The Price of an Answer
“Nonlinear means no shortcuts” is one of those phrases everyone repeats and nobody measures. So I measured it. Take 35 famous nonlinear PDEs — Burgers, KdV, chaos, solitons — and ask: as parameters vary, how many independent directions do the solutions actually explore?
The diagnostic is effective rank (Φ₁). The result: 33 of the 35 equations sit at Φ₁ ≈ 1. Their parametric solution families trace a simple one-dimensional curve through high-dimensional space — walking along a wire. Which means about seven strategic solves are enough for machine-precision forward prediction, inverse parameter recovery, and field reconstruction, all in microseconds.
The equations stay nonlinear; their response to parameters is what’s simple. The true price of an answer is set by problem structure, not by solver assumptions. This work became LastSolve, and every demonstration is reproducible from the companion repository.