No-Signalling Constraints on Exponential Tilts in CHSH Scenarios

We characterize when exponential reweightings of a no-signalling CHSH probability box preserve no-signalling. While such tilts are automatically positive and normalized within each measurement setting, they can modify cross-setting marginals and thereby introduce signalling into the probability table. We identify a four-dimensional setting-only redundancy in the residual parametrization, derive the exact nonlinear compatibility conditions for no-signalling preservation, and obtain the linearized no-signalling constraint around a no-signalling reference box. For the unbiased Tsirelson CHSH box, we compute the linearized constraint in closed form and show that the admissible tangent space has dimension twelve before quotienting and dimension eight after quotienting by the setting-only redundancy, matching the standard dimension of binary no-signalling boxes. Exact-probability calculations confirm the predicted scaling: generic residual directions produce first-order no-signalling leakage, while admissible tangent directions suppress the leakage to second order. We further show that local-additive residuals, despite their algebraic locality, are not generically no-signalling safe. These results give a sharp first-order admissibility criterion for exponential tilts of Bell probability boxes.