BCa as Frequentist Inferential Geometry: A Rigorous Curvature Identity and Projective Normal Form

The BCa bootstrap interval can be understood not merely as a higher-order endpoint correction, but as a small piece of frequentist inferential geometry. We isolate a rigorous core for that interpretation. A simple exponential-tilt object built directly from jackknife data recovers the familiar BCa bias and curvature corrections exactly. The classical BCa adjustment is then shown to arise as the unique local transport law within a specific semigroup and factorization class once one imposes a natural composability condition on curvature correction, and this exact picture is connected back to the standard influence-function asymptotics. The same framework yields practical diagnostics for when BCa is acting as a controlled geometric correction and when it is becoming unstable, and it shows that the resulting structure is natural under ordinary changes of parameter scale. The philosophical upshot: BCa can be read as an operationally frequentist transport rule generated by bootstrap and jackknife structure.