The outgoing mode algebra is coordinate independent and acts

A global right derivative restricts in a formal coordinate $t_i$ to a local exact form $D_{t_i}(a\otimes fd{t}_i)=Ta\otimes fd{t}_i+a\otimes f^{\mathrm{\prime }}d{t}_i$. Its local mode class is therefore zero in the quotient of [F2]. Each finite-pole section has a Laurent expansion with a finite lower bound, so every local expansion is defined. Thus changing a global de Rham representative does not change any local component, and the map descends to the cokernel in [F1].

On a coordinate overlap, [F4] carries the state coefficient and relative coordinate by the same Aut O transition; the one-form factor transforms by the formal Jacobian. Its residue is unchanged by [F3]. Hence the two local mode classes are conjugate by the transition action, so the outgoing image is independent of the chosen formal coordinates.

Restrict the operation in [F5] to regular two-input sections and take its relative de Rham residue in the first variable: locally this quotients finite delta derivatives by right derivatives in that variable, killing every positive normal derivative and retaining the coefficient of $\delta$. For local expressions $a={\sum }_r{a}_r{f}_r(t)dt$ and $b={\sum }_s{b}_s{g}_s(t)dt$, it gives $B_t(a,b)={\sum }_{r,s,j\ge 0}((a_r{)}_{(j)}{b}_s)f_r^{(j)}{g}_{s}dt\mathrm{/}j!$. The state sums are finite, and uniform field truncation on their finite-dimensional coefficient spaces makes the sum over $j$ finite. Relative residue is invariant under coordinate change by [F3], with state transitions supplied by [F4] and the full operation in [F5]; hence these expressions glue. They have bounded energy and only finite-order poles at the marked points, because only finitely many derivatives and products occur. Right-$\mathcal{D}$ linearity makes a derivative in the first input a total first-variable derivative, whose relative residue is zero; a derivative in the second input remains a global right derivative on the output. Thus $B$ descends to the specified global-section cokernel, rather than requiring an identification with all analytic de Rham cohomology. Skew-symmetry and Jacobi follow by taking the same relative residues of the identities in [F5]. Restriction to a puncture gives exactly the bracket in [F2], so the outgoing map is a Lie homomorphism and its image is a Lie subalgebra.

By [F6], each local component acts on its corresponding module and its sum on any pure tensor is a finite sum over the finite set of marked points. The algebraic tensor product is spanned by pure tensors, so this rule extends uniquely by linearity. If there are no marked points, the target product is the zero Lie algebra and the empty sum is the zero action.