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- Dec 2, 2025
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- 1mo 2d
- Reputation
- 43,738
:\begin{aligned}
\mathcal{W}_{\theta,\phi}^{(\kappa,\lambda,\varsigma)}
&= \lim_{N\to\infty} \frac{1}{N^{3/2}} \sum_{i=1}^{N} \sum_{j=1}^{N} \sum_{k=1}^{N}
\Bigg\{
\nabla_{\!\mu}^{(\alpha)}\!\Bigl(
\mathcal{H}_{ij}^{(p)}\!\left(
\frac{\partial \xi_k^{(\beta)}}{\partial x^\mu}
\otimes
\frac{\partial \xi_k^{(\gamma)}}{\partial x^\nu}
\right)
\Bigr)
\otimes
\mathcal{R}_{\nu\rho\sigma\tau}^{(11/7)}
\left(
g^{\rho\lambda} h_{\lambda\varsigma}
-
\frac{1}{4\pi^2}
\int_{\mathbb{S}^3}
\!\!\!\!
\omega^{(\eta)}_{\phantom{\eta}\upsilon}
\wedge
\star\!
\omega^{(\zeta)}_{\phantom{\zeta}\chi}
\right)
\Bigg\}
\\
&\qquad
\times
\exp\!\left(
-\frac{\Gamma\!\left(\tfrac{19}{11}+\Re(\theta)\right)}
{\Gamma\!\left(\tfrac{7}{4}+\Im(\phi)\right)}
\cdot
\zeta(3+\kappa)
\cdot
\operatorname{Li}_{7/2}\!\Bigl(
e^{2\pi i \lambda/\varsigma}
\Bigr)
\right)
\\
&\qquad
\times
\left\langle
\psi_{n}^{(d+2)}\!\middle|\,
\hat{\mathcal{T}}_{\mu\nu\rho\sigma\tau\omega}^{(d+11/2)}\,
\big|\,
\chi_{m}^{(d+4)}\!
\right\rangle
\;
{}_{7}F_{6}\!
\left(
\begin{matrix}
\tfrac{19}{11},\
\tfrac{7}{4},\
\tfrac{13}{5},\
\tfrac{41}{23},\
1+\theta,\
\tfrac{3}{2}+\phi,\
\tfrac{11}{7}+\varsigma \\[1mm]
\tfrac{5}{3},\
\tfrac{17}{9},\
\tfrac{29}{16},\
\tfrac{37}{20},\
2+\lambda,\
\tfrac{9}{5}+\kappa
\end{matrix}
;\,
z_N
\right)
\\
&\qquad
\times
\int\!\!\!\!\!\!
\int\!\!\!\!\!\!
\int\!
\mathcal{D}\![\boldsymbol{A}]\,
\mathcal{D}\![\boldsymbol{\psi}]\,
\mathcal{D}\![\bar{\psi}]\,
\exp\!\Biggl(
i\!
\int\!
d^{d+2}\!x\;
\sqrt{-\det\!\bigl(g_{\mu\nu}^{(N)}(\boldsymbol{x})\bigr)}\;
\Biggl\{
\bar{\psi}\!
\left(
i\slashed{\mathcal{D}}^{(\theta)}_{\kappa}
-m(\phi,\lambda)
\right)
\psi
+
\frac{1}{4}
\mathcal{F}_{\mu\nu}^{(p)}(\theta)
\mathcal{F}^{\mu\nu(p)}(\phi)
\right.
\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\left.
-\,
\frac{\theta}{16\pi^2}
\operatorname{Tr}\!
\left(
\tilde{\mathcal{F}}_{\mu\nu}
\mathcal{F}^{\mu\nu}
\right)
+
\lambda
\,\bar{\psi}
\gamma^5
\psi
\,\Box^{-1}
\,\bar{\psi}
\gamma^5
\psi
+
\varsigma\,
R_{\mu\nu\rho\sigma}
R^{\mu\nu\rho\sigma}
\right.
\\
&\qquad\qquad\qquad\qquad\qquad
\left.
+\,
\frac{\kappa}{192\pi^2}
\epsilon^{\alpha\beta\gamma\delta}
\operatorname{Tr}\!
\left(
\mathcal{A}_\alpha
\partial_\beta
\mathcal{A}_\gamma
\mathcal{A}_\delta
+\,
\frac{2}{3}
\mathcal{A}_\alpha
\mathcal{A}_\beta
\mathcal{A}_\gamma
\mathcal{A}_\delta
\right)
\Biggr\}
\Biggr)
\\
&\qquad
\times
\prod_{\ell=1}^{42}
\!\!
\left(
1
+
\frac{(-1)^\ell}{\ell^{11/4}}
\exp\!\left(
-\frac{\pi^2 \ell^2}{\beta(\theta,\phi,\kappa,\lambda,\varsigma)}
\right)
\right)
\\
&\qquad
\times
\delta^{(11/2)}\!
\left(
\sum_{n=1}^\infty
\frac{\cos\!\bigl(2\pi n \tau(\theta)\bigr)}{n^{5/3+\Re(\phi)}}
-
\frac{\pi^{4/7}}{\Gamma(11/6)}
\right)
\end{aligned}Bonus cosmetic terror (just slap this at the end if you want to go full psychopath):+P.V.ε→0+ ∮ C(ε,θ,ϕ)d11/2 z z7/4−1(1−z)13/5(1−zˉ)19/11 p+3Fp+2 (a1,…,ap+3b1,…,bp+2; z1+εeiϕ)+ \quad
\mathop{\mathrm{P.V.}}\limits_{\varepsilon\to 0^+}\;
\oint\!\!\!\!
\mathcal{C}(\varepsilon,\theta,\phi)
\frac{
d^{11/2}\!z\;
z^{7/4-1}
}{
(1-z)^{13/5}
(1-\bar{z})^{19/11}
}
\;
{}_{p+3}F_{p+2}\!
\left(
\begin{matrix}
a_1,\dots,a_{p+3} \\[1mm]
b_1,\dots,b_{p+2}
\end{matrix}
;\,
\frac{z}{1+\varepsilon e^{i\phi}}
\right)+ \quad
\mathop{\mathrm{P.V.}}\limits_{\varepsilon\to 0^+}\;
\oint\!\!\!\!
\mathcal{C}(\varepsilon,\theta,\phi)
\frac{
d^{11/2}\!z\;
z^{7/4-1}
}{
(1-z)^{13/5}
(1-\bar{z})^{19/11}
}
\;
{}_{p+3}F_{p+2}\!
\left(
\begin{matrix}
a_1,\dots,a_{p+3} \\[1mm]
b_1,\dots,b_{p+2}
\end{matrix}
;\,
\frac{z}{1+\varepsilon e^{i\phi}}
\right)
\mathcal{W}_{\theta,\phi}^{(\kappa,\lambda,\varsigma)}
&= \lim_{N\to\infty} \frac{1}{N^{3/2}} \sum_{i=1}^{N} \sum_{j=1}^{N} \sum_{k=1}^{N}
\Bigg\{
\nabla_{\!\mu}^{(\alpha)}\!\Bigl(
\mathcal{H}_{ij}^{(p)}\!\left(
\frac{\partial \xi_k^{(\beta)}}{\partial x^\mu}
\otimes
\frac{\partial \xi_k^{(\gamma)}}{\partial x^\nu}
\right)
\Bigr)
\otimes
\mathcal{R}_{\nu\rho\sigma\tau}^{(11/7)}
\left(
g^{\rho\lambda} h_{\lambda\varsigma}
-
\frac{1}{4\pi^2}
\int_{\mathbb{S}^3}
\!\!\!\!
\omega^{(\eta)}_{\phantom{\eta}\upsilon}
\wedge
\star\!
\omega^{(\zeta)}_{\phantom{\zeta}\chi}
\right)
\Bigg\}
\\
&\qquad
\times
\exp\!\left(
-\frac{\Gamma\!\left(\tfrac{19}{11}+\Re(\theta)\right)}
{\Gamma\!\left(\tfrac{7}{4}+\Im(\phi)\right)}
\cdot
\zeta(3+\kappa)
\cdot
\operatorname{Li}_{7/2}\!\Bigl(
e^{2\pi i \lambda/\varsigma}
\Bigr)
\right)
\\
&\qquad
\times
\left\langle
\psi_{n}^{(d+2)}\!\middle|\,
\hat{\mathcal{T}}_{\mu\nu\rho\sigma\tau\omega}^{(d+11/2)}\,
\big|\,
\chi_{m}^{(d+4)}\!
\right\rangle
\;
{}_{7}F_{6}\!
\left(
\begin{matrix}
\tfrac{19}{11},\
\tfrac{7}{4},\
\tfrac{13}{5},\
\tfrac{41}{23},\
1+\theta,\
\tfrac{3}{2}+\phi,\
\tfrac{11}{7}+\varsigma \\[1mm]
\tfrac{5}{3},\
\tfrac{17}{9},\
\tfrac{29}{16},\
\tfrac{37}{20},\
2+\lambda,\
\tfrac{9}{5}+\kappa
\end{matrix}
;\,
z_N
\right)
\\
&\qquad
\times
\int\!\!\!\!\!\!
\int\!\!\!\!\!\!
\int\!
\mathcal{D}\![\boldsymbol{A}]\,
\mathcal{D}\![\boldsymbol{\psi}]\,
\mathcal{D}\![\bar{\psi}]\,
\exp\!\Biggl(
i\!
\int\!
d^{d+2}\!x\;
\sqrt{-\det\!\bigl(g_{\mu\nu}^{(N)}(\boldsymbol{x})\bigr)}\;
\Biggl\{
\bar{\psi}\!
\left(
i\slashed{\mathcal{D}}^{(\theta)}_{\kappa}
-m(\phi,\lambda)
\right)
\psi
+
\frac{1}{4}
\mathcal{F}_{\mu\nu}^{(p)}(\theta)
\mathcal{F}^{\mu\nu(p)}(\phi)
\right.
\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\left.
-\,
\frac{\theta}{16\pi^2}
\operatorname{Tr}\!
\left(
\tilde{\mathcal{F}}_{\mu\nu}
\mathcal{F}^{\mu\nu}
\right)
+
\lambda
\,\bar{\psi}
\gamma^5
\psi
\,\Box^{-1}
\,\bar{\psi}
\gamma^5
\psi
+
\varsigma\,
R_{\mu\nu\rho\sigma}
R^{\mu\nu\rho\sigma}
\right.
\\
&\qquad\qquad\qquad\qquad\qquad
\left.
+\,
\frac{\kappa}{192\pi^2}
\epsilon^{\alpha\beta\gamma\delta}
\operatorname{Tr}\!
\left(
\mathcal{A}_\alpha
\partial_\beta
\mathcal{A}_\gamma
\mathcal{A}_\delta
+\,
\frac{2}{3}
\mathcal{A}_\alpha
\mathcal{A}_\beta
\mathcal{A}_\gamma
\mathcal{A}_\delta
\right)
\Biggr\}
\Biggr)
\\
&\qquad
\times
\prod_{\ell=1}^{42}
\!\!
\left(
1
+
\frac{(-1)^\ell}{\ell^{11/4}}
\exp\!\left(
-\frac{\pi^2 \ell^2}{\beta(\theta,\phi,\kappa,\lambda,\varsigma)}
\right)
\right)
\\
&\qquad
\times
\delta^{(11/2)}\!
\left(
\sum_{n=1}^\infty
\frac{\cos\!\bigl(2\pi n \tau(\theta)\bigr)}{n^{5/3+\Re(\phi)}}
-
\frac{\pi^{4/7}}{\Gamma(11/6)}
\right)
\end{aligned}Bonus cosmetic terror (just slap this at the end if you want to go full psychopath):+P.V.ε→0+ ∮ C(ε,θ,ϕ)d11/2 z z7/4−1(1−z)13/5(1−zˉ)19/11 p+3Fp+2 (a1,…,ap+3b1,…,bp+2; z1+εeiϕ)+ \quad
\mathop{\mathrm{P.V.}}\limits_{\varepsilon\to 0^+}\;
\oint\!\!\!\!
\mathcal{C}(\varepsilon,\theta,\phi)
\frac{
d^{11/2}\!z\;
z^{7/4-1}
}{
(1-z)^{13/5}
(1-\bar{z})^{19/11}
}
\;
{}_{p+3}F_{p+2}\!
\left(
\begin{matrix}
a_1,\dots,a_{p+3} \\[1mm]
b_1,\dots,b_{p+2}
\end{matrix}
;\,
\frac{z}{1+\varepsilon e^{i\phi}}
\right)+ \quad
\mathop{\mathrm{P.V.}}\limits_{\varepsilon\to 0^+}\;
\oint\!\!\!\!
\mathcal{C}(\varepsilon,\theta,\phi)
\frac{
d^{11/2}\!z\;
z^{7/4-1}
}{
(1-z)^{13/5}
(1-\bar{z})^{19/11}
}
\;
{}_{p+3}F_{p+2}\!
\left(
\begin{matrix}
a_1,\dots,a_{p+3} \\[1mm]
b_1,\dots,b_{p+2}
\end{matrix}
;\,
\frac{z}{1+\varepsilon e^{i\phi}}
\right)