The Split Is a Property of Parity, Not of Schur Complements

Building Gram, covariance, kernel, or Toeplitz matrices from exponentials or hyperbolic functions? You're destroying 12 digits before you factorise anything. One line fixes it — 34,000× better. When your entries come from differences of large-argument exponentials, the intermediate products run 13 orders of magnitude above the values you actually want, and the subtraction eats your precision. Centring the argument first costs one line of code and cuts the worst entry-wise relative error from 240% down to 0.0073%. Measured against the largest entry, it's a factor of 300 million. If your matrix is "mysteriously" indefinite when the math says it must be positive, this is very likely why. Separately: the block structure that gets blamed on Schur complements is actually caused by parity, and knowing which one it is tells you when to expect it and when not to.