Logarithmic reconstruction and parallel transport of P(z)-products

For homogeneous generalized-weight inputs, project the formula to output weight $h_3$. F2 expresses it as $x^{h_3-h_1-h_2}$ times a finite polynomial in $\log x$ with coefficients in $W_{3,[h_3]}$. Its logarithmic degree is at most $K_{W_1}+K_{W_2}+K_{W_3}-3$. For fixed input vectors the finitely many nilpotent powers appearing on the inputs permit a common truncation bound in each integer weight string by F1. Thus the formula has precisely the truncation and grading compatibility of F3. Its evaluation at $a=0$ is $I$.

The mode bracket in F2 gives $e^{aL(0)}Y(v,t)e^{-aL(0)}=Y(e^{aL(0)}v,e^{a}t)$, coefficient by coefficient. Apply $P(z)$-Jacobi for $I$ to the inputs $e^{-aL(0)}w,e^{-aL(0)}v$, the algebra state $e^{-aL(0)}u$, and variables $e^{-a}{x}_0,e^{-a}{x}_1$, and conjugate the output by $e^{aL(0)}$. Cancelling the common scalar $e^a$ in the three delta prefactors changes the fixed insertion to $e^{a}z=x$; for example the first delta becomes $x_0^{-1}\delta ((x_1-x)\mathrm{/}{x}_0)$. The algebra-state conjugation changes its inserted field into $Y(u,x_0)$. The other two deltas use the same substitution and unchanged binomial conventions. This is exactly logarithmic intertwiner Jacobi for the displayed formula. The substitutions are identities in formal powers and finite logarithmic polynomials; they do not use analytic convergence.

Differentiate the formula with respect to $a$. F1's $L(0)$ identity gives ${\mathrm{\partial }}_a{\mathcal{Y}}_I(w,x)v=z{e}^{aL(0)}I(L(-1)e^{-aL(0)}w\otimes e^{-aL(0)}v)$. The commutator $[L(0),L(-1)]=L(-1)$ implies $L(-1)e^{-aL(0)}=e^a{e}^{-aL(0)}L(-1)$, so the right side is $z{e}^a{\mathcal{Y}}_I(L(-1)w,x)v=x{\mathcal{Y}}_I(L(-1)w,x)v$. As ${\mathrm{\partial }}_x=x^{-1}{\mathrm{\partial }}_a$ on every output monomial and logarithmic polynomial, this proves the derivative axiom. Similarly $L(j)e^{aL(0)}=e^{aj}{e}^{aL(0)}L(j)$; conjugating the three brackets of F1 multiplies their $z^i$ terms by $e^{ai}$, producing exactly the $x^i$ terms of F3. All required axioms hold. Conversely F3's $L(0)$ bracket is the same differential equation in $a$ with initial value $I$; on each generalized weight the integrating-factor identity forces the displayed formula. This proves uniqueness.

Evaluation of ${\mathcal{Y}}_I$ at any chosen logarithm of $\zeta \mathrm{\ne }0$ gives a $P(\zeta )$-map: substitute the logarithm in the identities proved above, keeping the other variables formal. The change $I\mapsto e^{aL(0)}I(e^{-aL(0)}\otimes e^{-aL(0)})$, $a=\mathrm{\ell }(\zeta )-\mathrm{\ell }(z_0)$, has inverse with $-a$ and commutes with postcomposition by module maps. Thus it is a natural bijection between the two spaces of intertwining maps. Applying that bijection to F4 shows that the same representing object at $z_0$, with the changed universal map, represents the $P(\zeta )$-maps. This proves existence at every nonzero point. For chosen products at $z_1,z_2$, apply F6 to the covering in F5 and the reversed path ${\gamma }^{-1}$ starting at the chosen logarithm of $z_2$; denote its terminal value by ${\mathrm{\ell }}_{\gamma }(z_1)$. F4 gives a unique module map $T_{\gamma }$ with $\overline{T_{\gamma }}{\iota }_{z_1}={\mathcal{Y}}_{{\iota }_{z_2}}{\mathrm{\mid }}_{\log x={\mathrm{\ell }}_{\gamma }(z_1)}$. Because module maps commute with $L(0)$, reconstruction commutes with postcomposition. Thus this equation also proves naturality in both input objects, by applying F4 to the two compositions with the universal map.

To check composition without assuming an analytic connection, use the explicit formula. If $a_{12}={\mathrm{\ell }}_{\gamma }(z_1)-{\mathrm{\ell }}_{z_2}$ and $a_{23}={\mathrm{\ell }}_{\eta }(z_2)-{\mathrm{\ell }}_{z_3}$, then sequential conjugation by their exponentials equals conjugation by $a_{12}+a_{23}$. Uniqueness of path lifting in F6 identifies this sum with the logarithmic change obtained by lifting $\eta \ast \gamma$ backwards from $z_3$: lift $\eta$ backwards first, and then lift $\gamma$ with the resulting logarithm, shifting the previously chosen lift by ${\mathrm{\ell }}_{\eta }(z_2)-{\mathrm{\ell }}_{z_2}\in 2\pi i\mathbb{Z}$. Consequently $\overline{T_{\eta }\circ T_{\gamma }}{\iota }_{z_1}=\overline{T_{\eta \ast \gamma }}{\iota }_{z_1}$, and F4 implies equality. The constant path has logarithmic change zero by F6, so its map is the identity; reversing a lifted path gives the unique reverse lift, with opposite change, so both transport compositions are identities. Finally F7 applied to F5's covering and the reversed endpoint-fixed homotopic paths gives equal backwards lift endpoints. The defining equality in step 4.1 and uniqueness in F4 give identical transports. This proves every asserted transport property.