Transcendental Law-Form as Ambient-Relative Classification: A Euclidean Case Study

Debates over Kant’s Second Analogy often turn on whether its law-form—“necessary connection according to a rule”—is an independent postulate or a tacit import of Newtonian structure. We reframe that issue by treating transcendental law-form as an ambient-relative classification problem. The method is modular: fix admissible symmetries, state structural guardrails on rule-application, classify symmetry-compatible generators, and interpret necessity through invariance together with preservation of invariant cores under lawful evolution. The Euclidean/quaternionic setting serves as a worked instance. Under locality, homogeneity, full rotational symmetry, identity preservation, compositionality, and an explicit minimality principle, the minimal scalar second-order generator is uniquely the Laplacian. The result is premise-relative: it shows how law-form follows from declared assumptions rather than from undeclared physical imports. On the concept side, the same architecture yields an invariant-core projector and a universal factorization account of admissible causal rules.