Quiver stability scattering diagram

Work first modulo total dimension greater than k. The HN identity of [F2] gives $1_A=g_{+}(\theta )1_{ss}(\theta )g_{-}(\theta )$, the unique positive/zero/negative factorization in the nilpotent Hall group. Assign its middle factor to the walls of the finite rational hyperplane subdivision. The source path convention has exponent the negative crossing sign and multiplies the last crossing first; its path product is g_+(end)^(−1)g_+(start). It therefore depends only on endpoints and is the identity on closed generic paths.

The diagrams for increasing k agree after truncation because their Hall factors are the truncations of the same HN factors. They define the required consistent diagram in the inverse-limit Hall group. Its support is exactly the asserted semistability set, including joints: a stable Jordan–Hölder factor supplies a full wall cone through every semistable weight, while the finite subobject inequalities preserve semistability on that cone's faces. Nonempty Hall labels remain nonzero by pullback to a geometric semistable object, which sends their class to 1.

Put ε(d,θ)=((t²−1)log1_ss(θ))_d. The actual framed-object/BCH argument of [F2] proves these classes regular, so the completed Poisson homomorphism is applicable to every wall logarithm. It sends (t²−1)^(−1)ε(d,θ) to $J(d,\theta )x^d$, where J is its Euler specialization. It preserves all truncated BCH products; hence it preserves consistency and gives exactly $exp({\mathrm{\Sigma }}_{\theta (d)=0}J(d,\theta )x^d)$ for the integrated wall element. This is the source unweighted potential Euler normalization. No triangulated slicing or unrelated weighted DT integration is inferred.