Existence and uniqueness of complete filtered Maurer–Cartan gauge flow

Apply [F2] to the identity Taylor morphism of $W$. In its completed coalgebra, multiplication $M_w(x)=e^{w}x$ is a continuous coalgebra automorphism with inverse $M_{-w}$, because $\mathrm{\Delta }(e^{w}x)=(e^w\otimes e^w)\mathrm{\Delta }x$. Consequently $Q^w=M_{-w}Q{M}_w$ is a coderivation and $(Q^w{)}^2=0$. It may have a constant term, since $M_w$ does not preserve coaugmentation. Expanding the unshuffle formula and choosing the inputs in the distinguished $q$-block gives $q_0^w=C(w)$ and $q_n^w(v_1,\dots ,v_n)={\sum }_{j\ge 0}{q}_{n+j}(w^j,v_1,\dots ,v_n)\mathrm{/}j!$ for $n\ge 1$: the indistinguishable degree-zero $w$ inputs give exactly $1\mathrm{/}j!$. Every expression is finite modulo a fixed total weight, and hence converges.

If $x-y\in F^{r}P[t{]}^{\wedge }$ and $a\in F^1{P}^{-1}[t{]}^{\wedge }$, then $q_1^x(a)-q_1^y(a)\in F^{r+1}P[t{]}^{\wedge }$. The unary terms cancel; each term in the multilinear telescoping expansion of every remaining difference contains $x-y$ and $a$, giving filtration at least $r+1$. The other inputs have positive filtration. This calculation is finite modulo each $F^p$.

In particular $Q^w(1)=C(w)$ and $Q^w(a)=C(w)a+q_1^w(a)$. Applying $(Q^w{)}^2$ to $a$ and projecting to length one gives $(q_1^w{)}^2(a)+q_2^w(C(w),a)=0$: the length-two term $C(w)a$ contributes $q_2^w(C(w),a)$ and the length-one term contributes $(q_1^w{)}^2(a)$; other lengths do not survive projection. Applying the same operation to $1$ gives $q_1^w(C(w))=0$. These identities also hold over completed polynomial coefficients, quotient by quotient.

Set $w_0(t)=w_{\ast }$ and $w_{j+1}(t)=w_{\ast }+{\int }_0^t{q}_1^{w_j(u)}(a(u))du$. The first difference lies in $F^1$; step 1.2 and induction give $w_{j+1}-w_j\in F^{j+1}$. Thus the sequence stabilizes in each quotient, where all iterates are genuine polynomials. Completeness defines $w\in P^0[t{]}^{\wedge }$. Reduction, integration and differentiation commute, so $w(0)=w_{\ast }$ and $w^{\mathrm{\prime }}=q_1^w(a)$.

If two solutions have the same initial value, their difference obeys the difference of the integral equations. It first lies in $F^1$, and step 1.2 forces it successively into every $F^r$. Separation gives equality.

Termwise differentiation in each finite quotient gives $\frac{d}{dt}C(w)=q_1^w(w^{\mathrm{\prime }})$. Hence $E=C(w)$ satisfies $E^{\mathrm{\prime }}=-q_2^w(E,a)$ by step 2.1, and $E(0)=0$. Its integral equation is $E(t)=-{\int }_0^t{q}_2^{w(u)}(E(u),a(u))du$. Initially $E\in F^1$; if $E\in F^r$, the right side lies in $F^{r+1}$ because $a\in F^1$. Induction and separation give $E=0$, completing both existence on the Maurer–Cartan locus and uniqueness.