Gopakumar-Vafa Invariants and Macdonald Formula

We use a derived constructible Chow exponential and prove that its coefficients are perverse minimal extensions from reduced cycles.

With compatible orientations, we formulate the cohomological PT/GV relation by this exponential.

If the relation holds over reduced cycles, its extension to the full Chow varieties is equivalent to the absence of nonreduced strict supports, and it implies the refined and numerical formulas.

For local $\Pp^2$ and $0\le n\le d+1$, we identify the stable-pair space with the smooth relative Hilbert scheme and the vanishing-cycle sheaf with its intersection complex.

We determine the first reducible summand and give del Pezzo examples.

In degree $(2,4)$, we determine the incidence--ribbon union, its canonical PT critical germ, and the attachment triangle, and we formulate the primitive degree-two KKV comparison.