Condition (C) requires that, for every finite ordered family of Lie polynomials $P_1,\ldots,P_q$, there are $\alpha,C>0$ such that

The constants may depend on the family, but not on the exponents.

The inequalities $I!\leq |I|!\leq q^{|I|}I!$ show that replacing $I!$ by $|I|!$ gives the same condition. Empty families are allowed.

For a finite alphabet, a real noncommutative series admits an analytic differential representation if and only if it has finite Lie rank and satisfies (C). If its Lie rank is $d$, a representation of dimension exactly $d$ exists.

The realization is obtained in the normalized residual coordinates $T_i=g\circ Q_i-(g,Q_i)$. Convergence of the output and of the vector-field coefficients follows from their convergent coordinate expansions.

A series has Lie rank $d$ and satisfies (C) if and only if $d$ is the least dimension of its analytic differential representations. The least dimension is attained. Constant series are included with dimension zero.

Two analytic differential representations of a series of Lie rank $d$, both of dimension $d$, are related by an analytic change of coordinates fixing the origin. This change sends the first output to the second and intertwines the action of every word on every state series. Its inverse is also convergent and formally continuous.

The coefficient of $w$ in the shuffle of $S$ and $T$ is the sum of $S(w|I)T(w|I^c)$ over subsets of the positions of $w$. Repeated letters retain their multiplicities. This product is associative and commutative, with unit the series supported at the empty word. No concatenation product on series is used here.

Lemmas 1–5. The $p$-fold coproduct assigns each position of a word to one of $p$ tensor factors. Tensor polynomials are represented by finite linear combinations of $p$-tuples of words. Its value on a letter is the sum of the $p$ placements of that letter, and Lie polynomials are primitive.

Pairing with this coproduct computes the shuffle product. Letter residuals are derivations. A shuffle product of $i$ zero-constant series vanishes on a product of fewer than $i$ Lie polynomials. For equal numbers of factors, the pairing is the sum over permutations of pairwise evaluations.

For normalized coordinate series, equal-degree monomials have diagonal pairing $I!=\prod_i I_i!$ and off-diagonal pairing zero. This corrects the printed $|I|!$ in Lemma 5(ii).

Given any totally ordered basis of the Lie polynomials, the nondecreasing products of its basis elements, including the empty product, form a basis of the concatenation algebra $\mathbb R\langle X\rangle$. Linear independence and spanning are stated separately in the conjunction.

Suppose the output and all vector-field coefficients admit the same bound $\alpha C^{|I|}$, where $\alpha,C>0$. For a word of length $p$, its generated coefficient has absolute value at most $\alpha(2d\alpha C)^p p!$. Consequently every analytic representation on a finite alphabet has a uniform factorial bound on its word coefficients.

The central binomial estimate used in the majorant calculation is $\binom{2p}{p}\leq 4^p$, rather than the printed $2^p$.

The generating series of an analytic differential representation satisfies condition (C). A finite family of Lie polynomials acts by analytic vector fields. Applying the factorial estimate to these new fields, and comparing the total factorial with the product of factorials, gives the assertion.

Proposition 1. Let $L_0$ be a Lie subalgebra of the Lie polynomials and let $P_1,\ldots,P_d$ represent a basis modulo $L_0$. Set $J=\mathbb R\langle X\rangle L_0$. If the zero-constant series $S_i$ vanish on $J$ and satisfy $(S_i,P_j)=\delta_{ij}$, substitution of the $S_i$ into commutative formal series is a bijection onto $J^\perp$.

Substitution is defined coefficient by coefficient using a finite truncation: at a word of length $n$, only monomials of degree at most $n$ contribute. There exist such coordinates dual to the degree-one PBW monomials. For these coordinates the coefficient of $S^I$ in $T\in J^\perp$ is $(T,P^I)/I!$. The annihilator is closed under the shuffle and all right residuals.

Let $P_1,\ldots,P_d$ be a basis modulo a Lie subalgebra $L_0$, and let $S_1,\ldots,S_d$ be the associated PBW coordinates of $J^\perp$, where $J=\mathbb R\langle X\rangle L_0$. If $T\in J^\perp$ satisfies (C), its coordinate expansion is convergent. The coefficient of $S^I$ is $(T,P^I)/I!$, so (C) is precisely the required geometric bound for this expansion.

If $g$ satisfies (C), so do $g\circ Q$ and its centered series $g\circ Q-(g,Q)$ for every noncommutative polynomial $Q$. In particular the normalized observation coordinates used in part b) and all their letter residuals satisfy (C).

An analytic change of coordinates is a formally continuous real algebra automorphism fixing the origin whose coordinate images, in both directions, are convergent power series.

The analytic inverse function theorem implies that a formal change of coordinates with convergent coordinate images has convergent inverse images. This is the implicit function theorem used in part b) of the paper.

Let $T_1,\ldots,T_d$ be normalized, zero-constant coordinates on $J^\perp$ as in Proposition 1, each satisfying (C). Every element of $J^\perp$ satisfying (C) is a convergent series in the $T_i$.

Indeed, the $T_i$ are convergent series in PBW coordinates and have identity linear part. Their inverse change of coordinates is convergent by the analytic inverse function theorem.