We show that structural grammatical priors produce targeted, linguistically specific effects on grammatical learning: improving filler-gap dependencies — which require long-distance hierarchical tracking — by 9–13 percentage points beyond structural regularisation alone (d = 2.41–2.82), while damaging locally cued phenomena regardless of whether the grammar is real or random. This phenomenon-specificity, revealed by a random grammar control, suggests the right question is not whether structural priors help, but for which constructions and why. We test this by augmenting BabyBERTa (7.4M parameters) with a differentiable PCFG auxiliary loss derived from Minimalist Grammar, trained on AO-CHILDES (893K sentences of child-directed speech). In a pre-registered study of 190 experimental runs spanning 7 constraint strengths, 3 data scales, 5 random seeds, and 3 independent lexicon permutations, our confirmatory hypotheses about overall accuracy and sample efficiency are falsified. However, a random grammar control (n = 15 runs per condition; three independent lexicon permutations) reveals that linguistically accurate category assignments specifically drive filler-gap gains: real grammar outperforms both a structurally equivalent random grammar and the no-grammar baseline, while both conditions equally damage subject-verb agreement. These results show that structural priors function as targeted interventions rather than global boosters: they help specifically the constructions, specifically long-distance dependencies, whose computational demands align with what phrase-structure representations encode. We release code and pre-registered materials.