair $(q, \Phi_q)$, where $\Phi_q$ is the sum of all fluxes associated with the charge $q$ in the original collection $x$.

For a given collection of 10-dimensional representation quanta $x$ (a `TenQuanta`), the reduction operation—which sums all fluxes associated with the same $U(1)$ charge—is idempotent. That is, reducing an already reduced collection results in the same collection: $x.\text{reduce}.\text{reduce} = x.\text{reduce}$.

Let $x$ be a collection of 10-dimensional representation quanta of type `TenQuanta` $\mathcal{Z}$, which consists of pairs $(q, \phi)$ where $q \in \mathcal{Z}$ is a $U(1)$ charge and $\phi$ is a flux. Let $M$ be an additive commutative monoid, and let $f$ be a function that assigns to each charge $q$ an additive monoid homomorphism $f_q: \text{Fluxes} \to M$. Then, the sum of $f_q(\phi)$ over all pairs in the original collection $x$ is equal to the sum of $f_q(\Phi)$ over all pairs $(q, \Phi)$ in the reduced collection $\text{reduce}(x)$, where each $\Phi$ represents the accumulated flux for a unique charge $q$. Mathematically, this is expressed as: $$\sum_{(q, \Phi) \in \text{reduce}(x)} f_q(\Phi) = \sum_{(q, \phi) \in x} f_q(\phi)$$

Let $x$ be a collection of 10-dimensional representations (quanta) in an $SU(5) \times U(1)$ model, represented as a set of pairs $(q, \phi)$ where $q \in \mathcal{Z}$ is a $U(1)$ charge and $\phi$ is a flux. If the multiset of charges $x.\text{toCharges}$ contains no duplicate elements (i.e., each charge $q$ appears at most once in the collection), then the reduction of $x$, denoted $x.\text{reduce}$—which sums the fluxes for each distinct charge—is equal to the original collection $x$.

Let $x$ be an object of type `TenQuanta` $\mathcal{Z}$, representing a collection of pairs $(q, \phi)$ where $q \in \mathcal{Z}$ is a $U(1)$ charge and $\phi$ is a flux. The function `toChargeMap` associates each charge $z \in \mathcal{Z}$ with the sum of all fluxes associated with that charge in the collection. The theorem states that the charge map of the reduced collection $x.\text{reduce}$ (where all fluxes belonging to the same charge are pre-summed) is equal to the charge map of the original collection $x$. Mathematically, for any $x$, it holds that: $$x.\text{reduce}.\text{toChargeMap} = x.\text{toChargeMap}$$

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`), and let $F_{\text{reduce}}$ be its reduction obtained by summing the fluxes for each distinct $U(1)$ charge. If a flux $f$ is an element of the multiset of fluxes associated with $F_{\text{reduce}}$, then $f$ is equal to the sum of some sub-multiset of the fluxes in the original collection $F$.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`) and let $\Phi(F)$ be the multiset of its associated fluxes. For any flux $f$ that is an element of the multiset of fluxes of the reduced collection $F_{\text{reduce}}$, it holds that $f$ is equal to the sum of the elements of some non-empty sub-multiset of $\Phi(F)$.

Let $F$ be a collection of 10-dimensional representation quanta (`TenQuanta`) and let $\Phi(F)$ be its multiset of flux pairs $(M, N) \in \mathbb{Z}^2$. If $\Phi(F)$ is one of the six configurations in `elemsNoExotics` (which are the flux configurations satisfying the "no chiral exotics" and "no zero flux" conditions), then the total number of chiral $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ representations in the reduced collection $F_{\text{reduce}}$ is 3. The total number of chiral $U$ representations is calculated as the sum of all non-negative chiral indices $\chi(U) = M - N$ associated with the fluxes in the multiset.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`). If the multiset of fluxes $\Phi(F)$ associated with $F$ is one of the six configurations in `FluxesTen.elemsNoExotics` (the configurations that satisfy the "no chiral exotics" and "no zero flux" conditions), then the total number of anti-chiral fermions in the $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ representation for the reduced collection $F_{\text{reduce}}$ is zero. Here, $F_{\text{reduce}}$ is obtained by summing the fluxes of all quanta in $F$ that share the same $U(1)$ charge.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`), and let $F_{\text{reduce}}$ be its reduction obtained by summing the fluxes for each distinct $U(1)$ charge. If the multiset of fluxes associated with $F$ is one of the six configurations in `elemsNoExotics` (which satisfy the "no chiral exotics" and "no zero flux" conditions), then the total number of chiral $Q = (3, 2)_{1/6}$ representations in the reduction $F_{\text{reduce}}$ is exactly 3.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`) over a charge lattice $\mathcal{Z}$. If the multiset of fluxes associated with $F$, denoted as $F.\text{toFluxesTen}$, belongs to the set of six flux configurations $\text{elemsNoExotics}$ that satisfy the "no chiral exotics" and "no zero flux" conditions, then the total sum of negative chiral indices for the representation $Q = (3, 2)_{1/6}$ in the reduced collection $F.\text{reduce}$ is zero. That is, \[ \sum_{M \in \text{chiralIndicesOfQ}(F.\text{reduce}.\text{toFluxesTen}), M < 0} M = 0 \] where $F.\text{reduce}$ is obtained by summing all fluxes corresponding to the same $U(1)$ charge.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`), consisting of pairs of $U(1)$ charges and flux pairs $(M, N) \in \mathbb{Z}^2$. Suppose that the multiset of fluxes associated with $F$ is one of the six configurations in `elemsNoExotics` that satisfy the "no chiral exotics" condition. Then, for the reduced collection of quanta $F_{\text{reduce}}$ (obtained by summing all fluxes corresponding to the same charge), the total number of chiral $E = (1, 1)_1$ representations is 3. Here, the number of chiral $E$ representations is defined as the sum of non-negative chiral indices $\chi_E = M + N$ over the fluxes in the multiset.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`). Suppose that the underlying multiset of fluxes associated with $F$ belongs to the set $\text{elemsNoExotics}$, which consists of the six flux configurations that satisfy the "no chiral exotics" and "no zero flux" conditions. Then, for the reduced collection $F_{\text{reduce}}$ (obtained by summing fluxes for identical $U(1)$ charges), the sum of all negative chiral indices $\chi_E = M + N$ for the Standard Model representation $E = (1, 1)_1$ is zero.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`) over a charge lattice $\mathcal{Z}$. Suppose that the multiset of fluxes associated with $F$, denoted $F.\text{toFluxesTen}$, is one of the six configurations in the set $\text{elemsNoExotics}$ (the configurations of flux pairs $(M, N) \in \mathbb{Z}^2$ that satisfy the "no chiral exotics" and "no zero flux" conditions). Then, the reduced collection $F_{\text{reduce}}$, obtained by summing all fluxes corresponding to the same $U(1)$ charge, also satisfies the `NoExotics` condition. Specifically, the total number of chiral generations is 3 and the number of anti-chiral generations is 0 for the Standard Model representations $Q = (3, 2)_{1/6}$, $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$, and $E = (1, 1)_1$.

Let $F$ be a collection of 10-dimensional representation quanta (a `TenQuanta`) over a charge lattice $\mathcal{Z}$, and let $\Phi(F)$ be the multiset of its associated fluxes. Suppose that $\Phi(F)$ belongs to the set $\text{elemsNoExotics}$, which consists of the six flux configurations (e.g., $\{(1, 0), (1, 0), (1, 0)\}$, $\{(3, 0)\}$, etc.) that satisfy the "no chiral exotics" and "no zero flux" conditions. Then, the multiset of fluxes of the reduced collection $F_{\text{reduce}}$, obtained by summing all fluxes corresponding to the same $U(1)$ charge, is also an element of $\text{elemsNoExotics}$.

For a given flux $f = \langle M, N \rangle$ (representing chirality and hypercharge fluxes), this function returns a multiset of fluxes by decomposing $f$ into specific basic components. The mapping is defined as follows: - $\langle 1, 0 \rangle \mapsto \{ \langle 1, 0 \rangle \}$ - $\langle 1, 1 \rangle \mapsto \{ \langle 1, 1 \rangle \}$ - $\langle 1, -1 \rangle \mapsto \{ \langle 1, -1 \rangle \}$ - $\langle 2, 1 \rangle \mapsto \{ \langle 1, 1 \rangle, \langle 1, 0 \rangle \}$ - $\langle 2, -1 \rangle \mapsto \{ \langle 1, -1 \rangle, \langle 1, 0 \rangle \}$ - $\langle 3, 0 \rangle \mapsto \{ \langle 1, 0 \rangle, \langle 1, 0 \rangle, \langle 1, 0 \rangle \}$ - $\langle 2, 0 \rangle \mapsto \{ \langle 1, 0 \rangle, \langle 1, 0 \rangle \}$ For any other flux $f$, the function returns the singleton multiset $\{ f \}$.

For any flux pair $f = (M, N) \in \mathbb{Z}^2$ (representing chirality and hypercharge fluxes), if $f$ belongs to one of the configurations of 10-dimensional matter curve fluxes that satisfy the "no chiral exotics" and "no zero flux" conditions (i.e., $f \in F$ for some $F \in \text{elemsNoExotics}$), then the sum of the basic flux components in the decomposition of $f$ is equal to $f$. Specifically, the sum of the multiset produced by $\text{decomposeFluxes}(f)$ yields the original flux $f$.

Given a `TenQuanta` $x$ over a charge space $\mathcal{Z}$, which is a collection of pairs $(q, f)$ consisting of a charge $q \in \mathcal{Z}$ and a flux $f$, the decomposition function returns a new `TenQuanta`. This is achieved by taking each element $(q, f) \in x$ and replacing it with a multiset of pairs $(q, f_i)$, where $\{f_i\}$ are the basic fluxes obtained by decomposing $f$ using the `decomposeFluxes` function. The resulting `TenQuanta` consists of fluxes that are only of the basic forms $\langle 1, 0 \rangle$ or $\langle 1, \pm 1 \rangle$, while maintaining the same total charge mapping when reduced.

For any two collections of 10-dimensional representation quanta $x$ and $y$ over a charge space $\mathcal{Z}$, the decomposition of their multiset sum $x + y$ into basic flux components is equal to the multiset sum of their individual decompositions. That is, $(x + y).\text{decompose} = x.\text{decompose} + y.\text{decompose}$.

For a `TenQuanta` $x$ over a charge space $\mathcal{Z}$ (a collection of charge-flux pairs $(q', f)$) and a specific charge $q \in \mathcal{Z}$, the result of decomposing the fluxes in $x$ and then filtering the collection to keep only those pairs with charge $q$ is equal to filtering $x$ for charge $q$ first and then decomposing its fluxes. That is, $$(x.\text{decompose}).\text{filter}(p \mapsto p.1 = q) = (x.\text{filter}(p \mapsto p.1 = q)).\text{decompose}$$ where the `decompose` function replaces each pair $(q', f)$ with a multiset of pairs $(q', f_i)$ using the basic flux components $\{f_i\}$ of $f$. This theorem assumes that equality of charges in $\mathcal{Z}$ is decidable.

For a `TenQuanta` $x$ over a charge space $\mathcal{Z}$ (a multiset of charge-flux pairs $(q, f)$), suppose that the multiset of its fluxes $x.\text{toFluxesTen}$ belongs to `FluxesTen.elemsNoExotics` (the set of six flux configurations that satisfy the "no chiral exotics" and "no zero flux" conditions). Then the charge map of the decomposition of $x$ is equal to the charge map of $x$ itself. That is, for any charge $z \in \mathcal{Z}$: $$(x.\text{decompose}).\text{toChargeMap}(z) = x.\text{toChargeMap}(z)$$ where the `toChargeMap` for a charge $z$ is the sum of all fluxes associated with $z$ in the collection, and the `decompose` operation replaces each flux in $x$ with its basic components while preserving its associated charge. This theorem assumes that equality of charges in $\mathcal{Z}$ is decidable.

For a `TenQuanta` $x$ over a charge space $\mathcal{Z}$ (a collection of charge-flux pairs $(q, f)$), suppose the multiset of its fluxes $x.\text{toFluxesTen}$ is one of the six configurations in `FluxesTen.elemsNoExotics` (the configurations that satisfy the "no chiral exotics" and "no zero flux" conditions). Then, the set of distinct charges in the decomposition of $x$ is equal to the set of distinct charges in $x$ itself. That is, $$(x.\text{decompose}).\text{toCharges}.\text{dedup} = x.\text{toCharges}.\text{dedup}$$ where `decompose` replaces each flux with its basic components while keeping the associated charge, and `dedup` denotes the set of unique elements in the multiset of charges.

Let $x$ be a `TenQuanta` over a charge space $\mathcal{Z}$, which is a collection of pairs $(q, f)$ consisting of a $U(1)$ charge $q \in \mathcal{Z}$ and a flux $f$. Suppose that the multiset of its fluxes, $x.\text{toFluxesTen}$, is an element of `FluxesTen.elemsNoExotics` (the set of six flux configurations for 10-dimensional matter curves that satisfy the "no chiral exotics" and "no zero flux" conditions). Then, the reduction of the decomposition of $x$ is equal to the reduction of $x$ itself: $$(x.\text{decompose}).\text{reduce} = x.\text{reduce}$$ where the `reduce` operation sums all fluxes associated with each unique charge into a single pair, and the `decompose` operation replaces each flux in $x$ with its basic components while preserving its associated charge. This theorem assumes that equality of charges in $\mathcal{Z}$ is decidable.

Let $x$ be a `TenQuanta` over a charge space $\mathcal{Z}$, representing a collection of pairs $(c, f)$ where $c \in \mathcal{Z}$ is a $U(1)$ charge and $f = \langle M, N \rangle$ is a flux. Let $x.\text{toFluxesTen}$ denote the multiset of fluxes associated with $x$. If $x.\text{toFluxesTen}$ is an element of `FluxesTen.elemsNoExotics` (the set of flux configurations for 10-dimensional matter curves that satisfy the "no chiral exotics" and "no zero flux" conditions), then the multiset of fluxes of the decomposed representation, $x.\text{decompose}.\text{toFluxesTen}$, is equal to either $\{\langle 1, 0 \rangle, \langle 1, 0 \rangle, \langle 1, 0 \rangle\}$ or $\{\langle 1, 1 \rangle, \langle 1, -1 \rangle, \langle 1, 0 \rangle\}$.

Given a finite set of $U(1)$ charges $c \subset \mathcal{Z}$, the function `liftCharge(c)` constructs a multiset of `TenQuanta` that correspond to valid 10-dimensional representations without exotic fluxes, duplicate charges, or zero fluxes. The construction considers two possible configurations for the underlying fluxes $\phi$: 1. A configuration where three charges from $c$ (forming a multiset of size 3) are each paired with the flux $\langle 1, 0 \rangle$. 2. A configuration where three charges $x, y, z \in c$ are paired with the fluxes $\langle 1, 1 \rangle$, $\langle 1, -1 \rangle$, and $\langle 1, 0 \rangle$ respectively. In both cases, the resulting collections are filtered to ensure that the multiset of charges covers all elements in the input set $c$. Finally, the `reduce` operation is applied to each collection, which sums the fluxes associated with the same charge, effectively merging duplicate charges into a single representation with a combined flux.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set. For any collection of 10-dimensional representations $x$ that is an element of the multiset $\text{liftCharge}(c)$, the set of unique charges associated with $x$ is exactly $c$.

For any finite set of $U(1)$ charges $c \subset \mathcal{Z}$ and any `TenQuanta` $x$ such that $x$ is an element of the multiset $\text{liftCharge}(c)$, the multiset of charges associated with $x$, denoted $x.\text{toCharges}$, contains no duplicate elements.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set of charges. For any collection of 10-dimensional representations $x$ that is an element of the multiset $\text{liftCharge}(c)$, there exists a collection $a$ such that: 1. $x$ is the reduction of $a$ (i.e., $a.\text{reduce} = x$), 2. the set of unique charges associated with $a$ is exactly $c$, and 3. the multiset of fluxes associated with $a$ is either $\{\langle 1, 0 \rangle, \langle 1, 0 \rangle, \langle 1, 0 \rangle\}$ or $\{\langle 1, 1 \rangle, \langle 1, -1 \rangle, \langle 1, 0 \rangle\}$.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set of charges. For any collection of 10-dimensional representations $x$ of type `TenQuanta`, if there exists a collection $a$ such that: 1. $x$ is the reduction of $a$ (i.e., $a.\text{reduce} = x$), 2. the set of unique charges associated with $a$ is exactly $c$ (i.e., $a.\text{toCharges.toFinset} = c$), and 3. the multiset of fluxes associated with $a$ is either $\{\langle 1, 0 \rangle, \langle 1, 0 \rangle, \langle 1, 0 \rangle\}$ or $\{\langle 1, 1 \rangle, \langle 1, -1 \rangle, \langle 1, 0 \rangle\}$, then $x$ is an element of the multiset $\text{liftCharge}(c)$.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set of charges. For any collection of 10-dimensional representations $x$ of type `TenQuanta`, $x$ is an element of the multiset $\text{liftCharge}(c)$ if and only if there exists a collection $a$ such that: 1. $x$ is the reduction of $a$ (i.e., $a.\text{reduce} = x$), 2. the set of unique charges associated with $a$ is exactly $c$ (i.e., $a.\text{toCharges.toFinset} = c$), and 3. the multiset of fluxes associated with $a$ is either $\{\langle 1, 0 \rangle, \langle 1, 0 \rangle, \langle 1, 0 \rangle\}$ or $\{\langle 1, 1 \rangle, \langle 1, -1 \rangle, \langle 1, 0 \rangle\}$.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set of charges. For any collection of 10-dimensional representations $x$ of type `TenQuanta`, if $x$ is an element of the multiset $\text{liftCharge}(c)$, then the multiset of fluxes associated with $x$ does not contain the zero flux $(0, 0)$.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set of charges. For any collection of 10-dimensional representation quanta $F$ (an object of type `TenQuanta`), if $F$ is an element of the multiset $\text{liftCharge}(c)$, then the multiset of fluxes associated with $F$, denoted $F.\text{toFluxesTen}$, satisfies the `NoExotics` condition. The `NoExotics` condition implies that the configuration yields exactly 3 chiral generations and 0 anti-chiral generations for each of the Standard Model representations $Q = (3, 2)_{1/6}$, $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$, and $E = (1, 1)_1$.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set of charges. Let $x$ be a `TenQuanta` over $\mathcal{Z}$, representing a collection of 10-dimensional representations as pairs $(q, \phi)$ of charges and fluxes. Suppose that: 1. The multiset of fluxes associated with $x$, denoted $x.\text{toFluxesTen}$, satisfies the "no chiral exotics" condition ($\text{NoExotics}$). 2. The multiset of fluxes $x.\text{toFluxesTen}$ contains no zero flux $(0,0)$ ($\text{HasNoZero}$). 3. The set of unique charges associated with $x$ is exactly $c$ ($x.\text{toCharges}.\text{toFinset} = c$). 4. The multiset of charges $x.\text{toCharges}$ contains no duplicate elements ($\text{Nodup}$). Then, $x$ is an element of the multiset $\text{liftCharge}(c)$, which constructs valid, reduced 10-dimensional representation configurations for the set of charges $c$.

Let $\mathcal{Z}$ be the type of $U(1)$ charges and $c \subset \mathcal{Z}$ be a finite set. For any collection of 10-dimensional representation quanta $x$ (an object of type `TenQuanta`), $x$ belongs to the multiset $\text{liftCharge}(c)$ if and only if the following three conditions are satisfied: 1. The multiset of fluxes associated with $x$, denoted $x.\text{toFluxesTen}$, is an element of $\text{FluxesTen.elemsNoExotics}$. This set contains the six specific flux configurations that satisfy both the "no chiral exotics" ($\text{NoExotics}$) and "no zero flux" ($\text{HasNoZero}$) conditions. 2. The set of unique $U(1)$ charges present in $x$ is exactly $c$ ($x.\text{toCharges}.\text{toFinset} = c$). 3. The multiset of charges associated with $x$, $x.\text{toCharges}$, contains no duplicate elements ($\text{Nodup}$).

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be commutative rings and $f: \mathcal{Z} \to \mathcal{Z}_1$ be a ring homomorphism. Let $c \subset \mathcal{Z}$ be a finite set of $U(1)$ charges. If a collection of 10-dimensional representation quanta $F$ (an object of type `TenQuanta`) belongs to the set of valid configurations $\text{liftCharge}(c)$, then the configuration obtained by mapping the charges of $F$ through $f$ and applying the `reduce` operation (which sums fluxes for any charges that become identical under $f$) is an element of $\text{liftCharge}(f(c))$, where $f(c)$ is the image of the set $c$ under $f$.

Given a `TenQuanta` $F$ over a commutative ring $\mathcal{Z}$, let each representation in the collection be associated with a $U(1)$ charge $q_i$ and a flux $N_i$. The anomaly coefficient is defined as the pair in $\mathcal{Z} \times \mathcal{Z}$: \[ \left( \sum_i q_i N_i, 3 \sum_i q_i^2 N_i \right) \] The first component represents the mixed $U(1)$-MSSM gauge anomaly coefficient, and the second component represents the mixed $U(1)_Y-U(1)-U(1)$ gauge anomaly coefficient.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be commutative rings and $f: \mathcal{Z} \to \mathcal{Z}_1$ be a ring homomorphism. Let $F$ be a `TenQuanta` object over $\mathcal{Z}$, representing a collection of 10d representations where each representation is associated with a $U(1)$ charge $q_i$ and a flux $N_i \in \mathcal{Z}$. The anomaly coefficient of $F$ is defined as the pair $(\mathcal{A}_1, \mathcal{A}_2) = (\sum_i q_i N_i, 3 \sum_i q_i^2 N_i)$. If we construct a new `TenQuanta` over $\mathcal{Z}_1$ by mapping the charges and fluxes of $F$ through $f$ (i.e., $q_i \mapsto f(q_i)$ and $N_i \mapsto f(N_i)$), then the anomaly coefficient of the mapped quanta is equal to $(f(\mathcal{A}_1), f(\mathcal{A}_2))$.

For a collection $F$ of 10-dimensional representation quanta (`TenQuanta`) over a commutative ring $\mathcal{Z}$ with decidable equality, let each quantum be associated with a $U(1)$ charge $q_i$ and a flux $\phi_i = (M_i, N_i)$. The anomaly coefficient is defined as the pair $\left( \sum q_i N_i, 3 \sum q_i^2 N_i \right)$. If $F.\text{reduce}$ is the collection obtained by summing the fluxes for each unique charge $q$ in $F$, then the anomaly coefficient of the reduced collection is equal to the anomaly coefficient of the original collection: \[ \text{anomalyCoefficient}(F.\text{reduce}) = \text{anomalyCoefficient}(F) \]