Conditional Entropy and Mutual Information of Locomotor Affordances in Screw-Structured Haptic Flow

Affordances in ecological psychology are relational properties of organism–environment systems, yet their quantitative characterization remains an open problem. Shannon information theory measures uncertainty and statistical dependence but does not by itself specify what environmental structure means for an acting organism. This article proposes a limited operational bridge between the two frameworks by treating Gibsonian specification as a relation to be quantified, rather than identifying it with Shannon entropy. Let A denote a locomotor affordance, such as stable continuation of walking, and let S denote a screw-structured haptic state derived from the coupled motion and loading of the body–environment system. The central quantities are the conditional distribution p(A | S), the residual conditional entropy H(A | S), and the mutual information I(A;S) = H(A)−H(A | S): stronger ecological specification corresponds to lower residual uncertainty and greater mutual information. Screw theory supplies the mechanical geometry through twists, helical axes, pitch, Plücker coordinates, and reciprocal screw conditions, including the projective reciprocity condition (K1 + K2) sinϕ + d cosϕ = 0. We hypothesize that screw-structured representations preserve affordance-relevant information more efficiently than Cartesian kinematics, yielding lower H(A | S_screw) and higher I(A;S_screw). A single-case gait demonstration and an explicit estimation protocol support the framework as a testable basis for quantifying ecological specificity in stable gait, perturbation, recovery, and fall-risk analysis.