Determining whether one symbolic expression can be transformed into another by repeatedly applying rewrite rules from a predefined set is a basic problem in automated reasoning and symbolic computation. A natural way to view this task is as search in a rewrite graph whose vertices are terms and whose edges are one-step rewrites. For arbitrary first-order term rewriting systems, however, this graph may be infinite, so uninformed search quickly becomes impractical. Our work studies directed term reachability in that general setting and introduces, to our knowledge, the first universally applicable admissible heuristic for this problem, requiring no assumptions such as orthogonality or left-linearity. The heuristic is based on an abelianized root-to-variable ILP formulation and is easily computable: root-to-variable paths with argument indices are extracted from a term, translated into an induced path-rewriting system, and compared with the target through an integer linear program that yields a lower bound on the number of rewrites still required. This bound can guide A* search for shortest rewrite sequences, while relaxed variants support the search for any witness derivation. A secondary tie-breaking score is derived from the same relaxation. Experiments on three benchmark rewriting systems show that, despite its nontrivial computational cost, the heuristic improves wall-clock performance over uninformed search and simpler syntactic baselines.