Existing video summarization methods mainly compress content for gist browsing, but they often break the prerequisite logic in instructional videos and induce logical inversions (e.g., conclusions before premises). We formalize this problem as Structure-Pedagogical Reconstruction (SPR). SPR raises two challenges: (1) Structure Hallucination, where retrieved knowledge is topologically valid but not evidence-grounded by the blackboard; and (2) Logical Inversion, where soft prompt-level graph injection fails to enforce prerequisite order during decoding. To address these challenges, we propose Knowledge-Centric Video Reconstruction (KCVR), a Plan-then-Generate neuro-symbolic framework that decouples epistemic planning from content generation. KCVR prunes a Dual-Layer Epistemic Graph into a minimal video-supported plan, then realizes the plan with visually anchored attention and topology-constrained decoding. We additionally release EduStruct, a 10-discipline benchmark for SPR and structure-centric evaluation. Experiments show that KCVR outperforms strong end-to-end baselines on Knowledge Progression Consistency and Learning Objective Coverage. Our code and data are available at https://github.com/mark1001-ljj/video_sum.