We prove and verify a quantitative law connecting the dynamical blow-up rate of a nonlinear evolution equation to an observable fingerprint that its routine diagnostics carry long before any indicator diverges. For any scalar diagnostic X(t) with algebraic blow-up X(t) ~ C(T*-t)^(-alpha) as t -> T*, the windowed residual variance grows as (T*-t)^(-2(alpha+1)), the windowed residual mean as (T*-t)^(-alpha), and the lag-one residual autocorrelation converges to unity. The onset time at which the residual-variance ratio crosses a fixed threshold scales as a power 1/(2(alpha+1)) of the baseline-variance floor, which accounts for the near-coincidence of early-warning timing across mechanistically unrelated equations. The identity itself is elementary, and we say so plainly; the contribution lies in its corollaries and in its inverse use. We derive the exponents rigorously for four structurally distinct singularity types: inviscid Burgers (gradient catastrophe, alpha = 1/4, 1, 7/4), focusing L2-critical quintic nonlinear Schrodinger (critical wave collapse, alpha = 1/2 with log-log correction), the semilinear heat equation (Type I reaction blow-up, alpha = 1/2), and Camassa-Holm (Riccati wave-breaking, alpha = 1, resolved by a Lagrangian integration where spectral methods fail). A central corollary covers the degenerate case alpha = 0: exactly conserved quantities carry no physical precursor, correcting a recurrent pitfall. Inverting the identity turns it into an instrument: the blow-up exponent is estimated from pre-divergence residual statistics without analytic access to the equation’s rate. A blind test recovers the exponent and singularity time from pre-divergence data alone; across a continuous family it recovers a smoothly varying hidden exponent over a full decade (0.2 <= alpha <= 2.0, R^2 > 0.99); and in an out-of-sample protocol with a randomly drawn, undisclosed exponent, the early-data estimate is confirmed by the system’s own later evolution to within a few percent in the exponent and better than one percent in the blow-up time.