The high-dimensional parameter space of deep neural networks -- the neuromanifold -- is endowed with a unique metric tensor defined by the Fisher information. Reliable and scalable computation of this metric tensor is valuable for theorists and practitioners. Focusing on neural classifiers, we return to a low-dimensional space of probability distributions, which we call the core space, and examine the spectrum and envelopes of its Fisher information matrix. We extend our discoveries there to deterministic bounds for the metric tensor on the neuromanifold. We introduce an unbiased random estimator based on Hutchinson's trace method and derive related bounds. It can be evaluated efficiently with a single backward pass per batch, with a standard deviation bounded by the true value up to scaling.