Coordinate Invariance of the Cubic Term at a Fixed-Point Family

Given a smooth family of self-maps of a Banach space with a fixed-point family, the thirdderivative at each fixed point defines a “cubic constant.” Such a constant is only geometricallymeaningful if it does not depend on the coordinates used to compute it. We show thecubic tensor transforms as a genuine rank-3 covariant tensor under any linear change ofcoordinates — a purely kinematic fact, true for every such family, with no smoothness ordynamical hypothesis actually needed beyond differentiability. Turning this into a coordinate-independent scalar ordinarily requires dividing by a determinant, which does not exist onan infinite-dimensional Banach space; we show the natural substitute — the one-parameterconformal subgroup, on which a determinant surrogate does exist — is exactly enough torecover a genuine scalar invariant, and prove this in full generality. The result is short,elementary, and complete: every step is either the chain rule or the homogeneity of amultilinear map.