Regular rank form of the finite-component phase dimension count

For each phase let ${\pi }_{\alpha }:U\to (0,\mathrm{\infty }{)}^2\times {\mathrm{\Omega }}_C$ select $(T,p,x^{\alpha })$, and write ${\tilde{\mu }}_i^{\alpha }={\mu }_i^{\alpha }\circ {\pi }_{\alpha }$. The same coordinate selection on increments is linear and satisfies $\mathrm{\parallel }{\pi }_{\alpha }h{\mathrm{\parallel }}_2\le \mathrm{\parallel }h{\mathrm{\parallel }}_2$. Substituting ${\pi }_{\alpha }h$ into the first-order expansion of ${\mu }_i^{\alpha }$ gives derivative $D{\tilde{\mu }}_i^{\alpha }(z)=D{\mu }_i^{\alpha }({\pi }_{\alpha }z){\pi }_{\alpha }$ and remainder $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, with zero remainder when ${\pi }_{\alpha }h=0$. Its derivative entries are selected entries of $D{\mu }_i^{\alpha }({\pi }_{\alpha }z)$ or zero; they are continuous because ${\pi }_{\alpha }$ does not increase distances and the original entries are continuous. For $P\ge 2$, $F$ has $q=C(P-1)$ scalar components, each a difference of these lifted maps. By [L2] their derivative rows are $D{\tilde{\mu }}_i^{\alpha }-D{\tilde{\mu }}_i^1$; the entries are continuous since the increment of each difference is bounded by the sum of the two entry increments. The finitely many scalar remainders assemble to a vector remainder $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, since its norm is at most $\sqrt{q}$ times their largest absolute value. Thus $F$ is $C^1$ by [F1]. The assumed $q\times q$ minor entails $1\le q\le n$, so [L1] gives local graph dimension $n-q=2+P(C-1)-C(P-1)=C-P+2$, including an isolated point if $q=n$.

If $P=1$, no phase-equality equations are indexed because there is no $\alpha \ge 2$. The local solution set is all of the open domain $U$, which has $n=2+(C-1)=C+1=C-P+2$ coordinates.

In the reactive variant put $q=0$ if $P=1$. The lifted maps constructed in step 1.1 are $C^1$ for every $P\ge 1$. By [L2], $D{A}_r(z)={\sum }_{i=1}^C{\nu }_i^{r}D{\tilde{\mu }}_i^1(z)$. For each derivative entry, its increment is bounded by the finite sum of $\mathrm{\mid }{\nu }_i^r\mathrm{\mid }$ times the corresponding continuous entry increments, which tends to zero. Thus the affinity derivative entries are continuous. As in step 1.1, the finitely many scalar remainders assemble to a vector remainder $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, so the combined map is $C^1$ by [F1] and has $q+R$ scalar equations. Since $1\le q+R\le n$, its stated minor lets [L1] give graph dimension $n-q-R=C-P+2-R$, with an isolated point if $q+R=n$. Linear independence of the ${\nu }^r$ identifies $R$ distinct reaction directions, while the nonsingular derivative minor remains a separate assumption.