Charge-Sector Construction of the Type-IIB Axion--Dilaton Wormhole Partition Function

I construct the Type-IIB axion–dilaton wormhole partition function from charge-sector data. In a chosen axion charge, equivalently form-field flux sector, the long-distance saddle calculation supplies a two-end operator term with coefficient matrix $C_{\nu }^{ij}$. The labels $i,j$ label end-insertion operators; the labels $A,B$ label parent universes. Reduction data $b$ convert this matrix into scalar coefficients $W_{\nu }[b]$. The wormhole partition function in the theta variable is $Z_{\mathrm{w}\mathrm{h}}(\theta ;b)={\sum }_{\nu }{W}_{\nu }[b]{\text{e}}^{i\nu \theta }$. I analyze properties and constraints this coefficients satisfy: discrete-symmetry covariance, phase, absolute bounds, moment positivity, Cauchy–Schwarz inequalities for the unreduced coefficient matrix, complex-$\theta$ domains, charge-lattice tails, and the dilute Bessel/Skellam limit. The $\theta$-dependence of the wormhole partition function is the Fourier transform of the charge-sector scalar coefficients.