Commutator-constrained factorizations in class-two exponent-p-groups

Groups of nilpotency class $2$ and exponent $p$ (with $p$ odd) admit concrete coordinate models governed by alternating ${\mathbb{F}}_p$-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 $g\in G$ and a target commutator value $h\in [G,G]$, count and construct pairs $(x,y)$ with $xy=g$ and $\left[x,y\right]=h$. In an explicit bilinear-data input model, we show that the decision, counting, and search variants reduce to solvability of a single linear system $T_w(u)=h$ over ${\mathbb{F}}_p$, where $T_w$: $u\mapsto b(u,w)$ is the contraction map determined by the $V$-projection $w$ of $g$. When solvable, the solution set $\mathcal{S}(g,h)$ is exhibited as an explicit torsor for $\ker (T_w)\times W$, 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 $\mathcal{H}(g)=\mathrm{im}(T_w)$, an exact uniformity law over attainable values, and a factor-swap bijection relating $\mathcal{S}(g,h)$ to a shifted product constraint. Finally, we define rank-profile polynomials $P_G(t)=\sum _{w\in V}{t}^{\mathrm{rank}(T_w)}$, 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 $1/2\in {\mathbb{F}}_p$, 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.