Continuity scaling: A rigorous framework for detecting and quantifying causality accurately

The paper states the key theorem and the continuity-scaling recipe, but defers the crucial mathematical arguments to the Supplementary Information; in the main PDF, the “key condition [DD]” and the proof of linear scaling of ⟨δ⟩ versus ln ε are not provided, so the logical chain is incomplete in the paper as given. The candidate solution supplies a plausible local-differential argument (normal coordinates, Jacobian bounds) that yields δ = Θ(ε) under D_y f ≠ 0 and δ essentially independent of ε if f is independent of y, supporting the directionality criterion qualitatively. However, it never derives the paper’s central log-scaling law; it assumes it “empirically,” and conflates vanishing Jacobian along an orbit with global y-independence. Hence both sides leave gaps: the paper’s main theoretical claim is unsubstantiated in the provided PDF, and the model’s proof does not establish the stated linear dependence on ln ε nor the precise monotonicity with “coupling strength.” Citations: the paper’s definitions and theorem are in the main text, with the theorem and scaling claims placed to SI ; the algorithmic slope estimation and chosen log-spaced ε-grid are given in Methods .