Declared local state-sum sewing input

A declared local state-sum sewing input for one closed collar consists of a complex graded intermediate state space $\mathcal{H}$, a nondegenerate bilinear pairing, and specified paired weight eigenvectors $(e_{\alpha },e^{\alpha }{)}_{\alpha \in I}$ with $⟨e_{\alpha },e^{\beta }⟩={\delta }_{\alpha }^{\beta }$. The index set $I$ is finite or countable. Completeness means that these paired vectors give the prescribed identity expansion on the model's stated state domain; for an infinite basis the topology and domain on which this expansion converges are supplied data. For a holomorphic family the vectors, pairings and domains are supplied at each parameter value. A model requiring an uncountable spectral integral must instead supply its own measure/integral sewing input.

Fix all other component moduli and external insertions. Supply an open continuation domain $D\subseteq {\mathbb{C}}^d$, a parameter $\lambda \in D$, the sector, holomorphic dimensionless eigenweights $h_{\alpha }(\lambda ),{\bar{h}}_{\alpha }(\lambda )$, and fixed intercepts $a,\bar{a}$. The bar labels the independent right-moving weight, not complex conjugation in $\lambda$. Set ${\delta }_{\alpha }=h_{\alpha }-a$, ${\bar{\delta }}_{\alpha }={\bar{h}}_{\alpha }-\bar{a}$, $d_{\alpha }={\delta }_{\alpha }-{\bar{\delta }}_{\alpha }$, and ${\mathrm{\Delta }}_{\alpha }=({\delta }_{\alpha }+{\bar{\delta }}_{\alpha })\mathrm{/}2$. A term is level matched when $d_{\alpha }=0$; then ${\mathrm{\Delta }}_{\alpha }={\delta }_{\alpha }={\bar{\delta }}_{\alpha }$ is its level-matched exponent. Supply linear component amplitude functionals $A_L(\cdot ;\lambda )$ and $A_R(\cdot ;\lambda )$ on the indicated paired state domains, with holomorphic paired coefficients $C_{\alpha }(\lambda ):=A_L(e_{\alpha };\lambda )A_R(e^{\alpha };\lambda )$. Any graded pairing signs are included in this declared contraction.

Choose $0<ϵ<1$ and a branch $q=r{e}^{i\theta }$ with $0<r<ϵ$ and ${\theta }_0\le \theta <{\theta }_0+2\pi$, omitting the cut of planar measure zero. Supply the scalar collar coefficient $F(q,\lambda )$ after all other insertions and differentials are fixed. The sewing assumption is the equality $F(q,\lambda )={\sum }_{\alpha \in I}{C}_{\alpha }(\lambda )P_{\alpha }(q,\lambda ),P_{\alpha }(q,\lambda ):=\exp (2{\mathrm{\Delta }}_{\alpha }(\lambda )\log r+i\theta d_{\alpha }(\lambda ))\mathrm{.}$ The full collar density is $F(q,\lambda )d^{2}q\mathrm{/}\mathrm{\mid }q{\mathrm{\mid }}^2$, with $d^{2}q=dxdy$ and dimensionless $q$; thus $P_{\alpha }=q^{{\delta }_{\alpha }}{\bar{q}}^{{\bar{\delta }}_{\alpha }}$ is exactly the supplied propagator coefficient. For a countable basis require absolute and locally uniform convergence jointly in $q$ and $\lambda$ on compact subsets of this punctured branch collar times $D$. For a single-valued closed channel additionally require integer $d_{\alpha }$; angular projection is used only under that condition.

For every compact parameter set $K\subset D$, require the explicit set $S_K:=\{\alpha \in I:{\mathrm{\Delta }}_{\alpha }(\lambda )=0\text{ for some }\lambda \in K\}$ to be finite. Define the remainder coefficient, with the same reference measure as $F$, by $R_K(q,\lambda ):=F(q,\lambda )-{\sum }_{\alpha \in S_K}{C}_{\alpha }(\lambda )P_{\alpha }(q,\lambda )={\sum }_{\alpha \mathrm{\notin }{S}_K}{C}_{\alpha }(\lambda )P_{\alpha }(q,\lambda )\mathrm{.}$ Require a measurable majorant $G_K(q)\ge 0$ with $\mathrm{\mid }{R}_K(q,\lambda )\mathrm{\mid }\le G_K(q)$ for every $\lambda \in K$ and ${\int }_{0<\mathrm{\mid }q\mathrm{\mid }<ϵ}{G}_K(q)d^{2}q\mathrm{/}\mathrm{\mid }q{\mathrm{\mid }}^2<\mathrm{\infty }$. The integrated remainder is the absolutely convergent scalar integral $H_K(\lambda ):={\int }_{0<\mathrm{\mid }q\mathrm{\mid }<ϵ}{R}_K(q,\lambda )d^{2}q\mathrm{/}\mathrm{\mid }q{\mathrm{\mid }}^2$ for $\lambda \in K$; require it to be the restriction of a supplied holomorphic function on an open neighborhood of $K$ in $D$, agreeing with the integral wherever that integral is defined there. These are separate hypotheses: avoiding zeros of ${\mathrm{\Delta }}_{\alpha }$ does not by itself imply integrability. For finite $I$ the series convergence is automatic, but the same remainder and continuation requirements remain. The sewing equality and all analytic conditions are per-model inputs, not consequences of the name CFT.

String channel propagator on a plumbing collar

This is an explicit conditional interface introduced to expose the planned theorem's local sewing hypothesis. It is not a duplicate proof of CFT sewing convergence and is not a global string-field or S-matrix construction.