This study proposes a methodology to systematize the logical connections among determination factors through the mathematical structuring of similarity assessments in trademark law. By defining marks and designated goods as manifold structures and introducing a metric tensor, this study formalizes existing legal principles governing trademark similarity assessment, such as the interactions among specific elements (appearance, pronunciation, and meaning) and cognitive asymmetry, into a problem of geometric distance measurement. As a theoretical application, this study presents a methodology that explores the theoretical lower bound of the threshold distinguishing similar from dissimilar marks by applying the similarity judgment jurisprudence of the Supreme Court of Korea as a mathematical constraints. In addition, using the tensor transformation law mediated by the Jacobian matrix, a methodology is proposed to verify the validity of distance-calculating metrics independently formed by the hidden layers of black-box artificial intelligence (AI) models by projecting them into a legal-doctrinal reference space. This study is academically significant because it provides a novel analytical framework by substituting the existing legal reasoning for trademark similarity assessment with a mathematical structure. The study has practical significance as it establishes a foundation for diagnosing and verifying whether the outputs generated by black-box AI models or practical legal judgments possess case-specific validity based on evidence and facts.