This is an AI-generated audio summary of an unrefereed mathematical candidate.

Imagine multiplying several polynomials together. If two factors acquire a common root, the multiplication map loses rank. The classical resultant detects exactly that collision. But multiplication also forgets how the individual factors were rescaled, provided their total product stays fixed.

The paper separates those two effects. It proves that every largest Jacobian minor of the multiplication map is the product of all pairwise resultants, multiplied by a coordinate describing the missing scaling directions. In short: resultants detect collisions, while torus characters record lost scale.

The strongest new candidate result concerns the integer lattice formed by those characters. The paper gives a local one-generator description at every prime. From that, it computes all Smith invariants, identifies every bad characteristic, proves that spanning trees already contain the full arithmetic information, and gives a fast greatest-common-divisor formula.

This extends the rank-one bordered-Jacobian result in the parent paper, Bordered Jacobian Foundations. The parent remains an immutable, separately published candidate, and is linked from the repository and Evidence Press page.

The work does not classify all factorisation normalisers or affine slices, and it does not prove a Keller-map, Hessian-conjecture, or Jacobian-conjecture result. The proofs, exact computations, and review records were produced in one AI-assisted workflow. Public release, hashes, and replay checks establish availability and integrity, not independent reproduction or peer-reviewed correctness.