Finite-quiver Hall counts and the limits of DT/BPS interpretation

The coefficients of $J(S_Z(\varphi ))$ are finite-field, Euler-normalized semistable groupoid masses: the weight of $[M]$ is $t^{{\chi }_Q(\mathbf{d}(M),\mathbf{d}(M))}\mathrm{/}\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }$. The wall crossing proved on this page is an arithmetic finite-field Hall identity and its stated Euler/skew integrations. It does not identify these coefficients with Behrend-weighted numerical Donaldson–Thomas invariants of a Calabi–Yau-threefold, motivic vanishing-cycle invariants, cohomological BPS spaces, or dimensions of physical Hilbert spaces.

The four required preceding pages provide distinct reading contexts. The derived definition uses dg enhancements and higher homotopy weights; the motivic definition uses stack-function coefficients; the cohomological definition uses rational intrinsic quotient-stack Borel–Moore homology and the exact-sequence correspondence operations; a Hall-module definition is an algebra action on a separate module. These coefficient theories remain distinct from the arithmetic groupoid masses. Their operation and integration theorems are not used in this finite-field proof.

Any physical BPS identification is conditional on a specified supersymmetric quiver model, charge assignment, central charge, superpotential, relevant Calabi–Yau-threefold enhancement and orientation data, together with a proved state-counting comparison. The finite-field argument establishes none of those additional identifications.