A formal chiral operation glues over a pointed curve

Use the full local map of [F7] on every section of the localized two-input module. Its coefficients are finite principal parts for each bounded-weight input and fixed pole order. On regular constant inputs its principal part is precisely the finite sum defining ${\chi }_t$: [F1] makes $a_{(j)}b=0$ for large $j$, and the definition of ${\mathrm{PP}}_u$ gives its coefficients $a_{(j)}b$ directly, with the normalized delta notation fixed by [F3]. No identification of states with operator-valued commutator coefficients is made. On polar inputs one uses the full formula in [F7], including negative vertex products; singular coefficients alone are not substituted for a map on the localization. For a holomorphic finite-pole input, only finitely many normal Taylor coefficients contribute to each output coefficient, by the truncation and weight bounds in [F7]. Those coefficients are holomorphic along the diagonal. Thus these are actual maps on the meromorphic sheaves specified in the Statement, not merely maps on formal completions.

The right-$\mathcal{D}$ convention in [F7] has negative formal adjoint of ${\mathrm{\partial }}_t+T$, with $T=L(-1)$ by [F2]. Its skew-symmetry includes the canonical-bundle and normal-coordinate reversal signs. Its unit is the horizontal map ${\omega }_X\to V_X^r$, $fdt\mapsto fdt\otimes \mathbf{1}$: $T\mathbf{1}=0$ by [F1], and ${\mu }_t((t_1-t_2{)}^{-1}\mathbf{1},b)=b\delta (t_1,t_2)$ is the canonical residue quotient, not merely vanishing on regular unit inputs. Consequently both the full skew and unit axioms hold in each coordinate.

On an overlap write $t^{\mathrm{\prime }}=\rho (t)$ and put ${\rho }_y(u)=\rho (y+u)-\rho (y)$. Apply [F4] to the adjoint field, coefficientwise over the base ring in $y$. Its coefficients depend on finite jets on bounded weight stages, so the same identity is valid with these formal base coefficients. Since $({\rho }_y{)}_u={\rho }_{y+u}$, it gives the complete OPE covariance formula $R_V({\rho }_y)Y(a,u)b=Y(R_V({\rho }_{y+u})a,\rho (y+u)-\rho (y))R_V({\rho }_y)b\mathrm{.}$ Pair the principal-part map of [F7] with any regular normal test function and take the residue in the first point. Under the substitution, its first one-form factor transforms by $d{t}_1^{\mathrm{\prime }}={\rho }^{\mathrm{\prime }}(y+u)du$ and its remaining output one-form by $d{t}_2^{\mathrm{\prime }}={\rho }^{\mathrm{\prime }}(y)dy$. The state factors transform by exactly the operators in the displayed full-field identity. Residue invariance [F6], applied coefficientwise, therefore makes every normal-jet pairing agree in the two coordinates, including for polar source sections. These pairings separate finite principal parts, as in [F7]; hence the full local maps agree on overlaps. This uses full field covariance, not uniqueness of the singular commutator as an asserted determination of polar values.

The connection also agrees on overlaps in the declared sign. Apply the same covariance identity to the vacuum, fixed by $R_V$, and use [F1] to obtain $R_V({\rho }_y)e^{uT}a=e^{(\rho (y+u)-\rho (y))T}{R}_V({\rho }_{y+u})a\mathrm{.}$ Differentiation in $u$ at zero gives ${\mathrm{\partial }}_y{R}_V({\rho }_y)+{\rho }^{\mathrm{\prime }}(y)T{R}_V({\rho }_y)=R_V({\rho }_y)T\mathrm{.}$ This is exactly the transition identity for the left connection ${\mathrm{\partial }}_t+T$; tensoring with the canonical line and taking its negative formal adjoint gives the right action used in [F7]. The finite-jet coefficients of the local maps vary holomorphically in the base coordinate, and the torsor transitions satisfy the cocycle by [F5]. Thus the local right-$\mathcal{D}$ maps glue consistently, including on triple overlaps, to the stated global operation.

The full three-input chiral Jacobi identity in [F7] compares exactly the three localized composites, with the right-$\mathcal{D}$ signs established there from the all-integer vertex Jacobi residue identity. Since the local maps agree on overlaps by step 2.1, their composites agree on overlaps as well, and the local identity descends to the global operation. Step 1.2 supplies its skew-symmetry and the actual residue-unit axiom. This establishes every claimed global property with the local Jacobi identity proved by the separate supplier, retaining the Statement's full source, target and convention.