A Scale Invariant Curvature Diagnostic from Coupled Derivative

Deep neural networks possess reparameterization symmetries that systematically alter parameter space Hessian curvature despite the underlying function remaining entirely static. Prior work addresses this variance by proposing invariant flatness measures that rely on explicit weight magnitude scaling. We introduce a scale invariant diagnostic constructed entirely from coupled first and second order loss derivatives, bypassing explicit parameter norms. The diagnostic couples the layer restricted Hessian Rayleigh quotient, evaluated along the layer’s own parameter direction, with the squared gradient norm of that same layer. Under continuous node wise rescaling Wl → cWl , Wl+1 → c −1Wl+1 in a positively homoge￾neous network, both the Rayleigh quotient of the layer restricted Hessian and the squared gradient norm of that same layer scale by c −2 ; dividing the former by the latter cancels this common factor exactly, for every c > 0. We analyze the same construction under general reparameterizations, characterize exactly the symmetry group for which the cancellation remains exact, and prove that it cannot be extended further. We show that replacing the Hessian with its pseudoinverse in the same coupling yields a second exactly invariant diagnostic that decreases rather than increases with curvature, resolving a sign obstruction that otherwise blocks connecting either quantity to a Laplace based generalization bound. We verify exact invariance on trained networks of up to fourteen hidden layers and one hundred twenty six thousand parameters, on two real datasets, across multiple architectures and seeds, and we report an exploratory multi seed experiment in which the inverse coupled diagnostic tracks the generalization gap across network widths. We present this as a precise, narrow invariance result: we do not establish a generalization bound, and we do not evaluate the diagnostic beyond the small scale settings reported here.