Completeness of the Bethe Ansatz\\through Partition Ideals and Weighted Point Sums

We study an algebraic counting approach to completeness for the periodicspin-$1/2$ XXX chain. A partition-lattice formula defines depermutationwith local intersection multiplicities, and a binomial alternating sumcorrects the singular-pair contributions. Normalized inhomogeneousfamilies give a local complex proving nonnegativity of the correctedlengths without spectral input. At a simple fully reduced zero, directcluster elimination also identifies this index with the length of anexplicit residual ideal; a weighted local hypothesis gives a jet-spaceKoszul model for that length. Finite flatness of the remainderfamilies also proves the affine degree formula under the partition-idealfiniteness hypotheses. A spectral resolution further identifies thecorrected sum with the normalized ordered-root pullback of thepolynomial-kernel ideal. Identifying its pointwise coefficients with theusual physical-root weights still requires root-simplicity statements;in particular, the uniform exclusion of repeated singular pairs remainsunproved, although several families are treated explicitly.