Support preservation from classical to quantum scattering diagrams

Prove that for every skew‑symmetrizable matrix B, the support of a quantum cluster scattering diagram Dq(B) equals the support of the corresponding classical cluster scattering diagram D(B). Resolve the potential cancellations in the classical limit and determine whether support preservation holds universally without assuming positivity.

Background

In the classical setting, a cluster scattering diagram D(B) admits a minimal-support realization whose support is entirely determined by the associated G‑fan. Quantum synchronicity suggests that quantum mutations mirror classical ones, hinting that QCSD supports should match their classical counterparts.

Positivity of wall exponents in QCSDs would prevent cancellations in the classical limit, implying support preservation. While this has been shown under positivity assumptions, the general case remains unsettled, as cancellations could occur when both positive and negative exponents are present.