Z(\wvec{\vec{s}\,''}{\vec{r}\,''};\vep).
\mlabel{eq:qsh1}$$ We thus obtained an algebra homomorphism $$Z: \calh_\frakM \to \sum_{{\wvec{\vec{s}}{\vec{r}}\in \cup_{n\geq 0} \frakM^n}}\
    \CC\, Z(\wvec{\vec{s}}{\vec{r}};\vep),
    \wvec{\vec{s}}{\vec{r}} \mapsto Z(\wvec{\vec{s}}{\vec{r}};\vep).
\mlabel{eq:regz}$$ With this map, $\calh_\frakM$ is a parametrization of the directional regularized MZVs that also reflects their multiplication property.

Log Laurent series of directional regularized multiple zeta values
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We first construct Laurent series with log coefficients. We then show that the nested sums from directional regularized MZVs are such log Laurent series.

Let $\CC\{\{\vep,\vep^{-1}\}$ be the algebra of convergent Laurent series, regarded as a subalgebra of the algebra of (germs of) complex valued functions meromorphic in a neighborhood of $\vep=0$. Take $\ln \vep$ to be analytic on $\CC\backslash (-\infty,0]$.

$\ln (-\vep) $ is transcendental over $\CC\{\{\vep,\vep^{-1}\}$.

We give a simple proof for the lack of references. Assume $\ln (-\vep) $ is algebraic over the field $\CC\{\{\vep,\vep^{-1}\}$ with the monic minimal polynomial $$\ln ^n(-\vep)+a_{n-1}(\vep)\ln ^{n-1}(-\vep)+\cdots +a_0(\vep)=0.$$ Differentiating the above equation, we have $$\sum _{i=0}^n \big(a'_i(\vep )\ln ^i(-\vep)+\frac i{\vep}a_i(\vep)\ln ^{i-1}(-\vep )\big)=0.$$ The highest power term in $\ln (-\vep)$ is $(\frac
n{\vep}+a'_{n-1}(\vep ))\ln ^{n-1}(-\vep)$. Because of the minimality, $n/\vep+a'_{n-1}(\vep )$ has to be 0, which is impossible for $a_{n-1}(\vep )\in \CC\{\{\vep,\vep^{-1}\}$.

$\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$ is closed under the differential operator $d/d\vep$. It 