Local string amplitude sewing and channel factorization

Assume the local state-sum sewing input of Declared local state-sum sewing input on one closed node and the declared amplitude form, pairing, intercepts, and analytic continuation. Let $\mathrm{\Delta }$ be the local holomorphic continuation coordinate for a closed level-matched state's weight, with $\mathrm{\Delta }=L_0^m-a={\bar{L}}_0^m-\bar{a}$, and let $C_{\alpha }(\mathrm{\Delta })=A_L(e_{\alpha };\mathrm{\Delta })A_R(e^{\alpha };\mathrm{\Delta })$ be its holomorphic paired component coefficient. For $\mathrm{Re}\mathrm{\Delta }>0$, the collar integral over $0<\mathrm{\mid }q\mathrm{\mid }<ϵ$ contributes $C_{\alpha }(\mathrm{\Delta })\pi {ϵ}^{2\mathrm{\Delta }}\mathrm{/}\mathrm{\Delta }$, whose meromorphic continuation has at most a simple pole at $\mathrm{\Delta }=0$ with residue $\pi C_{\alpha }(0)$. Only level-matched terms contribute.

For a separate open real collar, assume its own declared complete open state-pairing sewing equality, with plumbing factor $dq\mathrm{/}q{q}^{\mathrm{\Delta }}$, where $\mathrm{\Delta }$ is the local holomorphic continuation coordinate for the open $L_0^m-a$ weight exponent. Require its complete dual basis $e_{\alpha },e^{\alpha }$ to be finite or countably infinite; for a countable basis require absolute and locally uniform convergence on compact subintervals of the stated collar. Define $C_{\alpha }^{\mathrm{o}\mathrm{p}}(\mathrm{\Delta }):=A_L^{\mathrm{o}\mathrm{p}}(e_{\alpha };\mathrm{\Delta })A_R^{\mathrm{o}\mathrm{p}}(e^{\alpha };\mathrm{\Delta })$ and require it to be holomorphic near $\mathrm{\Delta }=0$. On each compact continuation parameter set $K$, require the set of open states whose exponent can vanish on $K$ to be finite. After removing those terms, require the remaining open collar density $R^{\mathrm{o}\mathrm{p}}(q,\mathrm{\Delta })$ to satisfy $\mathrm{\mid }{R}^{\mathrm{o}\mathrm{p}}(q,\mathrm{\Delta })\mathrm{\mid }\le G_K(q)$ for $\mathrm{\Delta }\in K$, with measurable $G_K$ and ${\int }_0^{ϵ}{G}_K(q)dq\mathrm{/}q<\mathrm{\infty }$, and require its integrated remainder to extend holomorphically near $K$. These are additional open-channel inputs, not supplied by the closed collar hypothesis. The open collar term then integrates to $C_{\alpha }^{\mathrm{o}\mathrm{p}}(\mathrm{\Delta }){ϵ}^{\mathrm{\Delta }}\mathrm{/}\mathrm{\Delta }$ for $\mathrm{Re}\mathrm{\Delta }>0$ and has at most a simple pole at $\mathrm{\Delta }=0$ with residue $C_{\alpha }^{\mathrm{o}\mathrm{p}}(0)$. This is a one-node local result, not S-matrix unitarity.

Declared local state-sum sewing input

The A-page local sewing input is an adopted per-model hypothesis because the required general analytic sewing result is not a supplied carrier here. The theorem's local state-pairing deduction is complete conditional on it.