13 The four-point residue: fusion, area, and an explicit density · Polyhedra, Moduli, and Quantum Geometry

13.1 The two spectral pinches and their coefficients

Put the fields with charges $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ at $0,1,z,\infty$ , respectively. Throughout this section $0<b<1,\qquad Q=b+b^{-1},\qquad \alpha_i=\frac{\mu_i}{2b},\qquad 0<\mu_i<1,\qquad \sum_i\mu_i=2.$ (13.1) To take a transverse residue, vary the charges near this locus and write $\varepsilon=\sum_i\alpha_i-Q+b,\qquad a_s=\alpha_1+\alpha_3.$ (13.2) The $s$ -channel pairs the marks $0,z$ ; its spectral integrand contains $C(\alpha_1,\alpha_3,Q-\alpha) C(\alpha,\alpha_2,\alpha_4) \mathcal F_\alpha(z)\mathcal F_\alpha(\bar z).$ For real charges the last product is the absolute square. For continuation in complex charges, both chiral factors are continued analytically, rather than conjugating the variable charges. We use the DOZZ/KRV normalization fixed in Section 12.

The bootstrap identity is established in an initial convergence domain by [60, Theorem 1.1]. Its continuation requires the discrete terms created by crossing poles, as the paragraph following that theorem also explains. The relevant contour mechanism and pole families are described in [95, Section 7.2, eqs. (87)–(90)].

Analytic input for the spectral derivation.

The infinite integral requires a separate continuation statement. In the present charge convention and ordering, the initial real bootstrap domain has $\alpha_i<Q/2$ , $\alpha_1+\alpha_3>Q/2$ , and $\alpha_2+\alpha_4>Q/2$ . The one-screening locus $\sum_i\alpha_i=Q-b$ lies outside that domain. Meromorphic dependence of each integrand factor alone does not justify continuing the integral. The spectral derivation below uses the following sufficient input, at fixed $0<b<1$ .

Continue from the initial domain along a specified path of complex charges, retaining every residue of a pole crossed by the contour. Locally, the deformed contours are required to coincide with $Q/2+i\mathbb R$ outside a fixed compact set. Near a generic target point, separate the finitely many pinch contributions, including their reflected partners, and write the remaining term using a locally fixed contour $\Gamma_{\mathrm{reg}}$ avoiding all poles. Its integrand $\mathcal I_{\mathrm{reg}}(\alpha;\varepsilon,\eta,z)$ must be holomorphic in the charge coordinates through $\varepsilon=0$ ; here $\eta$ denotes the remaining charge coordinates. Any additional discrete terms left in this remainder must also be holomorphic there. Require normal absolute convergence: on every compact set $K$ of these complex charge parameters and of noncolliding $z$ in the chosen branch chart, there is $G_K\in L^1(\mathbb R)$ such that on the common contour ends $\sup_{(\varepsilon,\eta,z)\in K} \left|\mathcal I_{\mathrm{reg}}(Q/2+iP;\varepsilon,\eta,z)\right| \le G_K(P).$ (13.3) The same normal-convergence requirement is imposed on compact continuation neighborhoods along the path, after their crossed-pole terms are separated. If another contour prescription moves the ends, the connecting pieces at infinity must additionally have integrals tending to zero uniformly on those parameter sets.

Under these conditions, the truncated regular integrals are holomorphic in $\varepsilon$ and converge locally uniformly as the spectral cutoff tends to infinity. Their limit is therefore holomorphic through $\varepsilon=0$ , and Cauchy’s residue integral in that parameter commutes with removal of the cutoff. The regular term contributes no residue; only the finite pinch terms do.

This is an additional analytic input for the contour proof. No estimate (13.3) is proved here, and the cited bootstrap theorem in its initial domain is not being used as such an estimate at the one-screening locus. The residue identity below also has an independent first-moment and period derivation. The curvature identity and finite fusion formulas use that derivation. Balanced limits are taken in the finite period expressions; no uniform spectral estimate through further pole coalescences or as $b\to0$ is asserted.

The local pinch calculation.

Two opposite pole families are $\begin{aligned} \alpha&=a_s+mb+n/b,\\ \alpha&=Q-\alpha_2-\alpha_4-m'b-n'/b =a_s+b-\varepsilon-m'b-n'/b, \end{aligned}$ with nonnegative integer indices. At $\varepsilon=0$ they collide only if $(m+m'-1)b^2+(n+n')=0.$ Since $0<b^2<1$ , this forces $m+m'=1$ and $n=n'=0$ . Thus the two intermediate charges are $\boxed{\quad \alpha_s^{(0)}=a_s,\qquad \alpha_s^{(1)}=a_s+b.\quad}$ (13.5) Their reflected partners $Q-\alpha_s^{(j)}$ are included in the usual unfolded contour. We first work at generic weights and then take the regularized limits at coincident poles.

The coefficients require only elementary special-function identities. Define $\mathfrak g(x)=\frac{\Gamma(x)}{\Gamma(1-x)}.$ (13.6) For a three-point function with $\Sigma=\alpha_1+\alpha_2+\alpha_3$ , the zero- and one-screening residues, with respect to the displayed sum coordinate, are $\begin{aligned} \operatorname*{Res}_{\Sigma=Q}C&=1,\\ \operatorname*{Res}_{\Sigma=Q-b}C &=-\frac{\pi\Lambda_L} {\prod_{j=1}^3\mathfrak g(2b\alpha_j)}. \end{aligned}$ For completeness these follow directly from DOZZ. Its denominator is $\Upsilon_b(\Sigma-Q)\prod_j\Upsilon_b(\Sigma-2\alpha_j)$ ; use $\Upsilon_b(Q-x)=\Upsilon_b(x),\qquad \Upsilon_b(x+b)=\mathfrak g(bx)b^{1-2bx}\Upsilon_b(x).$ At $\Sigma=Q$ reflection cancels the other factors. At $\Sigma=Q-b=1/b$ , the product of shifted ratios is $b/\prod_j\mathfrak g(2b\alpha_j)$ . Combining it with the DOZZ prefactor and $\mathfrak g(b^2)\mathfrak g(-b^2)=-b^{-4}$ proves the second identity. These are the conventions of [95, eqs. (9), (11), (13)].

Set $\delta_0=1-\mu_1-\mu_3,\qquad \delta_1=1-\mu_2-\mu_3,\qquad \delta_\infty=1-\mu_3-\mu_4,$ (13.8) and, initially assuming $\delta_0\ne0$ , define $h_0=\frac{\pi\mathfrak g(\delta_0)} {\mathfrak g(\mu_2)\mathfrak g(\mu_4)}, \qquad h_1=\frac{\pi\mathfrak g(-\delta_0)} {\mathfrak g(\mu_1)\mathfrak g(\mu_3)}.$ (13.9) At $\alpha=a_s$ , the left three-point function has its zero-screening pole and the right one its one-screening pole, with coefficient $-\Lambda_Lh_0$ . At $\alpha=a_s+b$ their roles reverse, with coefficient $-\Lambda_Lh_1$ . In the local coordinate $x=\alpha-\alpha_s^{(j)}$ the singular product is $\frac{\Lambda_Lh_j}{x(x+\varepsilon)}.$ The contour separating $x=-\varepsilon$ and $x=0$ gives the coefficient $-\Lambda_Lh_j/\varepsilon$ . In the unfolded $1/(4\pi i)$ spectral normalization the reflection partner supplies the other half of the local $1/(2\pi i)$ residue. Under the continuation and tail-control input above, the remaining contour integral is holomorphic in $\varepsilon$ and contributes no residue. The contour calculation therefore gives $\boxed{\quad \operatorname*{Res}_{\varepsilon=0}C_4(z) =-\Lambda_L\left( h_0|\mathcal F_0(z)|^2+h_1|\mathcal F_1(z)|^2\right). \quad}$ (13.10) The identity itself also follows without continuing the infinite spectral integral: combine the first-moment residue (12.8), in the transverse coordinate $\varepsilon=b(s+1)$ , with the independently proved period pairing (13.19) and the block factorization (13.12). This fixes the normalization and sign independently of the conditional contour derivation. Conventions for the overall normalization of correlations differ in the literature; the bootstrap normalization is calibrated by (13.7).

13.2 Why the screening integrals are the Virasoro blocks

Let $F(a,b;c;z)={}_2F_1(a,b;c;z)$ and take branches positive for $0<z<1$ , continued first to $\mathbb C\setminus(({-\infty},0]\cup[1,\infty))$ . The normalized $s$ -channel periods are $\begin{aligned} f_0(z)&=F(1-\mu_4,\mu_3;\mu_1+\mu_3;z),\\ f_1(z)&=z^{\delta_0} F(\mu_2,1-\mu_1;1+\delta_0;z). \end{aligned}$ The full normalized chiral blocks are $\mathcal F_j(z)=E_b(z)f_j(z),\qquad E_b(z)=z^{-\mu_1\mu_3/(2b^2)} (1-z)^{-\mu_2\mu_3/(2b^2)}.$ (13.12) Here the conformal weight is $\Delta(\alpha)=\alpha(Q-\alpha)$ and the central charge is $c_L=1+6Q^2$ .

To verify the block identification, the screening field has weight $\Delta(b)=b(Q-b)=1$ . Integration over a closed twisted cycle therefore commutes with the Virasoro action: its Ward commutator is a total derivative. A screening outside a circle separating $0,z$ from $1,\infty$ gives intermediate charge $a_s$ , while a screening inside gives $a_s+b$ . The corresponding Euler arc integrals are $\begin{aligned} S_0&=\int_1^\infty w^{-\mu_1}(w-1)^{-\mu_2}(w-z)^{-\mu_3}\mathop{}\!\mathrm dw =B(1-\mu_2,1-\mu_4)f_0(z),\\ S_1&=\int_0^z w^{-\mu_1}(1-w)^{-\mu_2}(z-w)^{-\mu_3}\mathop{}\!\mathrm dw =B(1-\mu_1,1-\mu_3)f_1(z). \end{aligned}$ The positive arc branches are phase-normalized versions of the twisted periods; closed Pochhammer contours justify the Ward-identity argument, with the same normalized functions. Dividing out the beta constants sets the leading coefficient to one. The leading exponents agree because $\Delta(a_s)-\Delta(\alpha_1)-\Delta(\alpha_3) =-2\alpha_1\alpha_3, \qquad \Delta(a_s+b)-\Delta(a_s)=\delta_0.$ The screened intertwiner construction and uniqueness of the normalized Ward solution identify the functions at every level; see [95, Section 18.1, eq. (224)]. Here $0<a_s,a_s+b<Q$ , so these internal weights are positive and avoid the degenerate Verma weights at $c_L>25$ . No external degenerate field is assumed.

The two blocks solve a second-order hypergeometric equation after removal of the common factor $E_b$ . This reduction is a consequence of the specified one-screening charge locus, not a general second-order equation for arbitrary four-point Liouville blocks.

13.3 The finite fusion matrix

For $\delta_1\ne0$ , the normalized $t$ -channel periods, pairing $z,1$ , are $\begin{aligned} g_0(z)&=F(1-\mu_4,\mu_3;\mu_2+\mu_3;1-z),\\ g_1(z)&=(1-z)^{\delta_1} F(1-\mu_2,\mu_1;1+\delta_1;1-z). \end{aligned}$ Their internal charges are $\alpha_2+\alpha_3$ and $\alpha_2+\alpha_3+b$ , and their full chiral blocks are $E_b g_j$ . Thus the same matrix relates periods and normalized screened Virasoro blocks: $\begin{pmatrix}f_0\\f_1\end{pmatrix} =\mathsf M\begin{pmatrix}g_0\\g_1\end{pmatrix},$ (13.15) where $\boxed{\quad \mathsf M=\begin{pmatrix} \dfrac{\Gamma(1-\delta_0)\Gamma(\delta_1)} {\Gamma(\mu_1)\Gamma(1-\mu_2)} &\dfrac{\Gamma(1-\delta_0)\Gamma(-\delta_1)} {\Gamma(\mu_3)\Gamma(1-\mu_4)}\\[10pt] \dfrac{\Gamma(1+\delta_0)\Gamma(\delta_1)} {\Gamma(1-\mu_3)\Gamma(\mu_4)} &\dfrac{\Gamma(1+\delta_0)\Gamma(-\delta_1)} {\Gamma(1-\mu_1)\Gamma(\mu_2)} \end{pmatrix}. \quad}$ (13.16) This is the Gauss connection formula, obtained by substituting $a=1-\mu_4$ , $b=\mu_3$ , $c=\mu_1+\mu_3$ in [54, eqs. 15.10.21–22]. The Wronskians give $\det\mathsf M=-\frac{\delta_0}{\delta_1}.$ (13.17) The matrix is independent of the Liouville parameter $b$ when the deficits $\mu_i$ are fixed. All $b$ -dependence in these blocks is then in the common factor $E_b$ and in the overall correlation normalization.

There is a simpler expression in actual arc coordinates. For real $0<z<1$ , put $\begin{aligned} T_0&=\int_{-\infty}^0 (-w)^{-\mu_1}(1-w)^{-\mu_2}(z-w)^{-\mu_3}\mathop{}\!\mathrm dw =B(1-\mu_1,1-\mu_4)g_0,\\ T_1&=\int_z^1 w^{-\mu_1}(1-w)^{-\mu_2}(w-z)^{-\mu_3}\mathop{}\!\mathrm dw =B(1-\mu_2,1-\mu_3)g_1. \end{aligned}$ Gamma reflection reduces (13.16) to $\boxed{\quad \begin{pmatrix}S_0\\S_1\end{pmatrix} =\frac1{\sin(\pi\delta_1)} \begin{pmatrix} \sin(\pi\mu_1)&-\sin(\pi\mu_3)\\ \sin(\pi\mu_4)&-\sin(\pi\mu_2) \end{pmatrix} \begin{pmatrix}T_0\\T_1\end{pmatrix}. \quad}$ (13.18) These arc pairs cease to be bases at the corresponding balanced weights. The divergent coefficients in this formula cannot be specialized there without first changing basis.

13.4 The Hermitian pairing and its direct verification

We now prove the area pairing directly from periods: $\boxed{\quad \mathcal A_\mu(z)=h_0|f_0(z)|^2+h_1|f_1(z)|^2. \quad}$ (13.19) Together with the first-moment residue, this supplies an independent derivation of (13.10). The complex beta integral is $\int_{\mathbb C}|w|^{-2p}|1-w|^{-2q}\mathop{}\!\mathrm d^2w =\pi\frac{\Gamma(1-p)\Gamma(1-q)\Gamma(p+q-1)} {\Gamma(p)\Gamma(q)\Gamma(2-p-q)},$ (13.20) for $0<p,q<1$ and $p+q>1$ . Introduce positive Laplace parameters for the two powers, perform the Gaussian integral in $w$ , and then use their sum and ratio as variables. The remaining integrals are an ordinary Gamma integral and $B(1-q,1-p)$ , proving the formula.

The area is a constant Hermitian form in the period basis. For $\delta_0\ne0$ , local monodromy has distinct eigenvalues $1,e^{2\pi i\delta_0}$ , so the form is diagonal in $(f_0,f_1)$ . If $\delta_0>0$ , sending $z\to0$ in the area integral and using (13.20) gives $h_0$ . If $\delta_0<0$ , rescaling $w=zu$ gives the leading coefficient $h_1$ . To determine the other coefficient, first assume also $\delta_1\ne0$ . Monodromy at $1$ makes the Hermitian form diagonal in $(g_0,g_1)$ , so the off-diagonal entry after the real connection matrix vanishes: $h_0\mathsf M_{00}\mathsf M_{01} +h_1\mathsf M_{10}\mathsf M_{11}=0.$ Every matrix entry here is nonzero at these real nonresonant weights. Hence either already determined coefficient fixes the other; Gamma reflection gives precisely (13.9). Continuity at $\delta_1=0$ , with $\delta_0\ne0$ fixed, completes this argument without crossing the singular $s$ -channel basis. For this continuity the area integrals are locally dominated: on compact sets of admissible weights their singular exponents stay below $2$ , and their decay at infinity stays faster than $|w|^{-2}$ .

The signs of $h_0,h_1$ are opposite. The pairing has signature $(1,1)$ ; the positive value of $\mathcal A_\mu$ on the period line is compatible with that indefinite signature. It is not a positive probability decomposition into two channels. In arc coordinates it becomes $\mathcal A_\mu= \frac{\sin(\pi\mu_2)\sin(\pi\mu_4)|S_0|^2 -\sin(\pi\mu_1)\sin(\pi\mu_3)|S_1|^2} {\sin(\pi\delta_0)}.$ (13.21) Define $H_s=\mathop{\mathrm{diag}}(h_0,h_1)$ and $H_t=\mathop{\mathrm{diag}}\left( \frac{\pi\mathfrak g(\delta_1)} {\mathfrak g(\mu_1)\mathfrak g(\mu_4)}, \frac{\pi\mathfrak g(-\delta_1)} {\mathfrak g(\mu_2)\mathfrak g(\mu_3)}\right).$ The connection preserves the area exactly: $\mathsf M^*H_s\mathsf M=H_t.$ (13.22) This follows either from the two area decompositions or directly from Gamma recurrence and reflection. On the specified real branch $\mathsf M^*=\mathsf M^T$ .

13.5 A closed formula for the four-point density and total volume

The area expression gives an explicit density in the conformal cross-ratio. In this section the period notation is $\omega_{\mathrm{DM},\mu}=\omega_{\mathrm{CH},\mu}$ and $V_{\mathrm{DM}}(\mu)=V_{\mathrm{CH}}(\mu)$ , with the same curvature- $-1$ normalization as (7.6). For a holomorphic column $\boldsymbol f=(f_0,f_1)^T$ and constant Hermitian matrix $H$ , differentiation gives $\partial_z\partial_{\bar z}\log(\boldsymbol f^*H\boldsymbol f) =\frac{\det H\,|f_0f_1'-f_0'f_1|^2} {(\boldsymbol f^*H\boldsymbol f)^2}.$ (13.23) The hypergeometric equation yields $W(f_0,f_1)=\delta_0 z^{-(\mu_1+\mu_3)}(1-z)^{-(\mu_2+\mu_3)}, \qquad -h_0h_1\delta_0^2 =\frac{\pi^2}{\prod_i\mathfrak g(\mu_i)}.$ (13.24) Using $\mathop{}\!\mathrm dz\wedge\mathop{}\!\mathrm d\bar z=-2i\mathop{}\!\mathrm dx\wedge\mathop{}\!\mathrm dy$ and the curvature- $-1$ convention therefore gives $\boxed{\quad \omega_{\mathrm{DM},\mu} =\frac{4\pi^2}{\prod_{i=1}^4\mathfrak g(\mu_i)} \frac{|z|^{-2(\mu_1+\mu_3)} |1-z|^{-2(\mu_2+\mu_3)}}{\mathcal A_\mu(z)^2} \mathop{}\!\mathrm dx\wedge\mathop{}\!\mathrm dy,\qquad z=x+iy. \quad}$ (13.25) The area in the denominator is explicitly the hypergeometric combination (13.19). Although that diagonal basis is singular at balanced weights, the density extends through them by its logarithmic limit.

Its integral can also be evaluated. At $z=0$ , if $\delta_0>0$ then $\mathcal A_\mu\to h_0$ , while if $\delta_0<0$ then $\mathcal A_\mu\sim h_1|z|^{2\delta_0}$ . In both cases the metric density has the form $\text{const.}\,|z|^{-2(1-|\delta_0|)}$ , giving cone angle $2\pi|\delta_0|$ . At $\delta_0=0$ the area has logarithmic growth and the end is a cusp. The same argument at $1,\infty$ gives the other two angles $2\pi|\delta_1|$ and $2\pi|\delta_\infty|$ . This agrees with the three-ended cone-sphere completion described in [66, Section 5].

Gauss–Bonnet, retaining the labels, gives the integral $\boxed{\quad V_{\mathrm{DM}}(\mu) =2\pi\bigl(1-|\delta_0|-|\delta_1|-|\delta_\infty|\bigr) =4\pi\min_{1\le i\le4}\{\mu_i,1-\mu_i\}. \quad}$ (13.26) For the last equality, order the weights as $x_1\le x_2\le x_3\le x_4$ and put $r=1-x_4$ . The sum constraint forces $x_2,x_3\ge r$ . The three absolute bipartition differences have sum $\sum_{j=1}^3|r-x_j| =1-r-x_1+|r-x_1|=1-2\min(r,x_1).$ This proves the stated minimum formula. Equal weights give $V_{\mathrm{DM}}=2\pi$ ; for example $\mu=(1/4,1/2,1/2,3/4)$ gives $V_{\mathrm{DM}}=\pi$ .

The integral here is of the smooth curvature on the open moduli space, with its cone/cusp completion. If one instead extends $\partial\bar\partial\log C_b^{\mathrm{uv}}$ as a distribution to the compactified sphere, the explicit pairwise powers have delta masses at collisions. Their logarithms are pluriharmonic only off those collisions.

13.6 Four equal weights: the logarithmic fusion calculation

When all $\mu_i=1/2$ , both $\delta_0$ and $\delta_1$ vanish. The functions $f_0,f_1$ coincide. On the representation side, $a_s+b=Q-a_s$ , so the two intermediate charges have the same Virasoro weight. The second solution is a regularized logarithmic, or parameter-derivative, block; it is not a second independent ordinary normalized highest-weight block at the same weight.

Use the independent periods $f(z)=F(1/2,1/2;1;z)=\frac2\pi K(z),\qquad g(z)=F(1/2,1/2;1;1-z)=\frac2\pi K(1-z).$ (13.27) Here $K$ uses the parameter convention of (7.10). Put $L=\log16$ and define $\ell_0=Lf-\pi g,\qquad \ell_1=Lg-\pi f.$ (13.28) The logarithmic expansion [54, Sections 15.10 and 19.12] gives $\ell_0(z)=f(z)\log z+ 4\sum_{m\ge1} \left(\frac{(1/2)_m}{m!}\right)^2 (H_{2m}-H_m)z^m,$ where $H_m$ is a harmonic number. Thus the holomorphic part has zero constant term. The analogous statement for $\ell_1$ uses $\log(1-z)$ . Solving (13.28) gives the finite connection matrix $\boxed{\quad \begin{pmatrix}f\\\ell_0\end{pmatrix} =\frac1\pi \begin{pmatrix}L&-1\\L^2-\pi^2&-L\end{pmatrix} \begin{pmatrix}g\\\ell_1\end{pmatrix}. \quad}$ (13.29) It has determinant $-1$ and its square is the identity.

The limiting area pairing is $\begin{aligned} \mathcal A(z) &=2\pi^2\operatorname{Re}(f\overline g) =8\operatorname{Re}(\overline{K(z)}K(1-z)), \\ &=2\pi L|f|^2-2\pi\operatorname{Re}(f\overline{\ell_0}). \end{aligned}$ The Hermitian matrices in the two displayed bases are $H_{fg}=\pi^2\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad H_{\log}=\begin{pmatrix}2\pi L&-\pi\\-\pi&0\end{pmatrix}.$ (13.31) The connection (13.29) preserves $H_{\log}$ exactly.

One can see the cancellation rather than just substitute into a singular formula. Take $\mu_1=\mu_3=(1-\delta)/2$ and $\mu_2=\mu_4=(1+\delta)/2$ . Then $h_0=\frac\pi\delta+\pi L+O(\delta),\qquad h_1=-\frac\pi\delta+\pi L+O(\delta),\qquad f_1-f_0=\delta\ell_0+O(\delta^2).$ The pole terms cancel in (13.19), leaving (13.30). This explains precisely how the regularized residue sector survives the coincidence of the ordinary blocks.

For counterclockwise loops about $0$ and $1$ , respectively, the period column $(f,g)^T$ has monodromies $\begin{pmatrix}1&0\\-2i&1\end{pmatrix},\qquad \begin{pmatrix}1&-2i\\0&1\end{pmatrix}.$ In the oriented torus-period basis $\Omega_1=4K(z)$ , $\Omega_2=4iK(1-z)$ , these become $N_0=\begin{pmatrix}1&0\\2&1\end{pmatrix},\qquad N_1=\begin{pmatrix}1&-2\\0&1\end{pmatrix}.$ (13.32) They give the expected projective $\Gamma(2)$ monodromy. The channel exchange $z\leftrightarrow1-z$ swaps $f,g$ and acts on $\tau=\Omega_2/\Omega_1$ by $\tau\mapsto-1/\tau$ . For oriented integral cycles this is a symplectic swap up to a common phase of the developing differential.

Finally, $\boxed{\quad \omega_{\mathrm{DM}} =\frac{4\pi^2}{|z(1-z)|^2\mathcal A(z)^2}\mathop{}\!\mathrm dx\wedge\mathop{}\!\mathrm dy, \qquad \mathcal A(z)=8\operatorname{Re}(\overline{K(z)}K(1-z)). \quad}$ (13.33) Indeed $W(f,g)=-1/[\pi z(1-z)]$ , or equivalently $\tau'=-i\pi/[4z(1-z)K(z)^2]$ . Near $z=0$ , $\mathcal A(z)=2\pi\log\frac{16}{|z|} +O\left(|z|\log\frac1{|z|}\right),\qquad \omega_{\mathrm{DM}}\sim \frac{\mathop{}\!\mathrm dx\wedge\mathop{}\!\mathrm dy} {|z|^2\log^2(16/|z|)}.$ This gives the integrable cusp density directly in the cross-ratio coordinate.

13.7 What the calculation establishes

The first-moment residue, the screened Virasoro blocks, and the independently computed Hermitian period pairing agree with all constants fixed. Under (13.3) and the stated contour-continuation conditions, the infinite spectral representation gives the same answer through its two pinches. The finite fusion transformation on the two-dimensional one-screening residue sector is the period connection matrix (13.16), with (13.29) at the fully balanced point. Its independent derivation requires no infinite-contour interchange. Matrix entries are not being identified with unspecified pointwise values of a bare $b$ - $6j$ function.

This completes the finite four-point residue calculation. Section 14 matches its quantum-group normalization explicitly, strengthens the Gram-matrix obstruction, and computes a classical cubic saddle for the separate compact-tetrahedron degeneration.