arXiv2026
Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticated result or paper draft therefore becomes weaker evidence of the student's own mathematical development. This creates a formation gap between output and personal capacity, and a trust gap between a convincing argument and warranted acceptance. The formation gap can persist even when the student understands the output: understanding a supplied argument does not by itself establish the capacity to initiate and direct inquiry. These gaps are not the whole story. AI can also help students explore examples, compare approaches, enter unfamiliar areas, and undertake ambitious research. The task is to design an apprenticeship that realizes these possibilities while developing substantive mathematical command. The central purpose of a mathematics PhD is to form mathematicians capable of original, independent, and responsible inquiry, including inquiry conducted with AI. This document develops that objective through four connected capacities: competence, judgment, independence, and responsibility. It distinguishes a work's contribution to mathematics from the evidence it provides of a student's formation; explains how a known answer can initiate rather than end creative inquiry; and proposes changes in learning activities, assessment, doctoral originality, advising, and institutional support. Purposeful independent work and ambitious AI-assisted research are complementary parts of the model. Its recommendations include proportionate contribution statements, recognition of advising costs, and staged pilots evaluating both mathematical ability and effective human--AI collaboration. The aim is not to preserve an inherited sequence of training, but to improve mathematical formation as mathematical practice changes.