Large Language Models (LLMs) encode substantial factual knowledge, but reliably extracting complete knowledge-base entries remains challenging when relations differ in cardinality, semantics, and missingness. We present COVER-KBC, our system for the AKBC Shared Task 2026, which formulates closed-book knowledge extraction as relation-typed, evidence-centric inference. Each relation is compiled into an inference program that controls elicitation, verification, stopping, and final selection. Model observations are accumulated in a provenance-aware evidence state and scored using independent support, calibrated verification, contradiction, and disagreement. To distinguish candidate correctness from answer-space completeness, COVER-KBC estimates residual search need and uses a budget-aware controller to adaptively acquire further evidence when necessary. Relation-specific finalization handles heterogeneous output structures and difficult numeric or nullable cases. Using a single Mistral-Small-3.2-24B checkpoint without additional training or external retrieval, COVER-KBC achieves a Macro-F1 of 0.5878 on the hidden test set. Our implementation is available at https://github.com/vquclinh/cover-kbc.