Explicit integer presentations for signed cactus critical groups

For a signed cactus with a chosen sink, we give an integer presentation with one coordinate per cycle. Its entries are explicit counts of overlapping root paths, with coefficients determined by the cycle signs. The presentation uses the standard integral Gram reduction and yields all invariant factors by determinantal divisors. For two unbalanced cycles joined by a path, we compute the invariant factors for every sink and every connecting path length. These formulas include an infinite family of cyclic groups whose unsigned counterparts require two generators. We also characterize when a signed reduced Laplacian has the unsigned critical group, and exhibit two cacti with identical marked abstract block-cut data but critical groups of orders 56 and 60. Thus the positions of attachment vertices within each cycle are essential data.