Groups of nilpotency class 2 and exponent p (with p odd) admit concrete coordinate models governed by alternating Fp-bilinear commutator data, a viewpoint central in both the structure theory of p-groups and algorithmic approaches to isomorphism testing and related problems. Motivated by decomposition primitives in computational group theory and by rank-based filters used in modern isomorphism pipelines, we introduce the commutator-constrained factorization problem: given g ∈ G and a target commutator value h ∈ [G, G], count and construct pairs (x, y) with xy = g and [x, y] = h. In an explicit bilinear-data input model, we show that the decision, counting, and search variants reduce to solvability of a single linear system Tw(u) = h over Fp, where Tw: u 7→ b(u, w) is the contraction map determined by the V-projection w of g. When solvable, the solution set S(g, h) is exhibited as an explicit torsor for ker(Tw)×W, yielding a closed counting formula and a certified witness construction by elementary linear algebra. We derive exact secondary laws: a complete description of the attainable commutator set H(g) = im(Tw), an exact uniformity law over attainable values, and a factor-swap bijection relating S(g, h) to a shifted product constraint. Finally, we define rank-profile polynomials PG(t) = Pw∈V trank(Tw), prove isomorphism invariance, and extract further invariants (radical size, extremal attainable-set size) directly from counting oracles. The odd-prime hypothesis is made explicit throughout: the centered coordinate law uses the scalar 1/2 ∈ Fp, whereas characteristic two requires a cocycle or quadratic-refinement formulation. We also record coordinate-invariance, conversion costs from power-commutator input, sparse implementation refinements, and limitations of the rank-profile invariant.