Principal-level arithmetic implies affine vacuum admissibility

Root heights. For a positive long root $\alpha ,\rho ({\alpha }^{\vee })\le \rho ({\theta }^{\vee })=h^{\vee }-1$ because theta-alpha is a nonnegative simple-root combination and rho pairs positively with every finite simple root under the normalized form. For a short root $\alpha ,\rho ({\alpha }^{\vee })=r(\rho ,\alpha )\le r(h^{\vee }-1)$. These bounds already suffice in the coprime case.

For the coprincipal case, the maximum short-coroot height is h-1. The finite irreducible classification leaves only $B_n,C_n,F4,G2$ when $r>1$. This numerical identity can be checked directly from the following simple-root and simple-coroot coordinates, in the usual arrow convention; the entries list theta, theta-coroot and theta_s-coroot respectively: B_n: (1,2,...,2), (1,2,...,2,1), (2,2,...,2,1); C_n: (2,...,2,1), (1,...,1), (1,2,...,2); F4: (2,3,4,2), (2,3,2,1), (2,4,3,2); G2: (3,2), (1,2), (2,3), with the first simple root short.

For $B_n$ use roots $\pm e_i,\pm e_i\pm e_j$; for $C_n$ use $\pm 2e_i,\pm e_i\pm e_j$; their standard simple differences and final simple root give the displayed coefficients by telescoping. For F4 use simple roots e2-e3,e3-e4,e4,(e1-e2-e3-e4)/2, with $\theta =e1+e2$ and ${\theta }_s=e1$; substitution gives all three rows. For G2 take alpha1 short, alpha2 long with Cartan off-diagonals -3,-1; the root-string list gives $\theta =3\alpha 1+2\alpha 2$ and ${\theta }_s=2\alpha 1+\alpha 2$, giving the last row. These are the corresponding finite classified diagrams, so they cover every non-simply-laced case, including B2=C2. For clarity, use $B_n$ for $n\ge 2$ and $C_n$ for $n\ge 3$, retaining $C_2=B_2$; their simple roots are $e_i-e_{i+1}$ followed by $e_n$ and $2e_n$, respectively. Their greatest positive short roots are $e_1$ and $e_1+e_2$. In $B_n$ the positive short roots are $e_i$, and $e_1-e_i$ is a nonnegative sum of simple differences. In $C_n$ they are $e_i-e_j$ and $e_i+e_j$ for $i<j$: their simple coordinates are respectively $1$ on $i,\dots ,j-1$, and $1$ on $i,\dots ,j-1$, $2$ on $j,\dots ,n-1$, $1$ on $n$. Each is coordinatewise bounded by $(1,2,\dots ,2,1)$, the coordinates of $e_1+e_2$. For $F_4$, the short roots are $\pm e_i$ and $\frac{1}{2}(\pm e_1\pm e_2\pm e_3\pm e_4)$ with all signs allowed. A vector with Euclidean coordinates $(x_1,x_2,x_3,x_4)$ has simple coordinates $(x_1+x_2,2x_1+x_2+x_3,3x_1+x_2+x_3+x_4,2x_1)$; for every short root these are bounded above by $(1,2,3,2)$, the coordinates of $e_1$. For $G_2$ with ${\alpha }_1$ short, its positive short roots are ${\alpha }_1,{\alpha }_1+{\alpha }_2,2{\alpha }_1+{\alpha }_2$. Hence the specified ${\theta }_s$ exists and dominates every positive short root in each case; antisymmetry of the simple-root order gives uniqueness. Coroot coordinates satisfy ${\beta }^{\vee }={\sum }_i{n}_i({\alpha }_i,{\alpha }_i)\mathrm{/}(\beta ,\beta ){\alpha }_i^{\vee }$ for $\beta ={\sum }_i{n}_i{\alpha }_i$. Thus the short-root coordinate bounds also bound every positive short coroot by ${\theta }_s^{\vee }$. Every long root is bounded by $\theta$, so its coroot is bounded by ${\theta }^{\vee }$; the displayed table gives ${\theta }^{\vee }\le {\theta }_s^{\vee }$. Therefore ${\theta }_s^{\vee }$ is the highest root of the entire dual root system, not just a candidate among its long roots. Thus every positive short-coroot height is bounded by that row's sum. The sums are 2n-1,2n-1,11,5, exactly h-1. Also $h^{\vee }\le h$ follows directly from the theta coefficients: theta-coroot has coefficient $a_i({\alpha }_i,{\alpha }_i)\mathrm{/}2\le a_i$.

Integral real roots and regularity. Normalize opposite finite root vectors by $[e_{\alpha },f_{\alpha }]={\alpha }^{\vee }$. Invariance gives $B(e_{\alpha },f_{\alpha })=2\mathrm{/}(\alpha ,\alpha )=r_{\alpha }$, and the current bracket gives $[e_{\alpha }\otimes t^n,f_{\alpha }\otimes t^{-n}]={\alpha }^{\vee }+r_{\alpha }nK$. This is the coroot of $\beta =\alpha +n\delta$, since $\beta$ evaluates to $2$ on it. Thus $⟨k{\mathrm{\Lambda }}_0+\widehat{\rho },{\beta }^{\vee }⟩=\rho ({\alpha }^{\vee })+r_{\alpha }np\mathrm{/}q$, with $r_{\alpha }=1$ on long roots and $r$ on short roots.

where $r_{\alpha }=1$ or r. The finite term is integral. Thus long roots occur exactly at n in qZ, short roots at n in (q/gcd(q,r))Z. Finite roots are all integral. Their span together with any long alpha+q delta contains q delta, so the rational span is full.

If $n=0$ and beta is positive, its pairing is positive. A nonintegral pairing is not a nonpositive integer. For the remaining positive integral roots $n>0$, the least positive affine term is p on long roots. In the coprime branch it is rp on short roots; the preceding bounds make both resulting pairings strictly positive. In the coprincipal branch it is p on short roots too, and $p\ge h$ exceeds the maximum short-coroot height h-1; the long bound is at most $h^{\vee }-1\le h-1$. This proves regular dominance and therefore vacuum admissibility.

In the coprime branch the real root system is the untwisted finite root system with delta replaced by q delta, so its simple roots are alpha_i and -theta+q delta. In the coprincipal branch take coroots: every finite coroot direction has loop period qK, because a short root's q/r step is multiplied by r in its coroot. The coroot system is therefore the standard untwisted affine system of the dual finite root system. Its highest root is theta_s-coroot, giving original simple roots alpha_i and -theta_s+(q/r)delta. This uses the dual finite ROOT DATUM, not an unproved existence of a dual Lie algebra. It also shows directly that the integral reflection group is affine and contains translations q Q-coroot.

Application to the actual level definition. The supplied principal-W nondegenerate level tests include exactly the p inequalities and gcd branches just used. Their additional q bounds do not change this implication. Positivity of p/q gives noncriticality. Hence every level in the actual definition satisfies the source admissible-number hypothesis, with the stated integral roots and simple system. Positive integral admissible levels $q=1$ demonstrate why the stronger nondegenerate predicate must not be advertised as equivalent to all admissible numbers.