Nodal Jacobians, Diagonal Nearby Cycles, and Local P=W for an Endoscopic Prym Sector

We prove an automatic orientation and saturation theorem for cohomological blocks in one-node Morse–Bott degenerations. Let a projective family have smooth total space, transverse model xy=t, and entire critical locus a torsor under an abelian variety of dimension G−1. Assume that its rational cohomology local systems over the punctured disk, compatibly with monodromy, agree with those of a genus-G Jacobian family whose central curve is integral with one node. Then the transverse orientation local system is trivial. In every degree, the canonical morphism is surjective, the logarithmic variation morphism is injective, and dim Φ^q = rank N_q = binom(2G−2,q−1). The resulting image–kernel filtration recovers the even monodromy and limit-weight steps. We establish all-degree rank formulas for nodal Jacobians and compare their monodromy filtration with the topological pinching filtration. An explicit transverse GL₂ Hitchin slice over a genus-two curve realizes the one-node criterion via the relative spectral correspondence and compactified-Jacobian deformation theory. For an explicit two-node slice, we prove fixed-order iterated saturation and exchange of the two orders of unipotent nearby cycles for the constant module. We also prove a categorical diagonal nearby-cycle comparison under an exact simultaneous product-chart hypothesis, identifying the diagonal logarithm with N₁+N₂. Finally, under (G1)–(G3) and the external assumption (NAH-lin), we obtain a local, sectorwise P=W comparison for a determinant-fixed endoscopic Prym sector. The Jacobian limit weights W′ remain distinct from Betti weights W; no global P=W theorem is claimed.