Inquiry Has Geometry: A Mathematical Theory of Recursive Question Transformation

Reasoning systems are usually evaluated after a question has already been supplied, although consequential failures can originate earlier in problem framing, information acquisition, and clarification. We introduce Spiral Inquiry Geometry (SIG), a mathematical theory in which questions are states in a metric space, inquiry modes act as transformations, and recursive questioning traces measurable trajectories. SIG defines path length, endpoint displacement, spiral ratio, radial dynamics, local geometric excess, operator noncommutativity, fixed points, cycles, and divergence. We prove a path-length lower bound, isometry and reversal invariance, path dependence under noncommuting operators, and convergence under contractive meta-inquiry. The conceptual seed is the recursive five-layer architecture of the Spiral Question Ladder; the formal theory is a distinct contribution. Two benchmark families test different consequences. On 126,500 candidate-question records from 11,790 QuestBench instances, nonlinear structural SIG features reach ROC-AUC .839 and top-1 .552 on Logic-Q, compared with .452 and .122 for proximity alone; Planning-Q reaches ROC-AUC .945 and average precision .902, while top-1 remains .365. On 3,226 genuine AskMind trajectories, rubric complexity is positively associated with path length (ρ = .176), spiral ratio (ρ = .211), and curvature (ρ = .177), but negatively with endpoint displacement (ρ = −.108). We then stress-test that result on the same trajectories under character n-gram hashing, word TF–IDF, LSA-256, binary Jaccard geometry, and a content-independent random-text control. Raw spiral correlations replicate across all substantive representations (ρ = .206–.219), but the random control also reveals a mechanical turn-count component. Crucially, ordered-logit models controlling clarification-turn count and question-token volume show incremental value for spiral ratio under TF–IDF, LSA, and Jaccard representations (FDR q ≤ .0033), whereas the random control is null (p = .151). Turn-adjusted spiral rankings agree across substantive representations (ρ = .522–.760) but not with the random control (ρ = −.038 to −.013). These results refine rather than merely confirm SIG: inquiry geometry contains representation-stable structure, but naive path statistics can confound semantic organization with trajectory length.