Geometric Coupling Topology on Free Correspondence: Zero Coupling Findings, One Stylometric Gradient, and Seven Confounds from Darwin's Letters

We apply Geometric Coupling Topology (GCT) to 1,867 consecutive letter pairs from Darwin's scientific correspondence (1843–1882) and 497 pairs from two control dyads (publisher, cousin). Using the refraction vector B = E_response - E_question computed on 4096-dimensional text embeddings, we initially report seven findings — and then systematically dismantle all of them through four rounds of internal review, control dyads, and cross-model validation. What does survive is a stylometric gradient: Darwin's directional variability scales monotonically with the intellectual demand of the relationship (Fox g = 0.69, Murray g = 0.47, Huxley g = 0.37, Hooker g = 0.16, Gray g = 0.10). The paper contributes a diagnostic checklist of seven confounds and a design principle: GCT measures the dominant variance source between question and answer, and that source is determined by what the experimental design leaves free to vary.