Physlib.QFT.PerturbationTheory.WickContraction.Sign.InsertSome

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on the indices of $\phi_s$ that is grading compliant, meaning for every contracted pair $\{i, j\} \in \Lambda$, the field operators $\phi_i$ and $\phi_j$ have the same field statistic ($\mathcal{F} \triangleright_s \phi_i = \mathcal{F} \triangleright_s \phi_j$). Let $a$ be a subset of the indices of $\phi_s$ such that: 1. Every index $i \in a$ is contracted in $\Lambda$ (i.e., $a$ contains no uncontracted indices). 2. For every index $i \in a$, its contracted partner $j$ in $\Lambda$ is also an element of $a$. Then the collective field statistic of the indices in $a$ is bosonic (the identity $1$ in the group of field statistics): $$\prod_{i \in a} (\mathcal{F} \triangleright_s \phi_i) = 1$$

Let $\mathcal{F}$ be a field specification and $\phi_s$ be a list of field operators. Let $\Lambda$ be a Wick contraction on $\phi_s$, and let $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ be the contraction obtained by inserting a field $\phi$ at index $i$ and contracting it with an uncontracted index $j$ of $\Lambda$. For any original indices $i_1, i_2$ of $\phi_s$, let $i_1' = \text{succAbove}_i(i_1)$ and $i_2' = \text{succAbove}_i(i_2)$ be the corresponding indices in the expanded list. The set $S_{\Lambda'}(i_1', i_2')$ of indices between $i_1'$ and $i_2'$ that contribute to the permutation sign in Wick's theorem is: $$S_{\Lambda'}(i_1', i_2') = \begin{cases} \text{lift}_i(S_{\Lambda}(i_1, i_2)) \cup \{i\} & \text{if } i_1' < i < i_2' \text{ and } i_1 < j \\ \text{lift}_i(S_{\Lambda}(i_1, i_2)) \setminus \{j'\} & \text{if } i_1 < j < i_2 \text{ and } i \le i_1' \\ \text{lift}_i(S_{\Lambda}(i_1, i_2)) & \text{otherwise} \end{cases}$$ where $j' = \text{succAbove}_i(j)$, $S_{\Lambda}(a, b)$ is the set of indices $k$ such that $a < k < b$ and $k$ is either uncontracted in $\Lambda$ or contracted with an index $d > a$, and $\text{lift}_i(A) = \{ \text{succAbove}_i(k) \mid k \in A \}$.

Given a field specification $\mathcal{F}$, a list of field operators $\phi_s$, and a Wick contraction $\phi_s \Lambda$ on these fields, this function calculates a sign in $\mathbb{C}$ associated with inserting a new field operator $\phi$ at index $i$ and contracting it with an existing uncontracted index $j$ in $\phi_s \Lambda$. The result is the product over all contracted pairs $\{a_1, a_2\} \in \phi_s \Lambda$ (with $a_1 < a_2$) of the following values: - $𝓢(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \phi_{a_2})$, if $a_1 < i \le a_2$ and $a_1 < j$. - $𝓢(\mathcal{F} \triangleright_s \phi_j, \mathcal{F} \triangleright_s \phi_{a_2})$, if $a_1 < j < a_2$ and $i \le a_1$. - $1$, otherwise. Here, $\phi_k$ denotes the $k$-th element of the list $\phi_s$, $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of operator $\phi$, and $𝓢(\cdot, \cdot)$ denotes the statistical sign (typically $-1$ if both arguments are fermionic and $1$ otherwise).

Given a field specification $\mathcal{F}$, a list of field operators $\phi_s$, and a Wick contraction $\Lambda$ on $\phi_s$, this function calculates the statistical sign coefficient associated with inserting a new field operator $\phi$ at index $i$ and contracting it with the operator at an originally uncontracted index $j$. Specifically, let $\phi_s' = \text{insertIdx}(i, \phi, \phi_s)$ be the new list of field operators and $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ be the resulting Wick contraction. In $\Lambda'$, the operator $\phi$ at index $i$ is paired with the operator originally at $j$, which is now at index $j' = \text{succAbove}_i(j)$. Let $i_1 = \min(i, j')$ and $i_2 = \max(i, j')$ be the indices of this newly contracted pair. The coefficient is defined as: $$𝓢(\mathcal{F} \triangleright_s \phi'_{i_2}, \text{stat}_{\mathcal{F}, \phi_s'}(S_{\Lambda'}(i_1, i_2)))$$ where: - $\mathcal{F} \triangleright_s \phi'_{i_2}$ is the field statistic (bosonic or fermionic) of the field at index $i_2$ in the new list. - $S_{\Lambda'}(i_1, i_2)$ is the set of indices $k$ such that $i_1 < k < i_2$ and $k$ is not contracted with any index $d \le i_1$ in $\Lambda'$. - $\text{stat}_{\mathcal{F}, \phi_s'}(A)$ is the collective statistic of the set of fields indexed by $A$ (fermionic if an odd number of fields in the set are fermionic, bosonic otherwise). - $𝓢(s_1, s_2)$ is the statistical sign, returning $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise.

Given a field specification $\mathcal{F}$, a list of field operators $\phi_s$, and a Wick contraction $\Lambda$ on $\phi_s$, the function computes the total statistical sign change associated with inserting a new field operator $\phi$ at index $i$ and contracting it with an existing uncontracted operator at index $j$. This sign factor is defined as the product of the statistical sign coefficient and the statistical sign product: $$\text{signInsertSome}(\phi, \phi_s, \Lambda, i, j) = \text{signInsertSomeCoef}(\phi, \phi_s, \Lambda, i, j) \cdot \text{signInsertSomeProd}(\phi, \phi_s, \Lambda, i, j)$$ where: - $\text{signInsertSomeCoef}$ is the statistical sign coefficient resulting from the interaction between the new contracted pair and the remaining uncontracted fields. - $\text{signInsertSomeProd}$ is the product of signs resulting from the interaction between the new field $\phi$ (and its partner at $j$) and the existing contracted pairs in $\Lambda$. The resulting value corresponds to the ratio between the sign of the new contraction (with $\phi$ inserted and paired) and the sign of the original contraction $\Lambda$.

Let $\mathcal{F}$ be a field specification, $\phi_s$ be a list of field operators of length $n$, and $\Lambda$ be a Wick contraction on $\phi_s$. Suppose we insert a field operator $\phi$ into the list at index $i \in \{0, \dots, n\}$ and contract it with a previously uncontracted index $j$ of $\Lambda$, resulting in a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ for the expanded list of length $n+1$. Then the statistical sign of the new contraction $\Lambda'$ is the product of the statistical sign of the original contraction $\Lambda$ and the sign factor $\text{signInsertSome}(\phi, \phi_s, \Lambda, i, j)$ associated with the insertion and new pairing: $$\text{sign}(\Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j) = \text{signInsertSome}(\phi, \phi_s, \Lambda, i, j) \cdot \text{sign}(\Lambda)$$

Let $\mathcal{F}$ be a field specification, $\phi$ a field operator, and $\phi_s$ a list of field operators. Let $\phi_s \Lambda$ be a Wick contraction on the indices of $\phi_s$. Suppose we insert the operator $\phi$ into $\phi_s$ at index $i \in \{0, \dots, |\phi_s|\}$ and contract it with an existing uncontracted index $j$ of $\phi_s \Lambda$. If the field statistic of $\phi$ is the same as that of the $j$-th field in the list $\phi_s$ (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_j$), then the sign factor $\text{signInsertSomeProd}$ is given by the product over all contracted pairs $\{a_1, a_2\} \in \phi_s \Lambda$ (where $a_1 < a_2$): $$\text{signInsertSomeProd}(\phi, \phi_s, \phi_s \Lambda, i, j) = \prod_{\{a_1, a_2\} \in \phi_s \Lambda} \text{val}(a_1, a_2)$$ where $\text{val}(a_1, a_2)$ is defined as the statistical sign $\mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \phi_{a_2})$ if the following condition holds: $$a_1 < j \quad \text{and} \quad (a_1 < i \le a_2 \quad \text{or} \quad (a_1 \ge i \quad \text{and} \quad a_2 > j))$$ Otherwise, $\text{val}(a_1, a_2) = 1$. Here, $\mathcal{F} \triangleright_s \phi$ denotes the statistic (bosonic or fermionic) of the operator $\phi$, and $\mathcal{S}(\cdot, \cdot)$ is the statistical sign (which is $-1$ if both arguments are fermionic and $1$ otherwise).

Let $\mathcal{F}$ be a field specification, $\phi$ a field operator, and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ a list of field operators. Let $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$ that is grading-compliant, meaning for every contracted pair $\{x, y\} \in \Lambda$, the field statistics satisfy $\mathcal{F} \triangleright_s \phi_x = \mathcal{F} \triangleright_s \phi_y$. Suppose we insert the operator $\phi$ into the list at index $i \in \{0, \dots, n\}$ and contract it with an existing uncontracted index $j$ of $\Lambda$. If the field statistic of $\phi$ is the same as that of the $j$-th field operator (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_j$), then the statistical sign factor $\text{signInsertSomeProd}$ can be expressed as a nested product over each contracted pair $a \in \Lambda$ and each index $x \in a$: $$\text{signInsertSomeProd}(\phi, \phi_s, \Lambda, i, j) = \prod_{a \in \Lambda} \prod_{x \in a} V(x)$$ where $V(x) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \phi_x)$ if the following condition holds: $$x < j \quad \text{and} \quad \left( (x < i \le \text{dual}(x)) \quad \text{or} \quad (x \ge i \quad \text{and} \quad \text{dual}(x) > j) \right)$$ Otherwise, $V(x) = 1$. Here, $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of operator $\phi$, $\mathcal{S}(\cdot, \cdot)$ denotes the statistical sign, and $\text{dual}(x)$ denotes the index paired with $x$ in the contraction $\Lambda$.

Let $\mathcal{F}$ be a field specification, $\phi$ a field operator, and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ a list of field operators. Let $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$ that is grading-compliant, meaning for every contracted pair $\{x, y\} \in \Lambda$, the field statistics satisfy $\mathcal{F} \triangleright_s \phi_x = \mathcal{F} \triangleright_s \phi_y$. Suppose we insert the operator $\phi$ into the list at index $i \in \{0, \dots, n\}$ and contract it with an existing uncontracted index $j$ of $\Lambda$. If the field statistic of $\phi$ is the same as that of the $j$-th field operator (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_j$), then the statistical sign factor $\text{signInsertSomeProd}$ can be expressed as a product over all indices $x \in \{0, \dots, n-1\}$: $$\text{signInsertSomeProd}(\phi, \phi_s, \Lambda, i, j) = \prod_{x=0}^{n-1} V(x)$$ where $V(x) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \phi_x)$ if $x$ is part of a contracted pair in $\Lambda$ and the following condition holds: $$x < j \quad \text{and} \quad \left( (x < i \le \text{dual}(x)) \quad \text{or} \quad (x \ge i \quad \text{and} \quad \text{dual}(x) > j) \right)$$ Otherwise, $V(x) = 1$. Here, $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of operator $\phi$, $\mathcal{S}(\cdot, \cdot)$ denotes the statistical sign, and $\text{dual}(x)$ denotes the index paired with $x$ in the contraction $\Lambda$.

Let $\mathcal{F}$ be a field specification, $\phi$ a field operator, and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ a list of field operators. Let $\Lambda$ be a Wick contraction on the indices $\{0, 1, \dots, n-1\}$ that is grading-compliant. Suppose we insert the operator $\phi$ into the list at index $i \in \{0, \dots, n\}$ and contract it with an existing uncontracted index $j$ of $\Lambda$. If the field statistic of $\phi$ is the same as that of the $j$-th field operator ($\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_j$), then the statistical sign factor $\text{signInsertSomeProd}$ is equal to the statistical sign between $\phi$ and the collective statistic of a subset of fields from $\phi_s$: $$\text{signInsertSomeProd}(\phi, \phi_s, \Lambda, i, j) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{ofFinset } (\mathcal{F} \triangleright_s \cdot) \text{ } \phi_s \text{ } S)$$ where $S \subseteq \{0, \dots, n-1\}$ is the set of indices $x$ such that $x$ is part of a contracted pair in $\Lambda$ with partner $y = \text{dual}(x)$, and the following conditions are satisfied: $$x < j \quad \text{and} \quad \left( (x < i \le y) \quad \text{or} \quad (x \ge i \quad \text{and} \quad y > j) \right)$$ Here, $\mathcal{F} \triangleright_s \phi$ denotes the field statistic (bosonic or fermionic) of $\phi$, $\mathcal{S}(\cdot, \cdot)$ is the statistical sign, and $\text{ofFinset}$ calculates the collective statistic (the product in the group $\text{FieldStatistic} \cong \mathbb{Z}_2$) of the fields indexed by $S$.

Let $\mathcal{F}$ be a field specification. Let $\phi$ be a field operator and $\phi_s$ be a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$, and let $j$ be an uncontracted index of $\Lambda$. Suppose we insert $\phi$ into $\phi_s$ at index $i \in \{0, \dots, n\}$ and contract it with the field operator originally at index $j$. Let $\phi_s'$ be the resulting list of length $n+1$ and $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$ be the new Wick contraction. Assume that the field statistic of the inserted operator $\phi$ is equal to the field statistic of the operator at index $j$ in the original list, i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_s[j]$. Let $j' = \text{succAbove}_i(j)$ be the index of the previously uncontracted operator in the new list $\phi_s'$. The statistical sign coefficient $\text{signInsertSomeCoef}(\phi, \phi_s, \Lambda, i, j)$ is given by: - If $i < j'$, the coefficient is $\mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s'}(S_{\Lambda'}(i, j')))$. - Otherwise (if $j' \le i$), the coefficient is $\mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s'}(S_{\Lambda'}(j', i)))$. where: - $\mathcal{S}(s_1, s_2)$ is the statistical sign function (returning $-1$ if both $s_1, s_2$ are fermionic, and $1$ otherwise). - $S_{\Lambda'}(i_1, i_2)$ is the set of indices strictly between $i_1$ and $i_2$ that are not contracted with any index $k \le i_1$ in the contraction $\Lambda'$. - $\text{stat}_{\mathcal{F}, \phi_s'}(A)$ is the collective field statistic of the set of operators in $\phi_s'$ indexed by $A$.

Let $\mathcal{F}$ be a field specification, $\phi$ a field operator, and $\phi_s$ a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$, and let $j$ be an uncontracted index of $\Lambda$. Suppose we insert $\phi$ into $\phi_s$ at index $i \in \{0, \dots, n\}$ and contract it with the operator originally at index $j$, resulting in a new list $\phi_s'$ of length $n+1$ and a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$. Let $j' = \text{succAbove}_i(j)$ be the shifted index of the contraction partner in the new list. The collective field statistic of the operators in $\phi_s'$ indexed by the sign-contributing set $S_{\Lambda'}(i, j')$ (the set of indices $k$ such that $i < k < j'$ and $k$ is not contracted with any index $\le i$ in $\Lambda'$) is equal to the collective field statistic of the operators in the original list $\phi_s$ indexed by the set of indices $x \in \{0, \dots, n-1\}$ satisfying the following conditions: 1. $i \le x < j$ 2. $x$ is either uncontracted in $\Lambda$, or its contraction partner $y$ in $\Lambda$ satisfies $y \ge i$. Mathematically, this identity is expressed as: $$\text{stat}_{\mathcal{F}, \phi_s'}(S_{\Lambda'}(i, j')) = \text{stat}_{\mathcal{F}, \phi_s}(\{x \mid i \le x < j \text{ and } (x \in \text{uncontracted}(\Lambda) \text{ or } \text{dual}_\Lambda(x) \ge i)\})$$ where $\text{stat}_{\mathcal{F}, \phi_s}$ denotes the product of the field statistics (bosonic or fermionic) of the selected operators.

Let $\mathcal{F}$ be a field specification, $\phi$ a field operator, and $\phi_s$ a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$, and let $j$ be an uncontracted index of $\Lambda$. Suppose we insert $\phi$ into $\phi_s$ at index $i \in \{0, \dots, n\}$ and contract it with the operator originally at index $j$, resulting in a new list $\phi_s'$ of length $n+1$ and a new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } j$. Let $j' = \text{succAbove}_i(j)$ be the shifted index of the contraction partner in the new list. The collective field statistic of the operators in $\phi_s'$ indexed by the sign-contributing set $S_{\Lambda'}(j', i)$ (the set of indices $k$ such that $j' < k < i$ and $k$ is not contracted with any index $\le j'$ in $\Lambda'$) is equal to the collective field statistic of the operators in the original list $\phi_s$ indexed by the set of indices $x \in \{0, \dots, n-1\}$ satisfying the following conditions: 1. $j < x < i$ 2. $x$ is either uncontracted in $\Lambda$, or its contraction partner $y$ in $\Lambda$ satisfies $y > j$. Mathematically, this identity is expressed as: $$\text{stat}_{\mathcal{F}, \phi_s'}(S_{\Lambda'}(j', i)) = \text{stat}_{\mathcal{F}, \phi_s}(\{x \mid j < x < i \text{ and } (x \in \text{uncontracted}(\Lambda) \text{ or } \text{dual}_\Lambda(x) > j)\})$$ where $\text{stat}_{\mathcal{F}, \phi_s}$ denotes the product of the field statistics (bosonic or fermionic) of the selected operators.

Let $\mathcal{F}$ be a field specification. Let $\phi$ be a field operator and $\phi_s$ a list of field operators of length $n$. Let $\Lambda$ be a Wick contraction on $\phi_s$, and let $j$ be an uncontracted index of $\Lambda$. Suppose we insert $\phi$ into $\phi_s$ at index $i \in \{0, \dots, n\}$ and contract it with the field operator originally at index $j$. Assume that the field statistic of the inserted operator $\phi$ is equal to the field statistic of the operator at index $j$ in the original list, denoted $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_s[j]$. Let $j' = \text{succAbove}_i(j)$ be the shifted index of the contraction partner in the new list. The statistical sign coefficient $\text{signInsertSomeCoef}(\phi, \phi_s, \Lambda, i, j)$ is given by: - If $i < j'$ (which implies $i \le j$ in the original indexing), the coefficient is $\mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s}(A))$, where $A$ is the set of original indices $x \in \{0, \dots, n-1\}$ such that: 1. $i \le x < j$ 2. $x$ is either uncontracted in $\Lambda$, or its contraction partner $y$ in $\Lambda$ satisfies $y \ge i$. - If $j' \le i$ (which implies $j < i$ in the original indexing), the coefficient is $\mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s}(B))$, where $B$ is the set of original indices $x \in \{0, \dots, n-1\}$ such that: 1. $j < x < i$ 2. $x$ is either uncontracted in $\Lambda$, or its contraction partner $y$ in $\Lambda$ satisfies $y > j$. Where: - $\mathcal{S}(s_1, s_2)$ is the statistical sign function, returning $-1$ if both statistics are fermionic and $1$ otherwise. - $\text{stat}_{\mathcal{F}, \phi_s}(X)$ is the collective field statistic (the product of statistics) of the operators in the original list $\phi_s$ indexed by the set $X$.

Let $\mathcal{F}$ be a field specification, $\phi$ be a field operator, and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\Lambda$ be a grading-compliant Wick contraction on the indices of $\phi_s$. Suppose $k$ is an uncontracted index in $\Lambda$, and the new operator $\phi$ is inserted into the list at index $i \in \{0, \dots, n\}$ to be contracted with the operator originally at index $k$. If the original index $k$ satisfies $k < i$ (which is equivalent to the condition $i.\text{succAbove}(k) < i$ in the shifted indexing) and the field statistic of $\phi$ is identical to that of $\phi_k$ (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_k$), then the statistical sign factor $\text{signInsertSome}$ satisfies the following identity: $$\text{signInsertSome}(\phi, \phi_s, \Lambda, i, k) \cdot \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s}(U_{\le k})) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s}(X_{< i}))$$ where: - $U_{\le k} = \{x \mid x \in \text{uncontracted}(\Lambda) \text{ and } x \le k\}$ is the set of indices of operators that are uncontracted in $\Lambda$ and appear at or before index $k$ in the original list. - $X_{< i} = \{x \mid 0 \le x < i\}$ is the set of all indices in the original list that appear before the insertion point $i$. - $\mathcal{S}(s_1, s_2)$ is the statistical sign function, returning $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise. - $\text{stat}_{\mathcal{F}, \phi_s}(A)$ is the collective field statistic (the product in the group of field statistics $\mathbb{Z}_2$) of the operators in $\phi_s$ indexed by the set $A$.

Let $\mathcal{F}$ be a field specification, $\phi$ be a field operator, and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\Lambda$ be a grading-compliant Wick contraction on the indices of $\phi_s$. Suppose $k$ is an uncontracted index in $\Lambda$, and the new operator $\phi$ is inserted at index $i \in \{0, \dots, n\}$ to be contracted with the operator originally at index $k$. If the insertion index $i$ is less than or equal to $k$ ($i \le k$) and the field statistic of $\phi$ is the same as that of $\phi_k$ (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_k$), then the statistical sign factor $\text{signInsertSome}$ satisfies the following identity: $$\text{signInsertSome}(\phi, \phi_s, \Lambda, i, k) \cdot \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s}(U_{<k})) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}_{\mathcal{F}, \phi_s}(X_{<i}))$$ where: - $U_{<k} = \{x \mid x \text{ is uncontracted in } \Lambda \text{ and } x < k\}$ is the set of indices of fields that are uncontracted in the original list and appear before $k$. - $X_{<i} = \{x \mid x < i\}$ is the set of all indices in the original list that appear before the insertion point $i$. - $\mathcal{S}(s_1, s_2)$ is the statistical sign function, returning $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise. - $\text{stat}_{\mathcal{F}, \phi_s}(A)$ is the collective field statistic (the product in $\mathbb{Z}_2$) of the operators in $\phi_s$ indexed by the set $A$.

Let $\mathcal{F}$ be a field specification, $\phi$ be a field operator, and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\Lambda$ be a grading-compliant Wick contraction on the indices of $\phi_s$. Suppose $k$ is an uncontracted index in $\Lambda$, and a new operator $\phi$ is inserted into the list at index $i \in \{0, \dots, n\}$ to be contracted with the operator originally at index $k$, forming a new contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } k$. If $k < i$ and the field statistic of $\phi$ matches that of $\phi_k$ (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_k$), then the statistical sign of the new contraction $\Lambda'$ is given by: $$\text{sign}(\Lambda') = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}(U_{\le k})) \cdot \mathcal{S}(\mathcal{F} \triangleright_s \phi, \text{stat}(X_{< i})) \cdot \text{sign}(\Lambda)$$ where: - $U_{\le k} = \{x \mid x \in \text{uncontracted}(\Lambda) \text{ and } x \le k\}$ is the set of indices of uncontracted operators in $\Lambda$ occurring at or before the original index $k$. - $X_{< i} = \{x \mid 0 \le x < i\}$ is the set of all indices in the original list before the insertion point $i$. - $\mathcal{S}(s_1, s_2)$ is the statistical sign function, which is $-1$ if both $s_1$ and $s_2$ are fermionic and $1$ otherwise. - $\text{stat}(A)$ is the aggregate field statistic (in $\mathbb{Z}_2$) of the operators in $\phi_s$ indexed by the set $A$.

Let $\mathcal{F}$ be a field specification, $\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators, and $\Lambda$ be a grading-compliant Wick contraction on $\Phi$. Let $\phi$ be a field operator to be inserted into $\Phi$ at index $i \in \{0, \dots, n\}$, and let $k$ be an index that was uncontracted in $\Lambda$. Suppose $i \le k$ (expressed formally as $\neg(\text{succAbove}_i(k) < i)$) and the field statistic of $\phi$ matches the field statistic of $\phi_k$ (i.e., $\mathcal{F} \triangleright_s \phi = \mathcal{F} \triangleright_s \phi_k$). The sign of the new Wick contraction $\Lambda' = \Lambda \hookleftarrow_\Lambda \phi, i, \text{some } k$, formed by inserting $\phi$ at index $i$ and contracting it with the shifted operator originally at index $k$, is given by: $$\text{sign}(\Lambda') = \mathcal{S}(\sigma(\phi), \sigma_{\text{unc}, <k}) \cdot \mathcal{S}(\sigma(\phi), \sigma_{\text{all}, <i}) \cdot \text{sign}(\Lambda)$$ where: - $\sigma(\phi)$ is the field statistic of $\phi$ ($\text{bosonic}$ or $\text{fermionic}$). - $\sigma_{\text{unc}, <k}$ is the collective field statistic of the operators in $\Phi$ that are uncontracted in $\Lambda$ and have indices strictly less than $k$. - $\sigma_{\text{all}, <i}$ is the collective field statistic of all operators in the original list $\Phi$ with indices strictly less than $i$. - $\mathcal{S}(s_1, s_2)$ is the statistical sign function, returning $-1$ if both statistics $s_1$ and $s_2$ are fermionic, and $1$ otherwise. - $\text{sign}(\Lambda)$ is the sign of the original Wick contraction.