The full Hall algebra is simple-generated exactly in componentwise finite Dynkin type

Restrict to a full induced subquiver on a vertex subset $I$. Objects of dimension supported on $I$ have zero vector spaces elsewhere, so their representations, flags, nilpotence and Euler factors are exactly those of the induced subquiver. A word in simples with this degree cannot involve a vertex outside $I$, since all simple degrees are nonnegative. Thus its Hall and composition homogeneous components agree with those of the subquiver. A proper composition component for an induced subquiver therefore proves properness for $Q$.

On an affine D or E subtree let $H_{\delta }$ be the positive total height of its null vector. There is a constant $C$ bounding the sum of root-space multiplicities at every total height: by [F5] each finite root $\alpha$ contributes at most one integer mode $n$ to a given height $\mathrm{ht}(\alpha )+n{H}_{\delta }$, and imaginary modes contribute at most the finite rank. PBW therefore bounds the total-height generating series of the classical positive enveloping algebra coefficientwise by $P(t)={\prod }_{h\ge 1}(1-t^h{)}^{-C}$. For $s>0$, $\log P(e^{-s})=C{\sum }_{j\ge 1}1\mathrm{/}[j(e^{sj}-1)]\le (C\mathrm{/}s){\sum }_{j\ge 1}{j}^{-2}$. The series and rearrangements converge absolutely because their terms are positive and $e^{-s}<1$. A coefficient of height $L$ is at most $e^{sL}P(e^{-s})$; putting $s=L^{-1\mathrm{/}2}$ gives a bound $\exp (C^{\mathrm{\prime }}\sqrt{L})$ for a fixed $C^{\mathrm{\prime }}$. In particular the classical PBW rank in root degree $m\delta$ is at most $\exp (C^{\mathrm{\prime }\mathrm{\prime }}\sqrt{m})$. By the surjective upper-bound input [F4], the same bound holds for the composition degree.

If two vertices have $r$ arrows one way and $s$ the other, with $r+s\ge 2$, use degree $(1,1)$. Nilpotence forces at least one of the two scalar-arrow vectors to be zero: a nonzero scalar in each direction would give a nonzero arbitrarily repeatable two-cycle. Conversely either one-direction vector is nilpotent. Changes of vertex bases scale a nonzero vector by a common scalar, so the isomorphism classes are the zero representation and the two projective sets of nonzero vectors. Their number is $1+(q^r-1)\mathrm{/}(q-1)+(q^s-1)\mathrm{/}(q-1)>2$. The composition degree is spanned by the two simple words. Hence it is proper, with no quantum rank input.

Suppose the simple underlying graph contains a chordless cycle of length $n\ge 3$. On that induced subquiver use dimension $1$ at every vertex. Each proper subset of the $n$ edges specifies a representation by setting those arrows to $1$ and the rest to zero. Its support is a forest, so it is nilpotent in every original orientation. The $2^n-1$ distinct zero/nonzero supports give distinct isomorphism classes. For two nonadjacent vertices an extension of one vertex simple by the other is supported only at those vertices and has no arrow coordinates, so it splits. Their Hom space is zero by disjoint support, their cross Euler values vanish, and the split sum has a unique subobject of either prescribed vertex-simple type. Thus both Hall products are the same direct-sum class, and nonadjacent vertex simples commute. Permutations of the $n$ letters modulo adjacent swaps of nonneighbors correspond to acyclic orientations of the cycle: orient an edge from its earlier letter to its later one. Every acyclic orientation has a topological ordering. Any two such orderings for it are linear extensions of the same partial order; move the first letter of one left in the other, crossing only incomparable letters, and induct to connect them by the allowed swaps. Exactly two of the $2^n$ edge orientations are cyclic. Thus the composition degree has dimension at most $2^n-2$, less than the Hall degree. This proves properness for every orientation, including the nilpotent oriented-cycle case.

It remains to classify connected simple trees not of finite ADE type. A vertex of degree at least $4$ contains the induced four-leaf star ${\tilde{D}}_4$. If there are two vertices of degree $3$, choose a closest pair. Their joining path has no branch vertex inside; attach two outward neighbors at each end. The induced subgraph is an affine D tree with two branch vertices and four leaves. If there are no branch vertices, the tree is a path of type A. In the remaining case there is exactly one branch vertex of degree $3$, with three arms of lengths $p-1\le q-1\le r-1$, where $2\le p\le q\le r$. The cases $(2,2,r)$ and $(2,3,3),(2,3,4),(2,3,5)$ are precisely finite D and E; if $p\ge 3$, truncating arms gives $(3,3,3)$; if $p=2,q\ge 4$, it gives $(2,4,4)$; and if $p=2,q=3,r\ge 6$, it gives $(2,3,6)$. These are respectively the affine E6,E7,E8 induced subtrees. Thus any remaining non-ADE tree contains an affine D or E induced subtree. An arbitrary graph either has the parallel-edge case, has an induced chordless cycle (take a shortest cycle), or is a forest to which this classification applies.

The Hall basis contains each of the pairwise nonisomorphic indecomposables from [F3], so its degree $m\delta$ has dimension at least $c{q}^m\mathrm{/}m$. For large $m$ this exceeds $\exp (C^{\mathrm{\prime }\mathrm{\prime }}\sqrt{m})$, since $m\log q-\log m-C^{\mathrm{\prime }\mathrm{\prime }}\sqrt{m}\to +\mathrm{\infty }$. Hence composition is proper on the affine subtree. Steps 1.1 and 3.1, together with steps 2.1–2.2, prove properness whenever any component is not finite ADE. This argument used only the surjection in [F4], not an injection or an equality of quantum and Hall dimensions.

If every component is finite ADE, [F2] gives equality of composition and full Hall. Otherwise step 4.1 gives properness in a supported degree. The torus extension is a direct sum of its declared toral normal-form copies of the positive Hall algebra, so adjoining the torus neither removes nor creates the difference between these positive subspaces. This proves the entire componentwise criterion and its extended version.