Efficient design of separating input banks

Develop an efficient method for designing finite input banks that separate the divided leading derivative functions and thereby realize the native contact-spectrum bounds with fewer or better-conditioned observations.

Background

The native contact-spectrum theorem establishes the vanishing orders and multiplicities of positive singular values of the observation Jacobian near simultaneous query/key collapse. Its proof shows that some finite input bank exists by selecting evaluations that preserve linear independence of a finite collection of leading derivative functions.

The construction is existential and does not optimize the number, placement, or conditioning of the inputs. Efficient bank design matters because poorly chosen or insufficiently diverse input sets can fail to distinguish identifiable directions or can make the resulting numerical observations unstable.